Lauri Hella

Logical Hierarchies in PTIME

We consider the problem of finding a characterization for polynomial time computable queries on finite structures in terms of logical definability. It is well known that fixpoint logic provides such a characterization in the presence of a built-in linear order, but without linear order even very simple polynomial time queries involving counting are not expressible in fixpoint logic. Our approach to the problem is based on generalized quantifiers. A generalized quantifier isn-ary if it binds any number of formulas, but at mostnvariables in each formula. We prove that, for each natural numbern, there is a query on finite structures which is expressible in fixpoint logic, but not in the extension of first-order logic by any set ofn-ary quantifiers. It follows that the expressive power of fixpoint logic cannot be captured by adding finitely many quantifiers to first-order logic. Furthermore, we prove that, for each natural numbern, there is a polynomial time computable query which is not definable in any extension of fixpoint logic byn-ary quantifiers. In particular, this rules out the possibility of characterizing PTIME in terms of definability in fixpoint logic extended by a finite set of generalized quantifiers.

Definability hierarchies of generalized quantifiers

logics (back-and-forth techniques), quantifiers, in: A.I. Arruda, R. Chuaqui and Latin America (North-Holland, Amsterdam, ias logicas de primer orden con cuantificadores cardinaies, Rev. logics: the general framework, in: J. Banvise and S. Feferman, (Springer, 1985) 25-76. of Malitz quantifiers, Notre Dame J. Formal Logic 19 (1978) [9] L.A. Henkin, Some remarks on infinitely long formulas, in: Anonymous, Infinitistic Methods: Symposium on Foundations of Mathematics (Paiistwowe Wydawnictwo 1961) 167-183. te-q&ntiger equivalence, in: J.W. Addision, L.A. Henkin and A. Tarski, eds., eIs (North-Holland, Amsterdam, 1965) 407-412. The quantifier “there exist uncountably many” and some of its relatives, in: Feferman, eds., Model-Theoretic Logics (Springer, 1985) 123-176. gic with the quantifier “there exist uncountably many”, Ann. Math. Logic 1 (1970) l-93. , in: J. Barwise and S. Feferman, eds., Model-Theoretic antifiers, Notre Dame

Logics with aggregate operators

We study adding aggregate operators, such as summing up elements of a column of a relation, to logics with counting mechanisms. The primary motivation comes from database applications, where aggregate operators are present in all real life query languages. Unlike other features of query languages, aggregates are not adequately captured by the existing logical formalisms. Consequently, all previous approaches to analyzing the expressive power of aggregation were only capable of producing partial results, depending on the allowed class of aggregate and arithmetic operations.We consider a powerful counting logic, and extend it with the set of all aggregate operators. We show that the resulting logic satisfies analogs of Hanfs and Gaifmans theorems, meaning that it can only express local properties. We consider a database query language that expresses all the standard aggregates found in commercial query languages, and show how it can be translated into the aggregate logic, thereby providing a number of expressivity bounds, that do not depend on a particular class of arithmetic functions, and that subsume all those previously known. We consider a restricted aggregate logic that gives us a tighter capture of database languages, and also use it to show that some questions on expressivity of aggregation cannot be answered without resolving some deep problems in complexity theory.

Notions of locality and their logical characterizations over finite models

Many known tools for proving expressibility bounds for rst-order logic are based on one of several locality properties. In this paper we characterize the relationship between those notions of locality. We note that Gaifmans locality theorem gives rise to two notions: one deals with sentences and one with open formulae. We prove that the former implies Hanfs notion of locality, which in turn implies Gaifmans locality for open formulae. Each of these implies the bounded degree property, which is one of the easiest tools for proving expressibility bounds. These results apply beyond the rst-order case. We use them to derive expressibility bounds for rst-order logic with unary quantiiers and counting. We also characterize the notions of locality on structures of small degree.

Almost everywhere equivalence of logics in finite model theory

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′ are two logics and μ is an asymptotic measure on finite structures, then L ≡ a.e. L′ (μ) means that there is a class C of finite structures with μ(C) = 1 and such that L and L′ 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.

Logical hierarchies in PTIME

A generalized quantifier is n-ary if it binds any finite number of formulas, but at most n variables in each formula. It is proved that for each integer n, there is a property of finite models which is expressible in fixpoint logic, or even in DATALOG, but not in the extension of first-order logic by any set of n-ary quantifiers. It follows that no extension of first-order logic by a finite set of quantifiers captures all DATALOG-definable properties. Furthermore, it is proved that for each integer n, there is a LOGSPACE-computable property of finite models which is not definable in any extension of fixpoint logic by n-ary quantifiers. Hence, the expressive power of LOGSPACE, and a fortiori, that of PTIME, cannot be captured by adding to fixpoint logic any set of quantifiers of bounded arity.<<ETX>>

Implicit Definability and Infinitary Logic in Finite Model Theory

We study the relationship between the infinitary logic L ∞ω ω with finitely many variables and implicit definability in effective fragments of L ∞ω ω on finite structures. We show that fixpoint logic has strictly less expressive power than first-order implicit definability. We also establish that the separation of fixpoint logic from a certain restriction of first-order implicit definability to L ∞ω ω is equivalent to the separation of PTIME from UP ∩ co-UP. Finally, we delineate the relationship between partial fixpoint logic and implicit definability in partial fixpoint logic on finite structures.

The hierarchy theorem for generalized quantifiers

The concept of a generalized quantifier of a given similarity type was defined in [12]. Our main result says that on finite structures different similarity types give rise to different classes of generalized quantifiers. More exactly, for every similarity type t there is a generalized quantifier of type t which is not definable in the extension of first order logic by all generalized quantifiers of type smaller than t. This was proved for unary similarity types by Per Lindstr6m [17] with a counting argument. We extend his method to arbitrary similarity types. ?

Definability of Polyadic Lifts of Generalized Quantifiers

AbstractWe study generalized quantifiers on finite structures.With every function