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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 ...
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).
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).
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. ..."
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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.
Logic and Complexity without Order
, 1997
"... Introduction Descriptive Complexity can be thought of as differing from the more common view of computational complexity (i.e. measuring resource bounds on a machine) in two important respects: 1. We measure the complexity of a collection of structures as opposed to a collection of strings. This ca ..."
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Introduction Descriptive Complexity can be thought of as differing from the more common view of computational complexity (i.e. measuring resource bounds on a machine) in two important respects: 1. We measure the complexity of a collection of structures as opposed to a collection of strings. This can be seen as a generalisation, in that strings are a special case. It is also a useful generalisation in that the problems we are concerned with (graph problems, for instance) are naturally thought of as classes of structures, and must be encoded into strings in order to fit the mould of machine computations. 2. We measure the complexity of describing the collection as opposed to the complexity of computing it. So, the resources that are measured are logical resources (number and kind of quantifier, number of variables, etc.) as opposed to space, time and so on. Of course, the interest in descriptive complexity stems in large part from the fact that there is a close co
On Complexity of EhrenfeuchtFraïssé Games
"... Abstract. In this paper we initiate the study of EhrenfeuchtFraïssé games for some standard finite structures. Examples of such standard structures are equivalence relations, trees, unary relation structures, Boolean algebras, and some of their natural expansions. The paper concerns the following q ..."
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Abstract. In this paper we initiate the study of EhrenfeuchtFraïssé games for some standard finite structures. Examples of such standard structures are equivalence relations, trees, unary relation structures, Boolean algebras, and some of their natural expansions. The paper concerns the following question that we call EhrenfeuchtFraïssé problem. Given n ∈ ω as a parameter, two relational structures A and B from one of the classes of structures mentioned above, how efficient is it to decide if Duplicator wins the nround EF game Gn(A, B)? We provide algorithms for solving the EhrenfeuchtFraïssé problem for the mentioned classes of structures. The running times of all the algorithms are bounded by constants. We obtain the values of these constants as functions of n. 1
Languages, Theory
"... Given a document D in the form of an unordered labeled tree, we study the expressibility on D of various fragments of XPath, the core navigational language on XML documents. We give characterizations, in terms of the structure of D, for when a binary relation on its nodes is definable by an XPath ex ..."
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Given a document D in the form of an unordered labeled tree, we study the expressibility on D of various fragments of XPath, the core navigational language on XML documents. We give characterizations, in terms of the structure of D, for when a binary relation on its nodes is definable by an XPath expression in these fragments. Since each pair of nodes in such a relation represents a unique path in D, our results therefore capture the sets of paths in D definable in XPath. We refer to this perspective on the semantics of XPath as the “global view. ” In contrast with this global view, there is also a “local view ” where one is interested in the nodes to which one can navigate starting from a particular node in the document. In this view, we characterize when a set of nodes in D can be defined as the result of applying an XPath expression to a given node of D. All these definability results, both in the global and the local view, are obtained by using a robust twostep methodology, which consists of first characterizing when two nodes cannot be distinguished by an expression in the respective fragments of XPath, and then bootstrapping these characterizations to the desired results.