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Finite Variable Logics In Descriptive Complexity Theory
 Bulletin of Symbolic Logic
, 1998
"... this article. ..."
Canonization for ...Equivalence is Hard
"... . Let L k be the kvariable fragment of firstorder logic, for some k 3. We prove that equivalence of finite structures in L k has no Pcomputable canonization function unless NP P=poly. The latter assumption is considered as highly unlikely; in particular it implies a collapse of the polynom ..."
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. Let L k be the kvariable fragment of firstorder logic, for some k 3. We prove that equivalence of finite structures in L k has no Pcomputable canonization function unless NP P=poly. The latter assumption is considered as highly unlikely; in particular it implies a collapse of the polynomial hierarchy. The question for such a canonization function came up in the context of the problem of whether there is a logic for P. Slight modifications of our result yield answers to questions of Dawar, Lindell, and Weinstein [4] and Otto [16] concerning the inversion of the socalled L k invariants. 1 Introduction Membership in a class of ordered finite structures can be tested in polynomial time if, and only if, the class can be defined in least fixedpoint logic. This is a fundamental result of Immerman [12] and Vardi [19]. In the terminology of descriptive complexity theory, it says that leastfixed point logic captures polynomial time on ordered structures. The Achilles' heel ...
On the Expressive Power of Logics on Finite Models
, 2003
"... Structures" [Mos74], where they are called inductive relations. It should also be pointed out that in Immerman's book on "Descriptive Complexity" LFP is denoted by FO(LFP) (the closure of FO under least fixedpoints) and LFP 1 is denoted by LFP(FO) (least fixedpoints of firstorder formulas). ..."
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Structures" [Mos74], where they are called inductive relations. It should also be pointed out that in Immerman's book on "Descriptive Complexity" LFP is denoted by FO(LFP) (the closure of FO under least fixedpoints) and LFP 1 is denoted by LFP(FO) (least fixedpoints of firstorder formulas).
WeisfeilerLehman refinement requires at least a linear number of iterations
 IN PROCEEDINGS OF THE INTERNATIONAL COLLOQUIUM IN AUTOMATA, LANGUAGES, AND PROGRAMMING (ICALP), SPRINGERVERLAG, LECTURE NOTES IN COMPUTER SCIENCE (LNCS
, 2001
"... Let Lk,m be the set of formulas of first order logic containing only variables from {x1,x2,...,xk} and having quantifier depth at most m. Let Ck,m be the extension of Lk,m obtained by allowing counting quantifiers ∃ixj, meaning that there are at least i distinct xj’s. It is shown that for constants ..."
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Let Lk,m be the set of formulas of first order logic containing only variables from {x1,x2,...,xk} and having quantifier depth at most m. Let Ck,m be the extension of Lk,m obtained by allowing counting quantifiers ∃ixj, meaning that there are at least i distinct xj’s. It is shown that for constants h ≥ 1, there are pairs of graphs such that hdimensional WeisfeilerLehman refinement (hdim WL) can distinguish the two graphs, but requires at least a linear number of iterations. Despite of this slow progress, 2hdim WL only requires O ( √ n) iterations, and 3h − 1dim WL only requires O(log n) iterations. In terms of logic, this means that there is a c> 0 and a class of nonisomorphic pairs (G h n,H h n) of graphs with G h n and H h n having O(n) vertices such that the same sentences of Lh+1,cn and Ch+1,cn hold (h + 1 variables, depth cn), even though G h n and H h n can already be distinguished by a sentence of Lk,m and thus Ck,m for some k>hand m = O(log n).
Finite Models and Finitely Many Variables
 Banach Center Publications
, 1999
"... We consider L  first order logic restricted to k variables, and interpreted in finite structures. The study of classes of finite structures axiomatisable with finitely many variables has assumed importance through connections with computational complexity. In particular, we investigate the relation ..."
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We consider L  first order logic restricted to k variables, and interpreted in finite structures. The study of classes of finite structures axiomatisable with finitely many variables has assumed importance through connections with computational complexity. In particular, we investigate the relationship between the size of a finite structure and the number of distinct types it realizes, with respect to L . Some open questions, formulated as finitary LöwenheimSkolem properties, are presented regarding this relationship. This is also investigated through finitary versions of an EhrenfeuchtMostowski property.