Regular Representations of Uniform TC 0

We extend bounds on the expressive power of first-order logic over finite structures and over ordered finite structures, by generalizing to the situation where the finite structures are embedded in an infinite structure M, where M satisfies some simple combinatorial properties studied in model-theoretic stability theory. We first consider first-order logic over finite structures embedded in a stable structure, and show that it has the same generic expressive power as first-order logic on unordered finite structures. It follows from this that having the additional structure of, for example, an abelian group or an equivalence relation, does not allow one to define any new generic queries. We also consider first-order logic over finite structures living within any model M that lacks the independence property and show that its expressive power is bounded by first-order logic over finite ordered structures. This latter result gives an enormous class of structures in which the expressive power of first-order logic is sharply limited; it shows that common queries such as parity and connectivity cannot be defined for finite structures living within structures from this huge class. It also gives a pure combinatorial property of an interpreted structure M that is sufficient to extend results on first-order logic on ordered structures to first-order logic on finite structures embedded in M.

First-order expressibility of languages with neutral letters or: The Crane Beach conjecture

Computable Queries for Object Oriented Databases

Ehrenfeucht-Fraisse games have been introduced as a means of characterizing the relation of elementary equivalence between structures, or relational database instances in first order logic (FO), or equivalently Relational Calculus. In∼the usual Ehrenfeucht-Fraisse games the rules are determined by a linear ordering of a fixed lenght or, equivalently, by a special kind of tree - a chain of a fixed length -, where each point of that ordering or node of that tree corresponds to a quantification operation. Here we consider Ehrenfeucht-Fraisse games whose rules are determined by arbitrary trees such that their nodes correspond either to quantification operations (“q-nodes”) or to connective operations (“c-nodes”). By playing games on trees, we can refine the class of sentences which are considered in a given game, since a tree represents a particular class of sentences. We define and study several variations of tree games, for first and second order logic (SO). We give a sufficient condition for FO and SO equivalence restricted to formulae with up to n connectives, and hence also a sufficient condition for the non expressibility of a given query in those logics with formulae whose number of logical connectives is less than a given integer. We also give a sufficient condition to prove simultaneous lower bounds in both the number of connectives and in the quantifier types needed to express a given property in FO. If we consider only quantifier types, we get a characterization of the relation of preservation of sentences in the fragment of FO with the given set of quantifier types. We also study tree games for Σ n and Π n formulae. To illustrate the use of our games we use them to prove lower bounds in the connective size for several FO queries, like size of a database, size of a clique in a graph, size of a unary relation, transitive property in a graph, and degree of a node in a graph. Regarding SO, we prove lower bounds for quantifier rank for the parity query.Finally, we give a precise characterization of the logic whose elementary equivalence is characterized by a given tree game, as well as several equivalent characterizations of the existence of a winning strategy for Duplicator in the classical Ehrenfeucht-Fraisse game.

A relational database can be considered as a finite structure for a finite relational signature in first-order logic. This allows finite model theory to be used for investigating database theory. However, database research in the last decade has shifted its emphasis from the relational datamodel to higher-order datamodels including object oriented databases, object-relational databases, and most recently the model of semi-structured data and XML. A common characteristic of these new datamodels of interest is their richness in structure. In order to find a suitable generalisation of database theory that can cope with this structural richness a suitable logical ground is needed. In this paper local set theory, which can be considered as a version of higher-order intuitionistic logic, is investigated for this purpose. The central notions of database schema and instance for a slighly restricted class of higher-order datamodels are re-defined in local languages using a finite set interpretation. Terms of “set type” define simple queries, which provide the analogue of relational calculus queries in the relational model. Then the work on computable queries can be generalised as well. An analogue of the reflective relational machine model called LQ machine is defined and proven to be complete, i.e., able to express exactly the computable queries over local languages. Finally, fixed-point operators are added to simple queries. However, adding fixed-points will not increase expressiveness.

A language L over an alphabet A is said to have a neutral letter if there is a letter e/spl isin/A such that inserting or deleting e's from any word in A* does not change its membership (or non-membership) in L. The presence of a neutral letter affects the definability of a language in first-order logic. It was conjectured that it renders all numerical predicates apart from the order predicate useless, i.e., that if a language L with a neutral letter is not definable in first-order logic with linear order then it is not definable in first-order. Logic with any set /spl Nscr/ of numerical predicates. We investigate this conjecture in detail, showing that it fails already for /spl Nscr/={+, *}, or possibly stronger for any set /spl Nscr/ that allows counting up to the m times iterated logarithm, 1g/sup (m)/, for any constant m. On the positive side, we prove the conjecture for the case of all monadic numerical predicates, for /spl Nscr/={+}, for the fragment BC(/spl Sigma/) of first-order logic, and for binary alphabets.

This paper considers the logic FOcard, i.e., first-order logic with cardinality predicates that can specify the size of a structure modulo some number. We study the expressive power of FOcard on the class of languages of ranked, finite, labelled trees with successor relations. Our first main result characterises the class of FOcard-definable tree languages in terms of algebraic closure properties of the tree languages. As it can be eectively checked whether the language of a given tree automaton satisfies these closure properties, we obtain a decidable characterisation of the class of regular tree languages definable in FOcard. Our second main result considers first-order logic with unary relations, successor relations, and two additional designated symbols < and + that must be interpreted as a linear order and its associated addition. Such a formula is called addition-invariant if, for each fixed interpretation of the unary relations and successor relations, its result is independent of the particular interpretation of < and +. We show that the FOcard-definable tree languages are exactly the regular tree languages definable in addition-invariant first-order logic. Our proof techniques involve tools from algebraic automata theory, reasoning with locality arguments, and the use of logical interpretations. We combine and extend methods developed by Benedikt and Segoufin (ACM ToCL, 2009) and Schweikardt and Segoufin (LICS, 2010).

Model theory is concerned mainly, although not exclusively, with infinite structures. In recent years, finite structures have risen to greater prominence, both within the context of mainstream model theory, e.g., in work of Lachlan, Cherlin, Hrushovski, and others, and with the advent of finite model theory, which incorporates elements of classical model theory, combinatorics, and complexity theory. The purpose of this survey is to provide an overview of what might be called the model theory of finite structures. Some topics in finite model theory have strong connections to theoretical computer science, especially descriptive complexity theory (see [26, 46]). In fact, it has been suggested that finite model theory really is, or should be, logic for computer science. These connections with computer science will, however, not be treated here.It is well-known that many classical results of ‘infinite model theory’ fail over the class of finite structures, including the compactness and completeness theorems, as well as many preservation and interpolation theorems (see [35, 26]). The failure of compactness in the finite, in particular, means that the standard proofs of many theorems are no longer valid in this context. At present, there is no known example of a classical theorem that remains true over finite structures, yet must be proved by substantially different methods. It is generally concluded that first-order logic is ‘badly behaved’ over finite structures.From the perspective of expressive power, first-order logic also behaves badly: it is both too weak and too strong. Too weak because many natural properties, such as the size of a structure being even or a graph being connected, cannot be defined by a single sentence. Too strong, because every class of finite structures with a finite signature can be defined by an infinite set of sentences. Even worse, every finite structure is defined up to isomorphism by a single sentence. In fact, it is perhaps because of this last point more than anything else that model theorists have not been very interested in finite structures. Modern model theory is concerned largely with complete first-order theories, which are completely trivial here.

Model theory is concerned mainly, although not exclusively, with infinite structures. In recent years, finite structures have risen to greater prominence, both within the context of mainstream model theory, e.g., in work of Lachlan, Cherlin, Hrushovski, and others, and with the advent of finite model theory, which incorporates elements of classical model theory, combinatorics, and complexity theory. The purpose of this survey is to provide an overview of what might be called the model theory of finite structures. Some topics in finite model theory have strong connections to theoretical computer science, especially descriptive complexity theory (see [26, 46]). In fact, it has been suggested that finite model theory really is, or should be, logic for computer science. These connections with computer science will, however, not be treated here.It is well-known that many classical results of ‘infinite model theory’ fail over the class of finite structures, including the compactness and completeness theorems, as well as many preservation and interpolation theorems (see [35, 26]). The failure of compactness in the finite, in particular, means that the standard proofs of many theorems are no longer valid in this context. At present, there is no known example of a classical theorem that remains true over finite structures, yet must be proved by substantially different methods. It is generally concluded that first-order logic is ‘badly behaved’ over finite structures.From the perspective of expressive power, first-order logic also behaves badly: it is both too weak and too strong. Too weak because many natural properties, such as the size of a structure being even or a graph being connected, cannot be defined by a single sentence. Too strong, because every class of finite structures with a finite signature can be defined by an infinite set of sentences. Even worse, every finite structure is defined up to isomorphism by a single sentence. In fact, it is perhaps because of this last point more than anything else that model theorists have not been very interested in finite structures. Modern model theory is concerned largely with complete first-order theories, which are completely trivial here.

In recent years several resesmch articles have been devoted to the study of inductive queries on finite relational structures, including the papers by Aho and Ullman [1979], Chandra and Harel [1982], Harel and Kozen [1984]. The inductive queries are the ones expressible in the language obtained from first-order logic by adding the least fixed point operator for positive formulas. These queries are always computable in I~IME and can capture interesting properties, such as connectivity and acyclicity, which in general are not expressible in first-order logic. At about the same time a different direction of research focused on the asymptotic probabilities of first-order properties on finite relational structures. Fagin [1976] established both a labeled and an unlabeled O-1 law for first-order sentences on the class of all finite graphs. Compton [1980, 1984a] investigated asymptotic probabilities of first-order sentences on restricted classes of finite structures which arise naturally in eombinatorics, such as equivalence relations, permutations, various classes of forests, and others. After this, in Comptou [198~b], he showed that in many of these classes a O-1 law holds for first-order queries if and only if a O-1 law holds for queries expressible in monadic second order logic. The recent paper by Blass, Gurevich and Kozen [1984] brings together the two areas of research mentioned above. It is shown there that both labeled and unlabeled O-1 laws hold for the inductive queries on the class of all finite graphs. Moreover, the problem of deciding the asymptotic probability of an inductive query on the class of all finite graphs is EX~TIME complete. The proofs of these results depend on the fact that the first-order almost sure theory of finite graphs is u-categorical, that is to say it has enly one countable model up to isomorphism. Our aim in this paper is to investigate inductive queries and their probabilities on certain classes of finite relational structures for which the firstorder almost sure theory is not u-categorical. More specifically, in §3 we show:

Queries and computation on the web

We introduce a new logic for finite first-order structures with a linear odering. We study its expressive power. In particular we show that it is strictly stronger than first-order logic on finite structures. We close with a list of open problems.KeywordsFunction SymbolExpressive PowerIterate LogicPredicate SymbolConstant SymbolThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Finite-model theory - a personal perspective

Logical Hierarchies in PTIME

There are properties of finite structures that are expressible with the use of Hilbert's ∈-operator in a manner that does not depend on the actual interpretation for ∈-terms. but not expressible in plain first-order. This observation strengthens a corresponding result of Gurevich, concerning the invariant use of an auxiliary ordering in first-order logic over finite structures. The present result also implies that certain non-deterministic choice constructs, which have been considered in database theory, properly enhance the expressive power of first-order logic even as far as deterministic queries are concerned, thereby answering a question raised by Abiteboul and Vianu.