Lower bounds are established on the computational complexity of the decision problem and on the inherent lengths of proofs for two classical decidable theories of logic: the first order theory of the real numbers under addition, and Presburger arithmetic -- the first order theory of addition on the natural numbers. There is a fixed constant c > 0 such that for every (non-deterministic) decision procedure for determining the truth of sentences of real addition and for all sufficiently large n, there is a sentence of length n for which the decision procedure runs for more than 2 cn steps. In the case of Presburger arithmetic, the corresponding bound is 22cn. These bounds apply also to the minimal lengths of proofs for any complete axiomatization in which the axioms are easily recognized.

. The declarative programming paradigms used in constraint languages can lead to powerful extensions of Codd's relational data model. The development of constraint database query languages from logical database query languages has many similarities with the development of constraint logic programming from logic programming, but with the additional requirements of data efficient, set-at-a-time, and bottomup evaluation. In this overview of constraint query languages (CQLs) we first present the framework of [41]. The principal idea is that: "the k-tuple (or record) data type can be generalized by a conjunction of quantifier-free constraints over k variables". The generalization must preserve various language properties of the relational data model, e.g., the calculus/algebra equivalence, and have time complexity polynomial in the size of the data. We next present an algebra for dense order constraints that is simpler to evaluate than the calculus described in [41], and we sharpen some of...

The linear quantifier elimination algorithm for ordered fields in [15] is applicable to a wide range of...

This article studies the complexity of evaluating functional query languages for complex values such as monad algebra and the recursion-free fragment of XQuery. We show that monad algebra with equality restricted to atomic values is complete for the class TA[2O(n) , O(n)] of problems solvable in linear exponential time with a linear number of alternations. The monotone fragment of monad algebra with atomic value equality but without negation is complete for nondeterministic exponential time. For monad algebra with deep equality, we establish TA[2O(n) , O(n)] lower and exponential-space upper bounds. We also study a fragment of XQuery, Core XQuery, that seems to incorporate all the features of a query language on complex values that are traditionally deemed essential. A close connection between monad algebra on lists and Core XQuery (with “child ” as the only axis) is exhibited, and it is shown that these languages are expressively equivalent up to representation issues. We show that Core XQuery is just as hard as monad algebra w.r.t. query and combined complexity, and that it is in TC0 if the query is assumed fixed. As Core XQuery is NEXPTIME-hard, it is commonly believed that any algorithm for evaluating Core XQuery has to require exponential amounts of working memory and doubly exponential time in the worst case. We present a property of queries – the lack of a certain form of composition – that virtually all real-world XQueries have and that allows for query evaluation in singly exponential time and polynomial space. Still, we are able to show for an important special case – Core XQuery with equality testing restricted to atomic values – that the composition-free language is just as expressive as the language with composition. Thus, under widely-held complexitytheoretic assumptions, the composition-free language is an exponentially less succinct version of the language with composition.

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We study the complexity of quantifier elimination and decision in first-order theories of temporal constraints. With the exception of Ladkin, AI researchers have largely ignored this problem. We consider the first-order theories of point and interval constraints over two time structures: the integers and the rationals. We show that in all cases quantifierelimination can be done in PSPACE. We also show that the decision problem for arbitrarily quantified sentences is PSPACE-complete while for 9 k sentences it is \Sigma p k -complete. Our results must be of interest to researchers working on temporal constraints, computational complexity of logical theories, constraint databases and constraint logic programming. 1 INTRODUCTION The study of temporal constraints has recently received much attention from the AI community [All83, LM88, Lad88, VKvB89, vBC90, DMP91, KL91, Mei91, vB92, Kou92, GS93, SD93]. Much of this work draws upon concepts and techniques from the literature of general co...

. Constraint databases generalize relational databases by finitely representable infinite relations. This paper surveys the state of the art in constraint databases: known results, remaining open problems and current research directions. The paper also describes a new algebra for databases with integer order constraints and a complexity analysis of evaluating queries in this algebra. In memory of Paris C. Kanellakis 1 Introduction There is a growing interest in recent years among database researchers in constraint databases, which are a generalization of relational databases by finitely representable infinite relations. Constraint databases are parametrized by the type of constraint domains and constraint used. The good news is that for many parameters constraint databases leave intact most of the fundamental assumptions of the relational database framework proposed by Codd. In particular, 1. Constraint databases can be queried by constraint query languages that (a) have a semantics ba...

In previous work we have developed the scheme of indefinite L-constraint databases where L, the parameter, is a first-order constraint language. This scheme extends the constraint database proposal of Kanellakis, Kuper and Revesz to include indefinite (or uncertain) information in the style of Imielinski and Lipski. In this paper we study the complexity of query evaluation in an important instance of this abstract scheme: indefinite temporal constraint databases. Our results indicate that the data/combined complexity of query evaluation does not change when we move from queries in relational calculus over relational databases, to queries in relational calculus with temporal constraints over temporal constraint databases. This fact remains true even when we consider query evaluation in relational databases with indefinite information vs. query evaluation in indefinite temporal constraint databases. In the course of our work, we provide precise bounds on the complexity of decision/quanti...

An important requirement of advanced temporal applications is the ability to deal with definite, indefinite, finite and infinite temporal information. There is currently no database model which offers this functionality in a single unified framework. We argue that the combination of relational databases and temporal constraints offers a powerful framework which addresses these needs. In this thesis we develop the foundations of a theory of temporal constraint databases and indefinite temporal constraint databases. At first, we study a hierarchy of parameterized database models: ML -relational databases, L-constraint databases and indefinite L-constraint databases. The language L, the parameter, defines the constraint vocabulary and ML is the structure over which L-constraints will be interpreted. The models of temporal constraint databases and indefinite temporal constraint databases can then be studied as instances of the last two of the above parameterized models. In the course of...

The decision complexity of Presburger Arithmetic PA and its variants has received much attention in the literature. We investigate the complexity of quantifier elimination procedures for PA -- a topic that is even more relevant for applications. First we show that the the author's triply exponential upper bound is essentially tight. This fact seems to preclude practical applications. By weakening the concept of quantifier elimination slightly to bounded quantifier elimination, we show, however, that the upper and lower bound for quantifier elimination in PA can be lowered by exactly one exponential. Moreover we gain uniformity in the coefficients, a property that we prove to be impossible for complete quantifier elimination in PA. Thus we have tight upper and lower complexity bounds for elimination theory in PA and uniform PA. The results are inspired by experimental implementations of bounded quantifier elimination that have solved non-trivial application problems e.g. in parametric i...