C. J. Fidge, I. J. Hayes, A. P. Martin, and A. K. Wabenhorst. A set-theoretic model for real-time specification and reasoning. In Mathematics of Program Construction, 1998.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....are those based on timed traces, i.e. where systems are modelled by functions whose domains range over all values of time. Examples include the Du ration Calculus [Hansen and Zhou Chaochen, 1997] the timed stream model [Broy, 1993] and the timed refinement calculus [Mahony and Hayes, 1992, Fidge et al. 1998b] These techniques provide a precise and intuitive approach to modelling timedependent behaviour, essentially the same as that used in the physical sciences. However, when it comes to modelling systems which, through digital control, move between various modes in which the time dependent ....

....it has been extended with a simple set theoretic notation for concisely expressing time intervals, i.e. non empty contiguous sets of times, and operators for accessing interval endpoints. In this section, we present a simplified subset of the extended notation based on that of Fidge et al. [Fidge et al. 1998b] which provides the syntax and semantics of a minimal set of operators in addition to those of standard set theory. This notation has been successfully employed to specify the requirements of a sizeable case study [Smith and Fidge, 2000] as well as those of the Nulka Active Missile Decoy, a ....

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Fidge, C., Hayes, I., Martin, A., and Wabenhorst, A. (1998b). A set-theoretic model for real-time specification and reasoning. In Jeuring, J., editor, Mathematics of Program Construc- tion (MPC'98), volume 1422 of Lecture Notes in Computer Science, pages 188-206. Springer-Verlag.

....features elemental function overloading, vector valued subscripts, and promotion, and how to resolve these kinds of overloading. Elemental overloading of operators to functions over time has been proposed for the Duration Calculus [2] and in a proposed extension to the Z specification language [5, 7]. A complete algorithm for this scheme has been presented [6] but the algorithm is reduced to a proliferation of case sensitive lifting processes [5, p. 11] In contrast, our algorithm is quite easy to understand, comprising only one rule schema, which describes how an expression is transformed. ....

C. J. Fidge, I. J. Hayes, A. P. Martin, and A. K. Wabenhorst. A Set-Theoretic Model for Real-Time Specification and Reasoning. In Johan Jeuring, editor, Proc. 4th Int. Conf. on Mathematics of Program Construction, number 1422 in Lecture Notes in Computer Science, pages 188--206, Marstrand, Sweden, June 1998. Springer Verlag.

....comprises a specification notation for specifying (and reasoning about) systems and a refinement notation for transforming specifications towards implementations. The aim of this paper is to illustrate the utility of a recent variant (a simplified subset) of the timed refinement calculus [6] through the specification and refinement of the well known Steam Boiler Control Problem [1, 10] The specification and refinement are based on an existing approach in the full timed refinement calculus [11] but extended to include system start up and run up times, acceptable errors and minimum ....

....by Z schemas, and accessibility for programmers already familiar with Z. The formalism has been extended with a simple set theoretic notation for concisely expressing time intervals [11] and operators for accessing interval endpoints [5] We adopt a simplified subset of this extended formalism [6] which, further to the aims of readability and accessibility, provides a minimal set of operators outside those of standard set theory. 2.1 Specification notation Absolute time T is modelled by real numbers. For the purposes of this paper, we use the non negative real numbers R . T= R ....

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C. J. Fidge, I. J. Hayes, A. P. Martin, and A. K. Wabenhorst. A settheoretic model for real-time specification and reasoning. In J. Jeuring, editor, Mathematics of Program Construction (MPC'98), volume 1422 of LNCS, pages 188--206. Springer-Verlag, 1998.

....in an environment that provides a reasonable automatisation for theorem proving. The only work in this direction we are aware of is an early attempt at implementing the Duration Calculus in the PVS theorem prover [12] In our work we give an axiomatisation of the Timed Interval Calculus (TIC) [5], a set theoretic notation for expressing properties of time intervals based on work by Mahony and Hayes [6] Many useful laws for reasoning about predicates expressed in TIC have been developed [5, 4, 14] and used in verifying a wide range of real time systems [2, 4, 14] However, these laws need ....

.... PVS theorem prover [12] In our work we give an axiomatisation of the Timed Interval Calculus (TIC) 5] a set theoretic notation for expressing properties of time intervals based on work by Mahony and Hayes [6] Many useful laws for reasoning about predicates expressed in TIC have been developed [5, 4, 14] and used in verifying a wide range of real time systems [2, 4, 14] However, these laws need a more precise characterisation to allow their implementation in a theorem prover. Our axiomatisation gives the infrastructure for such an implementation. The actual implementation has been carried out ....

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C. J. Fidge, I. J. Hayes, A. P. Martin, and A. K. Wabenhorst. A set-theoretic model for real-time specification and reasoning. In Mathematics of Program Construction (MPC'98), volume 1422 of Lecture Notes in Computer Science, pages 188--206. Springer, 1998.

....with refinement notions that are based on failures divergences and bisimulation. Assumptions upon the environment are commonly incorporated using a dummy process that specifies the desired property. Motivation for this paper came from ongoing work on real time processes with trace semantics [10, 12,11, 4, 2, 3, 18, 5, 6] and the search for a parallel operator which is consistent with the refinement and specification theory developed from Mahony s approach [9, 18, 5] We pursue two main goals: first, we recall and elaborate upon a promising specification and refinement theory [9] for dataflow processes with strict ....

C. J. Fidge, I. J. Hayes, A. P. Martin, and A. K. Wabenhorst. A set-theoretic model for real-time specification and reasoning. In J. Jeuring, editor, Mathematics of Program Construction (MPC'98), volume 1422 of Lecture Notes in Computer Science, pages 188--206. Springer-Verlag, 1998. 17

.... of Mahony and Hayes [13, 12] This has been extended with a simple set theoretic notation for concisely expressing time intervals by Millerchip et al. 14] and operators for accessing interval endpoints by Duddy et al. 5] We adopt a simplified subset of the formalism suggested by Fidge et al. [7] which aims at providing a minimal set of operators outside those of standard set theory. Absolute time Tis modelled by real numbers. T= R Observable variables of a system are modelled as total functions from the time domain to a type representing the set of all values the variable may assume. ....

....Volts: A system is specified by constraints on the time intervals over which properties hold. Sets of such intervals can be specified using the interval brackets and a. The combination of round and square brackets reminds us that the interval end point is allowed to be either open or closed [14, 7]. For example, the set of all intervals where the sample command is applied for the whole interval is specified as samplea and the set of all intervals where the sample command is applied and the input is greater than 10 volts for the whole interval is specified as sample in 10a: In ....

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C.J. Fidge, I.J. Hayes, A.P. Martin, and A.K. Wabenhorst. A set-theoretic model for real-time specification and reasoning. In J. Jeuring, editor, Mathematics of Program Construction (MPC'98), volume 1422 of Lecture Notes in Computer Science, pages 188--206. Springer-Verlag, 1998.

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Fidge, C., Hayes, I., Martin, A., and Wabenhorst, A. A settheoretic model for real-time specification and reasoning. In Jeuring, J., editor, Mathematics of Program Construction (MPC'98), volume 1422 of LNCS, pages 188--206. Springer-Verlag.

....occur at any time between these minimum and maximum times. The tick action body merely needs to advance the current time. Tick Delta(now : Time) now 0 = now 1 2. 3 A Trace Based Model The following trace based model is expressed using Mahony s approach for relating sets of time intervals [14, 6]. Variables Let input variable thermometer denote the temperature at every instant in time. It is thus a function of type Time Temperature. Similarly, let output variable heater tell us all times at which the heater is switched on. It is of type Time Switch. In both cases these timed ....

....in P , that is not explicitly indexed by a time, with v( 14] This allows timed variables to appear in predicate P as if they are simple variables of their range type. Furthermore, constants ff and can appear in P to refer to the start and end times of the current interval, respectively [6]. Timed trace specifications then consist of relationships between sets of time intervals, using conventional set theory operators. However, a frequently needed capability is an operator for connecting intervals end to end, to support reasoning about sequences of behaviours [14, 22, 18] In ....

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C. J. Fidge, I. J. Hayes, A. P. Martin, and A. K. Wabenhorst. A set-theoretic model for real-time specification and reasoning. In J. Jeuring, editor, Mathematics of Program Construction (MPC'98), volume 1422 of Lecture Notes in Computer Science, pages 188--206. Springer-Verlag, 1998.

....Time intervals are the basic modelling concept in a number of real time specification and reasoning formalisms. The Duration Calculus [14, 3] and its many derivatives, such as Temporal Algebra [12] are founded in interval temporal logic [8] and integral calculus. The Timed Interval Calculus [6, 2] is founded in Information Technology Division, Defence Science and Technology Organisation, Salisbury, South Australia 5108, Australia. 1 set theory, and is thus compatible with the Z specification language [11] However, despite their different bases, all of these interval calculi offer ....

....1 and S 2 , then their concatenation S 1 ; S 2 is the set of all intervals composed of an interval from S 1 concatenated with an interval from S 2 . The right hand endpoint of the interval from S 1 must equal the left hand endpoint 2 of the interval from S 2 and the intervals may not overlap [2]. See Section 3.2. Using this operator it is possible to express properties concerning state changes, or events. For example, a requirement that the gas must be switched off no more than three seconds after the flame goes out can be expressed as follows. j ( flame ) j ; j ( ....

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C. J. Fidge, I. J. Hayes, A. P. Martin, and A. K. Wabenhorst. A set-theoretic model for real-time specification and reasoning. In J. Jeuring, editor, Mathematics of Program Construction (MPC'98), volume 1422 of Lecture Notes in Computer Science, pages 188--206. Springer-Verlag, 1998.

.... Mahony developed the timed refinement calculus as a powerful specification and refinement method for real time systems, using a Z like notation [17] The language evolved via a series of case studies [14, 18, 16] to produce a simple and elegant set theoretic specification and reasoning technique [7]. To support specifications of continuously observable physical properties, the time domain Tis modelled by the real numbers [15] T = R Time intervals are represented as the set of all times between some infimum a and supremum z . For instance, the left open, right closed interval between ....

....(a . z ) left and right closed [a . z ] and left closed, right open [a . z ) intervals. The . separator [15] is used when defining sets of reals for consistency with Z s : operator which returns sets of integers. The set of all (finite, non empty) time intervals is denoted I[7]. I = fa; z : Tj a z ffl (a . z )g [ fa; z : Tj a z ffl [a . z )g [ fa; z : Tj a z ffl (a . z ]g [ fa; z : Tj a 6 z ffl [a . z ]g The central feature of timed trace formalisms is their use of functions from the time domain to model the dynamic behaviour of observable system ....

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C. J. Fidge, I. J. Hayes, A. P. Martin, and A. K. Wabenhorst. A settheoretic model for real-time specification and reasoning. In J. Jeuring, editor, Mathematics of Program Construction (MPC'98), volume 1422 of Lecture Notes in Computer Science, pages 188--206. Springer-Verlag, 1998.

....VERIFICATION RESEARCH CENTRE SCHOOL OF INFORMATION TECHNOLOGY THE UNIVERSITY OF QUEENSLAND Queensland 4072 Australia TECHNICAL REPORT No. 00 23 Incremental Development of Real Time Requirements: The Light Control Case Study Graeme Smith and Colin Fidge July 2000 Phone: 61 7 3365 1003 Fax: 61 7 3365 1533 http: svrc.it.uq.edu.au To appear in Journal of Universal Computer Science, Special Issue on Requirements Engineering, 2000. Note: Most SVRC technical reports are available via anonymous ftp, from svrc.it.uq.edu.au in the directory ....

....notation, coupled with formal rules for refinement and realisation. 3. 1 Specification Notation Our specification notation is based on Mahony s timed refinement calculus, a formalism specifically designed for describing embedded real time systems [Mahony and Hayes, 1992; Mahony, 1992; Fidge et al. 1998b] Let Tdenote the time domain. All time varying properties of the system are assumed to be represented by timed traces, i.e. functions from the time domain to the value of the variable at that time. For instance, given a variable property v capable of assuming values from set T , it is ....

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Fidge, C., Hayes, I., Martin, A., and Wabenhorst, A. A settheoretic model for real-time specification and reasoning. In Jeuring, J., editor, Mathematics of Program Construction (MPC'98), volume 1422 of LNCS, pages 188--206. Springer-Verlag.

....approach to modelling real time systems has been via the representation of physical variables as functions which vary over time. Examples include the Duration Calculus (DC) 11, 12] the Temporal Agent Model [10] and, more recently, Temporal Algebra [4] and the Timed Interval Calculus (TIC) [2]. These languages are used to express and reason about dynamic properties of variables. They and their predecessors [7] showed how concatenation of time intervals can form the basis of an effective real time modelling and reasoning capability. TIC and DC differ in that TIC is founded in set ....

....that there are no delays between detection of high water levels and low methane, and no flags. Conclusions are drawn in Section 4. 2 Notation and Laws for Timed Trace Predicates The Timed Interval Calculus is a simple set theoretic notation for concisely expressing properties of time intervals [2]. We present the existing foundations, the extensions and transformation laws. 2.1 Time Let Tbe the time domain, denoting the real numbers R. Time intervals will be specified as real intervals: for a; z : Twith a z , the left open, right closed interval between a and z is (a . z ] b = ft : ....

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C.J. Fidge, I.J. Hayes, A.P. Martin and A.K. Wabenhorst. A Set-Theoretic Model for Real-Time Specification and Reasoning. In J. Jeuring (Ed) Mathematics of Program Construction (MPC'98), Lecture Notes in Computer Science Vol. 1422, pages 188--206, Springer-Verlag, 1998.

....model systems by the way their observable properties change over time. Examples include the duration calculus [24] and the timed refinement calculus [16, 12] In this paper we present an integration of Object Z with the specification notation of a simplified subset of the timed refinement calculus [6]. In Section 2, we introduce the Object Z notation and discuss its limitations with respect to modelling continuous and real time systems. In Section 3, we present the specification notation of the timed refinement calculus and show how it can be integrated with Object Z in Section 4. In Section ....

....is a Z based notation for the specification and refinement of real time systems. It has been extended with a simple settheoretic notation for concisely expressing time intervals [13] and operators for accessing interval endpoints [3] In this section, we present a simplified subset of the notation [6] which provides a minimal set of operators outside those of standard set theory. Absolute time, T, is modelled by real numbers. T= R For the purposes of this paper, we assume the units of Tis seconds. Observable variables of a system are modelled as total functions from the time domain to a ....

[Article contains additional citation context not shown here]

C.J. Fidge, I.J. Hayes, A.P. Martin, and A.K. Wabenhorst. A set-theoretic model for real-time specification and reasoning. In J. Jeuring, editor, Mathematics of Program Construction (MPC'98), volume 1422 of LNCS, pages 188--206. Springer-Verlag, 1998.

No context found.

C. J. Fidge, I. J. Hayes, A. P. Martin, and A. K. Wabenhorst. A set-theoretic model for real-time specification and reasoning. In Mathematics of Program Construction, 1998.

No context found.

C. J. Fidge, I. J. Hayes, A. P. Martin, and A. K. Wabenhorst. A set-theoretic model for real-time specification and reasoning. In Mathematics of Program Construction, 1998.

No context found.

C. J. Fidge, I. J. Hayes, A. P. Martin, and A. K. Wabenhorst. A set-theoretic model for real-time specification and reasoning. In Mathematics of Program Construction, 1998.

No context found.

C. J. Fidge, I. J. Hayes, A. P. Martin, and A. K. Wabenhorst. A set-theoretic model for real-time specification and reasoning. In Mathematics of Program Construction, 1998.

No context found.

Fidge, C., Hayes, I., Martin, A., and Wabenhorst, A. (1998b). A set-theoretic model for real-time specification and reasoning. In Jeuring, J., editor, Mathematics of Program Construction (MPC'98), volume 1422 of Lecture Notes in Computer Science, pages 188--206. Springer-Verlag.

No context found.

C.J. Fidge, I.J. Hayes, A.P. Martin, and A.K. Wabenhorst. A set-theoretic model for real-time specification and reasoning. In J. Jeuring, editor, Mathematics of Program Construction (MPC'98), volume 1422 of Lecture Notes in Computer Science, pages 188--206. Springer-Verlag, 1998.

No context found.

C.J. Fidge, I.J. Hayes, A.P. Martin, and A.K. Wabenhorst. A set-theoretic model for real-time specification and reasoning. In J. Jeuring, editor, Mathematics of Program Construction, volume 1422 of LNCS, pages 188--206. Springer, 1998.

No context found.

C. Fidge, I. Hayes, A. Martin, and A. Wabenhorst. A Set-theoretic Model for Real-Time Specification and Reasoning. In Mathematics of Program Construction, Springer Verlag, 1998.

No context found.

C J Fidge, I J Hayes, A P Martin, A K Wabenhorst, A Set-Theoretic Model for Real-Time Specification and Reasoning. In J Jeuring (Ed), Proc. MPC'98, LNCS 1422:188--206, Springer, 1998.

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C. Fidge et al. A Set-theoretic Model for Real-Time Specification and Reasoning. In Mathematics of Program Construction, Springer-Verlag, 1998.

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