Erich Gradel, Colin Hirsch, and Martin Otto. Back and Forth Between Guarded and Modal Logics. ACM Transactions on Computational Logics, 3(3):418 -- 463, 2002. See also: conference version [GHO00].  Home/Search   Document Details and Download   Summary   Related Articles   Check

....can quantify over relations R of arbitrary arity with the restriction that every tuple a R is guarded, i.e. there is some relation S of the original structure that contains a tuple b S such that a b. Note that every singleton a is guarded by a = a. For a more detailed definition see [15]. L(# ) denotes the extension of the logic L by cardinality quantifiers , for every cardinal #, where stands for there are at least # many . A formula #(x) where each free variable is first order defines on a given structure A the relation # . Definition 1. Let A = A, R 0 ....

E. Gr adel, C. Hirsch, and M. Otto, Back and forth between guarded and modal logics, in Proc. 15th IEEE Symp. on Logic in Computer Science, 2000, pp. 217--228.

....GSO for short. Then we de ne guarded bisimulation which will relate tuples of elements of two structures and not just single elements as bisimulation did. Finally, we show that GSO sentences that are guarded bisimulation invariant are exactly GF sentences. The results sumarized below come from [22]. De nition 68 Let M be a structure over a signature Sig and with the universe M . A tuple (m 1 ; m k ) is guarded i there is a relation R and elements m l such that R(m l ) holds in M and fm 1 ; m k g fm l g . A relation S M is guarded if it consists of ....

.... (x 1 ; x n ) is invariant under bisimulation if it cannot distinguish between guarded bisimilar tuples, i.e. if M 1 (a 1 ; a n ) and (a 1 ; a n ) is guarded bisimilar to a tuple (b 1 ; b n ) 2 M 2 then M 2 (b 1 ; b n ) The following theorem from [22] ties together the expressive power of GSO and GF. Theorem 73 (Gr adel, Hirsch, Otto) Every formula of GSO invariant under guarded bisimulation is equivalent to a GF formula. 8 Traces In nite words, which are linear orders on events, are often used to model executions of systems. In nite ....

E. Gradel, C. Hirsch, and M. Otto. Back and forth between guarded and modal logics. In LICS'00, pages 217-228, 2000.

....can quantify over relations R of arbitrary arity with the restriction that every tuple a R is guarded, i.e. there is some relation S of the original structure that contains a tuple b S such that a b. Note that every singleton a is guarded by a = a. For a more detailed definition see [14]. L(# ) denotes the extension of the logic L by cardinality quantifiers , for every cardinal #, where stands for there are at least # many . A formula #(x) where each free variable is first order defines on a given structure A the relation # Definition 1. Let A = A, R 0 , ....

E. Gr adel, C. Hirsch, and M. Otto, Back and forth between guarded and modal logics, in Proc. 15th IEEE Symp. on Logic in Computer Science, 2000, pp. 217--228.

....theorem for the modal calculus, due to Janin and Walukiewicz [18] saying that the properties de nable in the modal calculus are precisely the properties that are de nable in monadic second order logic and invariant under bisimulation. And, as shown recently by Gr adel, Hirsch, and Otto [12], this result also carries over to the guarded world. Indeed, there is a natural fragment of second order logic, called GSO, which is between monadic secondorder logic and full second order logic, such that guarded xed point logic is 5 precisely the bisimulation invariant portion of GSO. ....

E. Gr adel, C. Hirsch, and M. Otto, Back and forth between guarded and modal logics, in Proc. 15th IEEE Symp. on Logic in Computer Science, 2000, pp. 217-228.

....properties of modal logics (and hence also description logics) Guarded logics have received considerable attention in the last years and it seems that the notion of guarded quanti cation is indeed very important for the design of logics that are both expressive and algorithmically manageable. See [1, 2, 3, 5, 6, 10, 9, 8, 12, 13] for background and further results on guarded logics and [11] for an informal discussion. 2 A general criterion for decidability Guarded second order logic, abbreviated GSO, has the same syntax as ordinary second order logic, but second order quanti ers are restricted semantically to range over ....

....that y is a guarded tuple. More explicitly, on structures with relations R 1 ; R t guarded(y) t i=1 9x(R i x j y j = x ) Note that GSO includes monadic second order logic MSO. The expressive power of GSO is strictly between MSO and full second order logic [12]. Tree width. The tree width of a structure measures how closely it ressembles a tree. Informally a structure has tree width k, if it can be covered by (possibly overlapping) substructures of size at most k 1 which are arranged in a tree like manner. Tree width is an important notion in graph ....

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E. Gr adel, C. Hirsch, and M. Otto, Back and forth between guarded and modal logics, in Proc. 15th IEEE Symp. on Logic in Computer Science, 2000.

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E. Gr adel, C. Hirsch, and M. Otto, Back and forth between guarded and modal logics, ACM Transactions on Computational Logic, 3 (2002), pp. 418-463.

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E. Gr adel, C. Hirsch, and M. Otto, Back and forth between guarded and modal logics, ACM Transactions on Computational Logics, 3 (2002), pp. 418 - 463.

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E. GR ADEL, C. HIRSCH, AND M. OTTO, Back and forth between guarded and modal logics, to appear in ACM Transactions of Computational Logic, 2002.

....has to encode all non degenerate quanti er free two types by new binary relations which can be interpreted so as to form a simple transition system which faithfully encodes the underlying relational structure. Similar considerations and translations for guarded logics on graphs are presented in [8]. More speci cally, a non degenerate 2 type over is a full description of the isomorphism type of a two element structure in variables x; y, which may be formalised as a conjunction over a maximally consistent set of atomic and negated atomic formulae in variables x and y including the ....

E. Gr adel, C. Hirsch, and M. Otto, Back and forth between guarded and modal logics, to appear in ACM Transactions of Computational Logic, preprint 2001.

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Erich Gradel, Colin Hirsch, and Martin Otto. Back and Forth Between Guarded and Modal Logics. ACM Transactions on Computational Logics, 3(3):418 -- 463, 2002. See also: conference version [GHO00].

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Erich Gradel, Colin Hirsch, and Martin Otto. Back and Forth Between Guarded and Modal Logics. In Proceedings of 15th IEEE Symposium on Logic in Computer Science LICS 2000, pages 217--228, Santa Barbara, 2000. See also: journal version [GHO02].

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