M. R. Hansen and C. Zhou. Semantics and completeness of Duration Calculus. In J.W. de Bakker, C. Huizing, W.P. de Roever, and G. Rozenberg, editors, Real-Time: Theory in Practice, number 600 in LNCS, pages 209--225. Springer, 1992. 33  Home/Search   Document Not in Database   Summary   Related Articles   Check

....particular logic used here, however, is extremely simple and it should be noted that there is still considerable work being done to investigate specification logics for hybrid systems. Recent work in [20] and [21] have augmented CTL to reason about time intervals and there is a duration calculus [24] [25] 26] 27] which also appears to form a very attractive specification language for hybrid systems. This paper has only presented some of the basic principles and ideas behind using logics to formally specify hybrid system behaviour. 6 Verification, Validation, and Synthesis Given a system ....

M.R. Hansen, M.R. and C. Zhou, Semantics and completeness of the duration calculus, in editors, Realtime: theory in practice, De Bakker, Huizing, and de Roever (editors), 1991.

....of these systems, for if they were, they would be adequate for arithmetic, which is impossible by Godel s Theorem. Therefore, if we want to choose the reals as time domain, we can only get the relative completeness of these systems. e.g. the relative completeness of DC has been proved in [15]. If we only quantify over global variables, duration calculi are complete on abstract domains shown in [14] The reason is explained in [21] But if we introduce quantifications over program variables into DC, since we interpret program variables as functions from time domain to duration domain, ....

....HDC is complete on abstract domains by reducing higher order DC to first order two sorted interval temporal logic hinted by [13] In the literature of DC, there are two completeness results. One is on abstract domains ( see [14] Unfortunately it requires rule. The other is on real domain (see [15]) It assumes that if F is a valid interval temporal logic formula, in which only the temporal variables v 1 ; v n occur, then D is an axiom, where D is obtained by replacing each temporal variable v i with P i where P i is a state variable. Hence it is a relative completeness. Up to ....

M.R. Hansen and Zhou Chaochen. Semantics and completeness of duration calculus. In J. W. de Bakker, C. Huizing, W.-P. de Roever, and G. Rozenberg, editors, Real-Time: Theory in Practice, REX Workshop, volume 600 of LNCS, pages 209--225. Springer-Verlag, 1992.

....reported elsewhere (e.g. see [2, 4, 30] A selected bibliography for both phases of the project is included. 1 Introduction The ESPRIT Provably Correct Systems project is underway again. After discouraging delays and cuts, wehave reformed a tightly focussed project with just four partners in 1992, dedicated to cover the fundamental technical aspects of a development process for critical embedded systems, from the original capture of requirements rightdown to the computers and special purpose hardware on which the programs run. This breadth of scope is inspired by the brilliantwork of Bob ....

....of the system. Anders P.Ravn is the site leader and is aided by Kirsten M. Hansen, Michael R. Hansen (currently visiting Oldenburg) Hans Henrik Lvengreen, Jens Nordahl, Hans Rischel, Jens U. Skakkebaek and E.V. Srensen. 3. University of Oldenburg, FB10 Informatik, Ammerlander Heerstrae 114 118, D2900 Oldenburg, Germany. Responsible for production of correct programs. Concentrates on the design process from specification to the code of programs executed perhaps on multiple computers# plans mechanical checks and aids for this process. Prof. Ernst Rudiger Olderog is the site leader, with ....

[Article contains additional citation context not shown here]

M.R. Hansen and Zhou Chaochen. Semantics and completeness of Duration Calculus. In WP. deRoever, editor, Proc. REX'91, Real-Time: Theory in Practice,volume 600 of Lecture Notes in Computer Science. Springer-Verlag, 1992.

.... on Abstract Domains: Abstract This paper presents a completeness theorem for Duration Calculus [ZHR91] and some of its application oriented extensions with respect to an abstractly specified class of frames, as a generalization of the result on the standard real time frame coped with in [HZ92]. The choice of abstract semantics gives the opportunity to prove completeness of Duration Calculus not relative to a semantically defined set of axioms, as needed for its completeness with respect to the standard frame. The abstract semantics captures the essential property of finite variability ....

....state expressions and their durations can be modelled by appropriate sentences and temporal variables in interval logic. We give axioms for these counterparts and prove that they satisfy the properties of state durations that are expressed in the proof system for duration calculus found in [HZ92]. In section 5 we formally introduce twodimensional interval logic and two dimensional duration calculus, which have duration calculus of weakly monotonic time as their special case. We give complete proof systems for these calculi too, and propose axioms that are sufficient in order to gain the ....

M. R. Hansen and Zhou Chaochen. Semantics and Completeness of Duration Calculus. In: Real-Time: Theory and Practice, LNCS 600, Springer-Verlag, 1992, pp. 209-225.

....3; 2 are defined as: 3D = true; D; true; 2D = 3:D. This means that 3D is true for an interval iff D holds for some subinterval of it, and 2D is true for an interval iff D holds for all subintervals of it. DC has a set of axioms about states and rules which is sound and (relatively) complete [HaZ91]. These axioms and rules are listed below. DA 1 0 = 0. DA 2 For an arbitrary state P , P 0. The additivity rule of durations is described as DA 3 For arbitrary states P and Q, Q = P Q) P Q) The following theorem is provable from these axioms. Theorem 3.1 For an arbitrary state ....

Hansen, M. R. and Zhou, C. C.: Semantics and completeness of duration calculus, in J. W. de Bakker, K. Huizing, W. P. de Roever, and G. Rozenberg, eds., Real-Time: Theory in Practice, LNCS 600, pages 209-225, 1991.

.... :3: b = 0 b = In this section, we propose a proof system for DC which consists of a complete Hilbert style proof system for first order logic (cf. e.g. Sho67] axioms and rules for interval logic (cf. e.g. Dut95] Duration Calculus axioms and rules ([HZ92]) and axioms about iteration ( Gue98b] We assume that the readers are familiar with Hilbert style proof systems for first order logic and do not give one here. Here follow the interval logic and DC specific axioms and rules. 6 Axioms and rules for Interval Logic (A1 l ) ....

.... interval logic is complete with respect to an abstract class of time domains in place of R [Dut95] The proof system for interval logic, extended with the axioms DC 1 DC 6 and the rules IR 1 , IR 2 is complete relative to the class of interval logic sentences that are valid on its real time frame [HZ92]. Taking the infinitary rule instead of IR 1 and IR 2 yields an complete system for DC with respect to an abstract class of time domains, like that of interval logic [Gue98a] Adding appropriate axioms about reals, and a rule like, e.g. 8k kx 1 x 0 where k stands for 1 : 1 ....

M. R. Hansen and Zhou Chaochen. Semantics and Completeness of Duration Calculus. In: Real-Time: Theory and Practice, LNCS 600, Springer-Verlag, 1992, pp. 209-225.

.... 2 b = 3: b = k b = z k times for k 0 The proof system for DC consists of a complete Hilbert style proof system for first order logic (cf. e.g. 16] axioms and rules for interval logic (cf. e.g. 5] Duration Calculus axioms and rules ([6]) and axioms about iteration ( 8] Since the proof system for Interval Logic and Duration Calculus have become well known, we list here axioms about iteration only. Axioms about iteration 1 ) 1 = 0 ) 2 ) 3 ) ....

Michael R. Hansen and Zhou Chaochen. Semantics and completeness of duration calculus. In J.W. de Bakker, C. Huizing, W.P. de Roever, and G. Rozenberg, editors, Real-Time Systems: Theory in Practice, Lecture Notes in Computer Science 600, pages 209--226. SprigerVerlag, 1992.

.... b = 1, ddSee b = P = 0) 3 b = true true, and 2 b = 3: The proof system for DC consists of a complete Hilbert style proof system for rst order logic (cf. e.g. 21] axioms and rules for interval logic (cf. e.g. 5] Duration Calculus axioms and rules ([6]) and axioms about iteration ( 8] We only recall here some axioms and rules of the proof system of DC . 0 = 0 1 = S 0 S 1 (S 1 S 2 ) S 1 S 2 ) DC5) S = x S = y) S = x y S 2 if S 1 , S 2 in propositional calculus. IR 1 ) 0=A] ....

M. R. Hansen and Zhou Chaochen. Semantics and Completeness of Duration Calculus. In Real-Time: Theory and Practice, LNCS 600, pages 209-225. Springer-Verlag, 1992.

....calculus was introduced by Zhou, Hoare and Ravn in 1991 [ZHR91] as an extension of first order interval temporal logic [Dut95] equipped with means to reason about state and duration. A relative completeness theorem for duration calculus was formulated and proved by Hansen and Zhou in 1992 [HZ92]. Since then, many extensions of the calculus have been designed and studied to cope with a variety of applications[LRSZ93, SRRZ92, ZHRR92, HZS92, ZRH92, LRSZ92, ZHS93, DW94, PD97] Among the extensions of duration calculus are systems that capture properties of temporal computation processes, ....

....in [Gue98] Originally, duration calculus was introduced in [ZHR91] with a concrete semantics, based on the frame hhR; i; hR ; 0i; mi, where m( a; b] b Gamma a for all a; b 2 R, a b. A relatively complete proof system for duration calculus with respect to this frame was first given in [HZ92]. The above frame is the most important one for applications. Here we give the abstract semantics of duration calculus, which has been introduced in [Gue98] 1.3 Duration calculus with iteration In 1994 Dang and Wang introduced an extension of duration calculus that contains iteration of ....

M. R. Hansen and Zhou Chaochen. Semantics and Completeness of Duration Calculus. In: Real-Time: Theory and Practice, LNCS 600, Springer-Verlag, 1992, pp. 209-225.

....to Elsevier Preprint 12 November 1999 They use real numbers for time, which has advantages for specification and compositionality. Several syntaxes are possible to deal with real time: freeze quantification [4,12] explicit clocks in a first order temporal logic [11,21] integration over intervals [10], and time bounded operators [17] We study logics with time bounded operators, because those logics are the only ones which have, under certain restrictions, a decidable satisfiability problem [5] The logic MetricTL R extends the operators of temporal logic to allow the specification of time ....

....logics are proposed. These axioms are given for first order extensions of our logics, but no relative completeness results are studied (note that no completeness result can be given for first order temporal logics. Finally, a relative completeness result is given for the duration calculus in [10]. The completeness is relative to the hypothesis that valid interval logic formulae are provable. 2 Models and logics for real time 2.1 Models As time domain T, we choose the nonnegative real numbers R 0 = fx 2 Rjx 0g. This dense domain is natural and gives many advantages detailed ....

M. R. Hansen and Zhou Chaochen. Semantics and completeness of duration calculus. In J. W. de Bakker, C. Huizing, W. P. de Roever, and G. Rozenberg, editors, Proceedings of Real-Time: Theory in Practice, volume 600 of LNCS, pages 209--225, Berlin, Germany, June 1992. Springer.

.... such a propositional logic are given in [25] Other complete deductive systems for propositional modal logics which include the operator chop can also be found in [23] and [27] 4 INTRODUCTION In the first order case, different deductive systems exist for both ITL [21] and the duration calculus [14, 26] but little is known about their power. Close links between the two logics have been established in [14] a complete proof system for a dense timed ITL would yield a complete deductive system for the duration calculus. Except for restricted fragments, the duration calculus (and ITL) are not ....

....which include the operator chop can also be found in [23] and [27] 4 INTRODUCTION In the first order case, different deductive systems exist for both ITL [21] and the duration calculus [14, 26] but little is known about their power. Close links between the two logics have been established in [14]: a complete proof system for a dense timed ITL would yield a complete deductive system for the duration calculus. Except for restricted fragments, the duration calculus (and ITL) are not decidable [7] In this document, we examine completeness problems for first order ITL in a variant similar to ....

[Article contains additional citation context not shown here]

M. R. Hansen and Z. Chaochen. Semantics and completeness of duration calculus. In Real-Time: Theory in Practice, REX Workshop. SpringerVerlag, LNCS 600, 1992.

.... b = R 1, ddSee b = R P = 0) 3 b = true true, and 2 b = 3: The proof system for DC consists of a complete Hilbert style proof system for rst order logic (cf. e.g. 20] axioms and rules for interval logic (cf. e.g. 5] Duration Calculus axioms and rules ([6]) and axioms about iteration ( 8] We only recall here some axioms and rules of the proof system of DC . DC1) R 0 = 0 (DC2) R 1 = DC3) R S 0 (DC4) R S 1 R S 2 = R (S 1 S 2 ) R (S 1 S 2 ) DC5) R S = x R S = y) R S = x y (DC6) R S 1 = R S 2 if S ....

M. R. Hansen and Zhou Chaochen. Semantics and Completeness of Duration Calculus. In Real-Time: Theory and Practice, LNCS 600, pages 209-225. Springer-Verlag, 1992.

....the sense of [Ben83] Our method of proof significantly relies on the exact form of the completeness of the proof system for HDC, which underlies the extension in focus. It is known that the original proof system of DC is relatively complete with respect to its standard, real time based semantics[HZ92]. However, this completeness theorem applies to the derivability of individual formulas, and we need to have equivalence between the satisfiability of the infinite sets of instances of our new axioms and the consistency of these sets together with some other formulas, i.e. we need an complete ....

M. R. Hansen and Zhou Chaochen. Semantics and Completeness of Duration Calculus. In: Real-Time: Theory and Practice, LNCS 600, Springer-Verlag, 1992, pp. 209-225.

....and software (C C) Wherever special behaviours of this (existing) C C (reactively) interferes with the remaining sub domains one also has to describe this C C properly. Papers out of the same set of research projects that are being actively pursued by UNU IIST present Duration Calculus [12, 14, 76, 33, 11, 15] formalizations of railway road level crossings with optical sensors and gates: 59] We can also refer to [40, 41] which uses CCS [38] to model solid state interlocks. In the present paper we shall therefore refrain from illustrating this issue. It may, however, be worth noting, that none of ....

.... derived requirements, software specifications and designs: 66, 65, 28, 31, 30, 9] Duration Calculi, continuous, and even infinite time extensions of interval temporal logic, which also allow expressing and reasoning within continuous domains, and are a hallmark of UNU IIST s in house research [12, 14, 76, 33, 11, 15], have also been used to analyze concepts of railway systems: 35, 68, 13] Report No. 37, March 31, 1998 Draft UNU IIST, P.O. Box 3058, Macau Acknowledgements 56 7 Acknowledgements The authors extend their deepest appreciations to Messrs. Jin DanHua [4, 18, 21, 19, 22, 20] and Liu Xin [67, ....

M.R. Hansen and Zhou Chaochen. Semantics and completeness of duration calculus. In J.W. de Bakker, C. Huizing, W.-P. de Roever, and G. Rozenberg, editors, Real-Time: Theory in Practice, REX Workshop, volume 600 of Lecture Notes in Computer Science, pages 209--225. Springer Verlag, 1992.

....there is no common clock. So, the continuous time is most natural and suitable for them. We found that the duration calculus (DC) with continuous time and (relative) complete proof system is among the best formalism to specify the relationship between components of synchronous systems (see [1] and [2]) Since in DC a system is modelled by a set of state variables which are a f0; 1g valued function of time, the relationship and interface between components (including the discretization) are modelled easily as an approximation of these kind of functions. In this paper we show a class of DC ....

Michael R. Hansen and Zhou Chaochen. Semantics and Completeness of Duration Calculus. In Real-Time: Theory and Practice, LNCS 600, pages 209-225. Springer-Verlag, 1992.

....S = 6= 0 Report No. 202, June 2000 UNU IIST, P.O. Box 3058, Macau Projection in DC 25 DC has been primarily studied with respect to its real time based frame hhR; i; hR ; 0; i; oe: max oe Gamma min oei. A relatively complete proof system for DC with respect to this frame was presented in [HZ92]. A comprehensive introduction to DC is given in [HZ97] An complete proof system for DC on the class of all ITL frames can be found in [Gue98] There is no established way of interpreting duration terms at ITL nn discrete intervals. One reasonable way is to refer to the value of the duration ....

Hansen, M. R. and Zhou Chaochen. Semantics and Completeness of Duration Calculus. In Real-Time: Theory and Practice, LNCS 600, Springer-Verlag, 1992, pp. 209-225.

....0 b = 0 k b = z k times for k 0 2. 3 Proof System of DC The proof system for DC consists of a complete Hilbert style proof system for first order logic (cf. e.g. 25] axioms and rules for interval logic (cf. e.g. 9] Duration Calculus axioms and rules ([12]) and axioms about iteration ( 20] We assume that the readers are familiar with Hilbert style proof systems for first order logic and do not give one here. Here follow the interval logic and DC specific axioms and rules. Axioms and rules for Interval Logic (A1 l ) ....

M. R. Hansen and Zhou Chaochen. Semantics and Completeness of Duration Calculus. In Real-Time: Theory and Practice, LNCS 600, pages 209--225. Springer-Verlag, 1992.

.... b = R 1, ddSee b = R P = 0) 3 b = true true, and 2 b = 3: The proof system for DC consists of a complete Hilbertstyle proof system for first order logic (cf. e.g. 10] axioms and rules for interval logic (cf. e.g. 5] Duration Calculus axioms and rules ([6]) and axioms about iteration ( 8] We only recall here some axioms and rules of the proof system of DC . DC1) R 0 = 0 (DC2) R 1 = DC3) R S 0 (DC4) R S 1 R S 2 = R (S 1 S 2 ) R (S 1 S 2 ) DC5) R S = x R S = y) R S = x y (DC6) R S 1 = R S 2 if S 1 , ....

M. R. Hansen and Zhou Chaochen. Semantics and Completeness of Duration Calculus. In Real-Time: Theory and Practice, LNCS 600, pages 209--225. Springer-Verlag, 1992.

.... ( 0 b = R 1 = 0 k b = z k times for k 0 4 The proof system for DC consists of a complete Hilbert style proof system for first order logic (cf. e.g. 15] axioms and rules for interval logic (cf. e.g. 4] Duration Calculus axioms and rules ([5]) and axioms about iteration ( 7] Since the proof system for Interval Logic and Duration Calculus have become well known, we list here axioms about iteration only. Axioms about iteration (DC 1 ) R 1 = 0 ) DC 2 ) DC 3 ) R 1 = 0 ) ....

Michael R. Hansen and Zhou Chaochen. Semantics and completeness of duration calculus. In J.W. de Bakker, C. Huizing, W.P. de Roever, and G. Rozenberg, editors, Real-Time Systems: Theory in Practice, Lecture Notes in Computer Science 600, pages 209--226. Spriger-Verlag, 1992. 13

.... z k times for k 0 3 A Proof System for DC In this section, we propose a proof system for DC which consists of a complete Hilbert style proof system for first order logic (cf. e.g. Sho67] axioms and rules for interval logic (cf. e.g. Dut95] Duration Calculus axioms and rules ([HZ92]) and axioms about iteration ( Gue98b] We assume that the readers are familiar with Hilbert style proof systems for first order logic and do not give one here. Here follow the interval logic and DC specific axioms and rules. Axioms and rules for Interval Logic (A1 l ) ....

.... logic is complete with respect to an abstract class of time domains in place of R [Dut95] The proof system for interval logic, extended with the axioms DC 1 DC 6 and the rules IR 1 , IR 2 is complete relative to the class of interval logic sentences that are valid on its real time frame [HZ92]. Taking the infinitary rule instead of IR 1 and IR 2 yields an complete system for DC with respect to an abstract class of time domains, like that of interval logic [Gue98a] Adding appropriate axioms about reals, and a rule like, e.g. 8k kx 1 x 0 where k stands for 1 : 1 ....

M. R. Hansen and Zhou Chaochen. Semantics and Completeness of Duration Calculus. In: Real-Time: Theory and Practice, LNCS 600, Springer-Verlag, 1992, pp. 209-225.

No context found.

M. R. Hansen and C. Zhou. Semantics and completeness of Duration Calculus. In J.W. de Bakker, C. Huizing, W.P. de Roever, and G. Rozenberg, editors, Real-Time: Theory in Practice, number 600 in LNCS, pages 209--225. Springer, 1992. 33

No context found.

M. R. Hansen and Zhou Chaochen. Semantics and completeness of the Duration Calculus. In J. W. de Bakker, K. Huizing, W.-P. de Roever, and G. Rozenberg, editors, Real-Time: Theory in Practice, volume 600 of Lecture Notes in Computer Science, pages 209--225. Springer-Verlag, 1992.

No context found.

Hansen, M. R. and Zhou Chaochen. Semantics and Completeness of Duration Calculus. In: Real-Time: Theory and Practice, LNCS 600, Springer-Verlag, 1992, pp. 209-225.

No context found.

M.R. Hansen and Zhou Chaochen. Semantics and Completeness of Duration Calculus. In Real-Time: Theory in Practice, LNCS 600, Springer-Verlag, 1992, pages 209--225.

No context found.

M.R. Hansen and Zhou Chaochen. Semantics and completeness of duration calculus. In de Bakker et al. [dB + 92].

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