W. Thomas. Elements of an automata theory over partial orders. In POMIV 1996, volume 29 of DIMACS. AMS, 1996.  Home/Search   Document Details and Download   Summary   Related Articles   Check

.... together with the fundamental theoretical interest in generalising the known theory has lead people to study the formal language, automata and logical theory of Mazurkiewicz traces, partial orders, events structures, grids, tree decomposable graphs and graphs, some recent references are Thomas [161, 162]. Whereas the theories of Mazurkiewicz traces is already well developed, see e.g. 43] the quest continues for finding larger subclasses of graphs with nice logical and automata characterisations, and for which, hopefully, some interesting questions are decidable. Yet another important direction ....

W. Thomas. Elements of automata theory over partial orders. In D.A. Peled, V.R. Pratt, and G.J. Holzmann, editors, Partial Order Methods in Verification. American Mathematical Society, July 1996.

....and fix some notation. In Section 3 we show the normal form theorems. In Section 4 we define the simplified games for first order logic and monadic 1 1 logic. The analogous results for automata are presented in Section 5. The ensuing automata models are also compared with the automata of [Tho91, Tho97b, Tho97a] and [Cou90] Section 6 contains a conclusion. 2 Definitions and Notations A relational signature is a finite set of relation symbols R, each with a fixed arity a(R) and constant symbols c. We do not use function symbols. A structure A consists of a universe U A (the vertices Local Normal ....

....is local around y. For classes of structures that have a connected Gaifman graph we can show even stronger normal forms. The basic idea is that global information about the structure can be transported along the relations and collected in a designated place. This generalizes a similar procedure in [Tho97b, Tho97a], where the transport is much more deterministic. Theorem 3.4 Let C be a class of structures with a connected Gaifman graph. Then the following hold. a) On C every monadic 1 1 formula is equivalent to a formula of the form 9X 1 ; X l 9x8y , where is local around y. b) If there ....

[Article contains additional citation context not shown here]

W. Thomas. Elements of an automata theory over partial orders. In D. A. Peled, V. R. Pratt, and G. J. Holzmann, editors, Partial Order Methods in Verification, number 29 in DIMACS Series in Discrete Mathematics and Theoretical Computer Science. American Mathematical Society, 1997.

....is local around y. For classes of structures that have a connected Gaifman graph we can show even stronger normal forms. The basic idea is that global information about the structure can be transported along the relations and collected in a designated place. This generalizes a similar procedure in [Tho96, Tho97], where the transport is much more deterministic. 3.4 Theorem Let C be a class of structures with a connected Gaifman graph. Then the following hold. a) On C every monadic Sigma 1 1 formula is equivalent to a formula of the form 9X 1 ; X l 9x8y , where is local around y. b) If ....

....; x g 8y9y 1 ; y l ( 1ml (y m = y) I ) 5.2 Corollary A class of oe structures is monadic Sigma 1 1 definable if and only if it is accepted by a monadic Sigma 1 1 automaton. For structures of bounded degree, the monadic Sigma 1 1 automata generalize those of Thomas [Tho91, Tho96, Tho97]. In this case a Hintikka type of a small neighbourhood boils down to an isomorphism type, called tile. The automata of Thomas check that each vertex possesses a neighbourhood of one among a finite number of allowed isomorphism types. Moreover, some of them must occur at least a certain number of ....

W. Thomas. Elements of an automata theory over partial orders. In D. Peled, editor, Proc. Workshop on Partial Order Methods in Verification. American Mathematical Society, 1996. To appear in DIMACS Series in Discrete Mathematics and Theoretical Computer Science.

....within the clusters. So the formula for duplicate graphs can be translated in a formula for label graphs which is true exactly for the label graphs of graphs G 2 C Decomp(k; H) that are models of the original formula. A well known automaton model for graphs is the model of tiling systems [Th91] [Th96]. This is a generalization of word automata and tree automata. Such an automaton guesses a state assignment to the vertices, checks the r sphere around every vertex of the graph (now labelled also by states) and counts the number of occurrences of r spheres up to a given threshold number. ....

W. Thomas, Elements of an automata theory over partial orders, in: Partial Order Methods in Verification (D.A. Peled et al., eds.), DIMACS Series in Discrete Mathematics and Theoretical Computer Science Vol. 29, Amer. Math. Soc. 1997, pp. 25-40.

....G if there is a mapping of states to the nodes of G such that every r sphere belongs to Delta. In the original definition there are also occurrence constraints, restrictions on the number of times a tile occurs in G. These are not needed for the classes of graphs which are considered here (see [Th96]) The class of languages which can be described by tiling systems of sphere radius r relative to a class K of graphs is called TS r (K) Let TS(K) be the union of the classes TS r (K) It is known that TS(K) EMSO(K) for an arbitrary class K of bounded width (cf. Th91] Thus, the equality ....

W. Thomas, Elements of an automata theory over partial orders, in: DIMACS Series in Discrete Mathematics and Theoretical Computer Science, Amer. Math. Soc. (to appear).

....is local around y. For classes of structures that have a connected Gaifman graph we can show even stronger normal forms. The basic idea is that global information about the structure can be transported along the relations and collected in a designated place. This generalizes a similar procedure in [Tho96, Tho97], where the transport is much more deterministic. 3.4 Theorem Let C be a class of structures with a connected Gaifman graph. Then the following hold. a) On C every monadic Sigma 1 1 formula is equivalent to a formula of the form 9X 1 ; X l 9x8y ; where is local around y. b) If ....

....C be a class of oe structures. The following are equivalent. 1. C is monadic Sigma 1 1 definable. 2. There exists a monadic Sigma 1 1 automaton such that C is the class of structures accepted by it. For structures of bounded degree, the monadic Sigma 1 1 automata generalize those of Thomas [Tho91, Tho96, Tho97]. In this case a Hintikka type of a small neighbourhood boils down to an isomorphism type, called tile. The automata of Thomas check that each vertex possesses a neighbourhood (of bounded size) of one among a finite number of allowed isomorphism types. Moreover, some of them must occur at least a ....

W. Thomas. Elements of an automata theory over partial orders. In D. Peled, editor, Proc. Workshop on Partial Order Methods in Verification. American Mathematical Society, 1996. To appear in DIMACS Series in Discrete Mathematics and Theoretical Computer Science.

....and fix some notation. In Section 3 we show the normal form theorems. In Section 4 we define the simplified games for first order logic and monadic 1 logic. The analogous results for automata are presented in Section 5. The ensuing automata models are also compared with the automata of [Tho91, Tho97b, Tho97a] and [Cou90] Section 6 contains a conclusion. 2 Definitions and Notations A relational signature is a finite set of relation symbols R, each with a fixed arity a(R) and constant symbols c. We do not use function symbols. A structure A consists of a universe U (the vertices of A) an ....

....is local around y. For classes of structures that have a connected Gaifman graph we can show even stronger normal forms. The basic idea is that global information about the structure can be transported along the relations and collected in a designated place. This generalizes a similar procedure in [Tho97b, Tho97a], where the transport is much more deterministic. Theorem 3.4 Let C be a class of structures with a connected Gaifman graph. Then the following hold. a) On C every monadic 1 formula is equivalent to a formula of the form 9X 1 ; X l 9x8y , where is local around y. b) If there ....

[Article contains additional citation context not shown here]

W. Thomas. Elements of an automata theory over partial orders. In D. A. Peled, V. R. Pratt, and G. J. Holzmann, editors, Partial Order Methods in Verification, number 29 in DIMACS Series in Discrete Mathematics and Theoretical Computer Science. American Mathematical Society, 1997.

....and fix some notation. In Section 3 we show the normal form theorems. In Section 4 we define the simplified games for first order logic and monadic # 1 1 logic. The analogous results for automata are presented in Section 5. The ensuing automata models are also compared with the automata of [Tho91, Tho97b, Tho97a] and [Cou90] Section 6 contains a conclusion. Local Normal Forms for First Order Logic 111 2 Definitions and Notations A relational signature # is a finite set of relation symbols R, each with a fixed arity a(R) and constant symbols c. We do not use function symbols. A # structure A consists ....

....is local around y. For classes of structures that have a connected Gaifman graph we can show even stronger normal forms. The basic idea is that global information about the structure can be transported along the relations and collected in a designated place. This generalizes a similar procedure in [Tho97b, Tho97a], where the transport is much more deterministic. Local Normal Forms for First Order Logic 115 Theorem 3.4 Let C be a class of structures with a connected Gaifman graph. Then the following hold. a) On C every monadic # 1 1 formula is equivalent to a formula of the form #X 1 , X ....

[Article contains additional citation context not shown here]

W. Thomas. Elements of an automata theory over partial orders. In D. A. Peled, V. R. Pratt, and G. J. Holzmann, editors, Partial Order Methods in Verification, number 29 in DIMACS Series in Discrete Mathematics and Theoretical Computer Science. American Mathematical Society, 1997.

....systems. Undecidability results such as for the emptiness of recognizable sets are easy to show, and the recently established monadic quantifier alternation hierarchy is also set up in this domain. The present paper is meant as an introduction, integrating results from [Th91] PST94] GRST96] [Th96a], and [MT96] Most arguments are presented on the intuitive level, assuming that the reader is familiar with basic automaton constructions and simple facts of logic. In Section 2, we collect the necessary terminology, and in Section 3 we recapitulate some well known results on tree automata and ....

....exhaust rather well the range of the first category; it seems that by any substantial generalization of trace dependency graphs one leaves the framework of classical automata theory. A candidate for the second category is, besides pictures and grids, the class of mirror concatenated trees ([Th96a]) they are obtained from two labelled ordered trees with identical numbers of leaves by identifying the frontiers (which is done order preserving) and by reversing the edge direction in one tree (so that the roots of the two trees give a smallest and a greatest element in the resulting partial ....

W. Thomas, Elements of an automata theory over partial orders, in: Proc. Workshop on Partial Order Methods in Verification (D. Peled, Ed.), DIMACS Ser. in Discr. Math. (to appear).

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W. Thomas. Elements of an automata theory over partial orders. In POMIV 1996, volume 29 of DIMACS. AMS, 1996.

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W. Thomas. Elements of an automata theory over partial orders. In Proceedings of Workshop on Partial Order Methods in Verification (POMIV 1996.

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W. Thomas. Elements of an automata theory over partial orders. In POMIV 1996, volume 29 of DIMACS. AMS, 1996.

No context found.

W. Thomas. Elements of an automata theory over partial orders. In POMIV 1996, volume 29 of DIMACS. AMS, 1996.

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