D.E. Muller, A. Saoudi, and P. E. Schupp. Weak alternating automata give a simple explanation of why most temporal and dynamic logics are decidable in exponential time. In Proceedings 3rd IEEE Symp. on Logic in Computer Science, pages 422--427, Edinburgh, July 1988.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....of the automata theoretic approach to logics of programs has prompted renewed inquiry into the complexity of automata on infinite objects, with considerable success. See [Courcoubetis and Yannakakis, 1988; Emerson, 1985; Emerson and Jutla, 1988; Emerson and Sistla, 1984; Manna and Pnueli, 1987; Muller et al. 1988; Pecuchet, 1986; Safra, 1988; Sistla et al. 1987; Streett, 1982; Vardi, 1985a; Vardi, 1985b; Vardi, 1987; Vardi and Stockmeyer, 1985; Vardi and Wolper, 1986b; Vardi and Wolper, 1986a; Arnold, 1997a; Arnold, 1997b] and [Thomas, 1997] Especially noteworthy in this area is the result of [Safra, ....

D. E. Muller, A. Saoudi, and P. E. Schupp. Weak alternating automata give a simple explanation of why most temporal and dynamic logics are decidable in exponential time. In Proc. 3rd Symp. Logic in Computer Science, pages 422--427. IEEE, July 1988.

.... The parity condition is of growing importance in automata theory, particularly in connection with games [6, 11] Alternating automata are very closely connected to Kozen s modal calculus [9] Because of this close correspondence, the calculus is a standard application for alternating automata [12, 2]. In the same spirit, the satis ability problem for the propositional calculus with backward modalities is proved decidable by Vardi [16] via a reduction to two way alternating automata. The calculus with backwards modalities augments the one way calculus with quanti cation over backward ....

A. Saoudi D. E. Muller and P. E. Shupp. Weak alternating automata give a simple explanation why most temporal and dynamic logics are decidable in exponential time. In Proceedings 3rd IEEE Symposium on Logic in Computer Science, pages 422427, 1988.

....2.2.57, Prop. 2.3.16, Prop. 2.3.17 and Theorem 3.3.5. 2 This result has been shown previously in [4] The alternating first recurrence automata in the class # 2 , i.e. the non alternating Kn formulae, have a close connection to the so called weak alternating automata of Muller, Saoudi and Schupp [68, 69]. In fact the only real di#erence is that FR automata are required to be tree like, whereas weak alternating automata are not. In this framework Muller, Saoudi and Schupp prove a correspondence theorem equivalent to the one above [68] Weak alternating automata have attracted interest, because ....

....equivalent to the one above [68] Weak alternating automata have attracted interest, because many programming logics are easily reducible to such automata, and they therefore provide a universal framework in which to study the decidability of these logics. For more discussion of this aspect, see [69]. Chapter 3 Automata 95 3.4 Fixpoints for ordinary automata As explained in last section, if we are able to devise a translation from alternating automata to ordinary ones, this will also yield an inductive translation from calculi like #Kn and SnS to ordinary automata and show their ....

Muller, D. E. & Saoudi, A. & Schupp, P.: Weak alternating automata give a simple explanation of why most temporal and dynamic logics are decidable in exponential time, in Proceedings of the 3rd IEEE Symposium on Logic in Computer Science, 1988, pp. 422-427

....complementation is relatively easy. Moreover, the task of connecting Buchi automata to weak alternating automata is not as complex as to show that Buchi automata and deterministic automata are equivalent. The idea of weak alternating automata is due to Muller, Saoudi, and Schupp ( MSS86] MS87] [MSS88]) However, in their work they emphasize complexity issues (especially regarding program logics and temporal logics) and not so much a reconsideration of the complementation problem. Recently, Vardi and Kupferman [KV97] have taken up the approach and supplied a self contained complementation proof ....

D.E. Muller, A. Saoudi, P.E. Schupp, Weak alternating automata give a simple explanation of why most temporal and dynamic logics are decidable in exponential time, in: Proc. 3rd IEEE Symp. on Logic in Computer Science, 1988, pp. 422-427.

....in direction 0 in s3 . one copy proceeds in direction 1 in state s2 and one copy proceeds in direction 1 in s1 . Weak alternating automata (WAA) introduced by Muller et al. 17] was one of the first types of alternating automata to be used for reasoning about temporal logic. For example, in [18] WAA are used to explain the complexity of decision procedures for certain temporal logics. More recently, WAA were used to define linear time algorithms for model checking CTL [1] In [2] Bernholtz et al. defined bounded alternation WAA, that also allow space efficient CTL model checking. In ....

....in the complete translation from a CTL formula into an HAA being exponential. 6 Willem Visser and Howard Barringer: Practical CTL Model Checking Note that we do not translate the linear time formula into an alternating Buchi word automaton, even though this translation is known to be linear [18]; this is because the reduction to a 1 letter nonemptiness problem, which we will see in the next section is crucial for efficient model checking, is impossible for alternating Buchi word automata, but possible for nondeterministic Buchi word automata [1] Example: Consider the CTL formula = ....

D.E. Muller, A. Saoudi, and P.E. Schupp. Weak Alternating Automata give a Simple Explanation of why Temporal and Dynamic Logics are Decidable in Exponential Time. In Third Symposium on Logic in Computer Science, pages 422--427, July 1988.

....approaches to the translation of formulas to automata. Streett [St90] suggested using so called formula checking automata. His approach eliminates the need for distinction between the local automaton and the eventuality automaton. Another approach, using weak alternating automata is described in [MSS88]. In that approach not only there 20 is no distinction between the local automaton and the eventuality automaton, but the automaton is constructed by a simple induction on the structure of the formula. We believe, however, that the distinction between the local automaton, which checks local ....

....even if it is avoidable for ETL f and ETL l . For example, for ETL r or the temporal calculus, the construction of the local automaton is straightforward, and the main difficulty lies in the construction of the global automaton [SVW87, Va88] Unlike our approach, the unitary approaches of [MSS88, St90] do not generalize to these logics. 4 Translations Among the Logics The results of the previous section show that the set of sequences describable by ETL f , ETL l or ETL r formulas are expressible by Buchi automata. In the case of ETL r , the converse is also clearly true. Thus, ETL r has ....

D.E. Muller, A. Saoudi, and P.E. Schupp, "Weak Alternating Automata Give a Simple Explanation of Why Most Temporal and Dynamic Logic are Decidable in Exponential Time", Proc. 3rd IEEE Symp. on Logic in Computer Science, 1988, pp. 422--427. 34

....to the atomic propositions. To check that the computations of a system are accepted by the LTL specification an approach is taken based on the theory that temporal formulas can be associated with finite state automata. Branching time temporal logic formulas correspond to automata on infinite trees [23] and alternating automata represent these formulas effectively [2, 18, 33] Linear time temporal logic formulas are associated with automata on infinite words [29, 30, 34] and specifically, Buchi automata can be used to represent these formulas effectively [14, 31] SPIN [16] is a well known ....

D.E. Muller, A. Saoudi, and P.E. Schupp. Weak alternating automata give a simple explanation of why most temporal and dynamic logics are decidable in exponential time. In Proceedings of the 3rd IEEE Symposium on Logic in Computer Science, pages 422--427, Edinburgh, July 1988.

....function, and F Q is the acceptance condition. An input word is accepted iff there is a run of the automaton on that word that visits F infinitely often. We assume that the transition function is complete, that is ffi(q; oe) 6= for all q 2 Q and oe 2 Sigma . A Buchi automaton is weak [20, 29] iff there exists a partition of Q into Q 1 ; Qn such that each Q i is either contained in F or disjoint from it; in addition, the blocks of the partition are partially ordered so that the transitions of the automaton never move from Q i to Q j unless Q i Q j . Theorem 1. The language of ....

D. E. Muller, A. Saoudi, and P. E. Schupp. Weak alternating automata give a simple explanation of why most temporal and dynamic logics are decidable in exponential time. In Proceedings of the 3rd IEEE Symposium on Logic in Computer Science, pages 422--427, Edinburgh, UK, July 1988.

.... Remarks and Related Systems The complexity of the decision problem for branching time logics has been studied by Emerson and Sistla [ES83] Temporal logics are known to be related to Buchi automata [VW86] and so their decision problems can be studied from an automata theoretic perspective as well [MSS88]. All these branching time logics exclude past time operators, but they can be added. The work of Gabbay [Gab87] and Gabbay et al. [GHR94] is particularly interesting because many temporal logics have the separation property : that is, any complicated formula A has a logically equivalent form A ....

David E Muller, Ahmed Saoudi, and Paul E Schupp. Weak alternating automata give a simple explanation of why most temporal and dynamic logics are decidable in exponential time. In Proc. Logics in Computer Science, pages 422--427, 1988.

....be used to derive truth and validity checking algorithms for both linear and branching temporal logics. The key observation is that while the translation from temporal logic formulas to nondeterministic automata is exponential [VW86b, VW94] the translation to alternating automata is linear [MSS88, EJ91, Var94, BVW94]. Thus, the advantage of alternating automata is that they enable one to decouple the logic from the algorithmics. The translations from formulas to automata handle the logic, and the algorithms are then applied to the automata. 2 Automata Theory 2.1 Words and Trees We are given a finite ....

....the alphabet 2 Prop . It turns out that the computations in which a given formula is true are exactly those accepted by some finite automaton on infinite words. The following theorem establishes a very simple translation between LTL and alternating Buchi automata on infinite words. Theorem 11. [MSS88, Var94] Given an LTL formula , one can build an alternating B uchi automaton A = Sigma; S; s 0 ; ae; F ) where Sigma = 2 Prop and jSj is in O(j j) such that L (A ) is exactly the set of computations in which the formula is true. Proof: The set S of states consists of all subformulas of ....

[Article contains additional citation context not shown here]

D. E. Muller, A. Saoudi, and P. E. Schupp. Weak alternating automata give a simple explanation of why most temporal and dynamic logics are decidable in exponential time. In Proceedings 3rd IEEE Symposium on Logic in Computer Science, pages 422--427, Edinburgh, July 1988.

....nondeterministic tree automata by allowing several successor states to go down along the same branch of the tree. It is known that, while the translation from branching temporal logic formulas to nondeterministic tree automata is exponential, the translation to alternating tree automata is linear [MSS88, EJ91]. In fact, Emerson stated that calculus formulas are simply alternating tree automata [Eme94] Muller et al. showed that this explains the exponential decidability of satisfiability for various branching temporal logics. We show here that this also explains the efficiency of model checking for ....

....unifying and optimal framework for both satisfiability and model checking problems for branching temporal logic. We first show how our automata theoretic approach unifies previously known results about model checking for branching temporal logics. The alternating automata used by Muller et al. in [MSS88] are of a restricted type called weak alternating automata. To obtain an exponential decision procedure for the satisfiability of CTL and related branching temporal logics, Muller et al. used the fact that the nonemptiness problem for these automata is in exponential time [MSS86] We prove that ....

[Article contains additional citation context not shown here]

D. E. Muller, A. Saoudi, and P. E. Schupp. Weak alternating automata give a simple explanation of why most temporal and dynamic logics are decidable in exponential time. In Proceedings 3rd IEEE Symposium on Logic in Computer Science, pages 422--427, Edinburgh, July 1988.

....automata can be used to derive modelchecking algorithms for both linear and branching temporal logics. The key observation is that while the translation from temporal logic formulas to nondeterministic automata is exponential [VW86b, VW94] the translation to alternating automata is linear [MSS88, EJ91, Var94, BVW94]. Thus, the advantage of alternating automata is that they enable one to decouple the logic from the combinatorics. The translations from formulas to automata handle the logic, and the algorithms that handle the automata are essentially combinatorial. 2 Automata Theory 2.1 Words and Trees We are ....

....over the alphabet 2 Prop . It turns out that the computations satisfying a given formula are exactly those accepted by some finite automaton on infinite words. The following theorem establishes a very simple translation between LTL and alternating Buchi automata on infinite words. Theorem 11. [MSS88, Var94] Given an LTL formula , one can build an alternating B uchi automaton A = Sigma; S; s 0 ; ae; F ) where Sigma = 2 Prop and jSj is in O(j j) such that L (A ) is exactly the set of computations satisfying the formula . Proof: The set S of states consists of all subformulas of and ....

[Article contains additional citation context not shown here]

D. E. Muller, A. Saoudi, and P. E. Schupp. Weak alternating automata give a simple explanation of why most temporal and dynamic logics are decidable in exponential time. In Proceedings 3rd IEEE Symposium on Logic in Computer Science, pages 422--427, Edinburgh, July 1988.

....papers that employed the automata theoretic approach used nondeterministic tree automata [9,39,51] The translation from formulas to nondeterministic automata is nontrivial; for example, the translation in [51] is exponential and consists of a sequence of successive translations. As argued in [28] and then utilized in [2,47] it is easier to translate formulas to alternating automata. Alternating tree automata generalize nondeterministic tree automata by allowing multiple successor states to go down along the same branch of the tree. It is known that while the translation from branching ....

....nondeterministic tree automata by allowing multiple successor states to go down along the same branch of the tree. It is known that while the translation from branching temporal logic formulas to nondeterministic tree automata is exponential, the translation to alternating tree automata is linear [2,28]. Similarly, there is a simple translation from calculus formulas to alternating tree automata [2,10] Alternating tree automata as defined in [29] however, cannot easily handle backwards modalities, since they are one way automata. To deal with backward modalities we introduce two way ....

D.E. Muller, A. Saoudi, and P. E. Schupp. Weak alternating automata give a simple explanation of why most temporal and dynamic logics are decidable in exponential time. In Proceedings 3rd IEEE Symposium on Logic in Computer Science, pages 422--427, Edinburgh, July 1988.

....is nonelementary, i.e. it may involve an unbounded stack of exponentials (that is, the complexity bound is of the form 2 : 2 n ; where the height of the stack is n. The following theorem establishes a very simple translation between LTL and alternating Buchi automata. Theorem 22. [MSS88, Var94] Given an LTL formula , one can build an alternating B uchi automaton A = Sigma; S; s 0 ; ae; F ) where Sigma = 2 Prop and jSj is in O(j j) such that L (A ) is exactly the set of computations satisfying the formula . Proof: The set S of states consists of all subformulas of and ....

D. E. Muller, A. Saoudi, and P. E. Schupp. Weak alternating automata give a simple explanation of why most temporal and dynamic logics are decidable in exponential time. In Proceedings 3rd IEEE Symposium on Logic in Computer Science, pages 422--427, Edinburgh, July 1988.

....0 in s3 . ffl one copy proceeds in direction 1 in state s2 and one copy proceeds in direction 1 in s1 . Weak alternating automata (WAA) introduced by Muller et al. MSS86] was one of the first types of alternating automata to be used for reasoning about temporal logic. For example, in [MSS88] WAA are used to explain the complexity of decision procedures for certain temporal logics. More recently, WAA were used to define linear time algorithms for model checking CTL [Ber95] In [BVW94] Bernholtz et al. defined bounded alternation WAA, that also allow space efficient CTL model ....

....is exponential in the size of the [VW94,GPVW95] This results in the complete translation from a CTL formula into an HAA being exponential. Note that we do not translate the linear time formula into an alternating Buchi word automaton, even though this translation is known to be linear [MSS88] this is because the reduction to a 1 letter nonemptiness problem, which we will see in the next section is crucial for efficient model checking, is impossible for alternating Buchi word automata, but valid for nondeterministic Buchi word automata [Ber95] Example: Consider the CTL formula ....

D.E. Muller, A. Saoudi, and P.E. Schupp. Weak Alternating Automata give a Simple Explanation of why Temporal and Dynamic Logics are Decidable in Exponential Time. In Third Symposium on Logic in Computer Science, pages 422--427, July 1988.

....nondeterministic tree automata by allowing several successor states to go down along the same branch of the tree. It is known that while the translation from branching temporal logic formulas to nondeterministic tree automata is exponential, the translation to alternating tree automata is linear [MSS88, EJ91]. In fact, Emerson stated that calculus formulas are simply alternating tree automata [Eme96] In [MSS88] Muller et al. showed that this explains the exponential decidability of satisfiability for various branching temporal logics. We show here that this also explains the efficiency of model ....

.... It is known that while the translation from branching temporal logic formulas to nondeterministic tree automata is exponential, the translation to alternating tree automata is linear [MSS88, EJ91] In fact, Emerson stated that calculus formulas are simply alternating tree automata [Eme96] In [MSS88], Muller et al. showed that this explains the exponential decidability of satisfiability for various branching temporal logics. We show here that this also explains the efficiency of model checking for those logics. The crucial observation is that for model checking, one does not need to solve the ....

[Article contains additional citation context not shown here]

D. E. Muller, A. Saoudi, and P. E. Schupp. Weak alternating automata give a simple explanation of why most temporal and dynamic logics are decidable in exponential time. In Proceedings 3rd IEEE Symposium on Logic in Computer Science, pages 422--427, Edinburgh, July 1988.

....to guess the truth of the quantified variable at each step. The overall complexity is determined by the determinization procedure and, as shown in [Saf88] it is single exponential. 5.2. 2 Via alternating automata The easy translation from PLTL to an alternating Buchi automaton is described in [MSS88] Var94] and [Var96] Suppose that we are given a formula OE using only atoms from the finite set P . The alternating Buchi automaton A = S; s 0 ; ae; F ) recognizes sequences of elements of 2 P . The set S of states of the corresponding automaton is just the set of subformulae of OE and ....

D. Muller, A. Saoudi, and P. Schupp. Weak alternating automata give a simple explanation of why most temporal and dynamic logics are decidable in exponential time. In 3rd IEEE Symposium on Logic in Computer Science, Proceedings, pages 422--427. IEEE, 1988.

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D.E. Muller, A. Saoudi, and P. E. Schupp. Weak alternating automata give a simple explanation of why most temporal and dynamic logics are decidable in exponential time. In Proceedings 3rd IEEE Symp. on Logic in Computer Science, pages 422--427, Edinburgh, July 1988.

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D.E. Muller, A. Saoudi, and P. E. Schupp. Weak alternating automata give a simple explanation of why most temporal and dynamic logics are decidable in exponential time. In Proceedings 3rd IEEE Symp. on Logic in Computer Science, pages 422--427, Edinburgh, July 1988.

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D. E. Muller, A. Saoudi, and P. E. Schupp. Weak alternating automata give a simple explanation of why most temporal and dynamic logics are decidable in exponential time. In Proceedings of the 3rd Annual Symposium on Logic in Computer Science (LICS 1988.

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David E. Muller, Ahmed Saoudi, and Paul E. Schupp. Weak alternating automata give a simple explanation of why most temporal and dynamic logics are decidable in exponential time. In 3rd IEEE Annual Symposium on Logic in Computer Science, pp. 422--427, 1988.

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David E. Muller, Ahmed Saoudi, and Paul E. Schupp. Weak alternating automata give a simple explanation of why most temporal and dynamic logics are decidable in exponential time. In LICS

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David E. Muller, Ahmed Saoudi, and Paul E. Schupp. Weak alternating automata give a simple explanation of why most temporal and 36 dynamic logics are decidable in exponential time. In 3rd IEEE Annual Symposium on Logic in Computer Science, pp. 422--427, 1988.

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D. E. Muller, A. Saoudi, and P. E. Shupp. Weak alternating automata give a simple explanation why most temporal and dynamic logics are decidable in exponential time. In Logic in Computer Science, pages 422-427, 1988.

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D. E. Muller, A. Saoudi, and P. E. Schupp. Weak alternating automata give a simple explanation of why most temporal and dynamic logics are decidable in exponential time. In Proceedings of the 3rd IEEE Symposium on Logic in Computer Science, pages 422-427, Edinburgh, UK, July 1988.

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