This is a survey on the relationship between logical definability and computational complexity on finite structures. Particular emphasis is given to game-based evaluation algorithms for various logical formalisms and to logics capturing complexity classes. In addition to the

The Ordered conjecture of Kolaitis and Vardi asks whether fixed-point logic differs from first-order logic on every infinite class of finite ordered structures. In this paper, we develop the tool of bounded variable element types, and illustrate its application to this and the original conjectures of McColm, which arose from the study of inductive definability and infinitary logic on proficient classes of finite structures (those admitting an unbounded induction). In particular, for a class of finite structures, we introduce a compactness notion which yields a new proof of a ramified version of McColm's second conjecture. Furthermore, we show a connection between a model-theoretic preservation property and the Ordered Conjecture, allowing us to prove it for classes of strings (colored orderings). We also elaborate on complexity-theoretic implications of this line of research.

We investigate the expressive power of various extensions of first-order, inductive, and infinitary logic with counting quantifiers. We consider in particular a LOGSPACE extension of first-order logic, and a PTIME extension of fixpoint logic with counters. Counting is a fundamental tool of algorithms. It is essential in the case of unordered structures. Our aim is to understand the expressive power gained with a limited counting ability. We consider two problems: (i) unnested counters, and (ii) counters with no free variables. We prove a hierarchy result based on the arity of the counters under the first restriction. The proof is based on a game technique that is introduced in the paper. We also establish results on the asymptotic probabilities of sentences with counters under the second restriction. In particular, we show that first-order logic with equality of the cardinalities of relations has a 0/1 law. 1 Introduction Counting is a fundamental operation of numerous algorithms. Cou...

We consider the problem of obtaining logical characterisations of oracle complexity classes. In particular, we consider the complexity classes LOGSPACE NP and PTIME NP . For these classes, characterisations are known in terms of NP computable Lindstrom quantifiers which hold on ordered structures. We show that these characterisations are unlikely to extend to arbitrary (unordered) structures, since this would imply the collapse of certain exponential complexity hierarchies. We also observe, however, that PTIME NP can be characterised in terms of Lindstrom quantifers (not necessarily NP computable), though it remains open whether this can be done for LOGSPACE NP . 1 Introduction Since Fagin showed that existential second order logic captures the class NP [7], and Immerman and Vardi characterised PTIME in terms of least fixed point logic [14, 25], a large number of complexity classes have been given logical characterisations, and a tight correspondence has been established betwe...

Databases provide one of the main concrete scenarios for finitemodel theory within computer science. This paper presents an informal overview of database theory aimed at finite-model theorists, emphasizing the specificity of the database area. It is argued that the area of databases is a rich source of questions and vitality for finite-model theory.

We give examples of classes of rigid structures which are of unbounded rigidity but Least fixed point (Partial fixed point) logic can express all Boolean PTIME (PSPACE) queries on these classes. This shows that definability of linear order in FO+LFP although sufficient for it to capture Boolean PTIME queries, is not necessary even on the classes of rigid structures. The situation however appears very different for nonzero-ary queries. Next, we turn to the study of fixed point logics on arbitrary classes of structures. We completely characterize the recursively enumerable classes of finite structures on which PFP captures all PSPACE queries of arbitrary arities. We also state in some alternative forms several natural necessary and some sufficient conditions for PFP to capture PSPACE queries on classes of finite structures. The conditions similar to the ones proposed above work for LFP and PTIME also in some special cases but to prove the same necessary conditions in general for LFP to c...

. We investigate the complexity of the fixed-points of bounded formulas in the context of finite set theory; that is, in the context of arbitrary classes of finite structures that are equipped with a built-in BIT predicate, or equivalently, with a built-in membership relation between hereditarily finite sets (input relations are allowed). We show that the iteration of a positive bounded formula converges in polylogarithmically many steps in the cardinality of the structure. This extends a previously known much weaker result. We obtain a number of connections with the rudimentary languages and deterministic polynomial-time. Moreover, our results provide a natural characterization of the complexity class consisting of all languages computable by bounded-depth, polynomialsize circuits, and polylogarithmic-time uniformity. As a byproduct, we see that this class coincides with LH(P), the logarithmic-time hierarchy with an oracle to deterministic polynomial-time. Finally, we dis...

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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 Löwenheim-Skolem properties, are presented regarding this relationship. This is also investigated through finitary versions of an Ehrenfeucht-Mostowski property.

We give a sufficient condition for the inexpressibility of the k-th extended vectorization of a generalized quantier Q in FO(Q k), the extension of first-order logic by all k-ary quantiers. The condition is based on a model construction which, given two FO(Q 1)-equivalent models with certain additional structure, yields a pair of FO(Q k)-equivalent models. We also consider some applications of this condition to quantifiers that correspond to graph properties, such as connectivity and planarity.