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An End-User Intelligible Specification Language and Its Execution

DOCTORAL THESIS

By Pathirage Gamini Wijayarathna This dissertation submitted to the Department of Information Systems Design, Graduate School of Information Systems, The University of ElectroCommunications in partial fulfillment of the requirements for degree of Doctor of Engineering.

Department of Information Systems Design Graduate School of Information Systems The University of Electro-Communications 1-5-1 Chofugaoka Chofu-Shi, Tokyo, 182, Japan email : gamini@maekawa.is.uec.ac.jp

September 1998

TABLE OF CONTENTS

Page CHAPTER I INTRODUCTION 1.1 Background 1.2 Objective 1 2 4

CHAPTER II ACCOUNTING INFORMATION SYSTEMS 2.1 Accounting systems 2.1.1 Accounting as an information system 2.1.2 Users of an accounting system 2.1.3 Accounting systems and its subsystems 2.2 Accounting language 2.2.1 Syntax 2.2.2 Semantics 2.3 The pilot system 2.3.1 Transactions generation system of a travel agent 2.4 Information systems’ requirements 2.5 View of a system 2.5.1 Entity-relationship view of a system 2.5.2 View of a system adopted in this research

6 7 7 8 9 11 11 14 14 15 16 17 18 18

CHAPTER III REPRESENTING TEMPORAL KNOWLEDGE 3.1 An incomplete temporal knowledge about events 3.2 Conventional methods to represent temporal knowledge 3.2.1 Allen’s method

21 22 23 23

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3.2.2 McDermott’s method 3.2.3 Vilain and Kautz’s method 3.2.4 Temporal operators 3.2.5 Time-sensitive Boolean operators 3.2.6 Problems with the above methods 3.3 The AND_THEN (TAND) connective 3.3.1 The standard logical connective AND 3.3.2 The TAND (∏) logical connective 3.3.4 Representation of temporal operators using TAND

26 29 29 32 33 34 34 35 50

3.3.3 Representation of Allen’s temporal relations using TAND 41

CHAPTER IV TEMPORAL LOGIC 4.1 Conventional temporal logic 4.1.1 First - order temporal logic 4.1.2 The logic USF 4.1.3 The Temporal Logic of Actions 4.1.4 Problems with conventional temporal logic 4.2 TANDTL temporal logic 4.3 Inconsistency proofs based on resolution 4.3.1 Classical resolution 4.3.2 Non-clausal resolution for propositional logic 4.3.3 Non-clausal resolution for first-order temporal logic 4.3.4 Non-clausal resolution for TANDTL logic

58 59 59 63 65 66 67 71 72 73 74 76

CHAPTER V EXECUTION OF TEMPORAL LOGIC 5.1 Logic and the view of a system

83 84

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5.2 Execution of TANDTL

86

CHAPTER VI SPECIFICATION LANGUAGES 6.1 Conventional specification languages 6.1.1 TELOS 6.1.2 Templar 6.2 GSL specification language 6.2.1 Structure of GSL 6.2.2 Features of GSL CHAPTER VII DISCUSSION

90 91 91 92 94 94 99 112

CHAPTER VIII CONCLUSION

116

ACKNOWLEDGEMNETS

119

BIBLIOGRAPHY

120

APPENDIX (A) AN EXAMPLE SYSTEM : CASE STUDY

126

APPENDIX (B) AN EXAMPLE SYSTEM WITH EXECUTION

154

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LIST OF FIGURES

Figure 2.1 Accounting as an information system Figure 2.2 The users of an accounting system Figure 2.3 A typical accounting system Figure 2.4 Interaction of accounting system with environment Figure 2.5 Entity-relationship representation Figure 2.6 Representation adopted in this research Figure 3.1 Allen’s temporal relations Figure 3.2 Allen’s temporal relations using PA algebra Figure 3.3 Possible occurrences of two events Figure 3.4 TAND versions of Allen’s temporal relations Figure 4.1 TAND with point based and interval based time models Figure 6.1 Role of major and minor relationships in a fact Figure A.1 Statement of income Figure A.2 Statement of financial position Figure A.3 Measures of activity Figure B.2 Overview of the outside sales system 8 10 12 13 19 20 25 31 35 57 71 97 151 152 153 162

Figure B.1 Conceptual Framework of Financial Information System 161

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LIST OF TABLES

Table 1 Truth table for distributive law (OR) Table 2 Truth table for distributive law (AND) Table 3 Truth table for u ∏ v = (u ∧ ?v) ∏ v 38 39 77

Table 4 Events, conditions and actions table for outside sales system 164 Table 5 A possible list of objects and relations for outside sales system 167

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CHAPTER I

INTRODUCTION

This chapter contains the background and objectives of this research. It discusses the importance of specifying requirements in software developments, importance of representing incomplete relative temporal knowledge, present approaches to both specification languages and temporal knowledge representation, and also execution of software requirements specifications. Finally it provides the objectives of the research.

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1.1 Background

Software developments commence with identifying, specifying and documenting requirements for an information system. Software requirements specification says, what the final system needs to perform, instead of how it should be done[Borgida 1985]. This is the most crucial and difficult step in information system development process. Accuracy of software requirements specification is largely responsible for the low cost and on-schedule completion of information system development projects[Vessey 1994]. Writing of software requirements specifications needs a requirements specification language. A variety of software requirements specification languages are available in the literature. They can be classified into two categories: formal and informal requirements specification languages. Formal requirements specifications have sound mathematical basis and use formal notation to produce concise, clear and precise software requirements specifications; whereas informal requirements specification languages employ graphics and semiformal textual grammars to provide end-user comprehensible software requirements specifications[Fraser 1991]. Intelligibility of the software requirements specification provides better communication between end-users and developers. Communication barrier between end-user and developer is one of the unresolved problems in requirements engineering[Hsia 1993]. Most formal requirements specification languages do not support user friendly representation. Hence Hall[Hall 1990] suggests that the formal requirements specification must be accompanied by a natural language description. Recent requirements specification languages employ an objectoriented approach to document requirements for an information system[Mylopoulos 1990][Jungclaus 1996]. A study shows that the object methodology is difficult to understand even by the novice systems

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analyst[Vessey 1994]. Hence it is obvious that the non-technical end-user will experience more difficulties with object-oriented representations. Although object-oriented techniques are well established in system designing and coding, its contribution to requirements engineering is not so apparent[Hsia 1993]. However, object-oriented principles (abstraction: aggregation & specialization, inheritance, complex objects) are helpful when dealing with complex systems[Sigfried 1996]. Therefore, the systems analyst will benefit from an object-oriented conceptual modeling technique. Representing and reasoning about temporal knowledge play an important role in information systems developments. We need to refer to what has happened already, what is happening now and what will happen in the future and then to make decisions based on them[Torsun 1995]. Situation calculus, Event calculus and First-Order Temporal logic (FTL) have been used for these purposes[Russell 1995]. FTL has temporal operators in addition to the logical connectives and quantifiers in first-order logic. Allen[Allen 1984] suggests thirteen possible temporal relationships. However, all the FTL temporal operators available in the literature[Tuzhilin 1995][Torsun 1995][Abadi 1990][Manna 1995] are not capable of representing Allen’s[Allen 1984] temporal relationships and all of them use the current state (now) as the origin in time line. Torsun[Torsun 1995] says that FTL is not complete as a first-order logic. In Templar [Tuzhilin 1995], temporal predicates can have temporal operators such as always_in_the_future, sometimes_in_the_future, for_time, within_time. However relative temporal knowledge has been represented using the clauses before, while and after. TROLL[Jungclaus 1996] employs existing temporal operators and the predicates “before” and “after”. TELOS[Mylopoulos 1990] uses a slightly modified version of Allen’s [Allen 1984] temporal relations to represent temporal knowledge. These show the importance of incomplete relative temporal knowledge in software requirements specifications. Once the requirements specification is ready, one has to consider how to implement the specification correctly. Therefore the program verification

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becomes very important. Temporal logic has been used for program verifications[Manna 1995]. However, software requirements specification languages based on temporal logic can produce an executable specification making the verification process easier and faster in automated program generations since the specification language itself can be used as a programming language[Barringer 1996]. As discussed, it is hard to find a requirements specification language that satisfies all the criteria (end-user understandable, executable specification which supports object-oriented principles and is capable to represent incomplete relative temporal information).

1.2 Objective

Since incomplete relative temporal knowledge play an important role in information systems, the proposed language should be capable of representing them in end user understandable form and should be flexible enough to introduce user defined temporal relations to the language. If not, systems analysts and end users have to use a predefined representation scheme which restricts the expressiveness of the language. On the other hand, object-oriented modeling techniques are capable to handle complex systems easily. Therefore the proposed language should provide facilities for object-oriented modeling techniques. Hence the objective of this research is to design a specification language which can be used for object oriented modeling and further it provides executable, intelligible specifications. This thesis can mainly be divided into three parts. First part discusses conventional approaches to represent incomplete relative temporal knowledge and introduces new logical connective TAND. The second part describes the development of a new temporal logic called TANDTL which employs TAND connective to represent temporal knowledge. The final part explains the features of new specification language called GSL which is based on the temporal logic TANDTL. GSL is executable and end-user intelligible. As

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discussed above, the proposed specification language should be able to provide object-oriented modeling. Therefore it is necessary to have a model of a system and examples to introduce the features of the language. Chapter 2 provides the model of a system adopted in this research and the pilot system which is going to be used to provide example throughout this thesis. Further the specification language GSL is designed giving special consideration to accounting information systems. Therefore Chapter 2 discusses about accounting information systems, its users, its subsystems and accounting language. Chapter 3 provides a detailed discussion about representing incomplete relative temporal information. This chapter can be considered as the main chapter of this thesis, as it provides the basis for the remaining chapters. It discusses about conventional temporal knowledge representation methods, problems with them and introduces a new logical connective TAND to represent incomplete relative temporal knowledge. Further it shows how TAND can be used to represent temporal knowledge in information systems. One of the objectives of this research is to develop an executable specification language based on temporal logic. Chapter 4 introduces the conventional temporal logic, their problems and proposes new temporal logic called TANDTL which removes all the existing temporal operators and uses only TAND connective to represent temporal knowledge. Chapter 5 discusses the similarities between logic and the view of a system adopted in this research and it shows how logic TANDTL can be executed. The purpose of this chapter is to show that a specification language based on TANDTL can be executed since TANDTL is executable. The remaining is to introduce the specification language GSL. Chapter 6 discusses about present specification languages and introduces GSL in detail. Chapter 7 provides the discussion and Chapter 8 concludes the thesis. Appendices A and B provide an application of GSL to real problems.

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CHAPTER II

ACCOUNTING INFORMATION SYSTEMS

This chapter introduces the accounting information systems which is the main application area of this research. It introduces what the accounting system is, its users, its subsystems and accounting language. Then it introduces the pilot system which is going to be used to provide examples. It is a kind of a subsystem of an accounting system of a travel agent who provides air line reservations to passengers. Then it discusses about requirements of an information system. Finally it presents the view of a system adopted in this research to model accounting systems and its subsystems. It further discusses why this particular view of a system is adopted and how it can solve the problems of other views of systems.

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2.1 Accounting systems

2.1.1 Accounting as an information system

Accounting is an information system that measures, processes and communicates financial information about an identifiable economic entity[Needles 1993]. Bookkeeping is not accounting but it is an important part of accounting. Bookkeeping is the process of recording financial transactions and keeping financial records. The modern accountant is concerned not only with record keeping but also with a whole range of activities that involve planning and problem solving; control and attention directing; and evaluation, review, and auditing. Accounting is generally known as the “language of business”[Eisen 1994]. According to the definition accounting has three major functions: 1. It measures business activities by recording data for future use. 2. The data are stored until needed and then processed to become useful information. 3. The information is communicated, through reports, to the decision makers. Figure 2.1 shows the process of a typical accounting information system. With the widespread use of the computers today, many of the information needs are being organized into a management information system (MIS). Accounting systems is the financial hub of the management information system. The management information systems consist of many inter connected subsystems including accounting system. Therefore accounting systems interact with many non-financial systems in order to provide the required information. These nonfinancial systems are represented in business activities in Figure 2.1.

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2.1.2 Users of an accounting system

The users of an accounting system can be divided into three: 1. Those who manage a business, 2. Outside business enterprises who have a direct financial interest in the business, and 3. People, organizations, and agencies that have an indirect financial interest in the business. These groups are shown in Figure 2.2. Behavior of these user groups are affected to the overall management information system. Action Decision Makers

Information Information Needs Business Activities data Measurement

Accounting Processing Communications

Figure 2.1 Accounting as an information system Management Management is the group of people who have overall responsibility for operating a business and for meeting the company’s goals. Success and survival in a competitive business environment require that management concentrate much of its effort on two important goals: profitability and liquidity. Managers must decide what to do, how to do it, and whether the results match their original plans. These decisions are based on analysis of accounting information. Therefore management is one of the most important users of accounting

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information. Their decisions may include modifications to the accounting systems. Users with direct financial interests Investors and potential investors are interested in the past performance of the company. Potential investors need accounting information before they invest on the business while those who already invested are interested to know about the success of the business. Many companies borrow money from other organizations. Creditors, those who lend money or deliver goods or services to company on credit basis are mainly interested to know whether the company is in a position to pay back their credits. Potential creditors evaluate financial stability of the company before they lend money or services. Modifications to the existing accounting system may be required to incorporate the creditors decisions. Users with indirect financial interests Society as a whole (such as government and public groups) is the most important user of accounting information. Government policy changes will effect an accounting system. Labor unions, employees are also important user groups. Their behavior will bring modifications to the system. These changes will come with short notice. Sometimes it may be few days or few hours. Therefore an accounting system should be flexible enough to deal with sudden changes to the system.

2.1.3 Accounting systems and its subsystems

An accounting system itself has several subsystems. Components of typical accounting system are given in Figure 2.3. The concept of subsystems is applied through various special purpose journals and subsidiary ledgers. For example, there can be Sales Journal, Purchase Journal,

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Business Activities Accounting

Users Management Owners Partners Directors Officers Managers Dept. Heads Supervisors Those with direct financial interest Investors Creditors Those with indirect financial interest Municipals Other Agencies Economic Planners Government Employees Labor Unions Customers

Activities that affect business activities

Figure 2.2 The users of an accounting system Cash Receipts Journal, Cash Payment Journal and General Journal etc. in the accounting system of a company. Each journal can be considered as a subsystem of the accounting information system. Special purpose journals are designed to record particular kind of transactions. Thus, all transactions in a special purpose journal result in debits and credits to the same account. Each special purpose journal has relevant subsidiary ledger to post the transactions. Finally at the end of the accounting period, as an example, at the end of the month, total of the transactions will be posted to the main ledger called general ledger. In a

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computerized accounting system, these special purpose journals and subsidiary ledgers represent subsystems. All these subsystems interact with the main sub system called general ledger sub system which produces the final accounting information. General ledger sub system is straightforward, easy to develop and maintain. However, since the other subsystems interact with the different user groups, they are subject to frequent changes. For example, let us consider the payroll subsystem of a company. When there is a demand for salary increase from the employees, management has to increase salaries in order to fulfill the demand. This may introduce modifications to the payroll sub system. In this case, the general ledger sub system is not subject to modifications since it gets only few journal entries from the payroll subsystem. Figure 2.4 depicts this situation. Therefore subsystems of an accounting system should be flexible enough to accommodate the changes with short notice, sometimes within few hours. The Pilot System, which is going to be explained in the section 2.3, is also a subsystem of an accounting system of a travel agent.

2.2 Accounting language

Accounting is the language of business. It has well defined syntax and semantics. Therefore manual accounting system can be considered as an executable accounting specification.

2.2.1 Syntax

An accounting system consists of several accounts, journals, ledgers and reports. Information is kept in journals and ledgers in account units till they are used to generate management information.

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Reports

Reports

Accounts Payable Subsystem Accounts Payable Ledger

Accounts Receivable Subsystem Accounts Receivable Ledger

Purchase Journal Vendor Invoice Customer Invoices

Sales Journal

Cash Receipts, Deposits Checks

Cash Payment Journal

General Ledger

Cash Receipts Journal

General Journal General Ledger Subsystem Financial Reports Other Transactions, Journal Entries Payroll Subsystem Reports

Attendance, Deductions

Figure 2.3 A typical accounting system

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General Ledger Subsystem Accounting System

Transactions

Other Subsystems

Environment Figure 2.4 Interaction of accounting systems with environment

A basic accounting language consists of 1. Set of accounts. 2. Set of journals (At least General Journal). 3. Set of ledgers (At least General Ledger). 4. Set of reports. 5. A process called journalizing. 6. A process called posting. 7. A process called closing. 8. An accounting equation : Assets = Liability + Capital. 9. Business transactions such as receipts, payments etc. 10. Each account consists of at least name, date column, description column, debit column and credit column. 11. Accounts are grouped into three categories namely, Assets, Liabilities and Owner accounts. 12. Ledgers contain set of accounts. 13. Each transaction should contain date, description, amount. 14. Journals should have at least date column, account debited column, account credited column and amount column.

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2.2.2 Semantics

Recording of information in various journals and ledgers are required to follow a well define procedure. These procedures give the necessary semantics for the accounting language. Semantics of a basic accounting system are as follows: 1. Each transaction recorded in a journal should satisfy the accounting equation. 2. For each transactions, total debits should equal to total credits. 3. Each transaction should at least be credited to one account and debited to one account. 4. Increases in assets are recorded in debit side and decreases in the value of assets are recorded in credit side of an asset account. 5. Increases in all liabilities and owner’s account are recorded as credits and decreases are recorded as debits. 6. Each transaction should be first recorded in a journal and then posted to a ledger. 7. All transactions should finally be posted to the general ledger.

2.3 The pilot system

This section explains the pilot system which is going to be used to explain the features of the specification language. The pilot system is a subsystem of an accounting system of a travel agent. This subsystem interacts with the passengers who request flight reservations and finally transactions are generated for the accounting system.

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2.3.1 Transactions generation system of a travel agent

Let us consider a travel agent who provides air line reservation services to passengers. The transactions generation system has to create transactions to the accounting system after going through various phases of air line bookings. Requirements for the system are as follows: 1. The system is activated when a request for a flight reservation is received from a passenger. 2. When a request for a reservation is made, the consultant should record the passenger details such as name, destination, intended departure date, intended air line and maximum cost. 3. If passenger specifies a particular air line, then consultant should contact the requested air line and get the reservation details such as possible departure date, flight number, departure time, destination, reservation status and cost. Reservation status will be one of the followings: “Confirmed”, “Waiting” or “Cancel”. 4. If the passenger does not specify an air line then the consultant has to search for a suitable air line and get the reservation details. 5. When the consultant gets the reservation details from the air line, he should contact the passenger and inform. 6. If the passenger agrees with the reservation details then consultant has to reserve the flight and generate a reservation transaction to the accounting system. 7. If the passenger is not satisfied about the reservation then cancel the reservation and ask him whether the consultant should try again. 8. If the passenger wants to try again, then the consultant has to follow the instructions starting either from 3 or 4. If not, remove the passenger details from the records. 9. If the reservation status is waiting, then ask the passenger whether he is ready to wait or not. If he is ready to wait then wait for confirmation. If not, follow the instruction in 7. 10. If the reservation status is cancel follow the instructions in 7.

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11. A passenger can cancel his request for reservation while he is waiting for confirmation without penalty. 12. Payment should be made after the reservation is confirmed and before departure in order to issue the ticket. 13. A passenger can cancel a reservation before payment. 14. If a passenger is made a cancel request after payment and 60 to 31 days before departure then cancel the reservation and charge him 10% of the cost. Generate the cancellation transaction to accounting system. 15. If a passenger is made a cancel request after payment and 30 to 15 days before departure then cancel the reservation and charge him 20% of the cost. Generate the cancellation transaction to accounting system. 16. If a passenger is made a cancel request after payment and 14 to 3 days before departure then cancel the reservation and charge him 30% of the cost. Generate the cancellation transaction to accounting system. 17. If a passenger is made a cancel request after payment and 2 days before departure then cancel the reservation and charge him 50% of the cost. Generate the cancellation transaction to accounting system. 18. If a passenger is made a cancel request after payment and on the day of departure then cancel the reservation and charge him 100% of the cost. Generate the cancellation transaction to accounting system.

2.4 Information systems’ requirements

Requirements of an information systems consist of three major components. 1. What. 2. Who. 3. When. These three components can be seen in any given information system. Therefore the specification language used to represent requirements should be capable of representing these three components. The component “What” considers the what should be done by a computerized system instead of how it can be done.

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This is normally the main topic of the requirement engineering. Even though the component “Who” is not essential for a functioning of a computerized information system, it is more useful to know about responsible person for an action in maintenance phase, specially when end-users and system analysts are new to the computerized system. However, it is necessary to know about the “When” component. Real world information systems consist of series of actions which are chronologically ordered. Activation of a particular action is entirely depended on actions already activated. Therefore without specifying temporal relations between event and activities in an information systems concisely, precisely and accurately, it is impossible to build a computerized information system. Therefore this research pays special attention on representing incomplete relative information.

2.5 View of a system

Building of a computer system requires a well design model for the application. Since accounting is the main application area of this research, the model selected should be flexible enough to represent any accounting system. As described above, accounting systems have lots of chronologically ordered external and internal events and activities. An event invoke one or more actions in accounting systems. That is the temporal order of the events and activities are very important. These actions may trigger another set of actions. On the other way, generation of one information in accounting system will lead to generation of several other information. The selected model should be capable to represent these characteristics of accounting systems. Entity-relationship models are widely used in data modeling than other models. However, entity-relationship modeling does not support the idea of creating new objects using one or more objects which is very important in accounting systems. For example, a particular journal can be considered as a collection of several similar transactions. In entity-relationship modeling, journals cannot be considered as a collection of transactions instead considered

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as two separate objects having some relation. Entity-relationship view of a system and another view of a system which can solve the problems with entityrelationship approach will be discussed in next sections.

2.5.1 Entity-relationship view of a system

In entity-relationship modeling approach, it is impossible to create a new entity with new attributes and attributes from parent entities as a result of a relationship between entities. It simply attaches entities to relationships. The new attributes are linked to the relationships and not to the entities [Chen 1976][Ullman 1982]. Bekke [Bekke 1992] shows that the entity-relationship modeling is not suitable for some complex problems. Figures 2.5 exhibits the entity-relationship representation of a relationship between Passengers and Flights, respectively.

2.5.2 View of a system adopted in this research

A system consists of objects which have properties as their attributes. Objects have relationships among them. These relationships will result in:

1. Modification to at least one attribute of the object, and/or 2. Creation of a new object with new attributes in addition to the original

attributes. New objects created will have relationships with other objects. However, Stefan Sigfried’s idea is little different [Sigfried 1996] in declining with above phenomena. He suggests complex classes contain object structures in addition to attributes and methods. His suggestion is not adopted in this research, because the idea of using objects as attributes in another object is supported by the temporal logic. This will be discussed in the section 5.1 of this thesis. Therefore, whatever the knowledge representation scheme adopted, it should be capable of representing 1) modification to at least one attribute of the object and 2) creation

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of a new object with new attributes in addition to the original attributes. However, the second requirement cannot be fulfilled if the well known entityrelationship representation (Figure 2.5) is used. Therefore, view of a system shown in Figure 2.6 which can satisfy both requirements is adopted in this research. The system starts functioning when an external or an internal event triggers an activity/activities under specific conditions. These activities will trigger other activities and so on [Martin 1993]. Events, activities, rules and conditions consist of objects and relationships. There are two types of events; namely external events and internal events. External events occur due to environmental changes while internal events occur due to execution of programs.

Reservation Date Name Number Destination

Passengers

Reservation

Flights

Figure 2.5 Entity-relationship representation

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Reservation Date

Name

Number

Destination

Passengers

Reservation

Flights

Figure 2.6 Representation adopted in this research

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CHAPTER III

REPRESENTING TEMPORAL KNOWLEDGE

This chapter discusses importance of incomplete relative temporal knowledge in information systems, the present approaches to represent temporal knowledge; Allen’s method, McDermott’s method, Vilain and Kautz’s method, Temporal operators, Time-sensitive Boolean operators in detail. Then the problems of present approaches to temporal knowledge representation (problems inherited with point based and interval time models and the necessity of complete time information of occurrence of events to represent temporal knowledge) will be discussed. Finally it introduces the new logical connective TAND which can solve the problems of above representation methods (because it is not based on any time models) and shows how TAND can be used to represent Allen’s temporal relations and existing temporal operators.

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3. 1 An incomplete temporal knowledge about events

While analyzing requirements of an information system, system analyst has to deal with what has already happened, what is happening now, what will happen and has to take some sort of actions according to them. However complete information about time of occurrence of an event may not be available. For example, in the transactions generating system, it is understood that a passenger will make a call for flight reservation but the exact time when he is going to call us is not available. For another example, there will be a cancellation after payments but the exact time when it will happen is not available. Therefore at the requirement analysis phase system analyst has to deal with the incomplete temporal knowledge about events rather than events with the complete information about exact time of occurrence. But the complete information about events may be required to answer a database query. However, to decide whether a passenger will make a cancellation request after the payment, it is not necessary to know the exact date of payment but it is enough to know that the payment is already made. Therefore it is sufficient to know that the passenger’s payment is found in the payment database. That means decisions can be made with relative time information. In software requirements specifications the requirements are stated with relative time information since the exact time information are not available at this stage. These show the importance of the incomplete relative temporal information about events in software requirements analysis phase. Therefore software requirements specification languages should be capable of representing incomplete relative temporal information about events.

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3.2 Conventional knowledge

methods

to

represent

temporal

Temporal knowledge representation models introduced so far can be grouped into two: namely point based and interval based temporal models. Point based time models represent time as a point in a real line while interval models represent time by time intervals. In point based models, time interval is represented as a set of time points which are bounded by start and end points. The following section describes conventional approaches to represent temporal knowledge.

3.2.1 Allen’s method

Allen [Allen 1983] has developed a temporal logic in which time intervals are the primitives. He points out the weaknesses of point based temporal models. Allen shows that there can be a time point in point based model where either two events could occur at the same time or non could occur. For example, consider the two events “Switch-On” and “Switch-Off”. Then according to Allen, there can be a time point where both “switch-On” and “Switch-Off” occur or none of them could occur. To solve this problem, he suggests time interval as the primitive for temporal representations. Time intervals can meet each other. Therefore “Switch-On” can meet “Switch-Off”. Further he suggests thirteen possible relations between time intervals, namely “before”, “after”, “during”, “contains”, “overlaps”, “overlapped-by”, “meets”, “met-by”, “starts”, “started-by”, “finishes”, “finished-by” and “equals”. Figure 3.1 depicts them graphically. Allen uses these temporal relations to define events and activities. In his temporal logic, he applies first-order predicates to represent temporal relations [Allen 1984].

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START(t1, t2)

: time interval t1 shares the same beginning as t2, but ends before t2 ends; FINISHES(t1, t2) : time interval t1 shares the same end as t2, but begins after t2 begins; BEFORE(t1, t2) : time interval t1 comes before interval t2, and they do not overlap in any way; OVERLAP(t1, t2) : interval t1 starts before t2, and they overlap; MEETS(t1, t2) : interval t1 comes before interval t2, but there is no any interval between them, i.e. t1 ends where t2 starts; EQUAL(t1, t2) : t1 and t2 are the same interval. To represent property holding over an interval T, Allen needs two other predicates; HOLDS( p, T) and IN(t, T). HOLDS(p ,T) means that the property p holds over an interval T and IN(t, T) means that the subinterval t is in interval T. He defines IN as follows: IN(t1, t2) ? (DURING(t1, t2) ∨ STARTS(t1, t2) ∨ FINISHES(t1, t2)). His definition for HOLDS(p, T) is HOLDS(p, T) ? (?t. IN(t, T) ? HOLDS(p, t)). The negation of a property p is defined as HOLDS(not(p), T) ? (?t. IN(t, T) ? ?HOLDS(p, t)). That means if a property p holds over an interval T, it holds over all subintervals of T. However, Allen could not define events and actions using HOLDS predicate. Therefore he defines another predicate OCCUR. The reason for use of another predicate is due to his definition of HOLDS predicate. HOLDS(p , T) says that property p holds over an interval T as well as all subinterval of T. OCCUR(e, t) says that the event e, occurs in the time interval t. Therefore to represent “event e1 occurs before event e2”. It is necessary to write

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p before q

p

q

q after p

p q p during q p p overlaps q q q overlapped-by p q contains p

p p meets q p p starts q q

q q met-by p

q started-by

p p finishes q p p equals q q q q finished- by p

Figure 3.1 Allen’s temporal relations

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OCCUR(e1, t1) ∧ OCCUR(e2, t2) ∧ BEFORE(t1, t2). However, Galton [Galton 1990] shows the weaknesses of Allen’s temporal interval in representing continues changes. According to Allen’s definition of HOLDS, to HOLDS(p, T) to be true p must be held in all subintervals of T. If p is continuously moving with time, p cannot be held in all subinterval at the same time. Hence if HOLDS(p, T) is true then p cannot move but has to rest over time interval T. Galton argues that in order to overcome this problem one must be prepared to treat time points and time intervals on an equal footing. Therefore we can think of three kind of time models: 1. Point based models, 2. Interval based models, and 3. Point-Interval based models.

3.2.2 McDermott’s method

There are two key concepts in McDermott’s temporal logic[McDermott 1982]. 1. More than one event can occur starting at a given instant. 2. Many events do not occur discontinuously. To capture these ideas, he defines universe as a collection of states. A state is an instantaneous snapshot of the universe. States are partially ordered by a relation “= <”. To represent “s1 comes before or identical to s2”, he writes (= < s1 s2). In this method, facts type and event types are used as building blocks. State (or time point) is used as the primitive. Mcdermott applies “Cambridge Polish” notation for logical formulas. Every term, atomic formula, and combination is of the form (p…), where p is a function, predicate or connective. His representation of universal quantifier, existential quantifier, “not” operator, “if” statement, “and” operator, “or” operator, “if and only if” statement and occurrence of events are as follows: Universal quantifier (forall) : (forall (-variables-) formula)

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Existential quantifier (exists) : (exists (-variables-) formula) “not” operator : (not formula) “if” statement : (if formula1 formula2) “and” operator : ( formula1 formula2…) “or” operator : (or formula1 formula2…) “if and only if” statement (iff) : (iff formula1 formula2) Occurrences of events (Occ) : (Occ state1 state2 event) “a” is an element of “x” : ( elt a x ). Occ( state1 state2 event) says that the event “event” happens between states “state1” and “state2”. Unbound variables are prefixed with “?”. Therefore, “?s1” represents the unbound variable “s1”. The expression (< s1 s s2 ) represents that the state “s” is in between states “s1” and “s2”. A CHRONICLE is a complete possible history of the universe, a totally ordered set of states extending infinitely in time. Definition of CHRONICLE : (iff (is CHRONICLE ?x) ; x is a CHRONICLE (and ; a set of states (forall (y) (if (elt y ?x) (is STATE y))) ; for all y, if y is an element of x then y is a STATE ; totally ordered (forall (s1 s2) ; for all s1, s2 (iff (and (elt s1 ?x) (elt s2 ?x)) ; if and only if s1 and s2 are elements of x then (or (< s1 s2) (> s1 s2) (= s1 s2)))) ; state s1 comes before state s2, state s2 comes ; before state s1 or state s1 and state s2 are ; identical ; infinite time (forall (t) ; for al t (exists (s) ; there exists s such that (and (elt s ?x) (= (d s) t)))))) ; state s is an element of x and date of state s is t

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where function “d” in (d s) gives the date of state “s”. Comments begin at “;”. Example : DAY is always followed by NIGHT and NIGHT by DAY. DAY and NIGHT never overlap. DAY and NIGHT are mutually exclusive (except at boundaries): (if (and (Occ ?s1 ?s2 DAY) (Occ ?s3 ?s4 NIGHT)) ; if DAY occurs between unbound states s1 and s2 and ; NIGHT occurs between unbound states s3 and s4 (forall (s) ; for all states s (if (and (= < ?s1 s ?s2) ( = < ?s3 s ?s4)) ; if state s is identical to unbound state s1 or s2 or ; in between unbound states s1 and s2 and ; state s is identical to unbound state s3 or s4 or ; in between unbound states s3 and s4 then (or (= s ?s2 ?s3) ; state s is identical to unbound state s2 or s3 or (= s ?s1 ?s4)))) ) ; state s is identical to unbound state s1 or s4. Each takes a nonzero amount of time : (if (or (Occ ?s1 ?s2 DAY) (Occ ?s1 ?s2 NIGHT) ; if DAY or NIGHT occurs between unbound states s1 and s2 then (< ?s1 ?s2)) ; unbound state s1 comes before unbound state s2. Each follow the other: (follows DAY NIGHT) ; DAY follows NIGHT (follows NIGHT DAY) ; NIGHT follows DAY where (iff (follows ?e1 ?e2) ; If and only if unbound event e1 follows unbound event e2 then (if (Occ ?s1 ?s2 ?e1) ; unbound event e1 occurs between unbound states s1 and s2 (forall (ch) (if (elt ?s2 ch)

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; for all CHRONICLE ch, if state s2 is an element of ch (exists (s3) ; there exists s3 such that (and elt s2 ch) ; state s2 is an element of ch and (Occ ?s2 s3 ?e2))))))) ; unbound event e2 occurs between ; unbound states s2 and s3.

3.2.3 Vilain and Kautz’s method

In Vilain and Kautz’s point algebra or PA networks[Vilain 1986] [ Peter 1992], there are three basic relations that can hold between two points: <, = and >. The relative information between two points is represented as a disjunction of the basic relations <, =, >. For example, the relation {<, =} between points A and B is represented by (A < B) ∨ (A = B). Start and end points of a time interval T are represented by T- and T+ . For example, “time interval T1 before interval T2” can be expressed as follows: T1 before T2 : (T1- < T1+) ∧ (T2- < T2+) ∧ (T1+ < T2-) .

However, all Allen’s temporal relations cannot be represented using PA algebra. For example, “T1 meets T2” cannot be represented properly. Therefore the subset of Allen’s temporal relations that can be represented using the relations {<, ≤, =, >, ≥, ≠} are allowed in PA networks. Representation of Allen’s thirteen interval relations using point algebra is shown in Figure 3.2.

3.2.4 Temporal operators

Temporal operators available in temporal logic [Kroger 1987][Barringer 1996][Manna 1995][Torsun 1995] are yet another approach to represent temporal information. However, application of these operators are limited to point based temporal models. Most computer science applications including

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program verifications [Manna 1995] employ these temporal operators. Standard temporal operators are as follows: Next : Next A is true at time T if and only if A is true at time T+1. Last : Last A is true at time T if and only if A is true at time T-1. Always in the future : Always in the future A is true at time T if and only if A is true at all time point > T. Always in the past : Always in the past A is true at time T if and only if A is true all time points < T. Sometimes in the future : Sometimes in the future A is true at time T if and only if A is true at a time point T1 where T1 > T. Sometimes in the past : Sometimes in the past A is true at time T if and only if A is true at a time point T1 where T1 < T. Until : A Until B is true at time T if and only if B is true at a time point T1 and A is true at all the time points t where T < t < T1. Since : A Since B is true at time T if and only if B is true at a time point T1 and A is true at all time points t where T1 < t < T. However, these temporal operators are not expressible enough to represent Allen’s temporal relations. Further they have weaknesses inherited from point based temporal models

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pp before q q after p

p

p+ q (p- < p+) ∧ (q- < q+) ∧ (p+ < q-)

p p during q q contains p (p- < p+) ∧ (q- < q+) ∧ (q- < p-) ∧ (p+ < q+) p p overlaps q q overlapped-by p (p- < p+) ∧ (q- < q+) ∧ (p- < q-) ∧ (p+ < q+) p meets q q met-by p p starts q q started-by p p q p q Cannot be represented properly q q

(p- < p+) ∧ (q- < q+) ∧ (p- = q-) ∧ (p+ < q+) p p finishes q q finished- by p p equals q q (p- < p+) ∧ (q- < q+) ∧ (p- > q-) ∧ (p+ = q+) p q (p- < p+) ∧ (q- < q+) ∧ (p- = q-) ∧ (p+ = q+) Figure 3.2 Allen’s temporal relations using PA algebra .

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3.2.5 Time-sensitive Boolean operators

The time-sensitive Boolean operators suggested by Bassiouni et al. [Bassiouni 1994] are a different approach to represent and reason about temporal information. They argue that the standard Boolean connectives based on the True/ False logic are not suitable for handling events/activities defined over periods of time. For example, they say that the events p and q in ( p AND q ) are not necessary to occur at the same time. They define a new time-sensitive operator ANDS , ORS and NOTS to represent the concurrent occurrence of events. Hence the events p and q in (p ANDS q) occur simultaneously. Their definition of ANDS is as follows: Definition: If S1 and S2 are two sets of time intervals, then the expression S1 ANDS S2 returns the null set φ if S1 and S2 do not have any overlapping intervals. Otherwise ANDS returns the time sub-intervals that are contained in both S1 and S2 (i.e. it returns their overlapping). Similarly, they define time-sensitive ORS and NOTS. Definition: The time-sensitive unary operator NOTS returns the set of intervals which is the complement of its operand with respect to the universal interval <0, NOW>. Notice therefore that if S is a set of time intervals, then the expression NOTS (S) returns φ only if the union of the intervals of S is the universal interval. Definition: If S1 and S2 are two sets of time intervals, then the expression S1 ORS S2 returns φ only if both S1 and S2 are null sets. Otherwise, it returns the union of the intervals of S1 and S2.

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Time intervals are defined over integer time points. Further they show that the time-sensitive Boolean operators satisfy the properties of standard Boolean connectives.

3.2.6 Problems with the above methods

All temporal knowledge representation methods discussed above except temporal operators, require complete time information to represent the temporal information about events. They need exact time of occurrence of events. However, as described in the section 3.1, information systems consist of incomplete relative temporal knowledge rather than having complete time information of occurrence of events. Therefore these methods can not be used in software specifications to represent incomplete relative temporal knowledge. The temporal operators available in the literature are not expressible enough to represent temporal relations suggested by J. F. Allen [Allen 1983]. On the other hand, temporal operators inherit the weaknesses of point based models. Temporal operators are more suitable in program verifications rather than software requirement specifications. Although Allen’s temporal relations require complete time information of events, software requirements specifications require them only to represent the temporal information. However, Allen’s temporal intervals have weaknesses in representing continues changes. Therefore a temporal knowledge representation method which can represent Allen’s relative temporal relations with incomplete time information of occurrence of events is required. This thesis suggests a new method to represent incomplete relative temporal knowledge [Wijayarathna 1997] which can overcome these problems.

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3.3 The AND_THEN (TAND) connective

3.3.1 The standard logical connective AND

Logical connective AND ( ∧ ) does not discriminate the temporal order of events. Even though two events occur in different time zones, as far as the standard logical connective AND is concerned, they occur in the same time zone. Hence it treats the events as concurrent events. The whole time line is a single time zone for the AND connective. The “time zone” represents both time points and time intervals. Therefore the standard logical connective AND is not suitable for applications where the temporal order of events play an important role. As describe in the previous section, Bassiouni et al.[Bassiouni 1994] argues that events p and q in “p ∧ q” are not necessary to occur simultaneously. However, “p ∧ ?p = false” is one of the universally accepted properties of the standard AND connective ( Idempotent law ). Whenever p is true, ?p becomes false and when p is false, ?p becomes true. Therefore in both cases “( p ∧ ?p ) = ( true ∧ false )= ( false ∧ true )” is false. According to their argument p and ?p are not required to occur simultaneously; i.e. it is not required for “p ∧ ?p” to be false. Because p can be true at one time and ?p can be true at another time giving “ true ∧ true = true”. This contradicts with the universally accepted property “p ∧ ?p = false”. However the standard logical connective AND considers events p and q in “p ∧ q” as having occurred simultaneously, even though they really occur in two different time zones. Hence it is more realistic and logical to restrict standard logical connective AND to represent simultaneous events and to introduce the new temporal logical connective to represent other cases.

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3.3.2 The TAND ( ∏ ) logical connective

For any given two events p and q, the truth state at any time “t” will be one of the following: 1. Only p occurs, 2. Only q occurs, 3. Both p and q occur: 3.1 Both p and q occur simultaneously, 3.2 p and q occur sequentially, 4. Neither p nor q occur. Figure 3.3 shows these situations graphically. Event p Time T Only p occurs Event q Only q occurs Event q Event p Time T Both p and q occur simultaneously Event p Event q Time T p and q occur sequentially Time T Time T

Neither p nor q occur Figure 3.3 Possible occurrences of two events

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That is at any given time t, 1. p = true , 2. q = true, 3.1 ( p ∧ q ) = true, 3.2 ??? (cannot represent using existing connectives) or 4. ( ?p ∧ ?q ) = true . Here ∧ represents concurrent events. Therefore we can represent any relative temporal relation using these four truth states provided that we have a method to represent the sequential occurrence of events. For example to represent “p meet q” , we have to express that there exists a state where “p = true ” and then another state where “q = true ”. That is “p meet q” is true, iff { p = true and then q = true} is true. In this representation one state is followed by another state. As discussed in the section 3.3.1, the standard logical connective AND does not distinguish whether p and q occur simultaneously or not. Hence what is required is a logical connective to represent and then in such a way that “true and then true” is equal to true. The and then is defined as TAND logical connective and is represented by ∏ in this thesis. Definition: If p and q are true in time zones T1 and T2 respectively. Then (p ∏ q ) is true if and only if a) q is true immediately after p is true. b) Both p and q are not true at the same time. c) At any given time, either p or q is true. Axioms Now three important axioms about TAND ( ∏ ) can be defined. A1. ( p ∏ q ) ? ( ( p ∨ q ) ∧ ?( p ∧ q) ) A2. ( p ∏ q ) ≠ ( q ∏ p ) A3. Time zone T2 follows time zone T1.

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Therefore at any given time t, if ( p ∏ q ) is true then either p or q must be true. However both p and q cannot be true or false simultaneously. Properties of AND and TAND 1. Idempotent Law p ∧ ?p = false ; either p is true or false. p ∏ ?p = false ; p is true and then p is true. p is false and then p is true. p is false and then p is false. p ∏ ?p = true ; p is true and then p is false. Therefore TAND does not obey the Idempotent law as AND does. 2. Commutative law p∧q=q∧p p ∏q ≠ q∏ p Since p ∏ q means that p is true and then q is true while q ∏ p means that q is true and then p is true. Two things can be ascertained from p ∏ q. First, p and q do not occur at the same time and second, if T1 and T2 are sets of time zones where q and p occur respectively then ?t1∈T1 > ?t2∈T2. Therefore TAND does not obey the commutative law as AND does. However both “p ∧ q” and “q ∧ p” express that p and q occur at the same time. Therefore the order of events is insignificant. 3. Distributive law p ∧ ( q ∨ s ) = ( p ∧ q ) ∨ ( p ∧ s) p∨(q∧s)=(p∨q)∧(p∨s) p ∏ ( q ∨ s ) = (p ∏ q ) ∨ ( p ∏ s) (table 1) p ∏ ( q ∧ s ) = (p ∏ q ) ∧ ( p ∏ s) (table 2)

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TAND obeys the distributive law as AND does. The truth tables, Table 1 and Table 2 show this in detail. 4. Associative law p∧(q ∧ s)= (p∧q)∧ s p∏(q ∏ s)= (p∏q)∏s TAND obeys the associative law as AND does because associative law does not change the order of the occurrence of events. 5. DeMorgan’s law ?( p ∧ q ) = ?p ∨ ?q ?( p ∏ q ) ≠ ?p ∨ ?q If TAND obeys DeMorgan’s law then ?( p ∏ q ) = ?p ∨ ?q. Since the conventional logical connective ∧ obeys the DeMorgan’s law, ?p ∨ ?q = ?( p ∧ q ). However, by the definition, ( p ∏ q ) should be false when ( p ∧ q ) is true. Therefore, TAND does not obey DeMorgan’s law. As shown above TAND obeys distributive and associative laws as AND does but not other laws. (q and s occur in time zone T2 and p occurs in time zone T1 ; T1 follows T2 ) p

T T T T F F F F

q

T T F F T T F F

s

T F T F T F T F

( q ∨ s ) ( p ∏ q) ( p ∏ s) p ∏ ( q∨ s ) ( p ∏ q )∨ ( p ∏ s )

T T T F T T T F T T F F F F F F T F T F F F F F T T T F F F F F T T T F F F F F

Table 1 Truth table for distributive law (OR)

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p

T T T T F F F F

q

T T F F T T F F

s

T F T F T F T F

( q ∧s )

T F F F T F F F

( p ∏ q) ( p ∏ s) p ∏ ( q∧ s ) ( p ∏ q ) ∧ ( p ∏ s )

T T F F F F F F T F T F F F F F T F F F F F F F T F F F F F F F

Table 2 Truth table for distributive law (AND) Case 1. Suppose ( p TAND q ) is true and p is true. Now for axiom A1 to be true, right hand side of axiom A1 must be true, since the left hand side is true. ( A ? B is true iff B is true or A is false.) That is, ( p TAND q ) ? ( q ∧ ?p ) ∨ ( p ∧ ?q )

? ( q ∧ ?p ) ∨ ( p ∧ ?q ) = T (since ( p TAND q ) = T) ? ( q ∧ F ) ∨ ( T ∧ ?q ) = T ? F ∨ ?q = T ? ?q = T ? q=F

(since ( p = T) ) (since ( q ∧ F) = F and ( T ∧ ?q ) = ?q ) (since F ∨ ?q = ?q )

Starting with both ( p TAND q ) and p are true, it is inferred that q is false. Therefore it proves that if ( p TAND q ) is true and p is true then q must be false. Case 2. Now let us assume that ( p TAND q ) is true and q is true. Then applying the same logic it can be shown that always p must be false. Case 3. Let us say that ( p TAND q ) is true and p is false. Then, ( p TAND q ) ? ( q ∧ ?p ) ∨ ( p ∧ ?q )

? ( q ∧ ?p ) ∨ ( p ∧ ?q ) = T (since ( p TAND q ) = T)

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? ( q ∧ T ) ∨ ( F ∧ ?q ) = T ? q∨ F =T ? q =T

(since ( p = F) ) (since ( q ∧ T ) = q and ( F ∧ ?q ) = F )

(since q ∨ F = q ) This means whenever ( p TAND q ) is true and p is false, q must be true. Case 4. Now let us say that ( p TAND q ) is true and q is false. Then it can be shown that p must be true. Case 5. In contrast to the other cases, suppose that (p TAND q), p and q are true. ( p TAND q ) ? ( q ∧ ?p ) ∨ ( p ∧ ?q )

? ( q ∧ ?p ) ∨ ( p ∧ ?q ) = T (since ( p TAND q ) = T) ? (T∧F)∨ (T∧F)=T ? F∨ F =T ? F=T

(since ( p = q = T) )

That is a contradiction. Therefore both p and q cannot be true at the same time when (p TAND q) is true. Case 6. Now let us assume that (p TAND q) is true and both p and q are false. ( p TAND q ) ? ( q ∧ ?p ) ∨ ( p ∧ ?q )

? ( q ∧ ?p ) ∨ ( p ∧ ?q ) = T (since ( p TAND q ) = T) ? (F∧T)∨ (F∧T)=T ? F∨ F =T ? F =T

(since ( p = q = F) )

This shows that both p and q cannot be false at the same time when (p TAND q) is true. Therefore considering above cases 1, 2, 3, 4, 5 and 6 it can be concluded that whenever ( p TAND q ) is true, either p or q should be true. However both p and q cannot be true or false simultaneously. This is a significant result as it

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proves that TAND can solve the problem that occur in point based temporal models. Allen[Allen 1983] says that there exists a time point in point based models where either ( p ∧ q ) is true or ( ?p ∧?q) is true. This means that the truth state of any given time t in point based temporal models can be one of the following: only p is true, only q is true, ( p ∧ q ) is true or ( ?p ∧?q) is true. However it is proved that there are only two truth states when ( p TAND q ) is true ( only p is true or only q is true ). This is Allen’s[Allen 1983] (p meets q) temporal relation. Since TAND is not based on any time models, it can be used to represent point based applications, interval based application as well as point-interval based applications.

3.3.3 Representation of Allen’s temporal relations using TAND

The important factor to remember when representing temporal relations with TAND is that the events p and q should be mutually exclusive. That is to say always ( p AND q ) should be false. Events p and q can contain more than one event. In this situation p and q can be considered as composite events. In this thesis events represent single events as well as composite events. Since it is already shown how to represent ( p meets q ) using ∏, in this section, representation of other temporal relations using TAND will be shown. First let us consider the case of ( A ∏ B ∏ C ) where A, B, and C are three mutually exclusive events. By definition of TAND, a) ( A ∧ B ∧ C) should be false. b) Time of occurrence of A< time of occurrence of B < time of occurrence of C. The ( A ∏ B ∏ C ) is equal to ( A ∏ B ) ∏ C because TAND obeys associative law. Therefore expressions having more than two mutually exclusive events can be decomposed into sub expressions having at most two events keeping the meaning of the expression unchanged.

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Then by axiom A1, ( A ∏ B ) ? ( A ∨ B ) ∧ ?( A ∧ B ). If ( A ∏ B ) is D, then ( A ∏ B ∏ C ) = ( D ∏ C ). Applying axiom A1 again we have ( D ∏ C ) ? ( D ∨ C ) ∧ ? ( D ∧ C ). Then ( A ∏ B ∏ C ) = ( D ∏ C )? (D∨C) ∧? (D∧C) = (D∧?C )∨ (?D∧C) ? [(( A ∧ ?B )∨ ( ?A ∧ B )) ∧?C ] ∨ [ ?(( A ∨ B ) ∧ ?( A ∧ B )) ∧ C ] ? ( A ∧ ?B ∧ ?C ) ∨ ( ?A ∧ B ∧ ?C ) ∨ (B ∧ A ∧ C ) ∨ (?A ∧ ?B ∧ C ) Since A, B and C are mutually exclusive events, (B ∧ A ∧ C ) is false. ? ( A ∧ ?B ∧ ?C ) ∨ ( ?A ∧ B ∧ ?C ) ∨ (?A ∧ ?B ∧ C ) ( F1 ) ? ( A=T, B=F, C=F ) or ( A=F, B=T, C=F ) or (A=F, B=F, C=T ) ? Only one of A, B and C can occur at any given time t Therefore it is enough to show that (X ∧ Y ∧ Z ) is false to prove the accuracy of any given temporal representation ( X ∏ Y ∏ Z ). p before q = p ∏ ( ?p ∧ ?q) ∏ q p before q is true, iff only p is true and then ( ?p ∧ ?q) is true and then only q is true. Since (p ∧ ?p ∧ ?q ∧ q ) is always false, at any given time t, only p can occur, only q can occur or both p and q cannot occur. In addition to this TAND gives the chronological order of the events. Therefore the representation of ( p before q ) is sound. If A= p, B = ( ?p ∧ ?q), C = q then applying F1 p before q ? [ p ∧?(?p ∧?q) ∧?q ]∨[?p ∧ (?p∧?q) ∧?q ] ∨ [?p ∧?( ?p ∧ ?q) ∧ q ] ( p before q ) ? [ p ∧ ?(?p∧?q)∧?q ] ∨ [?p∧?q ] ∨ [?p∧?(?p∧?q) ∧ q ]

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If ( p before q ) = T and p = T then, ( p before q ) ? [T∧?( F∧?q)∧?q ]∨ [F∧?q ]∨ [F ∧ ?( F ∧ ?q) ∧ q ] = T ? [ T ∧ ?F ∧ ?q ] ∨ F ∨ F] = T ? [ T ∧ T ∧ ?q ] ∨ F ∨ F] = T ? [?q ∨ F] = T ? ?q = T ? q=F If ( p before q ) = T and p = F then, ( p before q ) ? [ F∧ ?( T∧?q) ∧?q ]∨ [T∧?q ]∨ [T ∧ ?( T∧?q) ∧ q ] = T ? [ F∧ ?(?q) ∧?q ]∨ [T∧?q ]∨ [T ∧ ?(?q) ∧ q ] = T ? [ F∧ q ∧?q ]∨ [?q ]∨ [q ∧ q ] = T ? [ F ∨ ?q ∨ (q ∧ q) ] = T ? [ ?q ∨ q ] = T ? q = F or q = T If ( p before q ) = T and q = T then, ( p before q ) ?[p ∧?( ?p∧ F)∧ F ]∨ [?p ∧ F ]∨ [?p ∧?(?p ∧ F)∧ T ] = T ?[p ∧?(F)∧ F ]∨ [?p ∧ F ]∨ [?p ∧?(F)∧ T ] = T ?[p ∧T∧ F ]∨ [?p ∧ F ]∨ [?p ∧T∧ T ] = T ? [F ∨ F ∨ ?p ] = T ? ?p = T ? p=F If ( p before q ) = T and q = F then, ( p before q )? [ p ∧?(?p ∧ T) ∧ T ]∨ [?p ∧ T ]∨ [?p ∧?(?p ∧ T)∧F] = T ? [ p ∧ ?( ?p ∧ T) ∧ T ] ∨ [?p ∧ T ] ∨ [?p ∧ ?( ?p ∧ T) ∧ F ] = T ? [ p ∧ ?( ?p) ∧ T ] ∨ [?p ∧ T ] ∨ [?p ∧ ?( ?p) ∧ F ] = T ? [ p ∧ p ∧ T ] ∨ [?p ∧ T ] ∨ [?p ∧ p ∧ F ] = T

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? [ p ] ∨ [?p] ∨ [F ] = T ? p = T or p = F If (p before q) = T and p = q = T, then ( p before q ) ? [ p ∧ ?(?p∧?q)∧?q ] ∨ [?p∧?q ] ∨ [?p∧?(?p∧?q) ∧ q ] = T ? [ T ∧ ?(F∧F)∧F ] ∨ [F∧F ] ∨ [F∧?(F∧F) ∧ F ] = T ?[F∨F∨F]=T ?F =T This means that both p and q cannot be true at the same time when (p before q) is true. If (p before q) = T and p = q = F, then ( p before q ) ? [ p ∧ ?(?p∧?q)∧?q ] ∨ [?p∧?q ] ∨ [?p∧?(?p∧?q) ∧ q ] = T ? [ F ∧ ?(T∧T)∧T ] ∨ [T∧T ] ∨ [T∧?(T∧T) ∧ F ] = T ?[F∨T∨F]=T ?T=T This means that both p and q can be false at the same time when (p before q) is true. This shows that at any given time t, only p occurs or only q occurs but both cannot occur. That is the combination of events required to occur ( p before q ) or ( q after p ). TAND connects them so as to give the required chronological order. p overlaps q = ( p ∧ ?q) ∏ ( p ∧ q ) ∏ ( q ∧ ?p ) p overlaps q is true, iff p is true and then ( p ∧ q ) is true and then q is true. If the expression “overlaps” is accurate, it would be able to show that for any given time t, only p occurs, only q occurs or both p and q occur. Since [ ( p ∧ ?q) ∧ ( p ∧ q ) ∧ ( q ∧ ?p ) ] is false, the representation of (p overlaps q) is accurate.

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Let us say A = ( p ∧ ?q), B = ( p ∧ q ) and C = ( q ∧ ?p ). Then from F1, ( p overlaps q) = (A∏B∏ C )?(A∧?B∧?C )∨ (?A ∧B∧?C)∨(?A∧?B∧C) ?((p∧?q)∧?(p∧q)∧?(q∧?p))∨ ?(p∧q)∧ (q∧?p)). If ( p overlaps q ) = T and p = T, then ( p overlaps q ) = T ? ? ((p∧?q)∧?(p∧q)∧?(q∧?p))∨(?(p∧?q)∧(p∧q)∧?(q∧?p)) ∨ (?(p∧?q)∧ ((T∧?q)∧?(T∧q)∧?(q∧F))∨(?(T∧?q)∧(T∧q)∧?(q∧F)) ∨ (?(T∧?q)∧ ?(p∧q)∧ (q∧?p)) =T ?(T∧q)∧ (q∧F)) =T ? ((?q)∧?(q)∧?(F))∨(?(?q)∧(q)∧?(F)) ∨ (?(?q)∧ ?(q)∧ (F)) =T ? (?q∧?q∧T) ∨ (q ∧ q ∧ T) ∨ (q ∧?q∧ F) =T ?(?q ∨ q ∨ F) =T ?q = T or q = F If ( p overlaps q ) = T and p = F, then ( p overlaps q ) = T ? ((p∧?q)∧?(p∧q)∧?(q∧?p))∨(?(p∧?q)∧(p∧q)∧?(q∧?p)) ∨ (?(p∧?q)∧ ∨ (?(F∧?q)∧ ?(p∧q)∧ (q∧?p)) =T ?((F∧?q)∧?(F∧q)∧?(q∧T))∨(?(F∧?q)∧(F∧q)∧?(q∧T)) ?(F∧q)∧ (q∧T)) =T ?(F ∧?(F)∧?(q)) ∨ (?(F)∧(F)∧?(q)) ∨ (?(F)∧ ?(F)∧ (q)) =T ?(F ∨ F ∨ (T ∧ T ∧ q) =T ? ( F ∨ F ∨ q ) =T ? q =T If ( p overlaps q ) = T and q = T, then ( p overlaps q ) = T ?((p∧F)∧?(p∧T)∧?(T∧?p))∨(?(p∧F)∧(p∧T)∧?(T∧?p)) ∨ (?(p∧F)∧ ?(p∧T)∧ (T∧?p)) =T (?(p∧?q)∧(p∧q)∧?(q∧?p))∨ (?(p∧?q)∧

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?(F∧?p∧p) ∨ (?F∧p∧p) ∨ (?F∧ ?p∧ ?p) =T ?( F ∨ p ∨ ?p) =T ? p = T or p = F If ( p overlaps q ) = T and q = F, then ( p overlaps q ) = T ?((p∧T)∧?(p∧F)∧?(F∧?p))∨(?(p∧T)∧(p∧F)∧?(F∧?p)) ∨ (?(p∧T)∧ ?(p∧F)∧ (F∧?p)) =T ?((p)∧?(F)∧?(F))∨(?(p)∧(F)∧?(F)) ∨ (?(p)∧ ?(F)∧ (F)) =T ?(p∧T∧T) ∨ (?p∧F∧T) ∨ (?p∧ T∧ F) =T ?(p ∨ F ∨ F) =T ?p=T If (p overlaps q) = T and p = q = T, then ( p overlaps q ) = T ? ((p∧?q)∧?(p∧q)∧?(q∧?p))∨(?(p∧?q)∧(p∧q)∧?(q∧?p)) ∨ (?(p∧?q)∧ ?(T∧T)∧ ?(p∧q)∧ (q∧?p)) =T ?((T∧F)∧?(T∧T)∧?(T∧F))∨(?(T∧F)∧(T∧T)∧?(T∧F))∨(?(T∧F)∧ (T∧F)) =T ?(F∧F∧T)∨(T∧T∧T)∨(T∧ F∧ F) =T ? F∨ T ∨ F =T ? T =T That is, both p and q can be true at the same time when (p overlaps q) is true. If (p overlaps q) = T and p = q = F, then ( p overlaps q ) = T ? ((p∧?q)∧?(p∧q)∧?(q∧?p))∨(?(p∧?q)∧(p∧q)∧?(q∧?p)) ∨ (?(p∧?q)∧ ?(p∧q)∧ (q∧?p)) =T

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?((F∧T)∧?(F∧F)∧?(F∧T))∨(?(F∧T)∧(F∧F)∧?(F∧T))∨(?(F∧T)∧ (F∧T)) =T ?(F∧T∧T)∨(T∧F∧T)∨(T∧ T∧ F) =T ? F∨ F ∨ F =T

?(F∧F)∧

? F =T That is, both p and q cannot be true at the same time when (p overlaps q) is true. This indicates that at any given time t only p occurs, both p and q occur or only q occurs. That is the situation where ( p overlaps q ) or ( q overlappedby p ) occur. Proper chronological order represented by TAND connective gives the required representations for ( p overlaps q ) or ( q overlapped-by p ).

p starts q = ( p ∧ q ) ∏ (q ∧ ?p) (p starts q ) is true, iff ( p ∧ q ) is true and then q is true. Hence at any time, either ( p ∧ q ) or (q ∧ ?p) should be true. From axiom A1, ( p starts q) ? [ ( p ∧ q ) ∨ (q ∧ ?p) ] ∧ ?[ ( p ∧ q ) ∧ (q ∧ ?p) ] ?[ ( p ∧ q ) ∨ (q ∧ ?p) ] ∧ ?[ F ] ? [ ( p ∧ q ) ∨ (q ∧ ?p) ] If ( p starts q) = T and p = T, then ( p starts q) ? [ ( p ∧ q ) ∨ (q ∧ ?p) ] ∧ ?[ ( p ∧ q ) ∧ (q ∧ ?p) ] = T ?[ ( T ∧ q ) ∨ (q ∧ F) ] = T ? q =T If ( p starts q) = T and p = F, then ( p starts q) ? [ ( p ∧ q ) ∨ (q ∧ ?p) ] = T ? [ ( F ∧ q ) ∨ (q ∧ T) ] = T ?q=T

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If ( p starts q) = T and q = T, then ( p starts q) ? [ ( p ∧ q ) ∨ (q ∧ ?p) ] = T ? [ ( p ∧ T ) ∨ (T ∧ ?p) ] = T ? [ ( p ∧ T ) ∨ (T ∧ ?p) ] = T ? [ p ∨ ?p ] = T ? p = T or p = F If ( p starts q) = T and q = F, then ( p starts q) ?[ ( p ∧ q ) ∨ (q ∧ ?p) ] = T ? [ ( p ∧ F ) ∨ (F ∧ ?p) ] = T ?F∨F=T This cannot happen. That is our assumption q = F is wrong. Hence if ( p starts q ) is true, always q should be true. If ( p starts q) = T and p = q = T, then ( p starts q) ?[ ( p ∧ q ) ∨ (q ∧ ?p) ] = T ? [ ( T ∧ T ) ∨ (T ∧ F) ] = T ?T=T That is, both p and q can be true at the same time when (p starts q) is true. If ( p starts q) = T and p = q = F, then ( p starts q) ?[ ( p ∧ q ) ∨ (q ∧ ?p) ] = T ? [ ( F ∧ F ) ∨ (F ∧ T) ] = T ?F=T That is, both p and q cannot be true at the same time when (p starts q) is true.

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These show that there can only be two possible occurrences of events when ( p starts q ) is true: p and q both occur or only q occurs. These are the required truth states for ( p starts q ) or ( p ends q ) to occur. TAND connects these events in chronological order. Depending on the order of the occurrences either ( p starts q ) or ( p ends q ) will occur. p during q = (q ∧ ?p) ∏ ( p ∧ q ) ∏ (q ∧ ?p) ( p during q ) is true iff (q ∧ ?p) is true and then ( p ∧ q ) is true and then (q ∧ ?p) is true. The temporal relation ( p during q ) can be represented as (q ∧ ?p) ∏ ( p ∧ q ) ∏ ( p ∧ q ) ∏ (q ∧ ?p). Because ( p ∧ q ) ∏ (p ∧ q) equals ( p ∧ q ). Let ( p ∧ q ) occur during time zone TL. The TL can be divided into two time zones: TL1 and TL2 such that TL1 + TL2 equals TL and TL1 follows TL2. Then ( p ∧ q ) occur in time zone TL1 and then occur in time zone TL2. Therefore ( p ∧ q ) can be expressed as ( p ∧ q ) ∏ ( p ∧ q ) without making any errors. Then (p during q ) = (q ∧ ?p) ∏ ( p ∧ q ) ∏ ( p ∧ q ) ∏ (q ∧ ?p) = [(q ∧ ?p) ∏ ( p ∧ q )] ∏ [( p ∧ q ) ∏ (q ∧ ?p)] = X ∏ Y if [(q ∧ ?p) ∏ ( p ∧ q )] = X and [( p ∧ q ) ∏ (q ∧ ?p)] = Y. Then X = (q ∧ ?p) ∏ ( p ∧ q ) ? [(q ∧ ?p) ∨ ( p ∧ q )] ∧ ?[ (q ∧ ?p) ∧ ( p ∧ q )] ? (q ∧ ?p) ∨ ( p ∧ q ) Y = ( p ∧ q ) ∏ (q ∧ ?p) ? [ ( p ∧ q )∨ (q ∧ ?p) ] ∧ ?[ ( p ∧ q ) ∧ (q ∧ ?p)] ?( p ∧ q )∨(q ∧ ?p) = (q ∧ ?p) ∨ ( p ∧ q ) . This means, it is sufficient to consider either only X or Y. X represents ( p ends q ) and Y represents( p starts q ). So( p during q ) can be represented as a combination of ( p ends q ) and ( p starts q). That is ( p during q ) = ( p ends and

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q) TAND ( p starts q). It is shown in the earlier section that only q occurs and then both p and q occur are the required events for ( p ends q) to occur. The reverse sequence of events, that is both p and q occur first and then only q occur are the necessary order of events for the event ( p starts q ) to occur. Therefore it can be concluded that the required order of events ( p during q ) to occur is, first only q occurs and then both p and q occur and then only q occurs. p equals q = (p ∧ q) ( p equals q ) is true, iff p and q are both true at the same time. Since standard logical connective AND is redefined to represent only concurrent situations the required representation is (p ∧ q). Throughout this section it is proved that TAND can be used to represent temporal relations suggested by J. F. Allen[Allen 1983] uniquely and properly. Figure 3.4 shows them graphically.

3.3.4 Representation of temporal operators using TAND

The logical temporal connective TAND is capable of representing other temporal operators available in temporal logic. Even though there are several temporal operators such as sometimes in the past, always in the past, sometimes in the future, always in the future, next, last, until and since available in the literature, Barringer et al. [Barringer 1996] show that all temporal operators can be represented using Until and Since operators. Therefore this section presents how TAND can be used to represent Until and Since operators. Let us first consider what M, i ├ p is. It means that formula p will be interpreted in model M at the state with index i [Torsun 1995][Barringer 1996]. In other words, it says that formula p holds in model M at state i or p is true at state i in model M. States are defined over the discrete linear time model. Detailed explanation about this can be found in the next chapter. This section simply employs the definition of M, i ├ p.

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p Until q = True ∏ (?q ? ( p ∏ q)) Proof Case 1. M, I ├ p Until q iff there is some k > i such that M, k ├ q and for every j, if i < j < k then M, j ├ p ( By the definition of Until ). That is, M, i ├ p Until q ? there is some k>i such that M, k ├ q and for every j, if i < j < k then M, j ├ p. ?[ M, i+1 ├ q or M, i+1├ p and M, i+2 ├ q or M, i+1├ p and M, i+2 ├ p and M, i+3 ├ q or …. …. M, i+1├ p and M, i+2 ├ p and ………. and M, k-1├ p and M, k ├ q]. ? [ M, i+1 ├ q or M, i+1 ├ (p ∏ q) or M, i+1 ├ (p ∏ p ∏ q) or …. …. M, i+1 ├ (p ∏ p ∏…………………….. ∏ p ∏ q)]. ? [ M, i+1 ├ q or M, i+1 ├ (p ∏ q) or M, i+1 ├ (p ∏ q) or …. …. M, i+1 ├ (p ∏ q)]; Since p ∏ p = p. ? M, i+1 ├ q or M, i+1 ├ [ (p ∏ q) ∨ (p ∏ q) ∨ ……………∨ (p ∏ q) ]. ? M, i+1 ├ q or M, i+1 ├ (p ∏ q). ? M, i+1 ├ [ q ∨ (p ∏ q)]. ? M, i+1 ├ [ ?q ? (p ∏ q)].

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? M, i ├ True and M, i+1 ├ [ ?q ? (p ∏ q)]. ? M, i ├ True ∏ [ ?q ? (p ∏ q)]. ? True ∏ [ ?q ? (p ∏ q)]. Case 2. M, i ├ True ∏ [ ?q ? (p ∏ q)] ? M, i ├ True and M, i+1 ├[ ?q ? (p ∏ q)]. ? M, i+1 ├ [ ?q ? (p ∏ q)]. ? M, i+1 ├ [ q ∨ (p ∏ q)]. ? M, i+1 ├ q or M, i+1 ├ (p ∏ q). ? M, i+1 ├ q or M, i+1 ├ [ (p ∏ q) ∨ (p ∏ q) ∨ ……………∨ (p ∏ q) ]. ? [M, i+1 ├ q or M, i+1 ├ (p ∏ q) or M, i+1 ├ (p ∏ q) or …. …. M, i+1 ├ (p ∏ q)]. ? [M, i+1 ├ q or M, i+1 ├ (p ∏ q) or M, i+1 ├ (p ∏ p ∏ q) or …. …. M, i+1 ├ (p ∏ p ∏………………….. ∏ p ∏ q)]; Since p = p ∏ p. ?(M, i+1 ├ q or M, i+1├ p and M, i+2 ├ q or M, i+1├ p and M, i+2 ├ p and M, i+3 ├ q or …. …. M, i+1├ p and M, i+2 ├ p and ………. and M, k-1├ p and M, k ├ q). ? there is some k > i such that M, k├ q and for every j, if i<j<k then M, j├ p. ? p Until q.

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p Since q = (?q ? q ∏ p ) ∏ True Proof Case 1. M, i ├ p Since q iff there is some k < i such that M, k ├ q and for every j, if k < j < i then M, j ├ p. That is, M, i ├ p Since q ? there is some k < i such that M, k ├ q and for every j, if k < j < i then M, j ├ p. ?(M, i-1 ├ q or M, i-1├ p and M, i-2 ├ q or M, i-1├ p and M, i-2 ├ p and M, i-3 ├ q or …. …. M, i-1├ p and M, i-2 ├ p and ………. and M, k+1├ p and M, k ├ q). ?[M, i-1 ├ q or M, i-1 ├ (q ∏ p) or M, i-1 ├ (q ∏ p ∏ p) or …. …. M, i-1 ├ (q ∏ p ∏…………………….. ∏ p ∏ p)]. ?[M, i-1 ├ q or M, i-1 ├ (q ∏ p) or M, i-1 ├ (q ∏ p) or …. …. M, i-1 ├ (q ∏ p)]; Since p ∏ p = p. ? M, i-1 ├ q or M, i-1 ├ [ (q ∏ p) ∨ (q ∏ p) ∨ ……………∨ (q ∏ p) ]. ? M, i-1 ├ q or M, i-1 ├ (q ∏ p). ? M, i-1 ├ [ q ∨ (q ∏ p)]. ? M, i-1 ├ [ ?q ? (q ∏ p)].

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? M, i ├ True and M, i-1 ├[ ?q ? (q ∏ p)]. ? M, i ├ [ ?q ? (q ∏ p)] ∏ True. ? [ ?q ? (q ∏ p)] ∏ True. Case 2. M, i ├ [ ?q ? (q ∏ p)] ∏ True ? M, i ├ True and M, i-1 ├[ ?q ? (q ∏ p)]. ? M, i-1 ├ [ ?q ? (q ∏ p)]. ? M, i-1 ├ [ q ∨ (q ∏ p)]. ? M, i-1 ├ q or M, i-1 ? (q ∏ p). ? M, i-1 ├ q or M, i-1 ├ [ (q ∏ p) ∨ (q ∏ p) ∨ ……………∨ (q ∏ p) ]. ?[M, i-1 ├ q or M, i-1 ├ (q ∏ p) or M, i-1 ├ (q ∏ p) or …. …. M, i-1 ├ (q ∏ p)]. ?[M, i-1 ├ q or M, i-1 ├ (q ∏ p) or M, i-1 ├ (q ∏ p ∏ p) or …. …. M, i-1 ├ (q ∏ p ∏ …………………. p ∏ p)]; Since p = p ∏ p. ?(M, i-1 ├ q or M, i-1├ p and M, i-2 ├ q or M, i-1├ p and M, i-2 ├ p and M, i-3 ├ q or …. …. M, i-1├ p and M, i-2 ├ p and …………. and M, k+1├ p and M, k ├ q). ? there is some k < i such that M, k ├ q and for every j, if k < j < i then M, j ├ p. ? p Since q.

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Since Until and Since operators can be defined using TAND and all other temporal operators can be represented using Until and Since operators as shown by Barringer et al.[Barringer 1996], all the point based temporal operators can be defined using TAND. Sometimes in the future A = True ∏ [?A ? (True ∏ A)] Proof Sometimes in the future A = True Until A ; by the definition [Barringer 1996]. = True ∏ [?A ? (True ∏ A)]; replacing p by True and q by A in (p Until q). Always in the future A = ?[True ∏ (A ? (True ∏ ?A))] Proof Always in the future A = ?(Sometimes in the future ?A) ; by the definition[Barringer 1996]. = ?( True Until ?A ) ; by the definition[Barringer 1996]. = ?[True ∏ (A ? (True ∏ ?A))] ; replacing p by True and q by ?A in (p Until q). Next A = True ∏ (?A ? ( false ∏ A)) Proof Next A

= False Until A ; by the definition[Barringer 1996]. = True ∏ (?A ? ( false ∏ A)) ; replacing p by False and q by A in (p Until q).

Sometimes in the past A = (?A ? A ∏ True ) ∏ True

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Proof Sometimes in the past A = True Since A ; by the definition[Barringer 1996]. = (?A ? A ∏ True ) ∏ True ; replacing p by True and q by A in (p Since q). Always in the past A = ?[(A ? ?A ∏ True ) ∏ True] Proof Always in the past A = ?( True Since ?A ) ; by the definition[Barringer 1996]. = ?[(A ? ?A ∏ True ) ∏ True] ; replacing p by True and q by ?A in (p Since q).

Last A

= (?A ? A ∏ False ) ∏ True

Proof Last A = False Since A ; by the definition[Barringer 1996]. = (?A ? A ∏ False ) ∏ True ; replacing p by False and q by A in (p Since q). As shown above TAND logical temporal connective is capable of representing all existing temporal operators as well as all relative temporal relations suggested by J. F . Allen [ Allen 1983]. Since TAND is not based on any time models, it can rectify the weaknesses of both point based and interval based temporal models. It is simple and easy to use. Further it can be used with any time model; purely point based, purely interval based or point-interval based time models. Application of TAND is not limited to temporal logic but it can be used where the standard logical connective AND is used. Since all the temporal operators and relations can be defined using TAND, now we can think about a new temporal logic which employs TAND instead of standard temporal operators or relations to represent temporal information.

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p p before q q after p p ?p ∧ ?q p p during q q contains p

q q q p ∏ ( ?p ∧ ?q ) ∏ q

?p ∧ q p∧q ?p ∧ q (?p ∧ q) ∏ ( p ∧ q ) ∏ (?p ∧ q) p q (p ∧?q) (p ∧ q) (q ∧?p)

p overlaps q q overlapped-by p

(p ∧?q) ∏ ( p ∧ q ) ∏ (q ∧?p) p p meets q q met-by p p p q p starts q q started-by p p ∧q (q ∧?p) p p p finishes q q finished- by (q ∧?p) q p ∧q p equals q q p∧q ( p ∧ q ) ∏ (q ∧?p) q q p∏q

(q ∧?p) ∏ ( p ∧ q ) Figure 3.4 TAND versions of Allen’s temporal relations

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CHAPTER IV

TEMPORAL LOGIC

This chapter discusses about the first-order temporal logic(FTL), the temporal logic USF and the temporal logic of actions (TLA) which are used in knowledge based specifications, and their problems. The last section introduces the new temporal logic TANDTL. The temporal logic TANDTL employs TAND connective to represent temporal knowledge. Therefore TANDTL does not have any temporal operators or temporal relations instead it has TAND connective in addition to standard logical connectives.

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4.1 Conventional temporal logic

Representing and reasoning about time play a critical role in many areas of studies. In computer science, temporal logic[Kroger 1987][Manna 1995][Torsun 1995][Barringer 1996] has been used in concurrent systems, realtime systems, specifications, program verification, temporal database, planning expert systems and natural languages. Temporal logic has shown to be a powerful and flexible formalism, capable of representing and reasoning in these areas. In this section, conventional temporal logic, its weakness in software requirements specifications and proposed temporal logic will be discussed.

4.1.1 First-order temporal logic

First order temporal logic can be considered as a extension of first order logic with temporal operators. Syntax A language of first order temporal logic consists of the following set of symbols: 1. A set P of predicate symbols. 2. A set F of function symbols. 3. The equality symbol =. 4. A set V of variable symbols. 5. A set C of constant symbols. 6. A set Q of propositions. 7. The set of logical connectives (?, ∧, ∨, ?, ?). 8. The set of temporal operators (Ο, ?, à,U, W, q, s, x, S, Z). 9. The set of quantifiers (?, ?). These sets are disjoint. Given a first order temporal language, formulae of the language are constructed using the following rules:

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1. All variable symbols are terms. 2. If f is a function symbol of arity, k, k ≥0 and t1, …, tk are terms, then f(t1,…..,tk) is a term, f() is a constant. 3. If p is a predicate symbol of arity, k, k ≥0 and t1, …, tk are terms, then p(t1, ……, tk) is an atomic formula. If k=0 then p() is a proportional constant. 4. If t1 and t2 are terms, then t1=t2 is an atomic formula. The formation rules for FTL are 1. All atomic formula are formulae. 2. Formulae are constructed from other formulae by application of connectives and temporal operators. 3. If x is a variable symbol and A is a formula, then ?xA, ?xA are formulae. Semantics A formal semantics is described in terms of Kripke’s possible worlds [Torsun 1995][Fisher 1995]. The possible worlds are set of states S in times which are temporally related via the accessibility relation R. More formally a frame is defined by as a pair <S, R> where S is a non-empty set of possible states and R is a binary relation over R ?S x S, called the accessibility relation. Given a first order temporal language L, a model M is a tuple M = (S, R, D, ∏,ν) where ? (S,R) is a linear temporal frame where S is a non-empty set of possible states; R is a binary relation over R ? S x S, called the accessibility relation. ? D is a non-empty set, called the interpretation domain. Domain D is constant. ? ∏ is an interpretation function for variables, which maps each variable x ∈ V (a set of variables) to an element ∏(x) ∈ D. ? ν is a valuation function for constants, which maps: each function symbol f ∈ F(a set of functions) of arity n to a relation ν(f) from Dn into D;

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each predicate symbol p ∈ P ( a set of predicate symbols) of arity m to a relation ν(p) from Dm into the set {T, F}; each constant c ∈ C ( a set of constants) to an element ν( c ) ∈ D; each proposition q ∈ Q ( a set of propositions) to a truth value ν(q) ∈ {T,F}. Given a temporal interpretation M = (S, R, D, ∏,ν) and state s ∈ S (index i indicates state i, future times are larger than i, past times are less than i), the following rules associate a truth value with each formula A, and associate an element of D with each term t. An interpretation for the logic is defined as a pair <M,i>, where M is the model and i is the index of the state at which the temporal statement is to be interpreted. <M, i> ├ A means that “A is true in the model M at the state i”. ?For terms: <M, x> = ∏(x) for x ∈ V ; says that the interpretation of each variable x in the model M at a given state s is ∏(x). <M, c> = ν(c) for c ∈ C. <M, f(t1,….,tn)> = ν(f)(<M,t1>,…..,<M,tn>) for f ∈ F and t1,….,tn are terms. Interpretation for well-formed formulae is defined in terms of the satisfaction relation F between models and formulae. ? For atomic formulae: <M, i> ├ p(t1, …..,tm) iff ν<i,p>(<M,t1>,……,<M,tm>)=T, <M, i> ├ t1=t2 iff <M,t1> = <M,t2>. If d1,….dn are elements of D, then Mo<x1←d1,…..xn←dn> denotes the model obtained from M by modifying its assignment function to map the variable symbols x1,……,xn to d1,….dn respectively. Given a first order temporal language FTL, a model structure M = (S, R, D, α, I), evaluation functions ρ and ∏, the interpretation rules for FTL are as follows. ? For terms:

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ρ(M, x) = α(x), ρ(M, f(t1,….,tn)) = I(s,f) (ρ(M, t1),…..ρ(M,tn)). ? For atomic formulae: <M, ρ, ∏, si> ├ p(t1,….tn) iff I(s,f) <(ρ(M, t1),….ρ(M,tn)) ∈ ∏(si), <M, ρ, ∏, si> ├ t1=t2 iff ρ(M, t1) = ρ(M,t2). ? For connective: If A and B are formulae then ?A, (A ∧ B), (A ∨ B), (A ? B) and (A ? B)are interpreted as follows: <M, i> ├ T <M, i> ├ ?A iff not <M, i> ├ A <M, i> ├ A ∧ B iff <M, i> ├ A and <M, i> ├ B <M, i> ├ A ∨ B iff <M, i> ├ A or <M, i> ├ B. The semantics for ? and ? defined in terms of ? and ∨. ? For temporal operators: If A and B are formulae then ΟA (Next A), ?A (Always in the future A), àA (Sometimes in the future A), AUB (A until B), qA (Last A), sA (Always in the past A), xA (Sometimes in the past A), ASB (A since b) are interpreted as follows: <M, i> ├ ΟA <M, i> ├ ?A <M, i> ├ àA <M, i> ├ AUB <M, i> ├ qA <M, i> ├ sA <M, i> ├ xA <M, i> ├ ASB iff <M, i+1> ├ A iff ?j ≥ i <M, j> ├ A iff ?j ≥ i <M, j> ├ A iff ?j ≥ i <M, j> ├ B and ?k if (i ≤ k < j) then <M, k> ├ A iff i = 0 or <M, i-1> ├ A iff ?j 0≤ j < i, <M, j> ├ A iff ?j 0≤ j < i, <M, j> ├ A iff ?j 0≤ j < i, <M, j> ├ A and ?k if (i < k < j) then <M, k> ├ A

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? For quantifiers: <M, i> ├ ?x.A <M, i> ├ ?x.A iff all d ∈ D[Mo<x←d>, i ├ A], iff some d ∈ D[Mo<x←d>, i ├ A].

4.1.2 The logic USF

This logic [Barringer 1996] uses temporal operators since (S-) and until (U+) together with a fixed point operator Ψ. All the other temporal operators are defined using since (S-) and until (U+) operators. There are four types of well formed formulae: pure past formulae (talking only about the strict past), pure present formulae (talking only about the present), pure future formulae (talking only about strict future) and mixed formulae (talking about the entire flow of time). Syntax Let Q be a sufficiently large set of atoms. Let ?, ∧, ∨, ?, true, false be the usual classical connectives and let S- and U+ be the temporal connectives and Ψ be the fixed point operator. 1. An atomic q ∈ Q is in wff0 (pure present wff) and in wff (well formed formula). Its set of atoms in {q}. 2. Assume A and B are wffs with atoms {q1,……,qm} and {r1,…..rm} respectively. Then A ∧ B, A ∨ B, A ? B, B U+ A and B S- A are all wffs with atoms {q1,….qn,r1,…..rm}. a. If both A and B are in wff0 or wff+( pure future wff) then B U+ A is in wff+. b. If both A and B are in wff0 or wff- (pure past wff), then B S- A is in wff-. c. If both A and B are in wff*, then so are A ∧ B, A ∨ B, A ? B, where wff* is one of wff+, wff-, wff0. 3. ?A is also a wff and it is of the same type as A with the same atoms as A. 4. True (Truth) and false (Falsity) are wffs in wff0 with no atoms.

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5. If A is a wff in wff- with atoms {q,q1,…..,qn} then (Ψq)A is a wffwith the atoms {q1,….,qn}. Semantics Let D be a non-empty set, called the domain, and g be a function assigning the following: 1. For each m-place function symbol f and each n ∈ N a function g(n,f) : Dm ? D. 2. For each variable x and each n ∈ N, an element g(n, x) ∈ D. 3. For each m-place function symbol Q and each n ∈ N a function g(n, Q): Dm ? {0, 1}. The function g can be extended to a function g(n, A), giving a value in {0, 1} for each wff A(x1,…..xn) of the predicate USF [Barringer 1996] as follows: 1. g(n, f(t1,….,tm)) = g(n, f)(g(n, t1),……,g(n, tm)). 2. g(n, Q(t1,….,tm)) = g(n, Q)(g(n, t1),……,g(n, tm)). 3. g(n, A ∧ B ) = 1 iff g(n, A) = 1 and g(n, B) = 1. 4. g(n, A ∨ B ) = 1 iff either g(n, A) = 1 or g(n, B) = 1 or both. 5. 6. 7. 8. g(n, A ?B ) = 1 iff either g(n, A) = 0 or g(n, B) = 1 or both. g(n, ?A) = 1 iff g(n, A) = 0. g(n, true ) = 1 and g(n, false) = 0 for all n. g(n, B U+ A ) = 1 iff for some m > n, g(m, A) = 1 and for all n < k < m, g(k, B) = 1. 9. g(n, B S- A ) = 1 iff for some m < n, g(m, A) = 1 and for all m < k < n, g(k, B) = 1.

10. g(n, ?x A(x) ) = 1 for a variable x iff for all g’ such that g’ gives the same values as g to all function symbols and all predicate symbols and all variables different from x, we have g’(n, A(x)) = 1. 11. g(n, ?x A(x) ) = 1 for a variable x iff for some g’ such that g’ gives the same values as g to all function symbols and all predicate symbols and all variables different from x, we have g’(n, A(x)) = 1. 12. Let (Ψq) A(q, q1,…….,qm) be a pure past formula of USF, and let Bi ∈ wffV*i for i = 1,…..,m. To define g(n, A’) where A’ = (Ψq) A(q, B1,…….,Bm), first choose an assignment h such that h(qi) = {n ∈ N |

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g(n,Bi) = 1}. Then define g(n, A’) = 1 iff n ∈ h((Ψq) A(q, q1,…….,qm)).

4.1.3 The Temporal Logic of Actions

The temporal logic of actions (TLA) is a logic for specifying and reasoning about concurrent systems introduced by Leslie Lamport [Lamport 1994]. It combines two logics : a logic of actions and conventional temporal logic. The logic of actions consists of values, variables, states, state functions, predicates and actions while conventional temporal logic consists standard logical connectives and point based temporal operators. The temporal logic of actions is obtained by letting the elementary temporal formula in conventional temporal logic be actions in logic of actions. : data items (“true” and “false” are not included). : unprimed variables (refer to old state ) and primed variables (refer to new state). States : assignment of values to variables. State functions : non Boolean expressions built from variables and constant symbols ( Example : x2 + y - 3 ). Predicates : Boolean expression built from variables and constant symbols ( Example : x2 = y - 3 ). Actions : Boolean- valued expressions formed from variables, prime variables and constant symbols. ( Example : x’ + 1 = y where y is a variable and x’ is a primed variable ). Example : Representing a simple program using TLA [Lamport 1994]. A simple program, written in a conventional language: var natural x, y = 0; do < true → x := x + 1> Values Variables

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< true → y := y + 1> od This program is written in a conventional language, using Dijkstra’s do construct, with angle brackets enclosing operations that are assumed to be atomic. An execution of this program begins with x and y both zero, and repeatedly increments either x or y ( in a single operation ), choosing non deterministically between them [Lamport 1994]. TLA formula φ describing above program: Initφ M1 M2 M φ ( The symbol ( x = 0 ) ∧ ( y = 0) (x’ = x + 1 ) ∧ ( y’ = y ) ( y’ = y +1 ) ∧ ( x’ = x ) M1 ∨ M2 Initφ ∧ M means equal by definition)

The execution of the program is represented by φ = true. The predicate Initφ asserts the initial conditions, that x and y are both zero. The semantic meaning of action M1 is a relation between states asserting that the value of x in the new state is one greater than its value in the old state, and the value of y is the same in the old and new states. Thus a M1 step represents an execution of the program’s atomic operation of incrementing x. Similarly, a M2 step represents an execution of the program’s other atomic operation, which increments y. The action M is defined to be the disjunction of M1 and M2, so a M step represents an execution of one program operation. φ asserts that the initial condition is true initially, and that every step of the behavior represents the execution of an atomic operation of the program [ Lamport 1994].

4.1.4 Problems with conventional temporal logic

Temporal operators namely ΟA (Next A), ?A (Always in the future A), àA (Sometimes in the future A), AUB (A until B), qA (Last A), sA

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(Always in the past A), xA (Sometimes in the past A), ASB (A since b) are available in conventional temporal logic. These temporal operators employ point based time model to represent and reason about temporal knowledge. Therefore conventional temporal logic inherits the problems discussed in section 3.2.6.

4.2 TANDTL temporal logic

This research proposes a new temporal logic TANDTL by removing existing temporal operators from the first-order temporal logic [Kroger 1987][Manna 1995][Torsun 1995][Barringer 1996] and including TAND logical connective to represent temporal information . Further, the standard AND connective is redefined to represent concurrent events[Wijayarathna 1997]. This section describes the temporal logic TANDTL which can solve the above discussed problems of conventional temporal logic. The temporal logic TANDTL will be used as the formal basis for software requirements specifications language GSL and its execution. Syntax 1. Symbols ? The truth symbols T and F. ? The equality symbol =. ? The possessions symbol ∈. ? A set C of constant symbols a, b, c, …. ? A set V of variable symbols u, v, w, x, y, z, ….. ? A set F of function symbols f, g, h, ….. Each function symbol has an a positive integer called, its arity, indicating number of arguments. ? A set P of predicate symbols p, q, r, ….. Each predicate symbols also has an arity. ? A set Q of propositions. ? The set of standard logical connectives (?, ∧, ∨, ?, ? ). ? The temporal logical connective ∏.

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? The set of quantifiers (?, ? ). The constants and variables will represent objects, and the functions and predicates will represent functions and relations in those objects. 2. Terms The terms are the objects of the logic. They are built up according to following rules; ? The constants a, b, c are terms. ? The variables x, y, z are terms. ? If t1, t2, …….tn are terms (n ≥ 1) then f(t1, t2, …..,tn) is also a term. Function f is a n-arity function. 3. Atomic formulae or atoms Atoms represents the relation between objects and are constructed according to following rules; ? The truth symbols T (true) and F (false) are atoms. ? If t1, t2, …….tn are terms (n ≥ 1) then P(t1, t2, …..,tn) is an atom. Predicate P is a n-arity predicate. 4. Formulae or sentences The sentences of the logic are built from atoms according to the following rules; ? Every atom is a sentence. ? If A and B are sentences then so are ?A, A ∧ B, A ∨ B, A ? B, A ?B and A ∏ B. ? If x is a variable, then ?x (A) and ?x (A) are sentences. 5. Priority order of the connectives The connectives are listed from highest to lowest. ? ?, ?, ? ? ∨

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? ∧ ? ∏ ? ?, ? 6. Expressions An expression is either a term or a formula. Thus x, f(x) and p(x) are expressions. 7. Sub-term, Sub-formula and sub-expression ? Sub-term is a intermediate term which is used to build a term or a formula. ? Sub-formula is also a intermediate formula which is build a term or a formula. ? Sub-terms and sub-formulas are called sub-expressions. 8. Free and bound variables The occurrence of a variable x in an expression is bounded if x is within the scope of quantifiers ?x and ?x. Otherwise it is free. Semantics Let us assume that A and B are formulae. Then ? T is always true. ? F is always false. ? ?A is true if and only if A is false. ? A ∧ B is true if and only if both A and B are true at the same time. That is A and B are true simultaneously. ? A ∨ B is true if and only if A is true or B is true, and false otherwise (if both A and B are false at the same time). ? A ∏ B is true if and only if A ∧ B is false (that is A and B cannot be true simultaneously) and A is true and then B is true. ? A ? B is true if and only if A is false or B is true.

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? A ? B is true if and only if both A and B have same logical truth value. ? The expression X ∈ A express that X belongs to A. If X ∈ A and A∈B then it implies that X ∈ B. That is (X∈A)∧(A∈B)?(X∈B). Note that TANDTL temporal logic is not based on any fixed time model. Instead all temporal aspects are represented using relative time. That is to say whether two events A and B occur concurrently or not. If not, one event should follow the other. However, if we need, point based version of TANDTL temporal logic can be defined using Kripke’s possible worlds[Kripke 1959]. Semantics of point based version of TANDTL ? T is true at any given time t. ? F is false at any given time t. ? ?A is true at any given time t if and only if A is false at time t. ? A ∧ B is true at any given time t if and only if both A and B are true at time t. That is A and B are true simultaneously. ? A ∨ B is true at any given time t if and only if A is true at time t or B is true at time t, and false otherwise that (if both A and B are false at time t). ? A ∏ B is true if and only if A is true at time t and B is true at time t+1. The informal meaning of A ∏ B will be “A next B”. In the same way if B is true at time t and A is true at time t-1, then A ∏ B is true at time t. ? A ? B is true if and only if A is false or B is true. ? A ? B is true if and only if both A and B have same logical truth value. Therefore temporal logic TANDTL can be used with both point based and interval based time models. In interval based models, A ∏ B means “A meets B” while in point based models it means “A next B”. Figure 4.1 shows these temporal relations graphically. Therefore we will be able to execute a

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specifications based on TANDTL in a point based computer while providing the expressive power to represent the incomplete relative temporal knowledge. A meets B A∏B A Time interval T1 B Time interval T2 Time

Interval based time model

A

A next B A∏B

B Time

Time t

Time t+1

Point based time model Figure 4.1 TAND with point based and interval based time models

4.3 Inconsistency proofs based on resolution

Temporal logic has been used in various applications such as program verification, specification, synthesis of concurrent systems, synthesis of robot plans, verification of hardware devices and verification of reactive systems [Abadi 1990][Manna 1995][Torsun 1995][Tuzhilin 1995]. Some of these applications require proofs within temporal logic[Abadi 1990].

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4.3.1 Classical resolution

Resolution is a simple proof technique used in classical logic to prove the inconsistency of logical formulae [Torsun 1995]. The use of the resolution rule for proving inconsistency is restricted to formulae in conjunctive normal form. A proposition A is in conjunctive normal form if it is a conjunction (C1 ∧….∧Cm) of disjunction of Ci = Bi,1 ∨…….∨ Bi,n where each Bij is a literal, that is a propositional symbol or the negation of a propositional symbol. The empty clause is the only inconsistent clause; it is represented by F [Torsun 1995]. The inconsistency of the set S can be checked by generating logical consequences of S ending up with F. A very simple reasoning scheme will be used to generate logical consequences. Let A, B and C be formulae. Let us suppose that both formulae (A ∨ C ) and (B ∨ ?C) are true. If C is also true, then B is obviously true; otherwise, if C is false, then A is true. In both cases, (A ∨ B) is true. This leads to the rule { A ∨ C, B ∨ ?C } ├ A ∨ B, which can also be written as { ?C? A , C ?B } ├ A ∨ B. In the special case where C is a single proposition and where A and B are clauses, this rule is called the resolution rule [Torsun 1995]. Let S1 and S2 be two clauses belonging to the normal form S and let L be a literal. If L ∈ S1 and ?L ∈ S2, then the inconsistency of the set S of clauses can be proved by the following algorithm: While F ? S, select L, S1, S2 such that L ∈ S1 and ?L ∈ S2; compute resolvent R; replace S by S ∪ {R}. Resolvent R is obtained by { S1- L } ∪ {S2 - ?L}. Example : Let us prove the set of clauses S = { P ∨ Q, P∨ R, ?Q∨ ?R, ?P} is inconsistent. It is convenient to number the clauses. (1) P ∨ Q

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(2) P∨ R (3) ?Q∨ ?R (4) ?P Then applying the above algorithm, (5) Q (1,4) ; S1 = P ∨ Q, S2 = ?P, L = P. (6) R (2,4) (7) ?Q (3,6) (8) F (5,7) Since F is obtained at the end of the process, the set of clauses S is inconsistent.

4.3.2 Non-clausal resolution for propositional logic

Transforming a formula into normal form can be a lengthy process and the use of the resolution principle for proving inconsistency is restricted to formulae in conjunctive normal form. Non-clausal resolution technique allows us to extend the resolution proof machinery to arbitrary formulae. This allows one to use the resolution method as a proof technique, based on the deduction principle, even if the hypotheses and the negation of the conclusions are not clauses. Deduction principle: The formula C is a logical consequences of the finite set S if and only if S ∪ {?C} is inconsistent. A set is inconsistent if and only if F (false) is a logical consequence of it. Formally the deduction principle is expressed as {H1,……., Hn} ├ C ? {H1,…..,Hn, ?C} ├ F (false) [Torsun 1995]. Notation : ⊥P(A1, A2) = A1P├F ∨ A2 P├T where A1P├ Q denotes the formula obtained by replacing each occurrences of P by Q in A. Example : {A1 : ( P ∧ R) ∨ ( Q ∧ ?R), A2 : (?P ∧ R) ∨ (?Q ∧ ?R)} (1) ( P ∧ R) ∨ ( Q ∧ ?R) (2) (?P ∧ R) ∨ (?Q ∧ ?R) (3) ( Q ∧ ?R) ∨ (?Q ∧ ?R) A1 A2 ⊥P(1, 2)

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(4) ( P ∧ R) ∨ (?P ∧ R)

⊥Q(1, 2)

(5) F (false) ⊥R(4, 3) That is logical formulae A1 and A2 are inconsistent.

4.3.3 Non-clausal resolution for first-order temporal logic

This section presents the non-clausal resolution method for first-order temporal logic introduced by Abadi and Manna [Abadi 1990] [Torsun 1995]. The non-clausal resolution rule for classical propositional logic can be stated as A<u,…,u>, B<u,….u> A<true> ∨ B<false> where formulae A<u,…,u> and B<u,….u> have common sub formula u and the resolvent can be obtained by substituting true for one or more occurrences of u in A<u,…,u> and false for one or more occurrences of u in B<u,….u> and then taking the disjunction of their results. The expression A<true> ∨ B<false> is called the resolvent of A and B. The deduction rules have the form u1,…..um v can be applied to the formula Si if the formulae u1,…um occur as conjuncts in Si. Then, in order to obtain si+1, the derived formula v is added to the conjunction. This rule cannot be used in propositional or first-order temporal logic since the occurrence of u may refer to different instants of time. Example : Consider the formula u in the formulae ?u and àu If the classical resolution rule is applied, (1) ?u (2) àu (3) ?true ∨ àfalse ⊥u(2, 1)

However, u in àu can be true at a time and false at another time. Hence , we cannot deduce that the resolvent ?true ∨ àfalse is always false. Therefore Abadi and Manna [Abadi 1990] extend the classical non-clausal resolution method

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introducing several modality rules so that it can be used in propositional and first-order temporal logic. Extended non-clausal resolution rule A<u,…,u>, B<u,….u> A<true> ∨ B<false> where the occurrences of u in A and B are replaced with true and false, respectively, are all in the scope of the same number of Ο (Next operator) ‘s and are not in the scope of any other temporal operator in either A or B. Example : Consider the formulae Ο?Ο(?p ∨ q) ∧ à?p and ΟΟ?p Ο?p. (1) Ο?Ο(?p ∨ q) ∧ à?p (2) ΟΟ?p Ο?p (3) [Ο?Ο(true ∨ q) ∧ à?p ] ∨ [ ΟΟfalse Ο?p ] ⊥?p(2, 1) In this case, occurrences of ?p in the scope of two Ο’s are replaced with true and false. However, ?p in à?p and Ο?p cannot be replaced with either true or false because the occurrences of ?p are in the scope of à and Ο temporal operators. Abadi and Manna [Abadi 1990] introduce six essential modality rules to substitute the Ο’s versions for other temporal operators. Modality Rules: ΟΟ rule : ? rule : à rule : u rule : s rule : Οu, Οv ?u àu uuv usv Ο( u ∧ v) u ∧ Ο?u u ∨ Οàu u ∨ ( u ∧ Ο( u u v )) ; u : Until operator ?v ∧ ( u ∨ Ο(u s v )) ; s : Since operator

In classical logic, all quantifiers can be removed from formulae by application of skolemization rules [Abadi 1990] [Torsun 1995]. However, in first-order logic, quantifiers can be removed only in certain cases. Abadi and Manna propose temporal skolemization rules for first-order temporal logic [Abadi 1990].

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The classical skolemization rules Suppose that A and B are formulae. Q1,….Qn are quantifiers, x, x1,…,xn are variables, x’ is a new variable, and the occurrences of Q?x.B[x] and Q?x.B[x] under consideration are not in the scope of any quantifiers in A. Classical Skolemization may be described as a two-stage process: (1) Quantifiers move : Q1x1…..QnxnA<Q?x.B[x]> is rewritten to Q1x1…..Qnxn?x’.A<B[x’]> and Q1x1…..QnxnA<Q?x.B[x]> is rewritten to Q1x1…..Qnxn?x’.A<B[x’]> (2) Then Skolem terms replace existentially quantified variables: ?x1…..?xn?x.A[x] is rewritten to ?x1…..?xnA[f(x1,…….xn)], where f is a new Skolem function symbol. Universal quantifiers stay in place.

4.3.4 Non-clausal resolution for TANDTL logic

As discussed in section 4.2, TANDTL temporal logic does not have any temporal operators. Therefore, modality rules cannot be used for non-clausal resolution in TANDTL. However, a formula u may refer to different time instants as in conventional temporal logic. Example : Consider the following inconsistent formulae u and v ∏ u. Scope of u is different in two formulae. Therefore sub formula u cannot be replaced with true and false in these formulae. (1) u (2) v ∏ u (3) [(true) ∨ ( v ∏ false)]⊥u(2, 1) ; This is not valid. However, we can show that u ∏ v = (u ∧ ?v) ∏ v This can be established using the truth table 3. (F2).

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T1 u

1 2 3 4 5 6 7 8 F F F F T T T T

v

F F T T F F T T

T2 v

F T F T F T F T

Time Zone T1 Follows Time Zone T2 u∏v (u ∧ ?v) ∏ v

F F F F F T F F F F F F F T F F

Table 3 Truth table for u ∏ v = (u ∧ ?v) ∏ v Therefore, equation F2 can be used to rearrange the formulae having TAND connective before applying non-clausal resolution. Now in the above example, the formula (v ∏ u) can be re-written as (v ∧ ?u) ∏ u . Then applying non-clausal resolution, (1) u (2) (v ∧ ?u) ∏ u (3) {(false) ∨ [(v ∧ ?true) ∏ u ]} (4) false ⊥u(1, 2)

Example : Let us prove that “p before q” and “p after q” are inconsistent. p before q = p ∏ ( ?p ∧ ?q) ∏ q = [p ∏ ( ?p ∧ ?q)] ∏ q Applying F2, p before q = {[p ∏ ( ?p ∧ ?q)] ∧ ?q} ∏ q = { (p ∧ ?q) ∏ ( ?p ∧ ?q) ∧ ?q} ∏ q ; using distributive law = (p ∧ ?q) ∏ (( ?p ∧ ?q) ∧ ?q) ∏ q = (p ∧ ?q) ∏ ( ?p ∧ ?q) ∏ q p after q = q ∏ ( ?p ∧ ?q) ∏ p = (q∧?p)∏(?p∧?q) ∏ (p∧?q); applying F2 and distributive law Then applying non-clausal resolution method, (1) (p ∧ ?q)∏ ( ?p ∧ ?q) ∏ q (2) (q ∧ ?p)∏ ( ?p ∧ ?q) ∏ p (3) [ (false ∧ ?q)∏ ( ?p ∧ ?q) ∏ q] ∨ [ (q ∧ ?true)∏ ( ?p ∧ ?q) ∏ p] ;⊥p(1, 2) (4) false

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That is the set of formulae { “p before q” and “p after q”} is inconsistent. Example : Let us consider a simple inventory system which has number of items in stocks. There are two external events in this system : “Sales” and “Purchases”. The event “Sales” decreases stock level while the event “Purchases” increases the stock level of an item. System will function properly if these two events occur sequentially, but inconsistency will occur if two events appear concurrently. This example shows how non-clausal resolution technique can be used to identify the occurrence of inconsistency before the execution. The TANDTL logical statements for the system specifications will be as follows: S1. Sales ? Increase(Item_Quantity, Sales_Quantity) S2. Purchases ? Decrease(Item_Quantity, Purchase_Quantity) We have to inform the system that both actions “Increase(Item_Quantity, Sales_Quantity)” and “Decrease(Item_Quantity, Purchase_Quantity)” cannot occur concurrently. We can do that by introducing a model rule S3. ?[ Increase(Item_Quantity, Sales_Quantity) ∧ Decrease(Item_Quantity, Purchase_Quantity) ] Let Item_Quantity be IQ, Sales_Quantity be SQ and Purchase_Quantity be PQ. Then the above logical statements can be rewritten as: S1. ?Sales ∨ Increase(IQ, SQ) S2. ?Purchases ∨ Decrease(IQ, PQ) S3. ?Increase(IQ, SQ) ∨ ?Decrease(IQ, PQ) Applying non-clausal resolution technique: S4. ?Sales ∨ ?Decrease(IQ, PQ) ; ⊥Increase(IQ, SQ) (1, 3) S5. ?Purchases ∨ ?Increase(IQ, SQ) ; ⊥Decrease(IQ, SQ) (2, 3) S6. Increase(IQ, SQ) ; ⊥Sales (4, 1) S7. ?Increase(IQ, SQ) ; ⊥Purchases (2, 5) S8. F ; ⊥Increase(IQ, SQ) (6, 7)

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That is, the set of logical statements are inconsistent. Therefore we can identify the occurrence of inconsistency and inform the user before the execution of logical statements. However, without the model rule S4, non-clausal resolution technique cannot identify the inconsistency among the above logical statements. Therefore model rule play an important role here. As shown, non-clausal resolution technique can be used in TANDTL logic to prove the inconsistency of set of temporal formulae (i.e. formulae with ∏ connective). However, TANDTL logic requires a method to remove the quantifiers in order to apply non-clausal resolution for inconsistency proofs. The skolemization rules are employed in first-order logic. These skolemization rules will be reconsidered in the following section and modified so that they can be used in the TANDTL logic. Finally non-clausal resolution technique in firstorder logic will be extended to TANDTL logic. Prenex normal form Every first-order logical formula has a normal form called prenex normal form. The reason for considering the prenex normal form of a formula is to simplify the proof procedure[Torsun 1995]. Definition [Torsun 1995] A formula is in prenex form iff either it contains no quantifiers, or it is of the form Q1x1,…..QnxnB, where B is a formula with no quantifiers (quantifierfree), x1,…..,xn (not necessarily distinct) variables, and Qi ∈ { ?, ?}, for i=1,……,n. B is sometimes called the matrix. The following lemma is used to show the every formula is equivalent to a prenex formula. Lemma [Torsun 1995]

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The following formulae are valid. ( a ) ?x ( A ? B ) ? ( ?x A ? ?x B). ( b ) ?x ( A ? B ) ? ( ?x A ? ?x B). Definition (Bound and free variables) Let x be a variable. E be an expression (i.e. a formula or a term) of predicate logic, and x occurs in E. The occurrence of x is bound in E if it is within the scope of a quantifier ?x or ?x. Theorem [Torsun 1995] For every formula in first order form, there exists an equivalent prenex form. Proof. There is a simple algorithm to transform a formula into an equivalent prenex form. (1) The connectives ? and ? are eliminated using the following rewriting rules: ( A ? B ) replaced by ( A ? B) ∧ ( B ? A) ( A ? B) replaced by ( ?A ∨ B ) (2) Bound variables are renamed (if necessary) in such a way that free and bound variables do not share common names. This is required not only for the whole formula, but also for every sub formula. (3) A quantification whose scope does not contain any occurrence of the quantified variable is suppressed. (4) All occurrences of the negation are transformed immediately before the atoms, by the use of the following rewriting rules: ??xA ≈ ?x?A. ??xA ≈ ?x?A. ?(A ∧ B) ≈ (?A ∨ ?B ). ?(A ∨ B) ≈ (?A ∧ ?B ). ??A ≈ A.

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(5) All quantifications are transferred in front of the formula, with the help of appropriate rewriting rules. The rules concerning the conjunctions are given below, those concerning the disjunction are obtained by the duality principle. ( ?xA ∧ ?x B) ≈ ?x (A ∧ B). ( ?xA ∧B) ≈ ?x (A ∧ B) if B does not contain x. ( A ∧?xB) ≈ ?x (A ∧ B) if A does not contain x. ( ?xA ∧B) ≈ ?x (A ∧ B) if B does not contain x. ( A ∧?xB) ≈ ?x (A ∧ B) if A does not contain x. In order to make this set of rules complete, one has to allow the renaming of bound variables. Example : ?xp(x) ∧?xq(x) will first be rewritten as ?xp(x) ∧?yq(y) before applying above rules. Above theorem can be extended to TANDTL logic by introducing rules (4a) and (5a). Rule (4a) ?(A ∏ B) ≈ (A∧B)∨(?A ∧?B )∨(B ∏ A) Rule (5a) ( ?xA ∏ ?x B) ≈ ?x (A ∏ B). ( ?xA ∏B) ≈ ?x (A ∏ B) if B does not contain x. ( A ∏?xB) ≈ ?x (A ∏ B) if A does not contain x. ( ?xA ∏B) ≈ ?x (A ∏ B) if B does not contain x. ( A ∏?xB) ≈ ?x (A ∏ B) if A does not contain x. As shown above every formula in TANDTL logic can be transformed to prenex normal form. Classical skolemization rules require the formulae in prenex normal form. Therefore classical non-clausal resolution technique can be extended to TANDTL logic. Hence the inconsistency of set of formulae in

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TANDTL logic can be determined using extended non-clausal resolution method.

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CHAPTER V

EXECUTION OF TEMPORAL LOGIC

This chapter discusses the similarities between logic and the view of the system adopted in this research. Then it shows how the temporal logic TANDTL can be executed. Finally it compares the execution of temporal logic USF with the TANDTL. The purpose of this chapter is to show that the view of a system employed in this research can be implemented using temporal logic TANDTL.

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5.1 Logic and the view of a system

Terms in temporal logic are objects while predicates represent relations between objects. Let us have a closer look at syntax 7 in TANDTL. It says that a term can be constructed using another terms, formulae or expressions. Expressions are either terms or formulae. Formulae consist of objects and relations between objects. Since term is an object, according to syntax 7 the following objects can be defined: 1. Atom objects, 2. Sentence objects. Atoms are relations between objects, such as P(x, y, z) where P is the relation while x, y, z are objects. Therefore P is a relation object. If Q and R are atoms then ?Q, Q ∧ R, Q ∨ R, Q ? R, Q ?R and Q ∏ R are sentences. Now consider the sentence Q ? R. It can be considered as a rule as it says that if Q holds then do R. Since syntax 7 allows us to assign this sentence to a object we can have rule objects. On the other way it can be used to create another object encapsulating rules into the object in addition to attribute and methods. Practitioners and researchers have identified the weaknesses in representing business rules in conventional object-oriented systems analysis and design methods. They suggest two solutions; ? One to represent rules as objects [Taylor 1995][Graham 1994][Zaho 1994] and ? Other to encapsulate business rules in to a class[Wu 1995]. These show that the logic can be applied not only with conventional object technologies but also with future object technologies. Finally syntax 7 says that an object can be created as a result of interaction between objects. The view of a system adopted in this research also says that the interaction between objects can either modify an attribute in an object/objects or create new object having new attributes as well as attributes inherited from the parent objects. Hence the view of a system is supported by the TANDTL logic.

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Composite rules Since the logical connectives can be used to connect sentences having the form Q ? R, there can be a sentence in a form [(Q1?R1)∧(Q2?R2)]. If Q1 ? R1 is defined as rule 1 and Q2 ? R2 as rule 2, our sentence will appear as [ Rule 1 ∧ Rule 2]. This is a composite rule. Therefore logic allow us to write composite rules. Of course, as described in the previous section, this composite rule can be considered as a new object. Composite events/ activities A sentence in the form Q ∏ R is already a composite event/ activity as it commands to do Q and then do R. The events Q and R are grouped together as composite events. As described in the above section, it is possible to create a new object to represent this composite event/ activity. As explained above, the temporal logic TANDTL is capable of 1. Solving problems appeared in point based and interval based time models, 2. Creating rule objects, 3. Encapsulating rules into an object, 4. Building composite rules and 5. Building composite events/ activities. Recall that the syntax and semantics of TANDTL is the same as the first - order temporal logic except temporal expressions. Syntax and Semantics of temporal expressions is redefined using TAND logical connective.

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5.2 Execution of TANDTL

Logical view of a software requirements specifications

As already discussed, a software requirements specifications states what actions to be invoked when the specified conditions become true. Conditions can be either temporal conditions or non-temporal conditions. Temporal conditions express the chronological order of events/ actions while non-temporal conditions states only the actions to be taken. There are two types of rules: rules which interact with the environment and rules which confine to the internal system. In TANDTL logic all these rules can be expressed by sentences in the form C ? A where C is a set of conditions and A is set of actions to be invoked. The set of conditions C can be further divided into two subsets CT and CNT which stand for temporal conditions and non-temporal conditions respectively. Hence sentences C ? A can be represented as ( CT ∧ CNT) ? A. Events/ Actions in specifications are relations between objects. Therefore events/ actions can be represented as R(O1, O2,……,On) in logic where R is a relation and O1, O2,……,On are objects. In addition to the set of rules, a software requirements specification may contain attributes of objects, composite rules and composite events/ activities. Logical representation of them are already discussed. Execution Software requirements specification is a set of rules which states actions to be taken when conditions are held. That is Hold the conditions ? Do the actions. Therefore the following steps are necessary to execute a specification: 1. Check whether the condition is held 2. If the condition is held then activate the actions specified.

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Checking the conditions Checking of non temporal conditions is easy since all the attribute values are available. However, checking of a temporal condition may not be easy. Since TANDTL represents all temporal conditions using TAND logical connective, if we have a methods to check whether A ∏ B is true then we can check the truth value of any given temporal condition. Application of temporal database can solve the problem since temporal databases can store records in chronological order and they are capable to answer a query which inquire the chronological order of events A and B. Hence, once an event is executed, we have to create a record with event identifier in the temporal database. The record format would be (E, T) where E is the event identifier and T is the time. Therefore our temporal database will look like (E1, T1), (E2, T2), (E3, T3), …….,(En, Tn). E1 ∏ E2 is true 1. If and only if E1 and E2 are found in the temporal database and 2. T2 comes just after T1. E1 ∏ E2 is false 1. One of E1 or E2 are not found in the temporal database or 2. T2 does not follow T1. Executing actions Let us define predicate Exec(A) to represent execution of event A. Execution rule are as follows: 1. Exec(T) = True. 2. Exec(F) = False. 3. Exec( A ? B) = Query the temporal database for truth value of A and if it is true execute B. That is Exec(B). If not, do not do anything. 4. Exec( A ∧ B) = Exec(A) ∧ Exec(B). 5. Exec( A ∨ B) = Exec(A) ∨ Exec(B). 6. Exec( A ∏ B) = Exec(A) ∏ Exec(B).

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Logically, “Exec(A) ∨ Exec(B)” is true either A is executed or B is executed. However, in this execution method, execution will be carried out from left to right. Therefore, A will executed first. If A cannot be executed then B will be executed. If both A and B cannot be executed, then the expression “Exec(A) ∨ Exec(B)” is false. Since we can write all the temporal knowledge with TAND logical connective, rule 6 provides the basis for execution of temporal events. As mentioned in the previous section, we have to create a record in the temporal database for each execution of events. Comparison of TANDTL executions with USF executions This section presents the comparison of TANDTL executions with logic USF [Barringer 1996]. The USF logic employs Until and Since temporal operators to represent all other future and past temporal operators. In USF, specification is considered as a collection of “If past then execute future” rules. That is hold C in the past ? execute B now. As expressed above, in USF rules, C is always specified using past temporal operators while B is specified using future temporal operators. However, in TANDTL, no past or future temporal operators are available instead all of them are represented using TAND logical connective. Execution rules of logic USF. 1. Exec(true, m) = true. 2. Exec(false, m) = false. 3. Exec(A ∨ B, m) = Exec(A, m) ∨ Exec(B, m). 4. Exec(A ∧ B, m) = Exec(A, m) ∧ Exec(B, m). 5. Exec(B Since A, 0) = false. 6. Exec(B Since A, m+1) = Exec(A, m) ∨ [Exec( B, m) ∧ Exec( B Since A, m)]. 7. Exec(B Until A, m) = Exec(A, m+1) ∨ [Exec( B, m+1) ∧ Exec( B Since A, m+1)]. Where m is a time point.

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Even though the logic USF has logical approach to execute specifications, execution of rules 6 and 7 are not logical but procedural. According to the rule 6, to execute B Since A at time m+1, we have to execute A at time m and only if we can not execute A at time m then execute B at time m together with the B Since A. However, logically both (A ∨ B) and ( B ∨ A ) are same. This shows that the execution of rule 6 is not logical but procedural. Rule 7 has the same weakness. Since TANDTL represents all temporal events using TAND connective and TAND itself possesses the execution order, TANDTL can be executed easily and logically. The objective of this research is to develop end-user intelligible and executable software requirements language. Now it is clear that a specifications language based on TANDTL is executable but the statements in TANDTL are not understandable to non-technical end-users. The chapter 6 of this thesis explains the language called GSL based on TANDTL and show its intelligibility. Therefore GSL can be considered as the End-user intelligible version of TANDTL. Syntax and semantics of TANDTL are also applied to GSL.

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CHAPTER VI

SPECIFICATIONS LANGUAGES

This chapter discusses about present specifications languages and finally introduces new software requirements specification language GSL which is based on temporal logic TANDTL. Since TANDTL is executable, the specifications written in GSL are also executable. Further GSL provides enduser intelligible specifications. Temporal knowledge is represented in GSL using And_Then relation which represents TAND connective in TANDTL. Examples have been provided using the pilot system.

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6.1 Conventional specifications languages

This section briefly discusses conventional approaches to software requirements specifications languages and representation of temporal knowledge. The discussion mainly concerns about the end-user intelligibility of the specifications written in these languages and their attempt to represent incomplete relative temporal knowledge. Finally the proposed specifications language GSL (General-purpose Specification Language) will be introduced.

6.1.1 TELOS

Telos[Mylopoulos 1990] is one of latest specifications language developed to represent temporal knowledge in information systems. It employs modified versions of Allen’s[Allen 1983] temporal relations (equals, meets, before, overlaps, during, starts and ends). Therefore it inherited the weaknesses of interval based time model. However, Telos was designed to represent incomplete temporal information. For example, infinite time interval is represented as (1986/10/25..*). Let us consider the following Telos specification. TELL TOKEN martin IN paper (at 1986/10..*); author firstAuthor : Stanly (at 1986/10..*); : LaSalle (at 1987/1..*); : Wong (before 1987/5) title : ‘The MARTIN system’ END

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This operation introduces the token martin in the knowledge base. The IN clause makes martin as an instance of the class paper for an unbounded time interval starting October 1986. Similarly, the WITH clause asserts that Stanley is the first author of martin during the interval 1986/10..*, LaSalle is an author during the interval 1987/1,..* while Wong was an author for some time before MAY 1987. Integrity constraints and deductive rules are resented as follows: TELL CLASS Paper IN SimlpeClass WITH integrityConstraint :$ (?y/Person) ( y ∈ this.author ? ?(?t/Time)y ∈ this.referee[at t]) $ deductiveRule :$ (?x/Paper) (?z/Address) ( z ∈ author.address ? z ∈ x.replyAddress) $ (at Alltime) END The infinite interval “Alltime” and the special interval “Now” represent the current system time. These show how Telos specifications are rigid. Specifications written in Telos are not always in end-user understandable form. Still end-user has to get some assistance from the system analysts. It is not so easy to use. Users cannot define their own temporal relations. Further, Telos specifications are not executable. Telos dose not support the view of a system adopted in this research.

6.1.2 Templar

The Templar specifications language[Tuzhilin 1995] provides user friendliness to a certain extent, still the end-user needs some assistance from the systems analyst to understand the software requirements specifications. However there is no guarantee that the sentences in Templar specifications are always in end-user intelligible form. It solely depends on the capabilities of the systems analyst. On the other hand, Templar does not employ object-oriented principles directly in its specifications, instead claims that Templar

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specifications can be converted to object-oriented specifications. Tuzhilin[Tuzhilin 1995] provides a discussion about other specifications languages. Templar specifications language was designed to provide: 1. Powerful and user friendly specifications to system analysts 2. End user understandable specifications 3. Temporal knowledge representation. Templar uses When clause to represent events, If -Then to represent rules, Then_Do to represent actions. Temporal knowledge is represented by While, Before and After clauses. Sample Templar specification is stated below. When arrives(customer, branch) While close(branch) If has_atm(branch) Then_Do use_atm(customer, branch) Since first-order predicates are used in Templar specifications, they are not always understandable to end users. Therefore end user has to get some assistance from system analysts to understand the specifications. On the other hand, temporal knowledge is sometimes distributed over clauses in the language and sometimes included in the predicates. Even though Templar provide user defined temporal constructs, it is not easy for users to develop them. For example, to define B Since C one has to write if C then X if B and Not C and previous X then X if B and not C and not previous X then not X if not B and not C then not X. This shows how difficult to define temporal constructs by users. Templar’s uses point based temporal operators in predicates and employs while, before and after clauses to represent interval based temporal knowledge. Since Templar is based on first-order temporal logic, it cannot use interval based temporal relations in predicates but as clauses in the language. Templar does not support the view of a system adopted in this research.

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6.2 GSL specifications language

This section introduces the new software specifications language proposed in this thesis. GSL is based on TANDTL temporal logic. Therefore it can provide wide range of temporal relations in end user understandable form. The specifications written in GSL is not always grammatically correct but understandable to end users as well as system analysts. Since it is based on executable TANDTL, GSL specifications are also executable.

6.2.1 Structure of GSL

Facts The basic unit of knowledge representation in GSL is the “Fact”. Facts are common to all knowledge representation methods but they are represented and grouped in different ways. Semantic networks, rules, frames, objects and logics have their unique ways to represent and group facts into a body of knowledge[Torsun 1995]. Facts in GSL have O1RO2 (Object1 Relationship Object2) or OR (object Relationship) form. These forms resemble SVO or SV form in elementary English sentences and can be easily transformed to R(O1,O2) or R(O) in TANDTL temporal logic. Some researchers[Agusa 1984] model requirements as R(X,Y) where R is the relationship and X,Y are the entities. This approach will inherit problems of the Entity-relationship model. In GSL, a fact by itself can be an object. This new object in conjunction with another object and a relationship can form a new fact and so on. Therefore we model a requirement as a series of objects and relationships. Fact (F) = OR or or O1RO2 O1R1O2[RkOk+1; k=2,n].

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These facts can be transformed to TANDTL atoms very easily. Therefore facts in GSL will be represented in TANDGL as R(O), R(O1, O2) or R1(O1, R2(O2, R3(O3, O4))). Relationships In GSL, there are two types of relationships; namely Major and Minor relationships ( Rmajor and Rminor ). All facts must have one major relationship but they may or may not have minor relationships. Minor relationships cannot appear in a fact without a major relationship. We use unary and binary relationships. Tertiary and higher order relationships can be replaced with binary relationships[Torsun 1995]. We replace higher order relationships with one major relationship and one or more minor relationships. The main verb in an elementary English sentence represents a major relationship. That is, a fact does not have a meaning without a major relationship. Therefore a fact (F) can be represented as follows: Fact (F) = ORmajor or or O1RmajorO2 O1RmajorO2[Rminor(k)Ok+1;k=2,n] BNF grammar [Russel 1995] for facts in GSL as follows: Fact → Fact | Object | Object Connective Object | Object Connective Fact | Fact Connective Fact | ( Fact ) | Object Major-relation | Object Major-relation Object | Object Major-relation Object Minor-relation Object…. | Fact GSL-Relation Fact | Fact User-Define-Relation Fact

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User-Define-Relation → If Fact User-Define-Relation Fact Then Fact Object → Object | Fact Connective → And | Or | And_Then | Not | Of Role of Rmajor in a fact The relationship Rmajor creates either the most stable or the most unstable fact or the object in a fact. This most stable fact or the object contains complete information while the most unstable fact or the object has incomplete information. Role of Rminor in a fact The most stable fact or the object created by Rmajor does not require any Rminor links with any other object. It is in the most stable state and represents complete information. A fact in the most stable state can have minor relationships with other facts/objects to provide additional information. However, the most unstable fact or the object needs minor relationship/relationships to link with other object/objects to change the most unstable state to the most stable state where complete information is available. Figure 6.1 shows these concepts graphically. Minor relationships are used only to increase the readability and the intelligibility of the specifications while major relationship represents the real actions. All minor relationships can be ignored when a fact is transformed to a TANDTL expression. Since we consider only elementary English sentences, systems analysts should transform the user statements into elementary sentences that contain only one main verb as suggested by Chen[Chen 1983][Burg 1996]. Present modeling techniques do not represent minor relationships between objects but only major relationships.

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(Mr. Tanaka reserve flight-CX505 ), Narita , Hongkong, 1997.8.28 from (Minor relationship)

Incomplete information

(Mr. Tanaka reserve flight-CX505 from Narita), Hongkong, 1997.8.28 reserve (Major relationship) to (Minor relationship) (Mr. Tanaka reserve flight-CX505 from Narita to Hongkong), 1997.8.28 on (Minor relationship)

(Mr. Tanaka reserve flight-CX505 from Narita to Hongkong on 1997.8.28) Mr. Tanaka, Flight-CX505, Narita, Hongkong, 1997.8.28 Complete information

Figure 6.1 Role of major and minor relationships in a fact

Example : the fact “A passenger requests for reservation of a flight from Narita to Hongkong ” will be represented in Templar[Tuzhilin 1995] as: Request(Passenger, Flight, Narita, Hongkong) where “Request” is the major relationship which is the main verb in the statement. As shown in Figure 6.1, this represents incomplete information. However, it will be represented in GSL as: Passenger request flight from Narita to Hongkong

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where “request” is the major relationship while “from” and “to” are minor relationships. This form can easily be transformed into the following:

1. request(Passenger, flight, Narita, Hongkong) 2. To( From( request(Passenger, flight ), Narita) , Hongkong).

The second form represents the syntax of the fact and it provides complete information. The requirements in the first form are partially complete because it does not give all the information about how terms in a sentence relate to each other. Instead it gives the main verb of statements leaving reader to think the other possible relationships. Therefore representing requirements in the first form, we show fewer information to end-users than what we can get from them. When the number of terms in the sentence is increased, the information loss will also increase. This will lead to a misunderstanding. The representation method used in GSL will be very useful when additional information needs to be associated with tertiary or higher order relationships. Example : let us consider the statement “A passenger requests for reservation of a flight from Narita to Hongkong ”. This can be represented by the sentence, “request(Passenger, flight, Narita, Hongkong)”. When we want to express the statement “A passenger requests for reservation of a flight from Narita to Hongkong on 1997.8.28”, it is necessary to modify the structure of the sentence as, “request(Passenger, flight, Narita, Hongkong, 1997.8.28)”. This requires changes to “request(Passenger, flight, Narita, Hongkong)” wherever it appears. However, in our representation the first statement, “ Passenger request flight from Narita to Hongkong” can be extended to

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“ Passenger request flight from Narita to Hongkong on 1997.8.28” to include the additional information without modifying “ Passenger request flight from Narita to Hongkong” . Requirements represented in GSL are not always grammatically correct but intelligible to end-users and also the representation scheme is easy to use by systems analysts. Such requirements representation scheme could not be seen in other requirements specifications languages. Statements The statements in GSL have the form S = CiFi or CiOi where Ci = i th clause and Fi = i th fact , Oi = i th object, i=1,n. BNF grammar for GSL statement is Statement → Clause Fact Clause Clause → When | If | Then | Then_Do | Define | End_Define Connective → And | Or | Not | And_Then GSL-Relation → Forall | Exists | As | Is_A | Of | + | - | * | / | % As shown above User-Define-Relation should always accompany with If-Then rule to the language.

6.2.2 Features of GSL

Logical Connectives The connectives available in GSL [Wijayarathna 1998] are

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And, Or, Not and And_Then. Truth values True and False are also available. If F1 and F2 are facts then so are F1 And F2, F1 Or F2 and F1 And_Then F2. However, the connective Not should be attached to a major relationship or fact but not to a minor relationship. Because minor relationships are used only to improve the intelligibility of the specifications. When transforming GSL facts into TANDTL atoms, minor relationships are going to be ignored. Therefore, we define, if R is a relationship and F is a fact then Not R is also a relationship and Not F is also a fact. Highlighted words in GSL specifications are GSL reserve words. Names of objects start with a capital letter. Italics words are major relations while simple letter words are minor relations. Example 3: let us consider the statement “A passenger does not request for a reservation” . This is represented in GSL as, “Passenger Not request Reservation”. In this representation, the Not connective is attached to the major relationship “request”. The requirement, “Tanaka requested a reservation, but the destination is not Colombo”,

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is represented in GSL as “Tanaka request Reservation to Not Colombo”. In this case the minor relationship “to” is not connected to Not. This will be represented in Templar[Tuzhilin 1995] as “request(Tanaka, reservation, ?Colombo)”. Temporal operators and relations And_Then (TAND) connective and redefined And connective overcome the weaknesses of temporal logic in representing relative temporal knowledge [Wijayarathna 1997]. Therefore in GSL, And connective represents concurrent events while And_Then ( ∏ ) connective represents sequential events. Classical logic with And_Then connective is strong enough to represent temporal knowledge. Users can define their own temporal relations or operators using And_Then connective. This makes the GSL software requirements specifications language simple but powerful tool in expressing temporal information. GSL establishes a priority order of the connectives and user defined temporal relations/ operators. Not has the higher priority than And and Or And and Or have the same priority And and Or have the higher priority than And_Then. All user defined temporal relations can be replaced by Not, And, Or and And_Then connectives. Example : Let us define the temporal relations Before, After, While, Until, At_the_start_of and At_the_end_of using And_Then Connective If p Before q Then p And_Then not p and not q And_Then q

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If p After q Then q And_Then not q and not p And_Then p If p While q Then q And Not p And_Then p and q And_Then q And Not p If p Until q Then True And_Then q Or (p And_Then q) If p At_the_start_of q Then p And q And_Then q And Not p If p At_the_end_of q Then q And Not p And_Then p And q As shown in the above examples, users can define their own temporal relation or a temporal operator by adding a simple rule to the language. Therefore users are not restricted to the pre-defined temporal relations. These user defined temporal relations will be transformed to TANDTL sentences as follows: (p Before q ) ? p ∏ (?p ∧ ?q) ∏ q. (p After q ) ? q ∏ (?q ∧ ?p) ∏ p. (p While q) ? (q ∧?p) ∏ (p ∧ q) ∏ (q ∧?p). (p Until q ) ? True ∏ ( ?q ? (p ∏ q)). (p At_the_start_of q ) ? (p ∧ q) ∏ (q ∧?p). (p At_the_end_of q ) ? (?p ∧ q) ∏ (q ∧p). BNF grammar for If-Then statement: If-Then statement → If Fact Then Fact Events (Temporal Conditions)

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According to the view of a system adopted in this research (section 2.5.2), system starts functioning when an external or an internal event triggers an action(s) under specific condition. Hence GSL specifications should be capable of representing this view of system. When clause in GSL does this. BNF grammar for When statement : When statement → When Fact Then_Do Fact | When Fact If Fact Then_Do This statement can be rewritten as: WhenEvent If Condition Then_Do Action Using When statement, GSL can represent the view of a system adopted in this research. Recall events, conditions and actions are facts. Example : Now let us consider the requirement number 2 in our pilot system. It says “When the request for reservation is made the consultant should record the passenger details such as name, destination, intended departure date, intended air line and maximum cost”. This will be represented in GSL as: When Consultant receive Reservation_request from Passenger Then_Do Consultant record (Name Of Passenger And Destination Of Passenger And Departure_date Of Passenger And Airline Of Passenger And Max_cost Of Passenger) in Request_register The same requirement will be represented in Templar[Tuzhilin 1995] as: Whenreceive(Consultant, Reservation_request, Passenger) Then_Do record( Consultant, Name, Destination, Departure_date, Airline, Max_cost, Request_register)

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Major relations in GSL are represented as predicates in Templar. Names of objects become terms in formulae in Templar. A fact in GSL is the expanded version of formulae in Templar with semantic relations. This improves the intelligibility of the specifications. The event in the above GSL specification is “Consultant receive Reservation_request from Passenger”. The same event appears in Templar as “receive(Consultant, Reservation_request, Passenger)” In the above example, Of relationship represents possession. That is, it stands for the “part-of” relationships in data modeling. The major relationship is “receive” while “from” and “in” are minor relationship. The others are objects. Activities According to the view of a system adopted, events trigger actions (activities) under specific conditions. As shown above, When clause in GSL represents events. The corresponding actions for the events stated in When clause are available in Then_Do clause. While is a temporal relation which can be defined using And_Then connective in GSL. However, While is a clause in Templar because they employ Event-Action-Condition-Action (EACA) model as the view of a system. If F1 and F2 are facts, then F1 While F2 is also a fact and it is true if and only if F2 is true for some time interval t1 to t2 and F1 is true for some time t3 where t1 ≤ t3 ≤ t2. This fact can appear in When clause in GSL. That is an activity can appear in When clause in GSL. Therefore it is better to name events in GSL as “temporal conditions”. However Templar distinguishes events and actions separately. Hence When statement can be represented as:

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When If Then_Do

Temporal Condition Non-temporal Condition Action.

There are no temporal operators available in Non-temporal condition. Example : the requirement number 11 says that “A passenger can cancel his request for reservation while he is waiting for confirmation without penalty”, This will be states in GSL as: When Passenger request Cancellation While Passenger waiting_for Confirmation Then_Do Consultant cancel Reservation_request And Consultant Not charge Penalty from Passenger The corresponding Templar specification is: Whenrequest(Passenger, Cancellation) While waiting_for(Passenger, Confirmation) Then_Do cancel(Consultant, Reservation_request) ?charge(Consultant, Penalty, Passenger). One may not be able to distinguish the difference of appearance of While in two specifications. However, “Passenger request Cancellation While Passenger waiting_for Confirmation” is a single fact in GSL. In the meantime, Templar represents it as two formulae. The GSL specification can be transformed to TANDTL as : request(Passenger, Cancellation) While waiting_for(Passenger, Confirmation) ? cancel(Consultant, Reservation_request) ∧ ?charge(Consultant, Penalty, Passenger).

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Since While is a user defined temporal relation in GSL and can be represented using TAND connective in TANDTL, the above formula can be rewritten with TAND connective. For example, suppose that While temporal relation is defined as (p While q) ? (q ∧?p) ∏ (p ∧ q) ∏ (q ∧?p). Then the above formula can be rewritten as: {request(Passenger, Cancellation)∧?waiting_for(Passenger, confirmation)) ∏ (request(Passenger, Cancellation) ∧ waiting_for(Passenger, Confirmation)) ∏(request(Passenger, Cancellation) ∧ ?waiting_for(Passenger, Confirmation)} ? cancel(Consultant, Reservation_request) ∧ ?charge(Consultant, Penalty, Passenger). When executing above TANDTL logical formula, system will check whether the condition in left hand side of “?” is true. If it is true, then it will execute both actions cancel(Consultant, Reservation_request) and ?charge(Consultant, Penalty, Passenger). In the same way, if F1 and F2 are facts then, F1 Before F2, F1 After F2, F1 Until F2, F1 At_the_start_of F2 and F1 At_the_end_of F2 are also facts. Example : “If a passenger cancels his reservation request before the reservation is confirmed then no penalty for the passenger” is stated in GSL as: WhenPassenger request Cancellation Before Reservation is Confirm Then_Do Consultant Not charge Penalty from Passenger

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Example : “If the passenger does not pay after the reservation is confirmed then the consultant does not issue the ticket” can be represented in GSL as: WhenPassenger Not pay Payment After Reservation is confirm Then_Do Consultant Not issue Ticket. Example : Requirement 9 in our pilot system says that “If the reservation status is waiting, then consultant asks the passenger whether he is ready to wait. If he is ready to wait then wait for confirmation”. This will be stated in GSL as: WhenReservation is Waiting And Passenger want Waiting Then_Do Passenger wait Until Reservation is Confirm. When executing the specification, first the condition in When clause is evaluated. If it is true, then the actions in Then_Do clause will be executed. That is the action “ Passenger wait” will be executed until the action “Reservation is Confirm” is executed. Hence, repetitive actions can be stated in GSL requirements specifications. Example : “If a reservation is confirmed then consultant bill the passenger and create the transaction for accounting” is represented in GSL as: WhenReservation is Confirm Then_Do Consultant bill Passenger And Consultant create Transaction for Account. The And connective in TANDTL and GSL represents events occurs simultaneously. Therefore GSL can represent parallel activities with the And logical connective.

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Example : “Validation is the first step in the transaction recording process.” is represented as: WhenAccount_clerk receive Transaction Then_Do Validation At_the_start_of Journalizing And_Then Journalizing In the above example, the actions in Then_Do clause are “Validation” and “Journalizing”. These are objects. The view of a system adopted in this research allows creation of new objects with new attributes in addition to the original objects. “Validation” and “Journalizing” are new objects created by some other objects and relations. Suppose that “Journalizing” refers to the following fact. Clerk record Transaction in Journal. In this fact, Clerk, Transaction and Journal are objects. “record” is the major relation and “in” is the minor relation. Then, “Journalizing” is the new object created by Clerk object, Transaction object, Journal object, “record” relation and “is” relation. Therefore GSL is capable of representing different level of abstraction. “Journalizing” is a representation of higher level of abstraction while the fact “Clerk record Transaction in Journal” represents more detail level of abstraction. In other words, “Journalizing” hides the implementation information. That is a main concept of object orientation. Hence using GSL specification language users can write different level of specifications, more abstract once and more detail ones. That is GSL is scaleable. Define clause Define clause is a feature in GSL which is not available in Templar [Tuzhilin 1995] or other specifications languages. It implements the concept of abstraction in object-oriented principles [Sigfried 1996] [Martin 1993]. The view of a system adopted in this research cannot be implemented only by When statement described in the previous section. For example, creation of a new

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object with new attributes in addition to the original attributes cannot be implemented by When statement. The role of Define clause is to implement that. The syntax of the Define clause is either Define < object | fact > As < fact > End_Define Define < object > Is_A < object > End_Define Define < object > As < other construct> End_Define BNF grammar for Define statement : Define statement → Define Fact As Fact End_Define | Define Object As Fact End_Define | Define Object As Object End_Define | Define Object As If -Then statement Case 1: Define < object > As < fact > End_Define In this case, the fact is renamed as an object. This form can be used either to give a name to a fact or to assign a fact/ group of facts to an object. Example : Let us say that the process of recording a passenger details in the registry is called “passenger_recording” is represented as: Define Passenger_recording As Passenger record_to Register End_Define In this example, the fact “Passenger record_to Register” is assigned to the object “Passenger_recording”. That is, new object “Passenger_recording” is created by objects Passenger and Register and the major relation “record_to”. This helps systems analysts to use single word “Passenger_recording” instead of “Passenger record_to Register” whenever he needs to refer to the recording process. That is, systems analysts will be able to specify requirements in a abstract level and later they can specify the detail implementation of the requirements using GSL specification language. Since “Passenger_recording” is or or

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an object by the definition, it can be used as an object in the design and implementation phases, if systems analyst would wish to do so. The corresponding TANDTL statement will be “passenger_recording = record_to(passenger, register)”. Therefore whenever “passenger_recording” is called it will replace “record_to(passenger, register)”. Case 2: Define < fact > As < group of fact > End_Define This is how GSL creates composite facts. It can be either a composite event or a composite activity. Example : “To record passenger details in the register copy name, destination, departure date, air line, and maximum cost of passenger into register” is represented as : Define Passenger record_to Register As Name Of Passenger copy_to Register And Destination Of Passenger copy_to Register And Departure_date Of Passenger copy_to Airline Of Passenger copy_to Register And Max_cost Of Passenger copy_to Register

Register

And

End_Define Case 3: Define < object > Is_A < object > End_Define Example : “A JAL is an air line” is represented as: Define JAL Is_A Airline End_Define This is used to implement the “is-a” relationship in data modeling. Inheritance in GSL is based on the Is_A hierarchy. The object JAL in this example is declared as an Airline. Therefore, object JAL can have attributes of

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the object Airline. The corresponding TANDTL statement will be the “Is_A(JAL, Airline)”. Case 4: Define < object > As < other construct > End_Define This creates rules or condition objects. Example : Let us define the rule for penalty charges in rule 14. Penalty10% As Departure_date - Cancel_date > 31 And Departure_date - Cancel_date < 60 Then 10 % Payment End_Define. Define If

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CHAPTER VII

DISCUSSION

This chapter provides a discussion about TAND connective, temporal logic TANDTL and the specifications language GSL.

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When specifying software requirements using GSL specification language, users are allowed to define their own temporal relation. All temporal relations can be defined using And_Then connective. For example one can combine Allen’s[Allen 1983] (p before q ) and (p meets q ) together to represent a temporal relation called “on-or-before”. The required representation is, If ( p on-or-before q) Then [p And_Then ( Not p And Not q) And_Then q ] Or [p And_Then q ]. Yet another example, let us assume that one wants to define a temporal relation “odd-relation” to represent “only p occurs and then only q occurs and then both p and q occur and then only q occurs and then only p occurs”. Then, If ( p odd-relation q) Then ( p And Not q ) And_Then ( Not p And q ) And_Then ( p And q ) And_Then ( Not p And q ) And_Then ( p And Not q). The And_Then connective can be used to define existing temporal operators in conventional logic. This will increase the expressive power of temporal logic as well as GSL. For example let us consider how to represent the necessary computational steps of a software module to compute the factorial of a given integer “n”. The possible steps would be : 1) let z = 1 2) let z = z * n 3) reduce n by 1 4) repeat steps 2 and 3 until n equals 1 5) return z. Then we can represent this module as, factorial(n) = (let z=1) ∏ [ ((let z = z* n) ∏ (reduce n by 1)) until (n=1)] ∏ (return z).

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In this representation, And_Then is combined with existing temporal operator “until”.

As discussed in this thesis, GSL provides end-user understandable software requirements specifications using object-oriented principles. Even though the statements in GSL are not always grammatically correct, end-user will be able to understand the specifications. Systems analysts will enjoy the application of object-oriented technique in conceptual modeling. GSL can be used not only in object-oriented conceptual modeling but also in structured systems analysis and design. Structured systems analysis and designing methods[Hoffer 1996] are still used ubiquitously while the object-oriented methods[Martin 1993] are becoming popular. Researchers and practitioners[Taylor 1995][Graham 1994][Zhao 1994][Wu 1995] have identified weaknesses in representing business rules in object-oriented systems analysis and designing methods. They suggest two solutions: one to encapsulate business rules into a class and the other to represent rules as rules objects. GSL specifications can be used for both methods since it defines rules as objects and rule objects as attributes of other objects. Therefore GSL is in position to satisfy both present and future conceptual modeling requirements. Speech interfaces are becoming a reality now[Smith 1994][Zue 1994][Zue 1995]. Systems analysts will also benefit with natural spoken language interfaces in requirements acquisition. They will be able to acquire requirements through natural language spoken interface while they engage in the real working environment. This requires some sort of resemblance between the specifications language and the natural spoken language. GSL can be used with a speech interface since its statements resemble that of English language. GSL has built-in sequence, condition and repetition constructs. Therefore automated program generation will also be possible with GSL specifications.

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Since GSL represents temporal knowledge in facts, reasoning about temporal knowledge is possible. Events in When clause will accompany While, Before and After temporal relations, in contrast to When clause in Templar[Tuzhilin 1995]. Therefore the implementation in the two models is apparently different. GSL uses Fact as a modeling construct. Facts in GSL is a grammatically incorrect basic English statements which consist only one main verb and the statements are in active form. Therefore GSL statement can represent three major information components of requirements of information systems; What, Who and When. Since a Fact is in active form, it always starts with the responsible person that is “Who”. A Fact itself represent “What” component. Temporal relations in GSL represents “When” component.

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CHAPTER VIII

CONCLUSIONS

This chapter concludes this thesis and points out importance of this research. The software requirement specifications language GSL is developed to provide executable, end-user intelligible specifications. Further GSL can represent incomplete relative temporal knowledge with TAND connective. The existing temporal knowledge representation methods need complete time information about occurrences of events which cannot be found in information systems. However, new logical connective TAND does not require complete time information. It can represent incomplete relative temporal knowledge without any difficulty. The executable temporal knowledge TANDTL is developed using TAND connective to execute GSL specifications. Therefore TANDTL has only TAND connective in addition to standard logical connectives. This simplifies the both logic and the execution process. Since GSL is based on TANDTL temporal logic, GSL specifications are executable. Hence, GSL can solve the problems with accounting systems computerization.

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The software requirements specifications language GSL (General-purpose Specification Language) is designed to satisfy following requirements: 1. End-user Intelligible, executable software requirements specifications 2. Application of object-oriented principles in requirements specifications 3. Representation of relative temporal knowledge. Requirements specifications written in GSL are easily understandable to endusers than specifications written in other specifications languages. Also systems analysts will be able to use object-oriented concepts in conceptual modeling. Programs which satisfy the requirements specifications can be generated quickly from the executable software requirements specifications. GSL is based on temporal logic TANDGL. The temporal logic TANDGL is developed removing all existing temporal operators and relations and using TAND connective. Therefore TANDTL consists of TAND logical connective in addition to standard logical connective. The logical connective TAND is proposed to represent temporal knowledge because the existing temporal operators and temporal relations require complete time information to represent temporal knowledge. However, introduction of TAND connective simplifies the representation scheme and it is capable of representing incomplete relative temporal information which can be found in any given information system. Since TANDTL is executable and GSL is based on it, specifications is also executable. Again TANDTL simplifies the process because it applies only TAND connective to represent knowledge. Therefore execution of temporal actions can be done by TAND. the GSL execution temporal executing

Finally, this thesis proposes a new logical connective TAND to represent incomplete relative temporal knowledge, a new temporal logic TANDTL using TAND connective and a new software specifications language GSL based on TANDTL. GSL specifications are executable and intelligible to end-users.

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When writing GSL specifications for example system, it was found that the translation of requirements from source document to GSL specifications is easy. However, in some cases, it was required to identify events in order to write the specifications since the source document itself does not provide them. Therefore a source document written in natural language may require some modifications before translating to GSL specifications.

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ACKNOWLEDGEMENT

I wish to express my heartiest gratitude to my academic supervisor Professor M. Maekawa for his guidance, encouragement and valuable advice given to me throughout the research project and making comments which enabled me to improve both the content and presentation of the thesis. My sincere thanks are due to Research Assistants Mr. Y. Kawata, Mr. Soshi Masakazu and all members of Maekawa laboratory for their kind help and assistance given to me during the course of project. I would like to extend my sincere gratitude towards Mr. Kinya Iwata and family for their numerous help and encouragement given to me during my course of studies especially on arrival to Japan. I would also like to convey a special word of thanks to Randeniya and Silva family for their encouragement and assistance given to me in order to overcome difficult situations. Finally I would like to thank my wife Dilrukshi for her utmost support given to me during my course of studies.

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APPENDIX (A)

AN EXAMPLE SYSTEM : CASE STUDY

This appendix shows an application of GSL specification language for a real problem. This assignment case appears in appendix A in “Accounting Information systems: Essential Concepts and Applications”[Wilkinson 1993].

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Datacruncher Officer Equipment, Inc. Statement Datacruncher Office Equipment, Inc. of Dallas, Texas, is a manufacturer of varied machines and devices for the modern office. Among its products are desk calculators, terminals, printer units, key-to-tape units, micro film reader, word processing systems, time stamping machines, and addressing devices. The firm distribute its products nationwide through 150 franchised dealers who also handle the product of competitors. In addition, the firm sells direct to large and medium-size business firms and other organizations having substantial data processing requirements. The firm also provides services to its customers. About 600 customers receive statements at the end of a typical month of sales or service. Datacruncher was started in the late 1970s. The founders were four employees - two sales persons and two engineers - from a long established office machines manufacturer. They foresaw the growing importance of the office in the modern firm. Their vision has been amply rewarded, since their firm has enjoyed an explosive growth. Of course, vision alone was not sufficient to generate this growth. A sound knowledge of the office equipment market and skillful designed products were the essential ingredients. The firm’s growth is reflected by several measures. Sales have reached $70 million during this year just ended; this amount represents a 20 percent increase over last year’s sales. The number of managers and employees has climbed to almost 1100 as of this year end. Approximately 500 suppliers provide materials and parts for the 120 products that the firm manufactures. The physical facilities consist of the home office building and production plant, located just off an express parkway in Dallas, plus three regional sites in San Francisco, St. Louis, and Philadelphia. Each regional site contains a sales office and a warehouse. A finished-goods warehouse is attached to the plant in Dallas.

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Besides growth in sales the firm’s founders have emphasized the need to increase the firm’s share of the office equipment market and the return on total assets. To achieve these objectives, they have stressed aggressive salesmanship, new-product development, prompt deliveries of ordered products, minimized production and inventory cost, and prudent cash management. Organization Datacruncher is organized as a corporation and has 2500 stockholders. The board of directors consists of the four founders plus four outside directors. The four founders occupy top positions in the firm: Bill Dixon is the president, Bert Sanders is the vice president of marketing, Judy Hollis is the vice president of engineering, and Jim Marshall is the vice president of production. The first two were formerly salespersons; the last two were engineers. Other high-level managers include Harry Myler, vice president of administration; Charles Dauten, vice president of finance; Barbara Fulton, controller; and Tim Baker, director of information systems. Financial status The income statement for the last two years (as shown in Figure A.1) reflect the sales growth mentioned earlier. They also show, however, that net income has grown less rapidly than net sales. In fact, net income for this year has declined from last year’s net income. There are indications that this decline stems from two factors: (1) rising costs in production, inventory, and other areas; and (2) necessary reductions in the prices of certain products to combat the new products of competitors. The balance sheets for the last two years, (shown in Figure A.2) indicate that the firm’s financial position is basically sound. However, there are certain adverse signs, such as a shrinking cash balance. Procedures

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Four broad activities at the operational level can be identified as the revenue cycle, the production cycle, the expenditure cycle, and inventory management. Other activities include engineering design, market research, personnel and payroll, cash management, and general ledger accounting. Processing of transactions pertaining to the first four preceding activities, plus payroll and general ledger accounting, is aided by two computers. One of these computers is located within the accounting function, the other within the production function. Magnetic disks are employed for on-line storage of files. The following sections describe briefly the current processing, including key documents, outputs, and files. (In addition, Figure A.3 contains measures of activity relating to the following procedures.) Revenue Cycle Salespersons periodically visit the dealers and prospective business firms and other organizations in their sales regions. As they obtain orders, they may mail or phone in the orders to their regional sales offices. Each sales office then records the orders on a register and prepares formal sales orders in quadruplicate. The original of each order is mailed to the customer, whereas the last copy is filed by customer name. At the end of each day the batch of orders (consisting of the middle two copies of all orders prepared that day) is mailed to the home office. When received in the sales order department, the orders are reviewed for completeness and accuracy by sales order clerks and numbers are assigned. The orders are then forwarded to the credit department for credit check. When credit is approved for the amount of the orders, one copy of each order is sent to the inventory control department and the other copy to the billing department. By reference to computer printouts of product status, inventory control clerks determine whether or not sufficient inventory is available to fill each order. If sufficient inventory is available at the warehouse in the region where the customer resides, one copy of the order is mailed there. If sufficient

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inventory is not available in the regional warehouse but is available in the main finished-goods warehouse in Dallas, the copy of the order is routed there instead. In either case the goods are picked and readied for shipment, based on the order. A shipping report and bill of landing are prepared and the order is shipped. A copy of the shipping report is enclosed with the shipment as a packing slip, and another copy is returned to the billing department. If sufficient inventory is not available to fill the order in its entirety, the inventory control clerk prepares a backorder, which he or she routes to the production planning and control department. On the receipt of a shipping report, a billing clerk pulls the department’s copy of the sales order from a file, verifies that the product numbers and quantities match, and notes the shipping date and prices on the order copy. Next he or she sends the order copy (together with the other orders processed that day) to the data preparation section in the accounting data processing department. The orders then are keyed onto magnetic tapes, edited, sorted, and processed against the accounts receivable and finished-goods inventory (product) master files. Sales invoices are generated as outputs from this processing, together with an open sales invoice file on magnetic disk. Two copies of each sales invoice are mailed to a customer, two other copies are sent to the sales order department and the appropriate regional sales office, and a fifth copy is filed alphabetically. All cash receipts from customers are received in the mailroom at the home office. There they are opened and listed on a special form. Then the cheques are routed to the cashier, together with a copy of the list. The cashier prepares a bank deposit slip in duplicate, endorse the cheques, and delivers the deposit to the bank the next morning. A copy of the deposit slip is returned to a file in her office. Another copy of the listed receipts is sent to the credit department, where a clerk enters the customer numbers that corresponds to the listed names and address. The clerk forwards the list to the data preparation section, which keys the receipts data onto magnetic tape. The transaction data are then sorted and processed against the accounts receivable master file once each day. At the end of the month the accounts receivable master file is

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processed to produce an accounts receivable aging schedule, which is sent to the credit manager, and statements, which are mailed to customers. Production Cycle Products are manufactured either for inventory or to fill backorders. The overall production level generally is based on a sales forecast made by the marketing function. However, backorders occur because of out-of-stock conditions, and must be fitted into the schedule. In fact, backorders are given priority in order to pacify unhappy customers. Production operations are triggered when the production planning and control department receives production authorizations or backorders. The production authorizations are prepared by comparing forecasted sales levels with current levels of finished-goods-inventory on hand and are issued jointly by the production superintended and the inventory control manager. Backorders are prepared as described earlier, on the basis of orders that cannot be filled. The production planning and control department then obtain the bills of material from the engineering function and explores the production requirements to determine materials and parts requirements. With the materials and parts requirements in hand, a production planning clerk checks a computer printout of materials and parts inventory on hand. If the materials and parts on hand are adequate for a particular product, the clerk schedules a production run (based on available labor and machines). As each scheduled date nears, the clerk sends the affected production authorizations and backorders to the data preparation section of the production data processing department. There the production requirements data are keyed onto magnetic tape, sorted by product number, and processed to produce numbered production orders, materials requisitions, and move ticket. Files used in this processing run (all on magnetic disk) are the bill-of-materials file, the operations list file, the open production order file, and the work-inprocess inventory master file.

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Copies of the materials requisitions are sent to the materials storeroom, which then delivers materials and parts to the designated production departments. Copies of the production orders and the move tickets are sent to the first production department involved in the manufacturing process (usually the fabrication department). Copies of production orders are also sent to the cost accounting department, and copies of materials requisitions are kept in the data processing department for inventory processing. As work is completed on an order in a department, a move ticket is returned to the production planning and control department. At the end of each day, all returned move tickets are batched and forwarded to the data preparation section. There the move ticket data are keyed onto a magnetic tape, sorted by production order number, and processed to produce a daily production status report. The open production order file is updated during the processing. In separate daily processing steps, the materials requisitions are batched, keyed onto a magnetic tape, sorted by material-part number, and processed to update the raw-materials inventory master file. Then the materials requisitions are resorted by production order number and processed (together with labor jobtime tickets forwarded from work centers and sorted in a like manner on a separate magnetic tape) to update the work-in-process inventory master file. When a production order has progresses through the fabrication and assembly departments, the units of completed products are inspected. Those units that pass inspection are released to the finished-goods warehouse, and copies of the order release are sent to the production planning and control department and the cost accounting department. From the central warehouse the finished products are shipped, via stock transfer notices, to the three remote warehouses as needed to replenish stocks. The production planning and control department records the completion and then sends the releases to the data preparation section. There they are keyed onto a magnetic tape, sorted by product number, and processed against the finished-goods and work-in-process inventory master file, as well as the open production orders file. A completed

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production orders report is also printed; it includes the costs charged to each order. When the materials and parts needed to manufacture particular products are not available, the production planning and control clerk prepares purchase requisitions. These requisitions are sent to the purchasing department.

Expenditure Cycle A wide variety of expenditures, ranging from utilities to insurance, are necessary. Expenditures for raw materials and parts, as well as subassemblies, are particularly significant, since the products manufactured by the firm require a high level of precision. Thus, the procedure pertaining to the purchases of such items and the disbursements for them is another of the critical transaction cycles within Datacruncher. Purchases are initiated by either production planning and control clerks or inventory control clerks. The former clerks issue purchase requisitions when they note that materials and parts are not adequate for upcoming production runs, whereas the later clerks issue similar documents when their experience suggests that the on-hand quantities of particular items have declined to reorderable levels. On the basis of these purchase requisitions, buyers in the purchasing department select suppliers who are known to be reliable and enter their codes on the requisitions, together with acceptable price for the items to be ordered. The requisitions are then forwarded to the data preparation section in the production data processing department. There they are keyed onto magnetic tape, sorted by supplier number, and processed to produce purchase orders. During subsequent runs, the raw-materials inventory master file and the open purchase order file are updated. The purchase orders are then signed by the purchasing manager and mailed to the suppliers. Copies of the purchase orders are forwarded to the receiving department and the accounts payable department, and a fourth copy is filed by supplier name in the purchasing department.

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When ordered materials and parts arrive at the receiving dock, receiving clerks pull the purchase order copies from their file. Then they count or weigh the items and prepare receiving reports. The items are next transferred to the materials storeroom and the initialed copies of the receiving reports are sent on to the accounts payable department and filed. Another copy of receiving report is sent to the data preparation section of the data processing department, and a third copy is filed numerically in the receiving department. In the data preparation section, the receiving reports are keyed onto a magnetic tape, sorted by materialpart number, processed to update the raw-materials inventory master file, resorted by purchase order number, and processed to update the open purchase order file. When suppliers’ invoices arrive in the accounts payable department, clerk pull the receiving reports and purchase orders from the file and compare the documents. After completing their vouching of the invoices, they prepare disbursement vouchers, record them in a voucher register, and file all the documents together by payment due date. Each day other clerks pull the vouchers due for payment that day and send them to the data processing department. There the payment data are keyed onto magnetic tape, sorted by supplier account number, and processed to produce cheque vouchers and a cheque register. The accounts payable master file is also updated during this run; in effect, each affected supplier’s account is credited to reflect the obligation and debited to reflect the payment. Inventory Management Three inventory files are maintained by the firm. The raw-materials inventory master file is updated to reflect orders for materials and parts, as well as receipts from suppliers and issues into production. The finished-goods inventory master file is updated to reflect the newly manufactured products and the sales of products to customers. The work-in-process inventory master file is

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updated to reflect the start in production of each production order, the issues of materials into production, the charges of labor (from job-time records) into production, the application of over-head to production, and the completion of production. Problems A number of specific problems have become apparent. Some of these problems relate to the procedures described earlier, whereas other problems arise from weaknesses in the organizational structure and in financial planning. Many of these problems stem from the fact that the founders have focused on selling and engineering. They have not given as much attention to the areas of accounting, finance, production, and inventory management. Most of the problems also arise from rapid growth in sales. Some of the significant problems, in addition to those noted earlier, should be mentioned. Interest costs are relatively high, as are costs in production and distribution. Backorders are fairly numerous, even though inventory levels have been rising. Promised delivery dates on customers’ orders are often missed, even though lead times of two weeks or more often are allowed. Processing backlogs are sustained in several of the accounting department. These backlogs lead to a variety of ill effects; for instance, purchase discounts are frequently lost and numerous errors are introduced into the transaction data. The percentage of products that do not pass inspection is rather high, perhaps at least in part because of fairly obsolete production equipment and a high labor turnover. Production schedules are difficult to keep up to date, and production jobs often fall behind their schedules. In fact, production clerks keep extremely busy “pushing” jobs, monitoring their progress, and answering phone calls from concerned customers and salespersons. Also, production rates tend to fluctuate, so that production employees are idled at other times and required to work overtime at other times. This problem stem in part from rush backorders; however, it also arises from sales forecasts that prove to be quite inaccurate and from planning procedures that are relatively weak. For instance, the budget process is fairly rudimentary. Budgets are not tied to carefully established cost

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standards, are not developed in detailed formats, and are not revised to reflect changed conditions. Finally, the reports provided to managers are rather inadequate; most are of the status variety, such as the weekly materials-and-parts status report, the daily product-status report, and the monthly report of budgeted costs versus actual costs. Initiation of Systems Development These problems, and their effects on the firm’s financial status, have been of concern to the founders for some time. Their view is that at least some of the problems are aggravated by an inadequate accounting information system. Recently, in fact, they created the position called director of information systems and hired Tim Baker, because they strongly felt that corrective measures were necessary. Perhaps, they thought, he could harmonize and update an accounting information system that is rather uncoordinated and somewhat obsolete at present. Highlighted words represent GSL clauses and relations, italic words are used for major relations. Words started with a capital letter indicate objects while the words started with a simple letter represent minor relations. GSL specifications for Revenue Cycle When Then_Do When Then_Do Salesperson obtain Order Salesperson mail Or Phone Order to RegionalSalesOffice RegionalSalesOffice receive Order RegionalSalesOffice record Order on Register And_Then RegionalSalesOffice prepare OriginalCopyOfOrder And SecondCopyOfOrder And ThirdCopyOfOrder And LastCopyOfOrder And_Then RegionalSalesOffice mail OriginalCopyOfOrder to Customer And_Then RegionalSalesOffice store LastCopyOfOrder

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by CustomerName Define Define Define Define Define When Then_Do When Then_Do OriginalCopyOfOrder Is_A Order End_Define SecondCopyOfOrder Is_A Order End_Define ThirdCopyOfOrder Is_A Order End_Define LastCopyOfOrder Is_A Order End_Define Backorder Is_A Order End_Define RegionalSalesOffice end Day RegionalSalesOffice mail Order to HomeOffice SalesOrderDepartment Of HomeOffice receive Order from RegionalSalesOffice SalesOrderClerk Of SalesOrderDepartment review Order for Completeness And Accuracy alesOrderDepartment review Order After SalesOrderDepartment Of HomeOffice receive Order from RegionalSalesOffice Order is Complete And Accurate SalesOrderClerk assign Number to Order And_Then SalesOrderClerk send Order to CreditDepartment for CreditCheck CreditDepartment approve Credit for amount Of Order After CreditDepartment receive Order from SalesOrderDepartment CreditDepartment send SecondCopyOfOrder to InventoryControlDepartment And_Then CreditDepartment send ThirdCopyOfOrder to BillingDepartment InventoryControlClerk Of InventoryControlDepartment receive SecondCopyOfOrder from CreditDepartment

WhenS

If Then_Do

When

Then_Do

When

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Then_Do Define

InventoryControlClerk perform checkForSufficientInventory Warehouse at Area Of Customer As Warehouse End_Define

Define MainFinishedGoodsWarehouse at Dallas As MainWarehouse End_Define Define CheckForSufficientInventory As InventoryControlClerk checkfor SufficientInventory in Warehouse Or MainWarehouse byreferringto ProductStatusReports After InventoryControlClerk Of InventoryControlDepartment receive SecondCopyOfOrder from CreditDepartment

End_Define When Then_Do InventoryControlClerk found SufficientInventory at Warehouse While CheckForSufficientInventory InventoryControlClerk mail Order to Warehouse

When Then_Do When If Then_Do

InventoryControlClerk Not found SufficientInventory at Warehouse While CheckForSufficientInventory InventoryControlClerk mail Order to MainWarehouse CheckForSufficientInventory OutOfStock InventoryControlClerk prepare Backorder And_Then InventoryControlClerk send Order to ProductionPlanningAndControlDepartment OutOfStock As Quantity = 0

Define End_Define

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When Then_Do

Warehouse Or MainWarehouse receive Order from InventoryControlDepartment Warehouse Or MainWarehouse prepare Shipment for Order And_Then Warehouse Or MainWarehouse prepare ShippingReport And BillOfLanding And_Then Warehouse Or MainWarehouse enclose ShippingReport with Shipment As PackingSlip And_Then Warehouse Or MainWarehouse enclose BillOfLanding with Shipment And_Then Warehouse Or MainWarehouse ship Shipment And_Then Warehouse Or MainWarehouse send ShippingReport to BillingDepartment

When Then_Do

BillingClerk Of BillingDepartment receive ShippingReport BillingClerk retrieve LastCopyOfOrder And_Then BillingClerk verify ProductNumber And Quantity And_Then BillingClerk write ShippingDate Of ShippingReport on LastCopyOfOrder And_Then BillingClerk write Price Of ShippingReport on LastCopyOfOrder And_Then BillingClerk send LastCopyOfOrder to DataPreparationSection Of AccountingDataProcessingDepartment AccountingDataProcessingDepartment receive LastCopyOfOrder from BillingDepartment AccountingDataProcessingDepartment retrieve LastCopyOfOrder And_Then

When Then_Do

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AccountingDataProcessingDepartment print FirstCopyOfSalesInvoice And SecondCopyOfSalesInvoice And ThirdCopyOfSalesInvoice And FourthCopyOfSalesInvoice And FifthCopyOfSalesInvoice And_Then AccountingDataProcessingDepartment send FirstCopyOfSalesInvoice And SecondCopyOfSalesInvoice to Customer And_Then AccountingDataProcessingDepartment send ThirdCopyOfSalesInvoice And FourthCopyOfSalesInvoice to SalesOrderDepartment And_Then AccountingDataProcessingDepartment store FifthCopyOfSalesInvoice in AlphabeticalOrder Define FirstCopyOfSalesInvoice And SecondCopyOfSalesInvoice And ThirdCopyOfSalesInvoice And FourthCopyOfSalesInvoice And FifthCopyOfSalesInvoi

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