Robert S. Street and E. Allan Emerson. An automata theoretic procedure for the propositional mu-calculus. Information & Computation, 81:249-- 264,  Home/Search   Document Not in Database   Summary   Related Articles   Check

....Centre of the Danish National Research Foundation. can be encoded into the calculus. On binary trees the logic is as expressive as monadic second order logic of two successors [18, 6] On the other hand the logic is manageable. Satisfiability problem for the logic was shown to be EXPTIME complete [15, 23, 4] which means that it is of the same complexity as for many much less expressive logics. The best known upper bound for the model checking problem is exponential but it is polynomial if nesting of fixpoints is bounded [3, 1] One of lacking elements in this picture was finitary complete ....

....that it is closely connected with a decision procedure for the logic. This shows a possibility of integrating the two in the way we now describe. Essentially the only know method for checking satisfiability of the calculus formulas comes from the tableau method and the use of automata theory [23]. For a given formula the algorithm constructs an automaton over infinite trees which accepts models of the formula. Then, if formula is satisfiable, a model is constructed from an accepting run of the automaton. In [17] it was shown that the process of checking satisfiability can be viewed as a ....

Robert S. Street and E. Allan Emerson. An automata theoretic procedure for the propositional mu-calculus. Information & Computation, 81:249-- 264,

....Hence it is good to know that the class of Kripke structures is a complete subclass of a quasi variety defined by Kozen s system. Let us review some methods used in previous approaches to the completeness problem. First step is a tableau method of model construction of Streett and Emerson [8]. For a given formula one constructs a tableau, if the formula is satisfiable then one can construct a model from a part of this tableau. It was shown in [6] that if the initial formula is unsatisfiable, and one cannot find a model in the tableau, then one can construct for the formula another ....

Robert S. Street and E. Allan Emerson. An automata theoretic procedure for the propositional mu-calculus. Information and Computation, 81:249--264, 1989.

....of the Danish National Research Foundation. 1 can be encoded into the calculus. On binary trees the logic is as expressive as monadic second order logic of two successors [18, 6] On the other hand the logic is manageable. Satisfiability problem for the logic was shown to be EXPTIME complete [15, 23, 4] which means that it is of the same complexity as for many much less expressive logics. The best known upper bound for the model checking problem is exponential but it is polynomial if nesting of fixpoints is bounded [3, 1] One of lacking elements in this picture was finitary complete ....

....that it is closely connected with a decision procedure for the logic. This shows a possibility of integrating the two in the way we now describe. Essentially the only know method for checking satisfiability of the calculus formulas comes from the tableau method and the use of automata theory [23]. For a given formula the algorithm constructs an automaton over infinite trees which accepts models of the formula. Then, if formula is satisfiable, a model is constructed from an accepting run of the automaton. In [17] it was shown that the process of checking satisfiability can be viewed as a ....

Robert S. Street and E. Allan Emerson. An automata theoretic procedure for the propositional mu-calculus. Information & Computation, 81:249-- 264, 1989. 39

....In this way we have means to compare different strategies by comparing signatures they induce. It turns out that there exists canonical, or the least possible, signature assignment. Most of the material presented here comes from [17] The notion of signature was proposed by Streett and Emerson [15]. The proof of the existence of memoryless strategies in parity games was given independently by Mostowski [13] and by Emerson and Jutla [7] Klarlund [10] proves a more general fact that a player has a memoryless strategy in a game if the has a strategy and his winning conditions are given as a ....

R. S. Street and E. A. Emerson. An automata theoretic procedure for the propositional mu-calculus. Information and Computation, 81:249--264, 1989.

....next section we present what we call operational semantics of formulas. We give a characterisation of satisfiability of a given formula in a given state by means of markings of infinite tableaux. This result makes very explicit the main tools used in the fundamental paper of Streett and Emerson [18]. After that we show some applications of operational semantics. We prove the small model theorem and the decidability result (i.e. the results from [18] 1 INTRODUCTION 2 We also define the notion of refutation and state the characterisation of validity (or rather unsatisfiability) by means of ....

....by means of markings of infinite tableaux. This result makes very explicit the main tools used in the fundamental paper of Streett and Emerson [18] After that we show some applications of operational semantics. We prove the small model theorem and the decidability result (i.e. the results from [18]) 1 INTRODUCTION 2 We also define the notion of refutation and state the characterisation of validity (or rather unsatisfiability) by means of refutations [14] We finish this section by presenting a new application of the operational semantics, namely tableau equivalence. We show that if two ....

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Robert S. Street and E. Allan Emerson. An automata theoretic procedure for the propositional mu-calculus. Information & Computation, 81:249-- 264, 1989. REFERENCES 54

....a propositional version of the calculus [8] was proven decidable in deterministic single exponential time [3] which makes this logic interesting for applications to verification of real life systems. Although the calculus, and especially the propositional calculus attracted much interest [8, 17, 1, 5] the attempts to find a complete finitary axiomatization for the calculus systematically failed. Even for strictly weaker modal logics such as PDL Delta no finitary complete axiom system has been known. A complete axiomatization of any calculus is still interesting even when it is decidable. The ....

....5 Characterization In this section we prove that fl is satisfiable iff there exists a pre model for it. From the results of the previous section, it will follow that fl is not satisfiable (viz :fl is valid) iff there exits a refutation for :fl. Similar results concerning pre model were proved in [17] and [16] It will be convenient to use a characterization of the extremal fixpoints in terms of possibly transfinite induction. We introduce two new constructs X:ff(X) and X:ff(X) where is any ordinal, with the following semantics: k 0 X:ff(X) k V al = k 0 X:ff(X) k V ....

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Robert S. Street and E. Allan Emerson. An automata theoretic procedure for the propositional mu-calculus. Information & Computation, 81:249--264, 1989.

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