The first order definability of graphs: Upper bounds for quantifier depth

Pikhurko, Oleg; Veith, Helmut; Verbitsky, Oleg

doi:10.1016/j.dam.2006.03.002

Cited by 25 publications

(63 citation statements)

References 25 publications

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“…Various aspects of descriptive complexity of graphs have recently been investigated in [10,27,33,34,41] with focus on the minimum quantifier depth of a first order sentence defining a graph. In particular, a comprehensive analysis of the definability of trees in first order logic is carried out in [10,33,41].…”

Section: Descriptive Complexity Of Graphsmentioning

confidence: 99%

Testing Graph Isomorphism in Parallel by Playing a Game

Grohe

Verbitsky

2006

Automata, Languages and Programming

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Our starting point is the observation that if graphs in a class C have low descriptive complexity, then the isomorphism problem for C is solvable by a fast parallel algorithm. More precisely, we prove that if every graph in C is definable in a finite-variable first order logic with counting quantifiers within logarithmic quantifier depth, then Graph Isomorphism for C is in TC 1 ⊆ NC 2 . If no counting quantifiers are needed, then Graph Isomorphism for C is even in AC 1 . The definability conditions can be checked by designing a winning strategy for suitable Ehrenfeucht-Fraïssé games with a logarithmic number of rounds. The parallel isomorphism algorithm this approach yields is a simple combinatorial algorithm known as the Weisfeiler-Lehman (WL) algorithm.Using this approach, we prove that isomorphism of graphs of bounded treewidth is testable in TC 1 , answering an open question from [13]. Furthermore, we obtain an AC 1 algorithm for testing isomorphism of rotation systems (combinatorial specifications of graph embeddings). The AC 1 upper bound was known before, but the fact that this bound can be achieved by the simple WL algorithm is new. Combined with other known results, it also yields a new AC 1 isomorphism algorithm for planar graphs.

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Section: Descriptive Complexity Of Graphsmentioning

confidence: 99%

Testing Graph Isomorphism in Parallel by Playing a Game

Grohe

Verbitsky

2006

Automata, Languages and Programming

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“…This is a qualitative extension of a result in [14], where the bound D 1 (M) ≤ n/2 + 2 is proved for any irredundant structure M with maximum relation arity 2. On the other hand, there are simple examples of irredundant structures with D (M) ≥ n/4 (see Remark 4.4).…”

Section: Introductionmentioning

confidence: 86%

“…In [14] we prove the following results. If M has only unary and binary relations, then I 1 (M) ≤ (n + 3)/2.…”

Section: Introductionmentioning

confidence: 97%

“…The following result was obtained in [14] for graphs with the proof easily adaptable for any structures (see Lemma 4.2 and Remark 4.9 in [14]). …”

Section: An Upper Bound For D (M)mentioning

confidence: 99%

“…Recall also that, if π = (v 1 v 2 ) is a transposition, then we may writeū (v 1 v 2 ) in place of πū. We now introduce an operation of expanding a class [v] M , i.e., adding to M new elements ∼-equivalent to v. This operation was considered in [14] in the particular case of uniform hypergraphs.…”

Section: Cloning An Element Of a Structurementioning

confidence: 99%

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Descriptive complexity of finite structures: Saving the quantifier rank

Pikhurko

Verbitsky²

2005

J. symb. log.

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We say that a first order formula Φ distinguishes a structure M over a vocabulary L from another structure M ′ over the same vocabulary if Φ is true on M but false onWe prove that every n-element structure M is identifiable by a formula with quantifier rank less than (1 − 1 2k )n + k 2 − k + 2 and at most one quantifier alternation, where k is the maximum relation arity of M . Moreover, if the automorphism group of M contains no transposition of two elements, the same result holds for definability rather than identification.The Bernays-Schönfinkel class consists of prenex formulas in which the existential quantifiers all precede the universal quantifiers. We prove that every n-element structure M is identifiable by a formula in the Bernays-Schönfinkel class with less than (1 − 1 2k 2 +2 )n + k quantifiers. If in this class of identifying formulas we restrict the number of universal quantifiers to k, then less than n − √ n + k 2 + k quantifiers suffice to identify M and, as long as we keep the number of universal quantifiers bounded by a constant, at total n − O( √ n) quantifiers are necessary.

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How complex are random graphs in first order logic?

Kim¹,

Pikhurko²,

Spencer³

et al. 2004

Random Struct Algorithms

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ABSTRACT:It is not hard to write a first order formula which is true for a given graph G but is false for any graph not isomorphic to G. The smallest number D(G) of nested quantifiers in such a formula can serve as a measure for the "first order complexity" of G. Here, this parameter is studied for random graphs. We determine it asymptotically when the edge probability p is constant; in fact, D(G) is of order log n then. For very sparse graphs its magnitude is (n). On the other hand, for certain (carefully chosen) values of p the parameter D(G) can drop down to the very slow growing function log * n, the inverse of the TOWER-function. The general picture, however, is still a mystery.

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The first order definability of graphs: Upper bounds for quantifier depth

Cited by 25 publications

References 25 publications

Testing Graph Isomorphism in Parallel by Playing a Game

Testing Graph Isomorphism in Parallel by Playing a Game

Descriptive complexity of finite structures: Saving the quantifier rank

How complex are random graphs in first order logic?

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