Finite Model Theory

The main result gives a sufficient condition for a classK of finite dimensional cylindric algebras to have the property that not every epimorphism in K is surjective. In particular, not all epimorphisms are surjective in the classes CAn of n-dimensional cylindric algebras and the class of representable algebras in CAn for finite n > 1, solving Problem 10 of (28)

The Church-Turing thesis is given a provable interpretation based on the idea that a computation by an idealized human agent must be a logically definable finite mathematical object. The argument is preserved under a large variation in the expressive power of the underlying logical language, thus providing a possible explanation of why the notion of effective computability is so robust.

The paper describes properties of Yablo sequences over growing domains of finite arithmetical models and over partial models of Kripke truth theory. We show that for any partial fixed-point model and for the Strong Kleene, Weak Kleene and Supervaluation valuation schema, all Yablo sentences Y (n) are neither true nor false under these schema or equivalently: the truth-value of all Yablo sentences Y (n) in fixed-point partial models under any of the above valuation scheme, is indeterminate. Furthermore, we show that under the logic of sufficiently large finite models (logic of potential infinity) all the Yablo sentences are false in the limit. The main philosophical conclusion is that a finitist, not accepting the notion of actual infinity may adopt the sl-theory of arithmetics as an adequate explication of the concept of potential infinity and simply give an answer to the Yablo paradox that in the limit (i.e. under the sl-semantics on the FM-domain of a given arithmetical model) all the Yablo sentences are false.

The main result gives a sufficient condition for a class K of finite dimensional cylindric algebras to have the property that not every epimorphism in K is surjective. In particular, not all epimorphisms are surjective in the classes CA n of n-dimensional cylindric algebras and the class of representable algebras in CA n for finite n > 1, solving Problem 10 of [28] for finite n. By a result of Németi, this shows that the Beth-definability property fails for the finite-variable fragments of first order logic as long as the number n of variables available is > 1 and we allow models of size ≥ n + 2, but holds if we allow only models of size ≤ n + 1. We also use our results in the present paper to prove that several results in the literature concerning injective algebras and definability of polyadic operations in CA n are best possible. We raise several open problems.

In this paper we present two systems for dealing with relations, the RelView and the Rath system. After a short introduction to both systems we exhibit their usual domain of application by presenting some typical examples. Cooperation for this paper was supported by European COST Action 274 "Theory and Applications of Relational Structures as Knowledge Instruments" (TARSKI).

One of the fundamental insights of mathematical logic is that our understanding of mathematical phenomena is enriched by elevating the languages we use to describe mathematical structures to objects of explicit study. If mathematics is the science of pattern, then the media through which we discern patterns, as well as the structures in which we discern them, command our attention. It is this aspect of logic which is most prominent in model theory, "the branch of mathematical logic which deals with the relation between a formal language and its interpretations" . No wonder, then, that mathematical logic, in general, and finite model theory, the specialization of model theory to finite structures, in particular, should find manifold applications in computer science: from specifying programs to querying databases, computer science is rife with phenomena whose understanding requires close attention to the interaction between language and structure.

Building upon the known generalized-quantifier-based first-order characterization of LOGCFL, we lay the groundwork for a deeper investigation. Specifically, we examine subclasses of LOGCFL arising from varying the arity and nesting of groupoidal quantifiers. Our work extends the elaborate theory relating monoidal quantifiers to NC 1 and its subclasses. In the absence of the BIT predicate, we resolve the main issues: we show in particular that no single outermost unary groupoidal quantifier with FO can capture all the context-free languages, and we obtain the surprising result that a variant of Greibach's "hardest context-free language" is LOGCFL-complete under quantifier-free BIT-free projections. We then prove that FO with unary groupoidal quantifiers is strictly more expressive with the BIT predicate than without. Considering a particular groupoidal quantifier, we prove that first-order logic with majority of pairs is strictly more expressive than first-order with majority of individuals. As a technical tool of independent interest, we define the notion of an aperiodic nondeterministic finite automaton and prove that FO translations are precisely the mappings computed by single-valued aperiodic nondeterministic finite transducers.

A notion of generalized quantifier in computational complexity theory is explored and used to give a unified treatment of leaf language definability, oracle separations, type 2 operators, and circuits with monoidal gates. Relations to Lindstroem quantifiers are pointed out.

We study natural language constructions which are deemed to express the existence of certain kinds of similarities between partial orderings. Specifically, we give examples of natural language sentences and their plausible logical forms that express the existence of homomorphism, embedding and variations of those. Semantically, we interpret the constructions as polyadic generalized quantifiers. We examine some of the quantifiers in question with respect to their F O-definability over finite models. Since they are definable in the existential fragment of SO, we investigate their completeness in the class N P . We prove that among the quantifiers under investigation, only the homomorphism quantifier is tractable. We stress methodological importance of our results for linguistics. Finally, we discuss some interconnections between computational and evolutionary semantics.

We answer the question of computational reasons for epistemic hardness of certain class of philosophically interesting mathematical concepts. We justify the statement that mathematical knowability may be identified with algorithmic learnability. We present framework of experimental logics equivalent to the notion of learnability. Then we prove the main result. By adjoining the minimal possible set of undecidable sentences to recursive axiomatization of arithmetics and closing it under logical consequence, we obtain a non-learnable theory. This gives an explanation to the fact that undecidable arithmetical sentences are cognitively difficult. We conclude that cognitively accessible mathematical concepts are exactly within the scope of learnablity.

Evaluating a Boolean conjunctive query Q against a guarded first-order theory F is equivalent to checking whether "F and not Q" is unsatisfiable. This problem is relevant to the areas of database theory and description logic. Since Q may not be guarded, well known results about the decidability, complexity, and finite-model property of the guarded fragment do not obviously carry over to conjunctive query answering over guarded theories, and had been left open in general. By investigating finite guarded bisimilar covers of hypergraphs and relational structures, and by substantially generalising Rosati's finite chase, we prove for guarded theories F and (unions of) conjunctive queries Q that (i) Q is true in each model of F iff Q is true in each finite model of F and (ii) determining whether F implies Q is 2EXPTIME-complete. We further show the following results: (iii) the existence of polynomial-size conformal covers of arbitrary hypergraphs; (iv) a new proof of the finite model property of the clique-guarded fragment; (v) the small model property of the guarded fragment with optimal bounds; (vi) a polynomial-time solution to the canonisation problem modulo guarded bisimulation, which yields (vii) a capturing result for guarded bisimulation invariant PTIME.

First-order logic is known to have a severely limited expressive power on finite structures. As a result, several different extensions have been investigated, including fragments of secondorder logic, fixpoint logic, and the infinitary logic Uz, in which every formula has only a finite number of variables. In this paper, we study generu~ize~ quu~t~~ers in the realm of finite structures and combine them with the infinitary logic .3'& to obtain the logics 9:,(Q), where Q = {Qi: in If is a family of generalized quantifiers on finite structures. Using the logics Y:,(Q), we can express polynomial-time properties that are not definable in LYE,, such as "there is an even number of x" and "there exists at least n/2 x" (n is the size of the universe), without going to second-order logic.

We consider the notion of intuitive learnability and its relation to intuitive computability. We briefly discuss the Church's Thesis. We formulate the Learnability Thesis. Further we analyse the proof of the Church's Thesis presented by M. Mostowski. We indicate which assumptions of the Mostowski's argument implicitly include that the Church's Thesis holds. The impossibility of this kind of argument is strengthened by showing that the Learnability Thesis does not imply the Church's Thesis. Specifically, we show a natural interpretation of intuitive computability under which intuitively learnable sets are exactly algorithmically learnable but intuitively computable sets form a proper superset of recursive sets.

Motivated by computer science challenges, we suggest to extend the approach and methods of finite model theory beyond finite structures. We study definability issues and their relation to complexity on metafinite structures which typically consist of (i) a primary part, which is a finite structure, (ii) a secondary part, which is a (usually infinite) structure that can be viewed as a structured domain of numerical objects, and (iii) a set of``weight'' functions from the first part into the second. We discuss model-theoretic properties of metafinite structures, present results on descriptive complexity, and sketch some potential applications.

Evaluating a boolean conjunctive query q over a guarded first-order theory T is equivalent to checking whether (T & not q) is unsatisfiable. This problem is relevant to the areas of database theory and description logic. Since q may not be guarded, well known results about the decidability, complexity, and finite-model property of the guarded fragment do not obviously carry over to conjunctive query answering over guarded theories, and had been left open in general. By investigating finite guarded bisimilar covers of hypergraphs and relational structures, and by substantially generalising Rosati’s finite chase, we prove for guarded theories T and (unions of) conjunctive queries q that (i) T implies q iff T implies q over finite models, that is, iff q is true in each finite model of T and (ii) determining whether T implies q is 2EXPTIME-complete. We further show the following results: (iii) the existence of polynomial-size conformal covers of arbitrary hypergraphs; (iv) a new proof of the finite model property of the clique-guarded fragment; (v) the small model property of the guarded fragment with optimal bounds; (vi) a polynomial-time solution to the canonisation problem modulo guarded bisimulation, which yields (vii) a capturing result for guarded-bisimulation-invariant PTIME.

Monadic least fixed point logic MLFP is a natural logic whose expressiveness lies between that of first-order logic FO and monadic second-order logic MSO. In this paper we take a closer look at the expressive power of MLFP. Our results are 1. MLFP can describe graph properties beyond any fixed level of the monadic second-order quantifier alternation hierarchy.

Evaluating a Boolean conjunctive query Q against a guarded first-order theory F is equivalent to checking whether "F and not Q" is unsatisfiable. This problem is relevant to the areas of database theory and description logic. Since Q may not be guarded, well known results about the decidability, complexity, and finite-model property of the guarded fragment do not obviously carry over to conjunctive query answering over guarded theories, and had been left open in general. By investigating finite guarded bisimilar covers of hypergraphs and relational structures, and by substantially generalising Rosati's finite chase, we prove for guarded theories F and (unions of) conjunctive queries Q that (i) Q is true in each model of F iff Q is true in each finite model of F and (ii) determining whether F implies Q is 2EXPTIME-complete. We further show the following results: (iii) the existence of polynomial-size conformal covers of arbitrary hypergraphs; (iv) a new proof of the fin...

A set A of vertices of a graph G is called d-scattered in G if no two d-neighborhoods of (distinct) vertices of A intersect. In other words, A is d-scattered if no two distinct vertices of A have distance at most 2d. This notion was isolated in the context of finite model theory by Gurevich and recently it played a prominent role in the study of homomorphism preservation theorems for special classes of structures (such as minor closed families). This in turn led to the notions of wide, almost wide and quasiwide classes of graphs. It has been proved previously that minor closed classes and classes of graphs with locally forbidden minors are examples of such classes and thus (relativized) homomorphism preservation theorem holds for them. In this paper we show that (more general) classes with bounded expansion and (newly defined) classes with bounded local expansion and even (very general) nowhere dense classes are quasi wide. This not only strictly generalizes the previous results but it also provides new proofs and algorithms for some of the old results. It appears that bounded expansion and nowhere dense classes are perhaps a proper setting for investigation of wide-type classes as in several instances we obtain a structural characterization. This also puts classes of bounded expansion in the new context. Our motivation stems from finite dualities. As a corollary we obtain that any homomorphism closed first order definable property restricted to a bounded expansion class is a restricted duality.

In this paper, we initiate the study of Ehrenfeucht–Fraïssé 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 the Ehrenfeucht–Fraïssé problem. Given n∈ωn∈ω as a parameter, and two relational structures AA and BB from one of the classes of structures mentioned above, how efficient is it to decide if Duplicator wins the nn-round EF game Gn(A,B)Gn(A,B)? We provide algorithms for solving the Ehrenfeucht–Fraïssé 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 nn.

formula with bounded number of universal quantifiers can express the negation of a transitive closure. This implies the solution of several open problems in finite model theory: On finite structures, positive transitive closure logic is not closed under negation. More generally the hierarchy defined by interleaving negation and transitive closure operators is strict. This proves a conjecture of Immerman. We also separate the expressive power of several extensions of Datalog, giving new insight in the fine structure of stratified Datalog. (FO + TC), denoted (FO + DTC). An important fragment of transitive closure logic is (FO + pos TC), which only allows positive applications of T C operators. In fact, Immerman's original result in [15] only identified (FO + pos TC) with NLOGSPACE. At that time most people believed that NLOGSPACE would not be closed under complementation, and therefore (FO + pos TC) not be equiva lent to (FO + TC). However, the closure of of NSPACE complexity classes under complementation [16,23] implied that on ordered structures, transitive closure logic collapses to its positive fragment. It has been one of the more notable open problems in finite model theory (posed in [15]) whether this is also true in the absence of order, i.e. whether (FO + pos TC) coincides with (FO + TC) on arbitrary finite

One of the fundamental insights of mathematical logic is that our understanding of mathematical phenomena is enriched by elevating the languages we use to describe mathematical structures to objects of explicit study. If mathematics is the science of pattern, then the media through which we discern patterns, as well as the structures in which we discern them, command our attention. It is this aspect of logic which is most prominent in model theory, "the branch of mathematical logic which deals with the relation between a formal language and its interpretations" [21]. No wonder, then, that mathematical logic, in general, and finite model theory, the specialization of model theory to finite structures, in particular, should find manifold applications in computer science: from specifying programs to querying databases, computer science is rife with phenomena whose understanding requires close attention to the interaction between language and structure. As with most branches of mathematics, the growth of mathematical logic may be seen as fueled by its applications. The very birth of set theory was occasioned by Cantor's investigations in real analysis, on subjects themselves motivated by developments in nineteenth-century physics; and the study of subsets of the real line has remained the source of some of the deepest results of contemporary set theory. At the same time, model theory has matured through the development of ever deeper applications to algebra. The interplay between language and structure, characteristic of logic, may be discerned in all these developments. From the focus on definability hierarchies in descriptive set theory, to the classification of structures up to elementary equivalence in classical model theory, logic seeks order in the universe of mathematics through the medium of formal languages. As noted, finite model theory too has grown with its applications, in this instance not to analysis or algebra, but to combinatorics and computer science. Beginning with connections to automata theory, finite model theory has developed through a broader and broader range of applications to problems in graph theory, complexity theory, database theory, computer-aided verification, and artificial intelligence. And though its applications have demanded

Unions of conjunctive queries, also known as select-project-join-union queries, are the most frequently asked queries in relational database systems. These queries are definable by existential positive first-order formulas and are preserved under homomorphisms. A classical result of mathematical logic asserts that the existential positive formulas are the only first-order formulas (up to logical equivalence) that are preserved under homomorphisms on all structures, finite and infinite. After resisting resolution for a long time, it was eventually shown that, unlike other classical preservation theorems, the homomorphism-preservation theorem holds for the class of all finite structures. In this paper, we show that the homomorphism-preservation theorem holds also for several restricted classes of finite structures of interest in graph theory and database theory. Specifically, we show that this result holds for all classes of finite structures of bounded degree, all classes of finite structures of bounded treewidth, and, more generally, all classes of finite structures whose cores exclude at least one minor.

In this paper we present two systems for dealing with relations, the RelView and the Rath system. After a short introduction to both systems we exhibit their usual domain of application by presenting some typical examples. Cooperation for this paper was supported by European COST Action 274 "Theory and Applications of Relational Structures as Knowledge Instruments" (TARSKI).

The concept of a generalized quanti er of a given similarity type was de ned in Lin66]. Our main result says that on nite structures di erent similarity types give rise to di erent classes of generalized quanti ers. More exactly, for every similarity type t there is a generalized quanti er of type t which is not de nable in the extension of rst order logic by all generalized quanti ers of type smaller than t. This was proved for unary similarity types by Per Lindstr om Wes] with a counting argument. We extend his method to arbitrary similarity types.

Monadic least fixed point logic MLFP is a natural logic whose expressiveness lies between that of first-order logic FO and monadic second-order logic MSO. In this paper we take a closer look at the expressive power of MLFP. Our results are 1. MLFP can describe graph properties beyond any fixed level of the monadic second-order quantifier alternation hierarchy.

The main result gives a sufficient condition for a class K of finite dimensional cylindric algebras to have the property that not every epimorphism in K is surjective. In particular, not all epimorphisms are surjective in the classes CA n of n-dimensional cylindric algebras and the class of representable algebras in CA n for finite n > 1, solving Problem 10 of [28] for finite n. By a result of Németi, this shows that the Beth-definability property fails for the finite-variable fragments of first order logic as long as the number n of variables available is > 1 and we allow models of size ≥ n + 2, but holds if we allow only models of size ≤ n + 1. We also use our results in the present paper to prove that several results in the literature concerning injective algebras and definability of polyadic operations in CA n are best possible. We raise several open problems.

"The Guarded Negation Fragment (GNFO) is a fragment of first-order logic that contains all unions of conjunctive queries, a restricted form of negation that suffices for expressing some common uses of negation in SQL queries, and a large class of integrity constraints. At the same time, as was recently shown,
the syntax of GNFO is restrictive enough so that static analysis problems such as query containment are still decidable. This suggests that, in spite of its expressive power, GNFO queries are amenable to novel optimizations. In this paper we provide further evidence for this, establishing that GNFO queries have distinctive features with respect to rewriting. Our results include effective preservation theorems for GNFO, Craig Interpolation and Beth Definability results, and the ability to express the certain answers of queries with respect to GNFO constraints within very restricted logics."

Locality notions in logic say that the truth value of a formula can be determined locally, by looking at the isomorphism type of a small neighborhood of its free variables. Such notions have proved to be useful in many applications. They all, however, refer to isomorphism of neighborhoods, which most local logics cannot test. A stronger notion of locality says that the truth value of a formula is determined by what the logic itself can say about that small neighborhood. Since expressiveness of many logics can be characterized by games, one can also say that the truth value of a formula is determined by the type, with respect to a game, of that small neighborhood. Such game-based notions of locality can often be applied when traditional isomorphism-based locality cannot. Our goal is to study game-based notions of locality. We work with an abstract view of games that subsumes games for many logics. We look at three, progressively more complicated locality notions. The easiest requires only very mild conditions on the game and works for most logics of interest. The other notions, based on Hanf's and Gaifman's theorems, require more restrictions. We state those restrictions and give examples of logics that satisfy and fail the respective game-based notions of locality.

In [9] we introduced a new framework for asymptotic probabilities, in which a σ-additive measure is defined on the sample space of all sequencesof finite models, where the universe of, is {1,2,…,n}. In this framework we investigated the strong 0-1 law for sentences, which states that each sentence either holds ineventually almost surely or fails ineventually almost surely.In this paper we define the strong convergence law for formulas, which carries over the ideas of the strong 0-1 law to formulas with free variables, and roughly states that for each formulaϕ(x), the fraction of tuplesain, which satisfy the formulaϕ(x), almost surely has a limit asntends to infinity.We show that the infinitary logic with finitely many variables has the strong convergence law for formulas for the uniform measure, and further characterize the measures on random graphs for which the strong convergence law holds.

Gaifman's normal form theorem showed that every ÿrst-order sentence of quantiÿer rank n is equivalent to a Boolean combination of "scattered local sentences", where the local neighborhoods have radius at most 7 n−1. This bound was improved by Lifsches and Shelah to 3 × 4 n−1. We use Ehrenfeucht-Fra ssà e type games with a "shrinking horizon" to get a spectrum of normal form theorems of the Gaifman type, depending on the rate of shrinking. This spectrum includes the result of Lifsches and Shelah, with a more easily understood proof and with the bound on the radius improved to 4 n−1. We also obtain bounds for a normal form theorem of Schwentick and Barthelmann.

We introduce a new framework for classifying logics on finite structures and studying their expressive power. This framework is based on the concept of almost everywhere equivalence of logics, that is to say, two logics having the same expressive power on a class of asymptotic measure 1. More precisely, if L, L 0 are two logics and is an asymptotic measure on finite structures, then L a.e. L 0 () means that there is a class C of finite structures with (C) = 1 and such that L and L 0 define the same queries on C. We carry out a systematic investigation of a.e. with respect to the uniform measure and analyze the a.e.-equivalence classes of several logics that have been studied extensively in finite model theory. Moreover, we explore connections with descriptive complexity theory and examine the status of certain classical results of model theory in the context of this new framework.

This book gives a comprehensive overview of central themes of finite model theory–expressive power, descriptive complexity, and zero-one laws–together with selected applications relating to database theory and artificial intelligence, especially constraint databases and constraint satisfaction problems. The final chapter provides a concise modern introduction to modal logic, emphasizing the continuity in spirit and technique with finite model theory. This underlying spirit involves the use of various fragments of and ...

The Guarded Negation Fragment (GNFO) is a fragment of first-order logic that contains all unions of conjunctive queries, a restricted form of negation that suffices for expressing some common uses of negation in SQL queries, and a large class of integrity constraints. At the same time, as was recently shown, the syntax of GNFO is restrictive enough so that static analysis problems such as query containment are still decidable. This suggests that, in spite of its expressive power, GNFO queries are amenable to novel optimizations. In this paper we provide further evidence for this, establishing that GNFO queries have distinctive features with respect to rewriting. Our results include effective preservation theorems for GNFO, Craig Interpolation and Beth Definability results, and the ability to express the certain answers of queries with respect to GNFO constraints within very restricted logics.

For a reasonable sound and complete proof calculus for first-order logic consider the problem to decide, given a sentence ϕ of first-order logic and a natural number n, whether ϕ has no proof of length ≤ n. We show that there is a nondeterministic algorithm accepting this problem which, for fixed ϕ, has running time bounded by a polynomial in n if and only if there is an optimal proof system for the set TAUT of tautologies of propositional logic. This equivalence is an instance of a general result linking the complexity of so-called slicewise monotone parameterized problems with the existence of an optimal proof system for TAUT.

Inspired by Fagin’s result that NP = Σ11, we have developed a partial framework to investigate expressibility inside Σ11 so as to have a finer look into NP. The framework uses interesting combinatorics derived from second-order Ehrenfeucht-Fraisse games and the notion of game types. Some of the results that have been proven within this framework are: (1) for any k, divisibility by k is not expressible by a Σ11 sentence where (1.i) each second-order variable has arity at most 2, (1.ii) the first-order part has at most 2 first-order variables, and (1.iii) the first-order part has quantifier depth at most 3, (2) adding one more first-order variable makes the same problem expressible, and (3) inside this last logic the parameter k creates a proper hierarchy with varying the number of second-order variables. Key Words: Ehrenfeucht-Fraisse games, divisibility, expressibility, first-order, second-order Category: F.4,F.1.3

We investigate the descriptive complexity of finite abelian groups. Using Ehrenfeucht-Frassé games we find upper and lower bounds on quantifier depth, quantifier alternations, and number of variables of a first-order sentence that distinguishes two finite abelian groups. Our main results are the following. Let G1 and G2 be a pair of non-isomorphic finite abelian groups. Then there exists a positive integer m that divides one of the two groups' orders such that the following holds: (1) there exists a first-order sentence ϕ that distinguishes G1 and G2 such that ϕ is existential, has quantifier depth O(log m), and has at most 5 variables and (2) if ϕ is a sentence that distinguishes G1 and G2 then ϕ must have quantifier depth Ω(log m). These results are applied to (1) get bounds on the first-order distinguishability of dihedral groups, (2) to prove that on the class of finite groups both cyclicity and the closure of a single element are not first-order definable, and (3) give a different proof for the first-order undefinability of simplicity, nilpotency, and the normal closure of a single element on the class of finite groups (their undefinability were shown by A. Koponen and K. Luosto in an unpublished paper).