We study two important extensions of first-order logic (FO) with iteration, the fixpoint and while queries. The main result of the paper concerns the open problem of the relationship between fixpoint and while: they are the same iff ptime = pspace. These and other expressibility results are obtained using a powerful normal form for while which shows that each while computation over an unordered domain can be reduced to a while computation over an ordered domain via a fixpoint query. The fixpoint query computes an equivalence relation on tuples which is a congruence with respect to the rest of the computation. The same technique is used to show that equivalence of tuples and structures with respect to FO formulas with bounded number of variables is definable in fixpoint. Generalizing fixpoint and while, we consider more powerful languages which model arbitrary computation interacting with a database using a finite set of FO queries. Such computation is modeled by a relational machine...

. Recently there was some attention on integration of description logics of the AL-family with rule-based languages for querying relational databases such as Datalog, so as to achieve the best characteristics of both kinds of formalisms in a common framework. Formal analysis on such hybrid languages has been limited to computational complexity: i.e., how much time/space it is needed to answer to a specific query? This paper carries out a different formal analysis, the one dealing with expressiveness, which gives precise characterization of the concepts definable as queries. We first analyze the applicability to hybrid languages of formal tools developed for characterizing the expressive power of relational query languages. We then present some preliminary results on the expressiveness of hybrid languages. In particular, we show that relatively simple hybrid languages are able to define all finite structures expressed by skolemized universally quantified second-order formulae with some ...

We present a formal framework for treating both incomplete information in the initial database and possible failures during an agent's execution of a course of actions.

In this paper we prove that the set of properties checkable by a Turing machine in DSPACE[n^k] is exactly equal to the set of properties describable by a uniform sequence of first-order sentences using at most k + 1 distinct variables. We prove that this is also equal to the set of properties describable using an iterative definition for a finite set of relations of arity k. This is a refinement of the theorem PSPACE = VAR[O[1]] [I82]. We suggest some directions for exploiting this result to derive trade-offs between the number of variables and the quantifier-depth in desciptive complexity. This has applications to parallel complexity.

This survey article introduces into the essential concepts and methods underlying rule-based query languages. It covers four complementary areas: declarative semantics based on adaptations of mathematical logic, operational semantics, complexity and expressive power, and optimisation of query evaluation. The treatment of these areas is foundation-oriented, the foundations having resulted from over four decades of research in the logic programming and database communities on combinations of query languages and rules. These results have later formed the basis for conceiving, improving, and implementing several Web and Semantic Web technologies, in particular query languages such as XQuery or SPARQL for querying relational, XML, and RDF data, and rule languages like the “Rule Interchange Framework (RIF) ” currently being developed in a working group of the W3C. Coverage of the article is deliberately limited to declarative languages in a classical setting: issues such as query answering in F-Logic or in description logics, or the relationship of query answering to reactive rules and events, are not addressed.

We say that a first order formula Φ distinguishes a structure M over vocabulary L from another structure M ′ over the same vocabulary if Φ is true on M but false on M ′. A formula Φ defines an L-structure M if Φ distinguishes M from any other non-isomorphic L-structure M ′. A formula Φ identifies an n-element L-structure M if Φ distinguishes M from any other non-isomorphic n-element L-structure M ′. We prove that every n-element structure M is identifiable by a formula with quantifier rank less than (1 − 1 2k)n+k2 −k+2 and at most one quantifier alternation, where k is the maximum relation arity of M. Moreover, if the automorphism group of M contains no transposition, the same result holds for definability rather than identification. The Bernays-Schönfinkel class consists of prenex formulas in which the existential quantifiers all precede the universal quantifiers. We prove that every n-element structure M is identifiable by a formula in the Bernays-Schönfinkel class with less than (1 − 1 2k2 +2)n + k quantifiers. If in this class of identifying formulas we restrict the number of universal quantifiers to k, then less than n − √ n + k2 quantifiers suffice to identify M and, as long as we keep the number of universal quantifiers bounded by a constant, at total n − O ( √ n) quantifiers are necessary. 1

The optimization of a large class of queries is explored, using a powerful normal form recently proven. The queries include the fixpoint and while queries, and an extension of while with arithmetic. The optimization method is evaluated using a probabilistic analysis. In particular, the average complexity of fixpoint and while is considered and some surprising results are obtained. They suggest that the worst-case complexity is sometimes overly pessimistic for such queries, whose average complexity is often much more reasonable than the provably rare worst case. Some computational properties of queries are also investigated. A probabilistic notion of boundedness is defined, and it is shown that all programs in the class considered are bounded almost everywhere. An effective way of using this fact is provided. 1 Introduction The complexity of query languages has traditionally been investigated using worst-case bounds. We argue that this approach provides an overly pessimistic picture o...

We explore an approach to developing Datalog parallelization strategies that aims at good expected rather than worst-case performance. To illustrate, we consider a very simple parallelization strategy that applies to all Datalog programs. We prove that this has very good expected performance under equal distribution of inputs. This is done using an extension of 0-1 laws adapted to this context. The analysis is confirmed by experimental results on randomly generated data. 1 Introduction The performance requirements of databases for advanced applications, and the increased availability of cheap parallel processing, have naturally lend great importance to the development of parallel processing techniques for databases. Much of the existing research in this direction has focused on parallelization of Datalog queries. In this paper we investigate parallel processing of Datalog from a probabilistic viewpoint. In contrast to existing work, we propose to guide the design and evaluation of para...

: This paper continues our work on infinite, recursive structures. We investigate the descriptive complexity of several logics over recursive structures, including first-order, second-order, and fixpoint logic, exhibiting connections between expressibility of a property and its computational complexity. We then address 0--1 laws, proposing a version that applies to recursive structures, and using it to prove several non-expressibility results. 0 Introduction Infinite recursive structures, with recursive graphs as a special case, have been studied quite extensively in the past. Most interesting properties of recursive graphs have been shown to be undecidable, and many are actually outside the arithmetic hierarchy; see, e.g., [AMS, Be1, Be2, BG, H, HH1]. In [HH2] we considered recursive structures to be generalizations of finite relational data bases, and investigated the class of computable queries over them, the motivation being borrowed from [CH1]. A computable query is a (partial) r...

We propose a framework for modelling and solving search problems using logic, and describe a project whose goal is to produce practically effective, general purpose tools for representing and solving search problems based on this framework. The mathematical foundation lies in the areas of finite model theory and descriptive complexity, which provide us with many classical results, as well as powerful techniques, not available to many other approaches with similar goals. We describe the mathematical foundations; explain an extension to classical logic with inductive definitions that we consider central; give a summary of complexity and expressiveness properties; describe an approach to implementing solvers based on grounding; present grounding algorithms based on an extension of the relational algebra; describe an implementation of our framework which includes use of inductive definitions, sorts and order; and give experimental results comparing the performance of our implementation with ASP solvers and another solver based on the same framework. 1.