Model Theory Makes Formulas Large

We investigate a model-theoretic property that generalizes the classical notion of "preservation under substructures". We call this property preservation under substructures modulo bounded cores, and present a syntactic characterization via Σ 0 2 sentences for properties of arbitrary structures definable by FO sentences. As a sharper characterization, we further show that the count of existential quantifiers in the Σ 0 2 sentence equals the size of the smallest bounded core. We also present our results on the sharper characterization for special fragments of FO and also over special classes of structures. We present a (not FO-definable) class of finite structures for which the sharper characterization fails, but for which the classical Łoś-Tarski preservation theorem holds. As a fallout of our studies, we obtain combinatorial proofs of the Łoś-Tarski theorem for some of the aforementioned cases.

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“…Theorem 3 strengthens the non-elementary lower bound given in [6]. Corollary 2 gives us the following.…”

Section: Theoremsupporting

confidence: 68%

Preservation under Substructures modulo Bounded Cores

Sankaran

Adsul

Madan

et al. 2012

Logic, Language, Information and Computation

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“…Hence, by [23], this isomorphism test can be performed in time polynomial in the size of the structures (the degree of the polynomial depends on f ). 4 Hence, the number p can indeed be computed within the given time bound. 2…”

Section: Lemma 32 From a Formula ϕ In Hanf Normal Form With Free Vamentioning

confidence: 99%

An optimal construction of Hanf sentences

Bollig

Kuske

2012

Journal of Applied Logic

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“…As we consider words with the less-than relation we also have unbounded branching degree. The result in [3] has recently been strengthened: Also for first order logic with three variables (FO 3 logic) the size of the decomposition is non-elementary [9]. There it was also shown that the size becomes double exponential for FO 2 logic.…”

Section: Introductionmentioning

confidence: 93%

“…However, as mentioned above, one common drawback of this method is the complexity: the number of formulas generated by the method is non-elementary in the size of the given formula. In [3] it has been shown that this is also a lower bound in general for FO logic and a disjoint sum of structures with unbounded branching degree. As we consider words with the less-than relation we also have unbounded branching degree.…”

Section: Introductionmentioning

confidence: 99%

LTL-Model-Checking via Model Composition

Felscher

2012

Lecture Notes in Computer Science

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Abstract. We develop a composition technique for linear time logic (LTL) over ordered disjoint sums of words. This technique allows to reduce the validity of an LTL formula in the sum to LTL formulas over the components. It is known that for first order logic (FO) and even its three variable fragment FO 3 the number of formulas generated for the components is at least non-elementary in the size of the input formula. We show that for LTL -expressively equivalent to FO logic -over an ordered disjoint sum of words the number of formulas for the components can be kept at an at most exponential growth in the size of the input formula.

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Model Theory Makes Formulas Large

Cited by 39 publications

References 18 publications

Preservation under Substructures modulo Bounded Cores

Preservation under Substructures modulo Bounded Cores

An optimal construction of Hanf sentences

LTL-Model-Checking via Model Composition

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