Results 1  10
of
10
Computing With FirstOrder Logic
, 1995
"... We study two important extensions of firstorder 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 obtaine ..."
Abstract

Cited by 55 (13 self)
 Add to MetaCart
(Show Context)
We study two important extensions of firstorder 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...
Topological Queries in Spatial Databases
 In Journal of Computer and System Sciences 58(1
, 1999
"... All intext references underlined in blue are linked to publications on ResearchGate, letting you access and read them immediately. ..."
Abstract

Cited by 48 (4 self)
 Add to MetaCart
(Show Context)
All intext references underlined in blue are linked to publications on ResearchGate, letting you access and read them immediately.
Fixpoint Logics, Relational Machines, and Computational Complexity
 In Structure and Complexity
, 1993
"... We establish a general connection between fixpoint logic and complexity. On one side, we have fixpoint logic, parameterized by the choices of 1storder operators (inflationary or noninflationary) and iteration constructs (deterministic, nondeterministic, or alternating). On the other side, we have t ..."
Abstract

Cited by 40 (5 self)
 Add to MetaCart
(Show Context)
We establish a general connection between fixpoint logic and complexity. On one side, we have fixpoint logic, parameterized by the choices of 1storder operators (inflationary or noninflationary) and iteration constructs (deterministic, nondeterministic, or alternating). On the other side, we have the complexity classes between P and EXPTIME. Our parameterized fixpoint logics capture the complexity classes P, NP, PSPACE, and EXPTIME, but equality is achieved only over ordered structures. There is, however, an inherent mismatch between complexity and logic  while computational devices work on encodings of problems, logic is applied directly to the underlying mathematical structures. To overcome this mismatch, we develop a theory of relational complexity, which bridges tha gap between standard complexity and fixpoint logic. On one hand, we show that questions about containments among standard complexity classes can be translated to questions about containments among relational complex...
Queries Are Easier Than You Thought (probably)
, 1992
"... 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 compl ..."
Abstract

Cited by 13 (6 self)
 Add to MetaCart
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 worstcase 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 worstcase bounds. We argue that this approach provides an overly pessimistic picture o...
Computing on Structures
"... this paper various devices operating directly on structures, without encoding. The motivation and benefits for doing this are manyfold. On a fundamental level, encodings of structures seem to be a technical device rather than an intrinsic feature. This point has already been made by several mathemat ..."
Abstract

Cited by 3 (1 self)
 Add to MetaCart
this paper various devices operating directly on structures, without encoding. The motivation and benefits for doing this are manyfold. On a fundamental level, encodings of structures seem to be a technical device rather than an intrinsic feature. This point has already been made by several mathematicians such as Tarski [Tar86], and Harvey Friedman [Fri71] (see Section 5). It has come up more recently in the context of databases, where devices computing on structures model more acurately database computation carried out against an abstract interface hiding the internal representation of data. Thus, the primary benefit of studying devices and languages computing on structures is that they clarify issues which are obscured in classical devices such as Turing machines. For example, they yield new notions of complexity, quite different from classical computational complexity. They reflect more acurately the actual complexity of computation, which, like database computation, cannot take advantage of encodings of structures. An example is provided by the query even on a set
Finite Models and Finitely Many Variables
 BANACH CENTER PUBLICATIONS
, 1999
"... We consider L^k  first order logic restricted to k variables, and interpreted in finite structures. The study of classes of finite structures axiomatisable with finitely many variables has assumed importance through connections with computational complexity. In particular, we investigate the rela ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
We consider L^k  first order logic restricted to k variables, and interpreted in finite structures. The study of classes of finite structures axiomatisable with finitely many variables has assumed importance through connections with computational complexity. In particular, we investigate the relationship between the size of a finite structure and the number of distinct types it realizes, with respect to L^k. Some open questions, formulated as finitary LöwenheimSkolem properties, are presented regarding this relationship. This is also investigated through finitary versions of an EhrenfeuchtMostowski property.
Computational Model Theory: An Overview
 LOGIC JOURNAL OF THE IGPL
, 1998
"... The computational complexity of a problem is the amount of resources, such as time or space, required by a machine that solves the problem. The descriptive complexity of problems is the complexity of describing problems in some logical formalism over finite structures. One of the exciting developmen ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
The computational complexity of a problem is the amount of resources, such as time or space, required by a machine that solves the problem. The descriptive complexity of problems is the complexity of describing problems in some logical formalism over finite structures. One of the exciting developments in complexity theory is the discovery of a very intimate connection between computational and descriptive complexity. It is this connection between complexity theory and finitemodel theory that we term computational model theory. In this overview paper we o#er one perspective on computational model theory. Two important observationsunderly our perspective: (1) while computationaldevices work on encodingsof problems, logic is applied directly to the underlying mathematical structures, and this "mismatch" complicates the relationship between logic and complexity significantly, and (2) firstorder logic has severely limited expressive power on finite structures, and one way to increase the...
FINITE MODELS AND FINITELY MANY VARIABLES
"... Abstract. This paper is a survey of results on finite variable logics in finite model theory. It focusses on the common underlying techniques that unite many such results. 1. Introduction. Finite variable logics have come to occupy a very important place in finite model theory. This survey examines ..."
Abstract
 Add to MetaCart
(Show Context)
Abstract. This paper is a survey of results on finite variable logics in finite model theory. It focusses on the common underlying techniques that unite many such results. 1. Introduction. Finite variable logics have come to occupy a very important place in finite model theory. This survey examines a number of the results that have been established and the techniques that have been used in this connection. Taking a broad enough view allows a picture to emerge which shows that essentially the same techniques have reappeared in differing guises in entirely different contexts. The questions that motivated Poizat's work on classification seem unconnected with McColm's conjectures, which in turn bear only an incidental relationship with the question of Chandra and Harel that motivated Abiteboul and Vianu's theorem and related work on relational complexity. The fact that finite variable logics play an important role in each case supports the view that they are in some way central to the model theory of finite structures. By focussing on the common techniques, this survey aims to expose the underlying connections between a variety of investigations. It is hoped that this will help to explain the importance of finite variable logics, as well as the breadth of applicability of the ideas that have been developed. The paper does not, however, aim to be comprehensive in its coverage of the work on finite variable logics in finite model theory as several strands of this work are omitted for lack of space. Significant among these is the work on finite variable logics and counting which has been covered in the excellent work by Otto [50], and the relation of finite variable logics to modal and temporal logics for which a good starting point is the survey by Hodkinson One of the central concerns of finite model theory is to study the limits of the expressive power of logical languages on finite structures. It is in this context that questions of a model theoretic nature arise naturally with respect to finite models. A large part
On the Computation of Approximations of Database Queries
, 2004
"... Reflective Relational Machines were introduced by S. Abiteboul, C. Papadimitriou and V. Vianu in 1994, as variations of Turing machines which are suitable for the computation of queries to relational databases. The machines are equipped with a relational store which can be accessed by means of dynam ..."
Abstract
 Add to MetaCart
Reflective Relational Machines were introduced by S. Abiteboul, C. Papadimitriou and V. Vianu in 1994, as variations of Turing machines which are suitable for the computation of queries to relational databases. The machines are equipped with a relational store which can be accessed by means of dynamically built first order logic queries. We initiate a study on approximations of computable queries, defining, for every natural k, the kapproximation of an arbitrary computable query q. This, in turn, motivates us to define a new variation of Reflective Relational Machines by considering two di#erent logics to express dynamic queries: one for queries and a possibly di#erent one for updates to the relational store. We prove several results relating kapproximations of queries with the new machines, and also with classes of queries defined in terms of preservation of equality of FO^k theories. Finally, we summarize a few open problems related to our work.