We characterize the polynomial time computable queries as those expressible in relational calculus plus a least fixed point operator and a total ordering on the universe. We also show that even without the ordering one application of fixed point suffices to express any query expressible with several alternations of fixed point and negation. This proves that the fixed point query hierarchy suggested by Chandra and Harel collapses at the first fixed point level. It is also a general result showing that in finite model theory one application of fixed point suffices. Introduction and Summary Query languages for relational databases have received considerable attention. In 1972 Codd showed that two natural languages for queries -- one algebraic and the other a version of first order predicate calculus -- have identical powers of expressibility, [Cod72]. Query languages which are as expressive as Codd's Relational Calculus are sometimes called complete. This term is misleading however becau...

this paper a series of languages adequate for expressing exactly those properties checkable in a series of computational complexity classes. For example, we show that a property of graphs (respectively groups, binary strings, etc.) is in polynomial time if and only if it is expressible in the first order language of graphs (respectively groups, binary strings, etc.) together with a least fixed point operator. As another example, a property is in logspace if and only if it is expressible in first order logic together with a deterministic transitive closure operator. The roots of our approach to complexity theory go back to 1974 when Fagin showed that the NP properties are exactly those expressible in second order existential sentences. It follows that second order logic expresses exactly those properties which are in the polynomial time hierarchy. We show that adding suitable transitive closure operators to second order logic results in languages capturing polynomial space and exponential time, respectively. The existence of such natural languages for each important complexity class sheds a new light on complexity theory. These languages reaffirm the importance of the complexity classes as much more than machine dependent issues. Furthermore a whole new approach is suggested. Upper bounds (algorithms) can be produced by expressing the property of interest in one of our languages. Lower bounds may be demonstrated by showing that such expression is impossible.

In order to study circuit complexity classes within NC¹ in a uniform setting, we need a uniformity condition which is more restrictive than those in common use. Two such conditions, stricter than NC¹ uniformity [Ru81,Co85], have appeared in recent research: Immerman's families of circuits defined by first-order formulas [Im87a,Im87b] and a uniformity corresponding to Buss' deterministic log-time reductions [Bu87]. We show that these two notions are equivalent, leading to a natural notion of uniformity for low-level circuit complexity classes. We show that recent results on the structure of NC¹ [Ba89] still hold true in this very uniform setting. Finally, we investigate a parallel notion of uniformity, still more restrictive, based on the regular languages. Here we give characterizations of subclasses of the regular languages based on their logical expressibility, extending recent work of Straubing, Th'erien, and Thomas [STT88]. A preliminary version of this work appeared as [BIS88].

process communication have been examined from a complexity pbint of view [SP, Y]. We study a new model, in which a collection of processes eo, "'', e~:~l

this paper, we consider circuits of bounded depth in the basis f; \Phig.

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Abstract. Cook and Reckhow defined a propositional formulation of the pigeonhole principle. This paper shows that there are Frege proofs of this propositional pigeonhole principle of polynomial size. This together with a result of Haken gives another proof of Urquhart's theorem that Frege systems have an exponential speedup over resolution. We also discuss connections to provability in theories of bounded arithmetic. $1. Introduction. The motivation for this paper comes primarily from two sources. First, Cook and Reckhow [2] and Statman [7] discussed connections between lengths of proofs in propositional logic and open questions in com-putational complexity such as whether NP = co-NP. Cook and Reckhow used the propositional pigeonhole principle as an example of a family of true formulae which

Bags, i.e. sets with duplicates, are often used to implement relations in database systems. In this paper, we study the expressive power of algebras for manipulating bags. The algebra we present is a simple extension of the nested relation algebra. Our aim is to investigate how the use of bags in the language extends its expressive power, and increases its complexity. We consider two main issues, namely (i) the impact of the depth of bag nesting on the expressive power, and (ii) the complexity and the expressive power induced by the algebraic operations. We show that the bag algebra is more expressive than the nested relation algebra (at all levels of nesting), and that the difference may be subtle. We establish a hierarchy based on the structure of algebra expressions. This hierarchy is shown to be highly related to the properties of the powerset operator. Invited to a special issue of the Journal of Computer and System Sciences selected from ACM Princ. of Database Systems,...

The representation of functions as low-degree polynomials over various rings has provided many insights in the theory of small-depth circuits. We survey some of the closure properties, upper bounds, and lower bounds obtained via this approach. 1. Introduction There is a long history of using polynomials in order to prove complexity bounds. Minsky and Papert [39] used polynomials to prove early lower bounds on the order of perceptrons. Razborov [46] and Smolensky [49] used them to prove lower bounds on the size of AND-OR circuits. Other lower bounds via polynomials are due to [50, 4, 10, 51, 9, 55]. Paturi and Saks [44] discovered that rational functions could be used for lower bounds on the size of threshold circuits. Toda [53] used polynomials to prove upper bounds on the power of the polynomial hierarchy. This led to a series of upper bounds on the power of the polynomial hierarchy [54, 52], AC 0 [2, 3, 52, 19], and ACC [58, 20, 30, 37], and related classes [21, 42]. Beigel and Gi...

: We study classes of infinite but finitely representable databases based on constraints, motivated by new database applications such as geographical databases. We formally define these notions and introduce the concept of query which generalizes queries over classical relational databases. We prove that in this context the basic properties of queries (satisfiability, containment, equivalence, etc.) are nonrecursive. We investigate the theory of finitely representable models and prove that it differs strongly from both classical model theory and finite model theory. In particular, we show that most of the well known theorems of either one fail (compactness, completeness, locality, 0/1 laws, etc.). An immediate consequence is the lack of tools to consider the definability of queries in the relational calculus over finitely representable databases. We illustrate this very challenging problem through some classical examples. We then mainly concentrate on dense order databases, and exhibit...