Eva Hoogland and Maarten Marx. Interpolation in guarded fragments. Studia Logica, 70(3):373--409, 2002.  Home/Search   Document Details and Download   Summary   Related Articles   Check

.... modules refine the language in response to concrete needs. These refinements can take the form of special decision methods for the whole GF, fragments or extensions (see, e.g. 27] novel logical core results on interpolation, finite model property, expressive 2 power, etc. see, e.g. [20]) applications living inside guarded fragments [3] and many more. Actually, we would like to push ideology even further: even the central component of the picture displayed in Figure 1 is just a module and can be exchanged by a di#erent fragment, as far as this new fragment enjoys some of the ....

....Returning to the main issue at stake what about interpolation in GF and PGF It turns out that matters are far more subtle here than in finite variable fragments. Let us try to give an indication of what s going on. First, using the notions of bisimulation just introduced, Hoogland and Marx [20] provide a number of counterexamples for Craig interpolation, both for GF and PGF. More precisely, they show the following: 1. There exist sentences #, # # GF using only 3 variables and predicate symbols of arity at most 3 such that = # # #, for which there does not exist an interpolant in ....

E. Hoogland and M. Marx. Interpolation in guarded fragments. Unpublished.

....b) Similarly for universal quanti ers. It is easy to see that, for nite vocabularies, this semantic de nition of CGF is equivalent to the one given above. An alternative possibility is to permit as guards any existential positive formula (x) that implies clique(x) This is what Maarten Marx [20,16] uses in his de nition of the packed fragment PF. The di erences between the cliqueguarded fragment and the packed fragment are purely syntactical. PF and CGF have the same expressive power. The work of Maarten Marx and ours have been done independently. Every LGF sentence is equivalent to a ....

E. Hoogland and M. Marx, Interpolation in guarded fragments. preprint.

....1995; van Benthem, 1996) One of the purposes of this paper is to show that these two routes are really two sides of the same coin. This has been observed several times but is never made precise. In fact strange anomalies exists. So does the interpolation property fail for the guarded fragment (Hoogland and Marx, 1999), while it holds for several relativised versions of rst order logic (Marx and Venema, 1997) We make the connection precise by showing that a certain guarded fragment (called here the packed fragment) of rst order logic forms precisely the set of rst order sentences which are invariant for ....

....characterisation of this fragment forms a slight generalisation of van Benthem s loosely guarded fragment 4 . We rst de ne the fragment. As before we consider a standard rst order language with equality with one restriction: terms are variables or constant symbols. 4 For a comparison see (Hoogland and Marx, 1999). jolli.tex; 8 12 1999; 14:09; p.13 14 We say that a formula packs a set of variables fx 1 ; x k g if FV ( fx 1 ; x k g and is a conjunction of formulas of the form t i = t j or R(t 1 ; t n ) or 9 yR(t 1 ; t n ) such that for every x i 6= x j , there is a ....

Hoogland, E. and M. Marx: 1999, `Interpolation in the guarded fragment'. manuscript.

....power with nice modal behavior. It consists of relational first order formulas whose quantifiers are relativized by atoms in a certain way. While GF has been established as a particularly well behaved fragment of first order logic in many respects, interpolation fails in restriction to GF, [HM99]. In this paper we consider the Beth property of first order logic and show that, despite the failure of interpolation, it is retained in restriction to GF. Being a closure property w.r.t. definability, the Beth property is of independent interest, both theoretically and for typical potential ....

....of the modal fragment can be seen as the main reason behind the robustness of the decidability of that fragment (cf. e.g. Var98] this gives hope as to the robustness of GF. And indeed, adding least and greatest fixed points to GF yields a decidable extension [GW99] However, as shown in [HM99], the interpolation theorem of first order logic fails for GF. In the present paper it will be shown that GF does have an alternative interpolation property, which closely resembles the interpolation property usually studied in modal logics. This result turns out to be strong enough to entail the ....

[Article contains additional citation context not shown here]

E. Hoogland and M. Marx. Interpolation in guarded fragments. Technical report, Institute for Logic, Language and Computation, University of Amsterdam, 1999.

....power with nice modal behavior. It consists of relational first order formulas whose quantifiers are relativized by atoms in a certain way. While GF has been established as a particularly well behaved fragment of first order logic in many respects, interpolation fails in restriction to GF, [HM99]. In this paper we consider the Beth property of first order logic and show that, despite the failure of interpolation, it is retained in restriction to GF. Being a closure property w.r.t. definability, the Beth property is of independent interest, both theoretically and for typical potential ....

....of the modal fragment can be seen as the main reason behind the robustness of the decidability of that fragment (cf. e.g. Var98] this gives hope as to the robustness of GF. And indeed, adding least and greatest fixed points to GF yields a decidable extension [GW99] However, as shown in [HM99], the interpolation theorem of first order logic fails for GF. In the present paper it will be shown that GF does have an alternative interpolation property, which closely resembles the interpolation property usually studied in modal logics. This result turns out to be strong enough to entail the ....

[Article contains additional citation context not shown here]

E. Hoogland and M. Marx. Interpolation in guarded fragments. Technical report, Institute for Logic, Language and Computation, University of Amsterdam, 1999.

No context found.

Eva Hoogland and Maarten Marx. Interpolation in guarded fragments. Studia Logica, 70(3):373--409, 2002.

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E. HOOGLAND AND M. MARX, Interpolation in guarded fragments, to appear in Studia Logica.

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