T. Hafer and W. Thomas. Computation tree logic CTLand path quantifiers in the monadic theory of the binary tree. In Proc. 14th International Coll. on Automata, Languages, and Programming, volume 267 of Lecture Notes in Computer Science, pages 269--279. Springer-Verlag, 1987.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....successor from right successor in a binary tree) Then, to allow these comparisons, branching time structures are restricted to binary trees and the other formalisms are restricted to those which have a symmetric behaviour, that is, they do not distinguish between left and right successors. In [HT87] the following result is shown. Theorem 3.9 CTL is as expressive as the monadic second order theory of two successors with set quantification restricted to infinite paths over infinite binary trees. Much more results have been established for comparisons among branching temporal logics and a ....

T. Hafer and W. Thomas. Computation tree logic ctl* and path quantifiers in the monadic theory of the binary tree. In Proceedings of the lJth International - 279. Springer-Verlag, 1987.

....In [32] Sistla and Clarke show that the satisfiability problem for PTL is PSPACE complete. As for branching time logic, the expressive power of CTL # and PCTL # is equivalent to the one of monadic second order logic on infinite binary trees with second order quantifiers over infinite paths [18]. Theorem 5.2 (Expressive completeness of CTL # and PCTL # ) CTL # and PCTL # are as expressive as MPL[ when interpreted over infinite binary trees. In [8] Emerson and Jutla prove that the problem of testing satisfiability for CTL # is 2EXPTIME complete. As pointed out by Hafer and Thomas ....

....Theorem 5.2 (Expressive completeness of CTL # and PCTL # ) CTL # and PCTL # are as expressive as MPL[ when interpreted over infinite binary trees. In [8] Emerson and Jutla prove that the problem of testing satisfiability for CTL # is 2EXPTIME complete. As pointed out by Hafer and Thomas [18], Theorem 5.2 can be generalized to CTL # k and PCTL # k with respect to MPL[ # i ) i=0 ] by incorporating successors into both temporal and monadic path logics [18] Theorem 5.3 (Expressive completeness of CTL # k and PCTL # k ) CTL # k and PCTL # k are as expressive as MPL[ # i ) i=0 ....

[Article contains additional citation context not shown here]

T. Hafer and W. Thomas. Computation tree logic CTL* and path quantifiers in the monadic theory of the binary tree. In T. Ottmann, editor, Automata, Languages and Programming, 14th International Colloquium, volume 267 of Lecture Notes in Computer Science, pages 269--279, Karlsruhe, Germany, 13--17 July 1987. Springer.

....blow up in the length of formulas. Ehrenfeucht games have been successfully exploited to deal with such a correspondence problem for first order monadic theories [8] and well behaved fragments of second order ones, e.g. the path fragment of the monadic second order theory of infinite binary trees [7]. As for the theories of time granularity, by means of suitable applications of Ehrenfeucht games, we obtained an expressively complete and elementarily decidable combined temporal logic counterpart of the path fragment of the monadic second order theory of DULSs [4] while Montanari et al. ....

T. Hafer and W. Thomas. Computation tree logic CTL* and path quantifiers in the monadic theory of the binary tree. In T. Ottmann, editor, Proceedings of the International Colloquium Automata, Languages and Programming, volume 267 of Lecture Notes in Computer Science, pages 269--279, Karlsruhe, Germany, 1987. Springer.

.... automata: given a PLTL formula # over of length n, one can construct an equivalent Buchi sequence automaton A# over 2 P with 2 states [86, 118] As for branching time logic, the expressive power of CTL # and PCTL # is equivalent to the one of monadic path logic over infinite binary trees [61]. Theorem 2.4.6 (Expressiveness of CTL # and PCTL # ) CTL # and PCTL # are expressively equivalent to MPLP [ pre ] when interpreted over infinite binary trees. As pointed out by Hafer and Thomas [61] Theorem 2.4.6 can be generalized to CTL # k PCTL # k with respect to MPLP [ pre , # i ) ....

....of CTL # and PCTL # is equivalent to the one of monadic path logic over infinite binary trees [61] Theorem 2.4.6 (Expressiveness of CTL # and PCTL # ) CTL # and PCTL # are expressively equivalent to MPLP [ pre ] when interpreted over infinite binary trees. As pointed out by Hafer and Thomas [61], Theorem 2.4.6 can be generalized to CTL # k PCTL # k with respect to MPLP [ pre , # i ) i=0 ] by incorporating successors into both temporal and monadic path logics. Theorem 2.4.7 (Expressiveness of CTL # k and PCTL # k CTL # k and PCTL # k are expressively equivalent to MPLP [ pre , # ....

[Article contains additional citation context not shown here]

T. Hafer and W. Thomas. Computation tree logic CTL* and path quantifiers in the monadic theory of the binary tree. In T. Ottmann, editor, Proceedings of the International Colloquium Automata, Languages and Programming, volume 267 of Lecture Notes in Computer Science, pages 269--279, Karlsruhe, Germany, 1987. Springer.

....blow up in the length of formulas. Ehrenfeucht games have been successfully exploited to deal with such a correspondence problem for first order monadic theories [22] and well behaved fragments of second order ones, e.g. the path fragment of the monadic second order theory of infinite binary trees [18]. Unfortunately, these techniques do not naturally lift to the full second order case. The existence of a correspondence between (combined) temporal logics and (combined) automata, satisfying the usual closure properties, allows one to reduce the task of finding a temporal logic counterpart of a ....

T. Hafer and W. Thomas. Computation tree logic CTL* and path quantifiers in the monadic theory of the binary tree. In T. Ottmann, editor, Automata, Languages and Programming, 14th International Colloquium, volume 267 of LNCS, pages 269--279. Springer, 1987.

....theory of countable linear orders is decidable, 3. The theory of countable boolean algebras with quantification over ideals is decidable. 38 Some Expressiveness Results We here mention some of the expressiveness results relating branching time temporal logics to S2S. ffl Hafer and Thomas [24] shows that when restricted to the infinite binary tree, CTL is exactly as expressive as S2S with quantification restricted to paths, and that ECTL is exactly as expressive as S2S with quantification restricted to chains. For n 2, however, SnS with quantification restricted to chains is ....

T. Hafer and W. Thomas. Computation tree logic CTL and path quantifiers in the monadic theory of the binary tree. In Proc. 14th ICALP, Lecture Notes in Computer Science, pages 269--279, 1987.

....may be finite and our quantifications over paths are thus over infinite and maximal finite paths. That is, we consider a branching notion of past which may be either finite, as in CTLbp [17] or infinte, as in POTL [26, 34] A branching past is more appropriate here than the linear past in PCTL# [8] which can also be used to augement branching time logics with past time operators. Second, propositions are generalized to predicates over free variables. A traditional atomic proposition is simply a predicate with no arguments. For example, the formula stmt(x : e) where stmt P r, has free ....

Th. Hafer and W. Thomas. Computation tree logic CTL* and path quantifiers in the monadic theory of the binary tree. In Automata, Languages and Programming Proceedings, ICALP'87, volume 267 of Lecture Notes in Computer Science, pages 267--279. Springer-Verlag, 1987.

.... easier to write and more natural [17] However, allowing past time makes verification algorithms harder to implement (though not necessarily from a complexity theoretic viewpoint) Additionally, all the main temporal logics with past time admit translations to their pure future fragment [11, 7, 6, 8, 24, 15, 26]. Forgettable past and the N modality. Being able to refer to past moments is often useful, but there also exist situations where it is convenient to forget the past. Consider for example, the following temporal formula G(alarm ) F problem) Spec1) where F means at some past time . ....

....can be done in polynomial time for NLTL formulae and is in fact PTIME complete (observe that the precise complexity of model checking a path is still an open problem for LTL and PLTL [3] Related works. Past time is more popular in linear time settings but branching time settings exist, e.g. [8, 15, 14, 16]. Automata theoretic methods for temporal logics were pioneered by Vardi and Wolper, and they were adapted for LTL Past and mu calculus Past in [17, 24, 12, 13] these logics have PSPACE complete verification problems) N can be encoded in richer formalisms, e.g. QPLTL (PLTL with arbitrary ....

T. Hafer and W. Thomas. Computation tree logic CTL and path quantifiers in the monadic theory of the binary tree. In Proc. 14th Int. Coll. Automata, Languages, and Programming (ICALP'87), Karlsruhe, FRG, July 1987.

....that there exists a computation starting from the current configuration which satisfies the path formula . Hence, when interpreting the past cannot go beyond the current configuration. This way of introducing past modalities in branching time logics corresponds to the interpretation adopted in [HT87]. The construction u: associates the current position (index) on the computation with the position variable u. Then, u can be used as a label allowing to refer to the position associated with it. The formula A u when interpreted at some position i means that the automaton A accepts the timed ....

....in the spirit of [VW86] We show that it is possible to construct for every BTATL p formula an equivalent formula in timed ECTL 9 , and then, we give a model checking algorithm for the logic timed ECTL 9 . 8. 1 The logic timed ECTL 9 Timed ECTL 9 is an extension a timed version of ECTL [HT87]. The set of timed ECTL 9 (state) formulas is given by: P j : j j 9A( 1 ; n ) where A is a timed Muller automaton over a finite nonempty alphabet V = fa 1 ; a n g. In timed ECTL 9 , timed Muller automata on infinite sequences are used to express path ....

[Article contains additional citation context not shown here]

T. Hafer and W. Thomas. Computation Tree Logic CTL and Path Quantifiers in the Monadic Theory of the Binary Tree. In ICALP'87. LNCS 267, 1987.

....Non determ. Finite Infinite Cumulative Non cumul. GPSS80] KVd83] LPZ85] Bar87] Zuc86] Gab89] Var88] MP92] MMKR94] ffl ffl ffl Linear time temporal logics [PW84] Wol89] Sti89] Sti92] ffl ffl non Ockhamist past [KP95] ffl ffl [ZC93] Zan96] ffl ffl ffl Ockhamist past [HT87] BLY96] ffl ffl ffl [LPS95] KP95] ffl ffl ffl [LS95] ffl ffl ffl ffl Figure 2: The semantics of past in the literature when written in PCTL. This example has been chosen because it is rich and realistic but still easy to understand. Our background hypothesis are: ffl The lift services n ....

T. Hafer and W. Thomas. Computation tree logic CTL and path quantifiers in the monadic theory of the binary tree. In Proc. 14th Int. Coll. Automata, Languages, and Programming (ICALP'87), Karlsruhe, FRG, July 1987, volume 267 of Lecture Notes in Computer Science, pages 269--279. Springer-Verlag, 1987.

.... for concurrency (it is often argued that concurrent systems giving rise to bisimulation equivalent computation trees are indistinguishable for all reasonable notions of observation) In [17] CTL was shown to be expressively equivalent to the bisimulation invariant fragment of monadic path logic [10]. The syntax of monadic path logic is the same as that of monadic second order logic. The bound set (monadic) variables ranges over all the paths and semantically this logic is very closely related to the first order logic [17] Thus at least CTL represents some objectively quantified expressive ....

....; xm ; X 1 ; Xn ) is satisfied in the tree T with x i interpreted by the node s i (1 i m) and X j interpreted by the set of nodes S j (1 j n) We shall denote by FOMLO the subset of first order formulas of MLO that do not have set quantification. We also consider Monadic Path Logic MPL [10]. Its syntax is the same as that of monadic second order logic. However, the bound set (monadic) variables range over all the paths (not over arbitrary sets of nodes) and semantically it is very closely related to the first order logic [17] Definition 1 (Future MLO Formula) A formula (x 0 ; X ....

T. Hafer and W. Thomas (1987). Computation tree logic CTL and path quantifiers in the monadic theory of the binary tree. In ICALP'87, LNCS 267:269--279, Springer-Verlag.

.... restricted class of models the yardstick is MSOL theory of the binary tree (S2S) It is known that the binary calculus is expressively complete with respect to full S2S [24, 11] and CTL is expressively complete with respect to the fragment of S2S where only quantification over paths is allowed [16]. As far as we are aware, the only expressive completeness results dealing with the general case of logics over all transition systems were given by van Benthem [5] and van Benthem and Bergstra [6] They show that a bisimulation closed class of transition systems is definable in first order logic ....

T. Hafer and W. Thomas. Computation tree logic CTL and path quantifiers in the monadic theory of the binary tree. In 14th Internat. Coll. on Automata, Languages and Programming, volume 267 of LNCS, pages 269--279, 1987.

....into the usual pure future calculus [Var88] In fact, Var88] gives a translation from some kind of backward and forward Buchi automata into usual Buchi automata, so that one has to translate from the calculus into Buchi automata, and backward. ffl CTL Past can be translated into CTL [HT87]. This is a simple corollary of Gabbay s proof for PTL. ffl PTL n X Past can be translated into PTL n X [MMKR93] This uses rewrite rules similar to Gabbay s rules. However these results certainly do not answer all questions. For example, if we want to add past time constructs to a ....

....by anonymous ftp, on machine ftp.imag.fr, in directory pub CONCUR (or by writing to the authors) 1 Temporal logics with Past 1.1 Syntax We define PCTL (for CTL with Past ) as an extension of CTL [EH86] with past time combinators. Our definition differs slightly from the PCTL used in [HT87] as we explain later. We assume a given set Prop = fa; b; problem; cause; g of atomic propositions. Definition 1.1 (Syntax of PCTL ) The formulas of PCTL are given by the following grammar PCTL 3 f; g : a j f g j :f j Ef j f U g j Xf j f S g j X Gamma1 f where a 2 Prop. ....

[Article contains additional citation context not shown here]

T. Hafer and W. Thomas. Computation tree logic CTL and path quantifiers in the monadic theory of the binary tree. In Proc. 14th ICALP, Karlsruhe, LNCS 267, pages 269--279. Springer-Verlag, July 1987.

....to the class of structures definable in the second order monadic theory of n successors, SnS . It would be nice if one could show that a branching time temporal logic has the same expressive power as SnS ; after all, branching time temporal logics are interpreted on computation trees. In [4] it is shown that a restricted version of SnS with set quantification restricted to paths is expressively equivalent to CTL for binary tree models. However, as was pointed out in [2] 1] and independently shown in [4] the full SnS can express properties which have no correlate in a ....

....branching time temporal logics are interpreted on computation trees. In [4] it is shown that a restricted version of SnS with set quantification restricted to paths is expressively equivalent to CTL for binary tree models. However, as was pointed out in [2] 1] and independently shown in [4]) the full SnS can express properties which have no correlate in a branching time temporal logic with a finite number of branching time operators. The reason for this is that the semantics of any branching time operator uses finitely many first order quantifiers, so the logic in question has a ....

Hafer, T., Thomas, W., Computation Tree Logic CTL and Path Quantifiers In The Monadic Theory Of The Binary Tree, Proc. 11th Int. Coll. on Autom. Lang. and Prog., Springer LNCS 267, pp. 269-279.

....two equivalent behaviours: a temporal property which holds of a particular system should hold for all equivalent systems. In this paper, we shall be interested specifically in bisimulation equivalence [19, 18] the branching time temporal logic CTL [2, 4] and so called monadic path logic MPL [11, 12]: monadic second order logic in which set This work was carried out while the second author was visiting Uppsala University supported by a grant from the Swedish STINT Fellowship Programme. quantification is restricted to paths. It is well known that CTL respects bisimulation equivalence in ....

....author was visiting Uppsala University supported by a grant from the Swedish STINT Fellowship Programme. quantification is restricted to paths. It is well known that CTL respects bisimulation equivalence in the above sense, while already first order logic does not. However, Hafer and Thomas [12] demonstrate that every CTL property can be expressed in MPL, and that the reverse is also true if you restrict attention to full binary trees. Two full binary trees are bisimulation equivalent only if they are isomorphic, and MPL respects isomorphism. In this paper we modify their argument ....

[Article contains additional citation context not shown here]

T. Hafer and W. Thomas (1987). Computation tree logic CTL and path quantifiers in the monadic theory of the binary tree. In Proceedings of ICALP'87: International Colloquium on Automata, Languages and Programming, Lecture Notes in Computer Science 267:269--279, Springer-Verlag.

....two equivalent behaviours: a temporal property which holds of a particular system should hold for all equivalent systems. In this paper, we shall be interested specifically in bisimulation equivalence [17, 16] the branching time temporal logic CTL [1, 3] and so called monadic path logic MPL [10, 11]: monadic second order logic in which set quantification is restricted to paths. It is well known that CTL respects bisimulation equivalence in the above sense, while already first order logic does not. However, Hafer and Thomas [11] demonstrate that every CTL property can be expressed in ....

....logic CTL [1, 3] and so called monadic path logic MPL [10, 11] monadic second order logic in which set quantification is restricted to paths. It is well known that CTL respects bisimulation equivalence in the above sense, while already first order logic does not. However, Hafer and Thomas [11] demonstrate that every CTL property can be expressed in MPL, and that the reverse is also true if you restrict attention to binary trees. Two binary trees are bisimulation equivalent only if they are isomorphic, and MPL respects isomorphism. In this paper we modify their argument to show ....

[Article contains additional citation context not shown here]

T. Hafer and W. Thomas (1987). Computation tree logic CTL and path quantifiers in the monadic theory of the binary tree. In Proceedings of ICALP'87: International Colloquium on Automata, Languages and Programming, Lecture Notes in Computer Science 267:269-- 279, Springer-Verlag.

....that there exists a computation starting from the current configuration which satisfies the path formula . Hence, when interpreting the past cannot go beyond the current configuration. This way of introducing past modalities in branching time logics corresponds to the interpretation adopted in [9]. The construction u: associates the current position (index) on the computation with the position variable u. Then, u can be used as a label allowing to refer to the position associated with it. The formula A u when interpreted at some position i means that the automaton A accepts the timed ....

....in the spirit of [15] We show that it is possible to construct for every BTATL p formula an equivalent formula in timed ECTL 9 , and then, we give a model checking algorithm for the logic timed ECTL 9 . 8. 1 The logic timed ECTL 9 Timed ECTL 9 is an extension a timed version of ECTL [9]. The set of timedECTL 9 (state) formulas is given by: P j : j j 9A( 1 ; n ) where A is a timed Muller automaton over a finite nonempty alphabet V = fa 1 ; ang. In timed ECTL 9 , timed Muller automata on infinite sequences are used to express path ....

[Article contains additional citation context not shown here]

T. Hafer and W. Thomas. Computation Tree Logic CTL and Path Quantifiers in the Monadic Theory of the Binary Tree. In ICALP'87. LNCS 267, 1987.

....support to our views. Structure of past Determined Non determ. Finite Infinite Cumulative Non cumul. 8] 12] 18] 1] 32] 7] 28] 20] 23] ffl ffl ffl Linear time temporal logics [24] 29] 26] 27] ffl ffl non Ockhamist past [14] ffl ffl [31] 30] ffl ffl ffl Ockhamist past [9], 2] ffl ffl ffl [15] 14] ffl ffl ffl [16] ffl ffl ffl ffl Fig. 2. The semantics of past in the literature Laroussinie and Schnoebelen 4 Specification of a lift system We use the classical example of a lift system (from [1,10] to experiment with the PCTL logic. We want to see whether ....

T. Hafer and W. Thomas. Computation tree logic CTL and path quantifiers in the monadic theory of the binary tree. In Proc. 14th Int. Coll. Automata, Languages, and Programming (ICALP'87), Karlsruhe, FRG, July 1987, volume 267 of Lecture Notes in Computer Science, pages 269--279. Springer-Verlag, 1987.

....is used in the proofs below. We have reduced the problem of establishing the k variable property for Sigma to checking the condition of Corollary 4(i) This will done using Ehrenfeucht Fraisse games [3, 5] Ehrenfeucht Fraisse games have been used widely in theoretical computer science; see e.g. [4, 6, 8, 10, 12, 13, 17, 18]. Here we use a modified version in which the number of pebbles is finite [9, 14, 10] Definition 5 Let A; B be structures for L and (u; v) a k configuration. We call (u; v) a local isomorphism if the map u(x) 7 v(x) x 2 u, is well defined and extends to an isomorphism of the substructures of ....

T. Hafer and W. Thomas, "Computation Tree Logic CTL and Path Quantifiers in the Monadic Theory of the Binary Tree," Proc. 14th Int. Colloq. Automata, Languages, and Programming, Karlsruhe, July 1987, Springer Lect. Notes Comp. Sci. 267, 269-279.

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T. Hafer and W. Thomas. Computation tree logic CTLand path quantifiers in the monadic theory of the binary tree. In Proc. 14th International Coll. on Automata, Languages, and Programming, volume 267 of Lecture Notes in Computer Science, pages 269--279. Springer-Verlag, 1987.

No context found.

T. Hafer and W. Thomas. Computation tree logic CTL* and path quantifiers in the monadic theory of the binary tree. In Thomas Ottmann, editor, Automata, Languages and Programming, 14th International Colloquium, volume 267 of Lecture Notes in Computer Science, pages 269--279, Karlsruhe, Germany, 13-- 17 July 1987. Springer-Verlag.

No context found.

T. Hafer and W. Thomas (1987). Computation tree logic CTL and path quantifiers in the monadic theory of the binary tree. In Proceedings of ICALP'87: International Colloquium on Automata, Languages and Programming, Lecture Notes in Computer Science 267:269--279, Springer-Verlag.

No context found.

T. Hafer and W. Thomas (1987). Computation tree logic CTL # and path quantifiers in the monadic theory of the binary tree. In Proceedings of ICALP'87: International Colloquium on Automata, Languages and Programming, Lecture Notes in Computer Science 267:269--279, Springer-Verlag.

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T. Hafer, W. Thomas, Computation tree logic CTL # and path quantifiers in the monadic theory of the binary tree, in: Proceedings of ICALP'87: International Colloquium on Automata, Languages and Programming, Lecture Notes in Computer Science, vol. 267, Springer-Verlag, Berlin, 1987, pp. 269--279.

No context found.

T. Hafer and W. Thomas. Computation tree logic CTL* and path quantifiers in the monadic theory of the binary tree. In Thomas Ottmann, editor, Automata, Languages and Programming, 14th International Colloquium, volume 267 of Lecture Notes in Computer Science, pages 269--279, Karlsruhe, Germany, 13--17 July 1987. Springer-Verlag.

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