V.P. Pratt. Models of program logics. In Proceedings 20th IEEE Symp. Foundations of Computer Science, pages 115-222, 1979.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....a model of Phi. 2 A.2 Discussion The logics D and DI share many characteristics, and many results for D extend to DI with no difficulties. For instance the proofs of finite model property and decidability for D in [56] are easily extended to DI, as well as the proof of EXPTIME completeness in [94]. However, while efficient in practical cases inference procedures have been successfully developed for D, extending them to DI has proved to be a difficult task, and to the best of our knowledge has been unsuccessful till now. To be more precise, the inference procedures for D based on the ....

....cases inference procedures have been successfully developed for D, extending them to DI has proved to be a difficult task, and to the best of our knowledge has been unsuccessful till now. To be more precise, the inference procedures for D based on the enumeration of models such as those in [56, 94] can be easily modified to accommodate converse programs. But these procedures are better suited for proving theoretical results than for being used in practice, since they are inherently exponential, not only in the worst case. 122 In contrast, inference procedures for D such as those in [93, ....

V. R. Pratt. Models of program logics. In Proceedings of the 20th IEEE Symposium on the Foundations of Computer Science, pages 115--122, 1979.

....that can be unwound in an actual model of the formula. A refinement of the quotient construction that exploits the above property is the tableau construction. In subsection 3.3.1, we will describe this technique for C L. Tableau based decision procedures for various logics are given in [BAHP83, EC82, EH85, HS84, LN, LN98, Pra79, Wo183, Wo185]. In subsection 3.3.2 we give the complexity results on C L, L L and some meaningful fragments of L L. A decision procedure for the satisfiability of formulae in C L involves an elaborate reduction to the emptiness problem for finite state automata on infinite trees. This will be discussed in ....

V.R. Pratt. Models of program logics. In Proceedings of the 20th IEEE Symposium on Foundations of Computer Science, pages 115-122, 1979.

....in [10] are easily extended to A preliminary version of this work appeared in [8] Communicating author: Fabio Massacci, Dipartimento di Informatica e Sistemistica, via Salaria 113, I 00198 Roma, Italy, email: massacci dis.uniroma1.it. CPDL, as well as the proof of EXPTIME completeness in [21]. However, efficient in practical cases inference procedures have been successfully developed for PDL, but their extension to CPDL has proved to be a difficult task and unsuccessful till now (to the best of our knowledge) To be precise, inference procedures based on model enumeration [10, ....

.... [21] However, efficient in practical cases inference procedures have been successfully developed for PDL, but their extension to CPDL has proved to be a difficult task and unsuccessful till now (to the best of our knowledge) To be precise, inference procedures based on model enumeration [10, 21] or on automata on infinite trees [27] have been extended to converse of programs. Yet, these procedures are more suited for proving theoretical results than for being used in applications. Tableaux procedures for PDL [20, 22] which are typically simpler in practice, have never been extended. In ....

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V. Pratt. Models of program logics. In Proc. of the 20th Annual Symp. on the Found. of Comp. Sci. (FOCS-79), pp. 115--122. IEEE Comp. Society Press, 1979.

....is highly inefficient. For one thing, the models constructed are of exponential size in the length of the given formula; for another, there are 2 O(j j) of them. Thus the naive satisfiability algorithm takes double exponential time in the worst case. There is a more efficient algorithm [Pratt, 1979b] that runs in deterministic single exponential time. One cannot expect to improve this significantly due to a corresponding lower bound. THEOREM 20. There is an exponential time algorithm for deciding whether a given formula of PDL is satisfiable. THEOREM 21. The satisfiability problem for PDL ....

....by a first order formula, but even L 1 cannot distinguish the latter from continuous separable dynamic algebras ( Kozen, 1981b] Equational theory of dynamic algebras. Many seemingly unrelated models of computation share the same equational theory, namely that of dynamic algebras ( Pratt, 1979b; Pratt, 1979a] In addition, many interesting questions arise from the algebraic viewpoint, and interesting connections with topology, classical algebra, and model theory have been made ( Kozen, 1979b; Nemeti, 1980] 15 BIBLIOGRAPHICAL NOTES Systematic program verification originated with ....

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V. R. Pratt. Models of program logics. In Proc. 20th Symp. Found. Comput. Sci., pages 115--122. IEEE, 1979.

.... checking problem non deterministic graphs deterministic graphs PDL P complete; see e.g. 18] P complete; see e.g. 18] P complete; this P complete; this paper, Corollary 10 paper, Corollary 10 Satis ability problem non deterministic graphs deterministic graphs PDL EXPTIME complete [22, 40] EXPTIME complete [39, 9] PDL with nominals EXPTIME complete [23] EXPTIME complete [23] CPDL EXPTIME complete [22, 40] EXPTIME complete [44] CPDL with nominals EXPTIME complete [20, 7] open EXPTIME complete; this open paper, Theorem 12 Table 1: A summary of results on logical reasoning ....

.... [18] P complete; this P complete; this paper, Corollary 10 paper, Corollary 10 Satis ability problem non deterministic graphs deterministic graphs PDL EXPTIME complete [22, 40] EXPTIME complete [39, 9] PDL with nominals EXPTIME complete [23] EXPTIME complete [23] CPDL EXPTIME complete [22, 40] EXPTIME complete [44] CPDL with nominals EXPTIME complete [20, 7] open EXPTIME complete; this open paper, Theorem 12 Table 1: A summary of results on logical reasoning tasks. 5 The Complexity of Reasoning with Path Constraints In order to characterize the complexity of reasoning problems ....

V.P. Pratt. Models of program logics. In Proc. 20th IEEE Symp. Foundations of Computer Science, pages 115-222, 1979.

.... checking problem non deterministic graphs deterministic graphs PDL P complete; see e.g. 18] P complete; see e.g. 18] P complete; this P complete; this paper, Corollary 10 paper, Corollary 10 Satisfiability problem non deterministic graphs deterministic graphs PDL EXPTIME complete [22, 40] EXPTIME complete [39, 9] PDL with nominals EXPTIME complete [23] EXPTIME complete [23] CPDL EXPTIME complete [22, 40] EXPTIME complete [44] CPDL with nominals EXPTIME complete [20, 7] open EXPTIME complete; this open paper, Theorem 12 Table 1: A summary of results on logical reasoning ....

.... [18] P complete; this P complete; this paper, Corollary 10 paper, Corollary 10 Satisfiability problem non deterministic graphs deterministic graphs PDL EXPTIME complete [22, 40] EXPTIME complete [39, 9] PDL with nominals EXPTIME complete [23] EXPTIME complete [23] CPDL EXPTIME complete [22, 40] EXPTIME complete [44] CPDL with nominals EXPTIME complete [20, 7] open EXPTIME complete; this open paper, Theorem 12 Table 1: A summary of results on logical reasoning tasks. Lemma 13 The problem below is in NLOGSPACE in jGj and jtj: instance: a finite L structure G, a path expression t, ....

V.P. Pratt. Models of program logics. In Proc. 20th IEEE Symp. Foundations of Computer Science, pages 115--222, 1979.

....Logic (ETL) that can express properties of a sequence de nable by a right linear grammar [Wol83, VW94] In our work, we are only dealing with automata on nite words. Furthermore, our work continues the line of research relating regular expressions and Propositional Dynamic Logic (see e.g. [BM75, Pra79, FL79, Pra81, HPS83, HKT00]) Plan of the paper. The rest of the paper is structured as follows. In Section 2 and Section 3, we de ne the class of grammar logics as well as other PDL like logics. Some basic de nitions about formal grammars are also recalled. In Section 4, we show that the general satis ability problem ....

....natural corresponding restriction on the class of standard L full models. Combinatory Propositional Dynamic Logic with Converse (converseCPDL) PT91] based on L full ; Combinatory Propositional Dynamic Logic (CPDL) PT91] based on L full without 7 Propositional Dynamic Logic (PDL) [FL79, Pra79]: based on L full without , U and fi i : i 0g; Test free Propositional Dynamic Logic (PDL ) Har84] based on L full without , U and fi i : i 0g; Test free Propositional Dynamic Logic with Identity (PDL ) based on L full without , U and fi i : i 0g but with id. ....

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V. Pratt. Models of program logics. In Proceedings of the 20th IEEE Symposium on Foundations of Computer Science, pages 115{ 122, 1979.

....is interpreted by (R ) I = i## , where (R I ) R I , if i =1 R I i 1 , otherwise Fo r bo th , concept satisfiability is an Exptime complete problem. This result is easily derived from the Exptime hardness proof for PDL in [ 18 ] and from the proof that PDL is in Exptime in [ 41 ] . Nevertheless, implementations of algorithms exhibit good performance in realistic applications [ 34 ] whereas, at the moment, this seems to be more problematical for . We believe that the main reason for this discrepancy, at least in the case of tableau algorithm implementations, lies in ....

V. R. Pratt. Models of program logic. Proc. of FOCS-79, pages 115--122, 1979.

.... 8 Model checking problem non deterministic graphs deterministic graphs PDL O(jjGjj j j) 11] O(jjGjj j j) 11] PDL path O(jjGjj j j) this O(jjGjj j j) this paper, Theorem 8 paper, Theorem 8 Satis ability problem non deterministic graphs deterministic graphs PDL EXPTIME complete [14, 22] EXPTIME complete [20, 5] PDL with nominals EXPTIME complete [15] EXPTIME complete [15] CPDL EXPTIME complete [14, 22] EXPTIME complete [25] CPDL with nominals EXPTIME complete [12, 4] open PDL path EXPTIME complete; this open paper, Theorem 10 Implication problem non deterministic graphs ....

.... path O(jjGjj j j) this O(jjGjj j j) this paper, Theorem 8 paper, Theorem 8 Satis ability problem non deterministic graphs deterministic graphs PDL EXPTIME complete [14, 22] EXPTIME complete [20, 5] PDL with nominals EXPTIME complete [15] EXPTIME complete [15] CPDL EXPTIME complete [14, 22] EXPTIME complete [25] CPDL with nominals EXPTIME complete [12, 4] open PDL path EXPTIME complete; this open paper, Theorem 10 Implication problem non deterministic graphs deterministic graphs inclusion constraints PSPACE hard, in EXPTIME; open this paper, Theorem 11 backward constraints ....

V.P. Pratt. Models of program logics. In Proceedings 20th IEEE Symp. Foundations of Computer Science, pages 115-222, 1979.

....satisfiability and subsumption of concept descriptions are EXPTIME complete problems. The reason for these DLs to be EXPTIMEhard is that they can simulate general TBoxes within concept descriptions (see below) The fact that they are in EXPTIME follows from results for PDL and converse PDL (Pratt, 1979; Harel, 1984) The tableau based algorithms that will be sketched below are NEXPTIME algorithms. The point in designing these algorithms was not to prove worst case complexity results, but rather to obtain practical algorithms, i.e. algorithms that are easy to implement and optimise, and which ....

Pratt, V. R.: 1979, `Models of Program Logic'. In: Proc. of the 20th Annual Sym. on the Foundations of Computer Science (FOCS-79). pp. 115--122.

....that ends on # ## # ####. ## # # , # while = ## # ####. ## # # # by Lemma 3 and the preservation of satisfiability. Finally, Lemma 2 shows that player II must win # (# 0 ) # Deciding satisfiability for PDL in a game based way matches the lower bound of its complexity given in [8]. Theorem 3. The complexity of deciding the winner of # (# 0 ) is in EXPTIME. Proof: During a play both players (co )nondeterministically choose to store one configuration they expect to perform a loop on. Furthermore, the actual configuration has to be stored together with a counter of size ....

V. Pratt. Models of program logics. In Proc. 20th Ann. Symp. on the Found. of Comp. Sci., FOCS'79, pages 115 -- 122. IEEE Comp. Soc. Press, 1979.

.... an exponential size structure that can be constructed in deterministic exponential time; the technique is similar to that used to show that logics of knowledge with common knowledge are decidable in deterministic exponential time [HM92] or that PDL is decidable in deterministic exponential time [Pra79] 2 Theorem 17: Let A be a subset of CONS,NORM,REF,SDP,UNIF,RANK containing CONS and either UNIF or SDP. For the case of one agent, the validity problem in structures satisfying A is co NP complete. PROOF. We show that the satisfiability problem is NP complete. It follows that the ....

V. R. Pratt. Models of program logics. In Proc. 20th IEEE Symp. on Foundations of Computer Science, pages 115--122, 1979.

....j= 2 . So (M 00 ; s) j= for every s 2 W . By (3) again, M 0 j= By (2) M j= as desired. Formulas of the form 2 oe ) 2 can be viewed as a fragment of propositional dynamic logic (PDL) The fact that there is an exponential time decision procedure for validity in PDL [Pra79] gives us an exponential time decision procedure for validity of these formulas. Furthermore, the proof of [FL79] that the validity problem for PDL is exponential time hard applies also to this fragment. Putting these results together, we see that the validity problem for formulas of the form 2 ....

V. R. Pratt. Models of program logics. In Proc. 20th IEEE Symp. on Foundations of Computer Science, pages 115--122, 1979.

.... in an exponential size structure that can be constructed in deterministic exponential time; the technique is similar to that used to show that logics of knowledge with common knowledge are decidable in deterministic exponential time [34] or that PDL is decidable in deterministic exponential time [52]. 2 Theorem 17: Let A be a subset of fCONS;NORM;REF;SDP;UNIF;RANKg containing CONS and either UNIF or SDP. For the case of one agent, the validity problem in structures satisfying A is co NP complete. Proof. We show that the satisfiability problem is NP complete. It follows that the validity ....

V. R. Pratt. Models of program logics. In Proc. 20th IEEE Symp. on Foundations of Computer Science, pages 115--122, 1979.

....context is often cleaner than in the framework of pure logic, since irrelevant syntactic details are suppressed. For example, Boolean algebra captures the essence of propositional logic at a better level of abstraction. The algebraic structure of PDL has been studied in the form of dynamic algebra [ 11 14, 29, 30] and has been found to aid insight and in some cases simplify proofs. Many of the results of this paper have natural algebraic and topological interpretations: Let L be the Boolean algebra of formulas of PL modulo the PL axioms of Section 4, and let rim= nXlXe Z , fL= fXlXe m . In Theorem ....

V. R. PRATT, Models of program logics, in "Proc. 20th IEEE Symp. on Foundations of Comp. Sci.," October 1979, pp. 115-122.

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V.P. Pratt. Models of program logics. In Proceedings 20th IEEE Symp. Foundations of Computer Science, pages 115-222, 1979.

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V. R. Pratt. Models of program logics. In Proceedings of the 20th Annual Symposium on Foundations of Computer Science, San Juan, Puerto Rico, 1979.

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V.P. Pratt. Models of program logics. In Proceedings 20th IEEE Symp. Foundations of Computer Science, pages 115-222, 1979.

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V. R. Pratt. Models of program logics. In Proceedings of the 20th Annual Symposium on the Foundations of Computer Science, pp. 115--122. IEEE Computer Society Press, 1979.

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Vaugham R. Pratt. Models of program logic. In Proc. of the 20th Annual Symp. on the Foundations of Computer Science (FOCS'79), pages 115-122, 1979.

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Vaughan R. Pratt. Models of program logics. In Proceedings of the 20th Annual Symposium on the Foundations of Computer Science, pages 115--122. IEEE Computer Society Press, 1979.

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V. R. Pratt. Models of program logics. In Proc. 20th Symp. on Foundations of Computer Science, FOCS'79, pages 115 -- 122. IEEE, 1979.

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V. R. Pratt. Models of program logics. In Proc. 20th IEEE Symp. on Foundations of Computer Science, pages 115-122, 1979.

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V. Pratt. Models of program logics. In Proceedings FoCS, pages 115--122, 1979.

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V. Pratt. Models of program logics. In Proceedings of the 20th IEEE symposium on Foundations of Computer Science, pages 115-122, 1979.

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