Erich Rosen. Modal logic over finite structures. Journal of Logic, Language and Information, 6:427--439, 1997.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....or Si has a winning strategy, i.e. the game G (A, X,B,Y, n) is deterministic. 6 The Expressive Power of Van Benthem [3] proved the following preservation result: a class of models defined by a first order sentence is closed under bisimulations i# it can be defined by a modal formula. Rosen [18] proved that this result remains true over the class of finite structures. Kurtonina and de Rijke [13] extended Van Benthem s result in a di#erent direction, by proving analogous preservation results for broad classes of description logics, including both restrictions and extension of the basic ....

E. Rosen. Modal logic over finite structures. Journal of Logic, Language and Information, 6(4):427--439, 1997.

....and the expressive power on finite models of particular logics. In addition, the study of finite models forms a natural task in computational linguistics as well as in database theory. In the context of modal logic the viewpoint of finite model theory has first been taken in a paper by Rosen [19]. The main result of this paper shows that van Benthems bisimulation result remains true over the class of finite models. Below we will adopt this result to the global setting. We need some new notions and notation. We write M n N to denote that M and N agree on all first order sentences of ....

....m n, if Zmw # v # , and Rw # w ## , then there is some v ## in N such that Rv # v ## and Zm 1w ## v ## (and vice versa) 3. For all m n, if Zmw # v # , then (M,w # ) # p n i# (N, v # ) # p n , for every #. 8 Next, we need a technical lemma that combines a number of results due to Rosen [19] into a single statement. Lemma 4.5 There is a unary function f with the following property. Let r #, and assume that (M,w) f(r) N, v) Then there exist models (M # , w # ) M,w) and (N # , v # ) M,w) with (M # , w # ) r (N # , v # ) as displayed in the following diagram: # ....

[Article contains additional citation context not shown here]

E. Rosen. Modal logic over finite structures. Journal of Logic, Language and Information, 6:427--439, 1997.

....has a winning strategy, i.e. the game 2 is deterministic. 6 The Expressive Power of Van Benthem [19] proved the following preservation result: a class of models defined by a first order sentence is closed under bisimulations iff it can be defined by a modal formula. Rosen [17] proved that this result remains true over the class of finite structures. Kurtonina and de Rijke [12] extended Van Benthem s result in different direction, by proving analogous preservation results for broad classes of description logics, including both restrictions and extension of the basic ....

....We write to denote that Si has winning strategies for both games 2 2 to denote that Si has winning strategies for the corresponding infinite games. The first key theorem in Rosen s paper is the following. Theorem 10 (Rosen [17]) Let be any class of models (each model node ) closed under isomorphism. Let be any subclass of , also closed under isomorphism. Then for all , the following conditions are equivalent: 2. There is a modal formula of quantifier rank that defines over We ....

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E. Rosen. Modal logic over finite structures. Journal of Logic, Language and Information, 6(4):427--439, 1997.

....with relationships between complexity classes and logics over finite structures. It is interesting to consider bisimulation invariance in the context of finite model theory. Rosen showed that Proposition 3 of the previous section remains true with the restriction to finite transition systems [24]. It is an open question whether Proposition 4 also remains true under this restriction. Part of the interest in relationships between M and 2M or 2OL with respect to finite transition systems is that within 2M and 2OL one can define NP complete problems: examples include 3 colourability on ....

Rosen, E. (1995). Modal logic over finite structures. Tech Report, University of Amsterdam.

....is concerned with relationships between complexity classes and logics over finite structures. It is interesting to consider bisimulation invariance in the context of finite model theory. Rosen showed that Proposition 1 (in section 2) remains true with the restriction to finite transition systems [17]. It is an open question whether Proposition 2 also remains true under this restriction. Part of the interest in relationships between M and 2M or 2OL with respect to finite transition systems is that within 2M and 2OL one can define NP complete problems: examples include 3 colourability on finite ....

Rosen, E. (1997). Modal logic over finite structures. Journal of Logic, Language and Information, 6, 427-439.

.... this setting (because of the finite model property) while firstorder logic does not even have interpolation when we restrict to finite models ( 10] 7] Yet, also in the finite model case, first order logic is as expressive as modal logic when we restrict to bisimulation invariant properties ([20]) 2 Preliminaries We define some notions necessary to our interpolation proof. First of all, we consider languages L built up from a set P of proposition constants and a set A of atomic actions as follows: there is a single constant sr (for source ) each p 2 P is a unary predicate symbol of L ....

E. Rosen. Modal Logic over Finite Structures. Report ML-95-08, Institute for Logic, Language and Computation, University of Amsterdam, 1995. ftp://ftp.cis.upenn.edu/pub/ircs/tr/95-27.ps.Z.

....(CSLI Publications, Stanford, 1996) contains many subsequent developments, of which I mention a few. i) Many different proofs have been found for the Modal Invariance Theorem by now, including techniques like elementary chains, saturated models, and Ehrenfeucht games. In particular, Eric Rosen proved in 1995 that the result also holds in finite model theory. I suspect that more generally, unlike with full first order logic, most of modal model theory is robust under the transition from ordinary model theory to finite model theory. ii) One can vary the expressive power of modal languages, and then ....

E. Rosen, 1995, 'Modal Logic over Finite Structures', Report ML-95-08, ILLC, Univ.

....to a modal program. Corollary 4.5 The same is true if we restrict to the class of finite models. Proof. Suppose OE(x; y) is safe for bisimulation in FIN, the class of finite models. Then 9y: OE(x; y) p(y) where p is fresh) is invariant for bisimulation in FIN. By a result of Rosen ([30]) the latter formula must be equivalent (in FIN) to a modal formula (p) p) must be completely additive in p (again, when restricting to finite models) So the following are valid equivalences in FIN: pq) j (p) q) and ( j . Due to the finite model property of modal logic (proved for ....

....has suggested to consider a fragment of L 1 that only has conjunctions and disjunctions over certain effective sets. A candidate for such a fragment is weak MSO, where second order quantification is restricted to finite sets. Another question suggested by the previous section and Rosen s work ([30]) is whether the invariant fragment of MSO also coincides with the calculus in the class of finite models. Even if this is so, the proof of safety would not automatically transfer, because complete additivity of formulas does not coincide with distributivity over finite unions. Finally, all of ....

E. Rosen. Modal logic over finite structures. Report ML-95-08, Institute for Logic, Language and Computation, University of Amsterdam, 1995. ftp://ftp.cis.upenn.edu/pub/ircs/tr/95-27.ps.Z.

....theorem remains open. For further information on preservation theorems on finite structures (and a collection of open problems) see [1] Preservation theorems also play a role in the study of fragments of first order logic, such as the finite variable fragments [2, 14, 16] the modal fragment [3, 15], and the guarded fragment [2] If important preservation theorems, along with model theoretic principles like Craig interpolation and Beth definability, hold for a logical language, this indicates that the language possesses good model theoretic properties and that there is a natural balance ....

.... theorem for FO 2 further confirms the view that the embedding of propositional modal logic into FO 2 does not really explain its good algorithmic and model theoretic properties (e.g. see [2, 8] Recall that the existential preservation theorem does hold for propositional modal logic (see [2, 15]) A preservation theorem links a semantic property P of definable model classes to a syntactic property of formulae. Thus the failure of a preservation theorem raises the possibility of finding some alternative syntactic characterization. Alechina and Gurevich [1] formulate this question as ....

E. Rosen. Modal logic over finite structures. Journal of Logic, Language and Information, 6:427--439, 1997.

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Erich Rosen. Modal logic over finite structures. Journal of Logic, Language and Information, 6:427--439, 1997.

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E. ROSEN, Modal logic over finite structures, Journal of Logic, Language and Information, 6 (1997), pp. 427--439.

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Rosen, E. (1997). Modal logic over finite structures. Journal of Logic, Language, and Information, 6:427--439.

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