M. Y. Vardi and L. Stockmeyer, Improved upper and lower bounds for modal logics of programs, Proceedings of the 17th acm symposium on theory of computing, 1985, pp. 240--251.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....has shown that even with the addition of the halt construct, PDL is strictly less expressive than the calculus. The propositional calculus satisfies a finite model theorem, as first shown in [Kozen, 1988] Progressively better decidability results were obtained in [Kozen and Parikh, 1983; Vardi and Stockmeyer, 1985; Vardi, 1985b] culminating in a deterministic exponential time algorithm of [Emerson and Jutla, 1988] based on an automata theoretic lemma of [Safra, 1988] Since the calculus subsumes PDL, it is EXPTIME complete. In [Kozen, 1982; Kozen, 1983] an axiomatization of the propositional ....

....the addition of the halt construct, PDL is strictly less expressive than the calculus. The propositional calculus satisfies a finite model theorem, as first shown in [Kozen, 1988] Progressively better decidability results were obtained in [Kozen and Parikh, 1983; Vardi and Stockmeyer, 1985; Vardi, 1985b] culminating in a deterministic exponential time algorithm of [Emerson and Jutla, 1988] based on an automata theoretic lemma of [Safra, 1988] Since the calculus subsumes PDL, it is EXPTIME complete. In [Kozen, 1982; Kozen, 1983] an axiomatization of the propositional calculus was ....

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M. Y. Vardi and L. Stockmeyer. Improved upper and lower bounds for modal logics of programs: preliminary report. In Proc. 17th Symp. Theory of Comput., pages 240--251. ACM, May 1985.

....# 0 is satisfiable, and (ii) # O( # ) Moreover, any model of # 1 can be used to define a model of # 0 and vice versa. Theorem 5.6 (CTL # k is elementarily decidable) The satisfiability problem for CTL # k is 2EXPTIME complete. Proof. Hardness follows from 2EXPTIME hardness of CTL # [35]. To show that it belongs to 2EXPTIME, we outline an algorithm for checking satisfiability for CTL # k of deterministic doubly exponential time complexity. Given a CTL # k formula # 0 , such an algorithm can be obtained as follows: 1. by exploiting Lemma 5.5, construct an equivalent formula # 1 ....

....the satisfiability problem for CTSL # k is 2EXPTIME complete, and thus elementarily decidable. Theorem 5.16 (CTSL # k is elementary decidable) The satisfiability problem for CTSL # k is 2EXPTIME complete. Proof. Hardness follows from 2EXPTIME hardness of the satisfiability problem for CTL # [35]. To show that the satisfiability problem for CTSL # k is in 2EXPTIME, we reduce it to the satisfiability problem for CTL # k , which has been shown to be 2EXPTIME complete in Section 5.1 (Theorem 5.6) To this end, let # be a finite alphabet and # # = # # # , is a new symbol not belonging ....

M. Y. Vardi and L. Stockmeyer. Improved upper and lower bounds for modal logics of programs. In ACM Symposium on Theory of Computing (STOC), pages 240--251, Baltimore, USA, May 1985. ACM Press. 34

....(ii) O( # ) Moreover, any model of # 1 can be used to define a model of # 0 and vice versa. The satisfiability problem for CTL # k over infinite k ary trees is 2EXPTIME complete. Proof. Let us assume k = 2. The general case is similar. Hardness follows from 2EXPTIMEhardness of CTL # [116]. To show that it belongs to 2EXPTIME, we outline an algorithm for checking the satisfiability of CTL # k formulas with deterministic doubly exponential time complexity. Given a CTL # k formula # 0 , such an algorithm is as follows: 1. by exploiting Lemma 2.4.10, construct an equivalent ....

....decidable. On the contrary, PLTL(CTL # k ) is elementarily decidable, and its exact complexity is shown in the following result. The satisfiability problem for PLTL(CTL # k ) over DULSs is 2EXPTIME complete. Proof. Hardness follows from 2EXPTIME hardness of the component logic CTL # [116]. Recall is the class of Buchi sequence automata (Definition 2.3.2) and k is the class of Rabin k ary tree automata (Definition 2.3.6) To show that the satisfiability problem for PLTL(CTL # k ) belongs to 2EXPTIME, we give a doubly exponential time algorithm that first embeds temporalized ....

M. Y. Vardi and L. Stockmeyer. Improved upper and lower bounds for modal logics of programs. In In Proceedings of the ACM Symposium on Theory of Computing, pages 240--251, Baltimore, USA, May 1985. ACM Press.

....A couple of examples: 44 1. Emptiness of Buchi automata is logspace complete for NLOGSPACE [ 2. Satisfiability for ETL is PSPACE complete [60] 3. Satisfiability for F(fl; is decidable in deterministic exponential time (this follows from a corresponding result for the modal calculus [58]. 23 Determinisation We have already seen that deterministic Buchi automata are strictly weaker than their nondeterministic counterpart. Nevertheless determinisation is a very important and useful property. It is used in decision procedures for various program logics such as 1. the modal ....

....23 Determinisation We have already seen that deterministic Buchi automata are strictly weaker than their nondeterministic counterpart. Nevertheless determinisation is a very important and useful property. It is used in decision procedures for various program logics such as 1. the modal calculus [53, 58]. 2. CTL [18] and not least the monadic 2nd order theory of the binary tree which we will return to later. So it is of interest to find alternative notions of automata for which determinisation is in fact possible. Three well known such notions exist, called Muller, Rabin, and Streett ....

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M. Y. Vardi and L. Stockmeyer. Improved upper and lower bounds for modal logics of programs. In Proc. 17th ACM Symp. Theory of Computing, 1985.

....of a combined temporal logic CTL (called Full Branching Time Logic) lead to a polynomial upper time bound on the number of states in a system and an exponential upper time bound on the length of a formula [6] Decidability is another fundamental algorithmic property of CTL . It is well known [8] that CTL has double exponential low time bound but a complete proof of the same upper time bound is published quite recently [15] The last cited result as well as an exponential upper time bound for the propositional Calculus [10] are proved in [15] on base of an improved upper time bound for ....

....in the next section 3. This exponential reduction together with exponential decidability of validity problem in Herbrand Models for SOPDL imply double exponential upper time bound for CTL . 2 Two Program Logics: CTL and SOPDL The Full Branching Time Logic (or Full Computation Tree Logic) CTL [7, 8, 9, 12, 13, 14, 15] is a powerful propositional temporal logic for reasoning about states and sequences of states of a program. The syntax of CTL is constructed from boolean values B and a finite alphabet of propositional variables P and consist of two parts: state formulae F stt and path formulae F pth . A ....

Vardi M.Y., Stockmeyer L. Improved upper and lower bounds for modal logics of programs. 17 th ACM Symposium on the Theory of Computing, 1985, p.240-251.

....are consistent with the transition diagram. We say that a binary structure M is contained in A provided M is generated by unwinding A starting starting at s 0 2 M and q 0 2 A and, moreover, a copy of) M is a subgraph of T . Linear Size Model Theorem The following theorem is from [Em85] cf. [VS85]) Its significance is that it provides the basis for our method of testing nonemptiness of pairs automata; 28 it shows the existence of a small binary structure accepted by the automaton contained in its transition diagram, provided the automaton is nonempty. Theorem 5.1 (Linear Size Model ....

....0 j= A Phi can plainly be unwound into a tree that is accepted by A. 2 We can now use the Linear Size Model Theorem to establish the following result (cf. EJ88] Theorem 5.2. The problem of testing nonemptiness of pairs tree automata is NP complete. proof sketch: Membership in NP (cf. Em85] [VS85]) Given a pairs tree automaton, if it accepts some tree, then a linear size modelexists contained within its transition diagram. Guess that model. Use the (efficient FairCTL) model checking algorithm of [EL87] to verify in deterministic polynomial time that A( i ( 1 Fgreen i 1 G:red i ) ....

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Vardi, M. and Stockmeyer, L., Improved Upper and Lower Bounds for Modal Logics of Programs, Proc. 17th ACM Symp. on Theory of Computing, pp. 240--251, 1985.

....of deterministic double exponential complexity in the length of the formula. 2 The proof of this theorem consists in showing that the satisfiability problem can be reduced to testing non emptiness of tree automata. A lower bound of deterministic double exponential time has also been established [42]. In [6] a deterministic double exponential algorithm for CTL interpreted over fair structures has been defined. 2.4.2 Model checking for CTL Model checking problem for CTL is solvable in deterministic linear time. The situation is different for CTL . Unfortunately, the method of assigning ....

M. Vardi, L. Stockmeyer (1985): Improved upper and lower bounds for modal logics of programs. Proc. of 7th Ann. Symp. on Theory of Computing, 240-251.

....theorem in a crucial way. Rabin s procedure for checking the emptiness of tree automata in [Rab69] is nonelementary, Hossley and Rackoff s procedure runs in doubly exponential time. Rabin improved the complexity of the decision procedure to exponential time in [Rab72] and Vardi and Stockmeyer ([VS85]) gave an NP decision procedure. Recently, independently, Emerson and Jutla ( EJ88] and Pnueli and Rosner ( PR89] gave a decision procedure for Rabin tree automata, whose running time is exponential in the number of accepting pairs, but only polynomial in the number of states. Pnueli and ....

....of PTL is in PSPACE ( HR83, SC85] However, for more general logics, such as Delta PDL ( Str80, Str82] propositional calculus in the branching case, and TL ( Var88] in the linear time case, just adding an acceptance condition to the tableau is not sufficient. Streett ( Str82] see also [VS85, ES84, SE84, VW86]) suggested that it is possible to express the hard parts of these logics, such as the negation of the Delta operator in Delta PDL, by a linear size nondeterministic Buchi automaton that has to reject any path of the tree. By Corollary 2.1, it is possible to obtain a deterministic Rabin ....

M. Y. Vardi and L. Stockmeyer. Improved upper and lower bounds for modal logics of program. In Proc. 17th ACM Symp. on Theory of Computing, pages 240--251, 1985.

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M. Y. Vardi and L. Stockmeyer, Improved upper and lower bounds for modal logics of programs, Proceedings of the 17th acm symposium on theory of computing, 1985, pp. 240--251.

No context found.

M.Y. Vardi and L. Stockmeyer. Improved upper and lower bounds for modal logics of programs. In Proc 17th ACM Symp. on Theory of Computing, pages 240--251, 1985.

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M.Y. Vardi and L. Stockmeyer. Improved upper and lower bounds for modal logics of programs. In Proc 17th ACM Symp. on Theory of Computing, pages 240-251, 1985.

....The satis ability problem for AqCTL is 3EXPTIME hard. Proof. We do a reduction from the problem whether a doubly exponential space alternating Turing machine T accepts an input word x. That is, given T and x, we construct an AqCTL formula 8q such that T accepts x i is satis able. In [VS85], the satis ability problem of CTL is proved to be 2EXPTIME hard by a reduction from an exponential space alternating Turing machine. Below we explain how universal quanti cation can be used to stretch the length of the tape that a polynomial CTL formula can describe by another ....

....to be 2EXPTIME hard by a reduction from an exponential space alternating Turing machine. Below we explain how universal quanti cation can be used to stretch the length of the tape that a polynomial CTL formula can describe by another exponential. As in the proof of Theorem 6, the formula in [VS85] maintains an n bit counter, and each cell of T s tape corresponds to a block of length n. In order to point on the letters i and 0 i simultaneously (that is, the letters that the atomic proposition q point on in the proof of Theorem 6) VS85] adds to each node of the tree a branch such ....

[Article contains additional citation context not shown here]

M.Y. Vardi and L. Stockmeyer. Improved upper and lower bounds for modal logics of programs. In Proc 17th ACM Symp. on Theory of Computing, pages 240-251, 1985.

....proof rules is that it enables one to apply model checking only to the underlying modules, which have much smaller state spaces. 3 Note also that while the satisfiability problem for LTL is PSPACE complete [91] the problem is EXPTIME complete for CTL [36,32] and 2EXPTIME complete for CTL [102,34]. A key observation, see [77,66,50,93,81] is that in modular verification the specification should include two parts. One part describes the desired behavior of the module. The other part describes the assumed behavior of the system within which the module is interacting. This is called the ....

M.Y. Vardi and L. Stockmeyer. Improved upper and lower bounds for modal logics of programs. In Proc 17th ACM Symp. on Theory of Computing, pages 240--251, 1985.

....alternatively (as in the proof above) the nondeterministic tree automaton for E: needs to be complemented. The doubly exponential size of the tree automaton then leads to EXPSPACE and 2EXPTIME upper bounds. On the other hand, typical EXPSPACE and 2EXPTIME lower bound proofs for temporal logic [VS85,KV95] require the use of temporal logic formulas that do not t into the restricted syntax that is present in the problems above (e.g. formulas of the form A d for some CTL formula ) VW86b] and let A 0 be nondeterministic B uchi word automaton that accepts exactly all words (i.e. ....

M.Y. Vardi and L. Stockmeyer. Improved upper and lower bounds for modal logics of programs. In Proc 17th ACM Symp. on Theory of Computing, pages 240-251, 1985.

....a module M and a CTL formula such that the size of M is quadratic in the length of , the length of is linear in the length of , and is satisfiable iff M 6j= r : The proof is the same for CTL . Here, we do a reduction from satisfiability of CTL , proved to be 2EXPTIME hard in [VS85] See [KV96] for more details. ut When analyzing the complexity of model checking, a distinction should be made between complexity in the size of the input structure and complexity in the size of the input formula; it is the complexity in size of the structure that is typically the computational ....

M.Y. Vardi and L. Stockmeyer. Improved upper and lower bounds for modal logics of programs. In Proc 17th ACM Symp. on Theory of Computing, pages 240--251, 1985.

....to expressive power. The more expressive a language is, the higher is the complexity of solving questions about it. For example, the complexities of the model checking and the satisfiability problems for the logic CTL are significantly higher than these for its less expressive subset CTL [SC85, VS85]. Similarly, while the containment problem for DBW can be solved in NLOGSPACE [WVS83, Kur87] it is PSPACE complete for BW [Wol82] Finally, while the complexity of the nonemptiness problem for BT can be solved in quadratic time [VW86b] it is NP complete for RT [Eme85, VS85, EJ88] The interested ....

.... expressive subset CTL [SC85, VS85] Similarly, while the containment problem for DBW can be solved in NLOGSPACE [WVS83, Kur87] it is PSPACE complete for BW [Wol82] Finally, while the complexity of the nonemptiness problem for BT can be solved in quadratic time [VW86b] it is NP complete for RT [Eme85, VS85, EJ88]. The interested readers can find more examples in [Eme90, Tho90] In the automata theoretic approach to verification, we translate specifications to automata. Which type of automata The answer, obviously, should be the weakest type that is still strong enough to express the required behaviors ....

M.Y. Vardi and L. Stockmeyer. Improved upper and lower bounds for modal logics of programs. In Proc 17th ACM Symp. on Theory of Computing, pages 240--251, 1985.

....6. The satis ability problem for AqCTL is 3EXPTIME hard. Proof. We do a reduction from the problem whether a doubly exponential space alternating Turing machine T accepts an input word x. That is, given T and x, we construct an AqCTL formula 8q such that T accepts x i is satis able. In [VS85], the satis ability problem of CTL is proved to be 2EXPTIME hard by a reduction from an exponential space alternating Turing machine. Below we explain how universal quanti cation can be used to stretch the length of the tape that a polynomial CTL formula can describe by another ....

....to be 2EXPTIME hard by a reduction from an exponential space alternating Turing machine. Below we explain how universal quanti cation can be used to stretch the length of the tape that a polynomial CTL formula can describe by another exponential. As in the proof of Theorem 4, the formula in [VS85] maintains an n bit counter, and each cell of T s tape corresponds to a block of length n. In order to point on the letters i and 0 i simultaneously (that is, the letters that the atomic proposition q point on in the proof of Theorem 4) VS85] adds to each node of the tree a branch such ....

[Article contains additional citation context not shown here]

M.Y. Vardi and L. Stockmeyer. Improved upper and lower bounds for modal logics of programs. In Proc 17th STOC, pages 240-251, 1985.

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M. Vardi, L. Stockmeyer (1985): Improved upper and lower bounds for modal logics of programs. Proc. of 7th Ann. Symp. on Theory of Computing, 240-251.

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M. Y. Vardi and L. Stockmeyer. Improved upper and lower bounds for modal logics of programs. In Proc. 17th Symp. on Theory of Computing, STOC'85, pages 240--251, Baltimore, USA, May 1985. ACM. 8

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M. Y. Vardi and L. Stockmeyer. Improved upper and lower bounds for modal logics of programs. In ACM Symposium on Theory of Computing (STOC '85), pages 240--251, Baltimore, USA, May 1985. ACM Press.

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M. Vardi and L. Stockmeyer. Improved upper and lower bounds for modal logics of programs. In Proc. Theory of Computing, pages 240--251. ACM Press, 1985.

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M. Y. Vardi and L. Stockmeyer. Improved upper and lower bounds for modal logics of programs. In Proc. 17th Symp. on Theory of Computing, STOC'85, pages 240--251, Baltimore, USA, May 1985. ACM.

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M. Y. Vardi and L. Stockmeyer. Improved upper and lower bounds for modal logics of programs. In Proc. 17th Symp. on Theory of Computing, STOC'85, pages 240-251, Baltimore, USA, May 1985. ACM.

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M. Y. Vardi and L. Stockmeyer. Improved upper and lower bounds for modal logics of programs. In ACM Symposium on Theory of Computing (STOC), pages 240--251, Baltimore, USA, May 1985.

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M. Y. Vardi and L. Stockmeyer. Improved upper and lower bounds for modal logics of program. In Proc. 17th ACM Symp. on Theory of Computing, pages 240--251, 1985.

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