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Finite Variable Logics In Descriptive Complexity Theory
 Bulletin of Symbolic Logic
, 1998
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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 firstor ..."
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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).
On the complexity of finding narrow proofs
 In Proceedings of the 53rd Annual IEEE Symposium on Foundations of Computer Science (FOCS ’12
, 2012
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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 ...
Lower bounds for existential pebble games and kconsistency tests
 In Proc. LICS’12
, 2012
"... Vol. 9(4:2)2013, pp. 1–23 www.lmcsonline.org ..."
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Finite Models and Finitely Many Variables
 BANACH CENTER PUBLICATIONS
, 1999
"... We consider L^k  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 rela ..."
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We consider L^k  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^k. Some open questions, formulated as finitary LöwenheimSkolem properties, are presented regarding this relationship. This is also investigated through finitary versions of an EhrenfeuchtMostowski property.
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).
Spectra of symmetric powers of graphs and the WeisfeilerLehman refinements, eprint: http://arxiv.org/pdf/0801.2322
"... The kth power of a nvertex graph X is the iterated cartesian product of X with itself. The kth symmetric power of X is the quotient graph of certain subgraph of its kth power by the natural action of the symmetric group. It is natural to ask if the spectrum of the kth power –or the spectrum of ..."
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The kth power of a nvertex graph X is the iterated cartesian product of X with itself. The kth symmetric power of X is the quotient graph of certain subgraph of its kth power by the natural action of the symmetric group. It is natural to ask if the spectrum of the kth power –or the spectrum of the kth symmetric power – is a complete graph invariant for small values of k, for example, for k = O(1) or k = O(log n). In this paper, we answer this question in the negative: we prove that if the well known 2kdimensional WeisfeilerLehman method fails to distinguish two given graphs, then their kth powers –and their kth symmetric powers – are cospectral. As it is well known, there are pairs of nonisomorphic nvertex graphs which are not distinguished by the kdim WL method, even for k = Ω(n). In particular, this shows that for each k, there are pairs of nonisomorphic nvertex graphs with cospectral kth (symmetric) powers. 1
FINITE MODELS AND FINITELY MANY VARIABLES
"... Abstract. This paper is a survey of results on finite variable logics in finite model theory. It focusses on the common underlying techniques that unite many such results. 1. Introduction. Finite variable logics have come to occupy a very important place in finite model theory. This survey examines ..."
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Abstract. This paper is a survey of results on finite variable logics in finite model theory. It focusses on the common underlying techniques that unite many such results. 1. Introduction. Finite variable logics have come to occupy a very important place in finite model theory. This survey examines a number of the results that have been established and the techniques that have been used in this connection. Taking a broad enough view allows a picture to emerge which shows that essentially the same techniques have reappeared in differing guises in entirely different contexts. The questions that motivated Poizat's work on classification seem unconnected with McColm's conjectures, which in turn bear only an incidental relationship with the question of Chandra and Harel that motivated Abiteboul and Vianu's theorem and related work on relational complexity. The fact that finite variable logics play an important role in each case supports the view that they are in some way central to the model theory of finite structures. By focussing on the common techniques, this survey aims to expose the underlying connections between a variety of investigations. It is hoped that this will help to explain the importance of finite variable logics, as well as the breadth of applicability of the ideas that have been developed. The paper does not, however, aim to be comprehensive in its coverage of the work on finite variable logics in finite model theory as several strands of this work are omitted for lack of space. Significant among these is the work on finite variable logics and counting which has been covered in the excellent work by Otto [50], and the relation of finite variable logics to modal and temporal logics for which a good starting point is the survey by Hodkinson One of the central concerns of finite model theory is to study the limits of the expressive power of logical languages on finite structures. It is in this context that questions of a model theoretic nature arise naturally with respect to finite models. A large part