J.R. Buchi. On a decision method in restricted second order arithmetic. In Proc. International Congress on Logic, Method and Philos. Sci. 1960, pages 1-11, 1962.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....The fair cycle detection problem is at the heart of many problems, namely in deciding emptiness of # automata like generalised B uchi and Streett automata, and in model checking of specifications written in linear and branching temporal logics like LTL and fair CTL. A generalised B uchi automaton [10] is provided together with several sets of accepting states. A run of such an automaton is accepting if it contains at least one state from every accepting set infinitely often. Accordingly, the language of the automaton is nonempty if and only if the graph corresponding to the automaton contains ....

J. R. B uchi. On a decision method in restricted second order arithmetic. In Proc. International Congress on Logic, Methodology and Philosophy Science, pages 1 -- 11. Stanford university Press, 1960.

....whether an arbitrary set of types is realizable. It is shown that is satis able in a model with ow of time F i there exists a quasimodel for over F. Thus, has a model with ow of time F i F j= If F is fhN; ig, fhZ; ig, or fhQ; ig, this last statement is decidable by results of [15, 46]. The other cases can now be obtained by reduction. Part (ii) is proved di erently, since the full monadic second order theory of hR; i is undecidable. The proof adapts the argument of the second part of [17] see [35] for details. As a consequence of Theorem 5 we obtain, for example, the ....

J. Buchi. On a decision method in restricted second order arithmetic. In Logic, Methodology, and Philosophy of Science: Proc. 1960 Intern. Congress, pages 1-11. Stanford University Press, 1962. 20

....We focus on the decision problems and their applications in system veri cation and synthesis. 1 Introduction In the last decades, a signi cant amount of literature has been devoted to the theory of nite automata on sequences and trees. In the sixties with their pioneering works B uchi [B uc62] McNaughton [McN66] and Rabin [Rab69] introduced this theory, which, more recently, turned out to be an important source of tools for the synthesis and veri cation of nonterminating computer programs (see [Tho90] for a survey on automata on sequences and trees) Exploring the connections ....

J.R. Buchi. On a decision method in restricted second-order arithmetic. In Proc. of the International Congress on Logic, Methodology, and Philosophy of Science 1960, pages 1 { 12. Stanford University Press, 1962.

....of automata with a tractable emptiness problem. 1 Introduction Since its early days the theory of nite automata had an astonishing impact in computer science. Several models of automata have been extensively studied and applied to many elds. In the sixties, with their pioneering work, B uchi [1, 2], McNaughton [11] and Rabin [12] enriched this theory by introducing nite automata on in nite objects. The connections between such automata and the logic theories have been fruitfully investigated and have originated automatatheoretic approaches to reduce decision problems in the eld of ....

J.R. Buchi. On a decision method in restricted second-order arithmetic. In Proc. of the International Congress on Logic, Methodology, and Philosophy of Science 1960, pages 1 - 12. Stanford University Press, 1962.

....a result, there has been a fair amount of interest in the construction of automata from temporal logical formulas, the history of which is actually fairly interesting. The starting point is clearly the work of B uchi on the decidability of the rst and second order monadic theories of one successor [B uc62]. These decidability results were obtained through a translation to in nite word automata, for which B uchi had to prove a very nontrivial complementation lemma. The translation is nonelementary, but this is the best that can be done. It is quite obvious that linear time temporal logic can be ....

....Fig. 3. From generalized B uchi to B uchi with nondeterministic automata; closure under intersection is obtained using a product construction similar to the one employed for nite word automata. Closure under complementation is much more tricky and has been the subject of an extensive literature [B uc62,SVW87,Saf88,KV97]. Checking that a (generalized) B uchi automaton is nonempty (accepts at least one word) can be done by computing its strongly connected components, and checking that there exists a reachable strongly connected component that has a non empty intersection with each set in F . 4 From Temporal Logic ....

J.R. Buchi. On a decision method in restricted second order arithmetic. In Proc. Internat. Congr. Logic, Method and Philos. Sci. 1960, pages 1-12, Stanford, 1962. Stanford University Press.

....cycle detection problem is at the heart of many problems, namely in deciding emptiness of automata like generalised B # uchi and Streett automata, and in model checking of speci Thetacations written in linear and branching temporal logics like LTL and fair CTL. A generalised B # uchi automaton [10] is provided together with several sets of accepting states. A run of such an automaton is accepting if it contains at least one state from every accepting set in Thetanitely often. Accordingly, the language of the automaton is nonempty if and only if the graph corresponding to the automaton ....

J. R. B # uchi. On a decision method in restricted second order arithmetic. In Proc. International Congress on Logic, Methodology and Philosophy Science, pages 1 # 11. Stanford university Press, 1960.

....we discuss how variants of SIM can be treated similarly although the decidability status of some of them is still unknown. Keywords: computational complexity, B uchi tree automaton, information logic, hybrid logic 1 Introduction From logic to automata. After the works of B uchi and Rabin [B uc62,Rab69] various classes of automata turned out to be well suited to solve decision procedures for logical problems, including some for temporal logics (see e.g. VW94,Var97,KVW00] for the calculus and its fragments (see e.g. EJ99,SE89,VW86,EJS01,Var98] and for description logics (see ....

R. Buchi. On a decision method in restricted second-order arithmetic. In International Congress on Logic, Method and Philosophical Science'60, pages 1-11, 1962.

....(RVA) symbolic representation has been introduced in [BBR97] Although satisfactory from a theoretical point of view (all rst order additive sets can be represented) it has been feared for a long time that those automata could not be used in practice. Indeed, they deal with in nite words [B uc62] which implies the use of considerably less ecient and more dicult to implement algorithms than for automata on nite words [SVW87,Saf89,KV97] However, using topological arguments, it has been shown in [BJW01] that handling the additive arithmetic over the reals and integers does not require ....

J. R. Buchi. On a decision method in restricted second order arithmetic. In Proceedings of the International Congress on Logic, Method, and Philosophy of Science, pages 1-12, Stanford, 1962. Stanford University Press.

.... B. f is said to be regular if and only if f is prefix regular and there is a finite set of non empty iterative functions iter(f) such that 8ff 2 f , 9u, 9g 2 iter(f) ff 2 u:g and g ae f(u) Such functions are called regular because of the analogy with regular sets of words of Buchi [6]. The only restriction imposed by the fact that we consider functions lies in the entry names, namely the entry names must be ultimately periodic to be finitely representable. Theorem 7. A function f is regular if and only if its entry names are ultimately periodic and the set of words in f is ....

B uchi, J. R. On a decision method in restricted second order arithmetics. In International Congress on Logic, Methodology and Philosophy of Science (1960), E. Nagel et al., Eds., Stanford University Press.

....keywords: temporal logic, xpoint calculus, clausal resolution. 1 Introduction Automata over in nite objects, termed automata, were originally introduced as a tool for investigating the decidability of restricted classical rst order and secondorder logics [2], but have also been extensively used in recent developments within wider areas of computer science. In particular, the success of the model checking approach [5, 10, 13, 17] when applied to the analysis of in nite computations is primarily due to the incorporation of automata related methods. ....

.... and ecient methods for proving decidability of varieties of temporal logic [4] Note that, in some cases, for example in the case of the branching time logic CTL , automata based methods are the only known methods for proving decidability of such expressive systems [5] Here, B uchi automata [2] over in nite words and in nite trees are especially important. It is known [4] that B uchi word automata, which are as expressive as propositional linear time calculus [14] are themselves strictly more expressive than propositional linear time temporal logic, PLTL [19] A well known example ....

J. R. Buchi. On a decision method in restricted second-order arithmetics. In Proc.of International Congress of Logic, Methodology and Philosophy of Science. Stanford University Press, 1962.

.... j= I x 2 X i I(x) 2 I(X) As usual, a sentence is a formula with no free variables. Each sentence de nes an language, denoted L , where: L = f j j= g: We say that L is MSO( de nable i there exists a sentence 2 MSO( such that L = L . A celebrated result of B uchi [4] shows that the class of languages expressible by sentences in MSO( coincides with the class of languages recognized by B uchi automata over . This class is the regular languages over . The rst order theory of in nite sequences over is denoted FO( and is obtained from MSO( by ....

Buchi, J. R.: On a decision method in restricted second order arithmetic. Proceedings of the International Congress on Logic, Methodology and Philosophy of Science, Stanford University Press (1960) 1-11

.... automata in a high level way. Technical proofs of the correctness of this representation are given. 1 Introduction Automata over in nite objects, termed automata, were originally introduced as a tool for investigating the decidability of restricted classical rst order and secondorder logics [2], but have also been extensively used in recent developments within wider areas of computer science. In particular, the success of the model checking approach [5, 10, 13, 17] when applied to the analysis of in nite computations is primarily due to the incorporation of automata related methods. ....

.... of varieties of temporal logic [4] Note that, in some cases, for example in the case of the branching time logic CTL , automata based methods are the only known methods for proving decidability of such expressive systems [5] In decidability results for temporal logic, B uchi automata [2] over in nite words and This work was partially supported by funding from EPSRC, under research grant GR L87491 1 in nite trees are especially important. It is known [4] that B uchi word automata, which are as expressive as propositional linear time calculus [14] are themselves strictly ....

[Article contains additional citation context not shown here]

J. R. Buchi. On a decision method in restricted second-order arithmetics. In Proc.of International Congress of Logic, Methodology and Philosophy of Science. Stanford University Press, 1962.

.... automata in a high level way. Technical proofs of the correctness of this representation are given. 1 Introduction Automata over in nite objects, termed automata, were originally introduced as a tool for investigating the decidability of restricted classical rst order and secondorder logics [3], but have also been extensively used in recent developments within wider areas of computer science. In particular, the success of the model checking approach [6, 12, 15, 20] when applied to the analysis of in nite computations, is primarily due to the incorporation of automata related methods. ....

.... that, in some cases, for example in the case of the branching time logic CTL , where the tableau construction is not directly applicable, automata based methods are essential for proving decidability of such expressive systems [6] In decidability results for temporal logic, B uchi automata [3] over in nite words and in nite trees are especially important. This work was partially supported by funding from EPSRC, under research grant GR L87491 1 It is known [5] that B uchi word automata, which are as expressive as propositional linear time calculus [16] are themselves strictly ....

[Article contains additional citation context not shown here]

J. R. Buchi. On a decision method in restricted second-order arithmetics. In Proc.of International Congress of Logic, Methodology and Philosophy of Science, pages 1-12. Stanford University Press, 1962.

....whether an arbitrary set of types is realizable. It is shown that is satis able in a model with ow of time F i there exists a quasimodel for over F. Thus, has a model with ow of time F i F j= If F is fhN; ig, fhZ; ig, or fhQ; ig, this last statement is decidable by results of [15, 46]. The other cases can now be obtained by reduction. Part (ii) is proved di erently, since the full monadic second order theory of hR; i is undecidable. The proof adapts the argument of the second part of [17] see [35] for details. As a consequence of Theorem 5 we obtain, for example, the ....

J. Buchi. On a decision method in restricted second order arithmetic. In Logic, Methodology, and Philosophy of Science: Proc.

.... the recognizable sets (those accepted by finite non deterministic automata) are exactly the regular (or rational) sets (those definable by regular expressions) An infinitary version of this theorem shows that regular sets of infinite sequences are exactly those recognized by B uchi automata [B uc60, McN66] To prove the timed analogues of these results we define timed regular and timed regular expressions and show that they denote exactly what timed automata can recognize. As in the classical theorem one direction, the construction of automata from expressions, is rather straightforward, ....

....the first part of B uchi McNaughton theorem. Theorem 8 ( Expressions # Automata) Every (generalized) timed regular language can be accepted by a timed automaton. 7. 3 From Automata to Expressions This construction is based on Theorem 6 and on the proof of the untimed theorem (see [B uc60, McN66] We assume that the automaton has gone through all the transformation described in Section 6.2 and also converted in a state reset form, as described below. A one clock timed automaton is state reset if the transitions entering a given state either all reset the clock, or all do not ....

J.R. B uchi. A decision method in restricted second order arithmetic. In E. Nagel, editor, Proc. Int. Congr. on Logic, Methodology and Philosophy of Science. Stanford University Press, 1960.

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J.R. Buchi. On a decision method in restricted second order arithmetic. In Proc. Internat. Congr. Logic, Method. and Philos. Sci. 1960, pages 1-12, Stanford, 1962. Stanford University Press.

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J.R. Buchi, On a decision method in restricted second-order arithmetic, in Proc.

....the literature, this logic is sometimes denoted by MSO. FO is the subclass of SOM restricted to languages de nable by rst order formulas. It is known [MP71] that FO is equal to the class of star free regular languages. In this paper we are mainly interested in the following earlier result (see [BE58, B uc62, Tra61]) Proposition 3 (B uchi Elgot Trakhtenbrot Theorem) The class SOM is equal to the class REG of regular languages. Next, we extend the logical language allowing generalized quanti ers. De nition 4. Consider a language L over an alphabet = a 1 ; a 2 ; a s ) Such a language gives ....

J. R. Buchi. On a decision method in restricted second-order arithmetic. In Proceedings Logic, Methodology and Philosophy of Sciences 1960, Stanford, CA, 1962. Stanford University Press.

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J.R. Buchi. On a decision method in restricted second order arithmetic. In Proc. International Congress on Logic, Method and Philos. Sci. 1960, pages 1-11, 1962.

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B uchi J. R. [1962], On a decision method in restricted second order arithmetic, in `Proc. Int. Congr. Logic, Method and Philosophy of Science 1960', Stanford University Press, Palo Alto, CA, USA, pp. 1-12.

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J. B uchi. On a decision method in restricted second-order arithmetic. In Proceedings of the Int. Congress on Logic, Methodology and Philosophy of Science, 1960.

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J.R. Buchi. On a decision method in restricted second order arithmetic. In Proc. Internat. Congr. Logic, Method. and Philos. Sci. 1960, pages 1-12, Stanford, 1962. Stanford University Press.

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J. R. B uchi. On a decision method in restricted second order arithmetic. In Logic, Methodology and Philosophy of Science: Proceedings of the 1960.

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J. R. B UCHI, On a decision method in restricted second-order arithmetic, in Proc. 1st International Congress on Logic, Methodology, and Philosophy of Science, Stanford University Press, 1962.

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J.R. Buchi. On a decision method in restricted second order arithmetic. In E.Nagel et al, editor, ICM'60, pages 1-11. Stanford University Press, 1960.

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J. R. Buchi, On a decision method in restricted second order arithmetic. Logic, Methodology and Philosophy of Science, Proc. 1960.

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J. R. Buchi. On a decision method in restricted second order arithmetic. In Proceedings of the 1960 International Congress on Logic, Methodology, and Philosophy of Science, pages 1-11. Stanford University Press, 1962.

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J. R. Buchi, On a decision method in restricted second-order arithmetic, in Proc. 1960 Int. Congr. for Logic, Methodology and Philosophy of Science, Stanford Univ. Press, Standford, (1962) 1-11.

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J.R. B uchi. On a decision method in restricted second order arithmetic. In Proc. International Congress on Logic, Method and Philos. Sci.

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