Finite Model Theory

Ebbinghaus, Heinz-Dieter; Flum, Jörg

doi:10.1007/978-3-662-03182-7

Cited by 237 publications

(313 citation statements)

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“…The semantics of atomic formulas, fixed-points and first-order operations are defined as usual (c.f., e.g., [EF99] for details), with comparison of number terms µ ≤ η interpreted by comparing the corresponding integers in [n + 1]. Finally, consider a counting term of the form # x φ, where φ is a formula and x an element variable.…”

Section: Logics and Structuresmentioning

confidence: 99%

Maximum Matching and Linear Programming in Fixed-Point Logic with Counting

Anderson

Dawar

Holm

2013

2013 28th Annual ACM/IEEE Symposium on Logic in Computer Science

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We establish the expressibility in fixed-point logic with counting (FPC) of a number of natural polynomial-time problems. In particular, we show that the size of a maximum matching in a graph is definable in FPC.This settles an open problem first posed by Blass, Gurevich and Shelah [BGS99], who asked whether the existence of perfect matchings in general graphs could be determined in the more powerful formalism of choiceless polynomial time with counting. Our result is established by showing that the ellipsoid method for solving linear programs can be implemented in FPC. This allows us to prove that linear programs can be optimised in FPC if the corresponding separation oracle problem can be defined in FPC. On the way to defining a suitable separation oracle for the maximum matching problem, we provide FPC formulas defining maximum flows and canonical minimum cuts in capacitated graphs.

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Section: Logics and Structuresmentioning

confidence: 99%

Maximum Matching and Linear Programming in Fixed-Point Logic with Counting

Anderson

Dawar

Holm

2013

2013 28th Annual ACM/IEEE Symposium on Logic in Computer Science

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“…Two-variable logics, and more generally finite variable logics, have been studied extensively in finite model theory (see, for example, [12,14,16,19]). …”

Section: Theorem 28mentioning

confidence: 99%

Tight Lower and Upper Bounds for the Complexity of Canonical Colour Refinement

Berkholz

Bonsma

Grohe

2013

Lecture Notes in Computer Science

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An assignment of colours to the vertices of a graph is stable if any two vertices of the same colour have identically coloured neighbourhoods. The goal of colour refinement is to find a stable colouring that uses a minimum number of colours. This is a widely used subroutine for graph isomorphism testing algorithms, since any automorphism needs to be colour preserving. We give an O((m + n) log n) algorithm for finding a canonical version of such a stable colouring, on graphs with n vertices and m edges. We show that no faster algorithm is possible, under some modest assumptions about the type of algorithm, which captures all known colour refinement algorithms.

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“…Hence in addition to allowing the use of set terms we shall also go be-yond FOL (First-Order Logic) by introducing an operation T C for transitive closure 6 . The corresponding language and semantics are defined as follows (see, e.g., [30,29,47,28,13]…”

Section: Interpreting and Implementing Principle (Nat)mentioning

confidence: 99%

A New Approach to Predicative Set Theory

Avron¹

2010

Ways of Proof Theory

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We suggest a new basic framework for the Weyl-Feferman predicativist program by constructing a formal predicative set theory P ZF which resembles ZF . The basic idea is that the predicatively acceptable instances of the comprehension schema are those which determine the collections they define in an absolute way, independent of the extension of the "surrounding universe". This idea is implemented using syntactic safety relations between formulas and sets of variables. These safety relations generalize both the notion of domain-independence from database theory, and Godel notion of absoluteness from set theory. The language of P ZF is type-free, and it reflects real mathematical practice in making an extensive use of statically defined abstract set terms. Another important feature of P ZF is that its underlying logic is ancestral logic (i.e. the extension of FOL with a transitive closure operation).

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Finite Model Theory

Cited by 237 publications

References 0 publications

Maximum Matching and Linear Programming in Fixed-Point Logic with Counting

Maximum Matching and Linear Programming in Fixed-Point Logic with Counting

Tight Lower and Upper Bounds for the Complexity of Canonical Colour Refinement

A New Approach to Predicative Set Theory

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