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GraphLog: a Visual Formalism for Real Life Recursion
 In Proceedings of the Ninth ACM SIGACTSIGMOD Symposium on Principles of Database Systems
, 1990
"... We present a query language called GraphLog, based on a graph representation of both data and queries. Queries are graph patterns. Edges in queries represent edges or paths in the database. Regular expressions are used to qualify these paths. We characterize the expressive power of the language a ..."
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Cited by 167 (18 self)
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We present a query language called GraphLog, based on a graph representation of both data and queries. Queries are graph patterns. Edges in queries represent edges or paths in the database. Regular expressions are used to qualify these paths. We characterize the expressive power of the language and show that it is equivalent to stratified linear Datalog, first order logic with transitive closure, and nondeterministic logarithmic space (assuming ordering on the domain). The fact that the latter three classes coincide was not previously known. We show how GraphLog can be extended to incorporate aggregates and path summarization, and describe briefly our current prototype implementation. 1 Introduction The literature on theoretical and computational aspects of deductive databases, and the additional power they provide in defining and querying data, has grown rapidly in recent years. Much less work has gone into the design of languages and interfaces that make this additional pow...
Counting Quantifiers, Successor Relations, and Logarithmic Space
 Journal of Computer and System Sciences
"... Given a successor relation S (i.e., a directed line graph), and given two distinguished points s and t, the problem ORD is to determine whether s precedes t in the unique ordering defined by S. We show that ORD is Lcomplete (via quantifierfree projections). We then show that firstorder logic with ..."
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Cited by 51 (2 self)
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Given a successor relation S (i.e., a directed line graph), and given two distinguished points s and t, the problem ORD is to determine whether s precedes t in the unique ordering defined by S. We show that ORD is Lcomplete (via quantifierfree projections). We then show that firstorder logic with counting quantifiers, a logic that captures TC 0 ([BIS90]) over structures with a builtin totalordering, can not express ORD. Our original proof of this in the conference version of this paper ([Ete95]) employed an EhrenfeuchtFraiss'e Game for firstorder logic with counting ([IL90]). Here we show how the result follows from a more general one obtained independently by Nurmonen, [Nur96]. We then show that an appropriately modified version of the EF game is "complete" for the logic with counting in the sense that it provides a necessary and sufficient condition for expressibility in the logic. We observe that the Lcomplete problem ORD is essentially sparse if we ignore reorderings of v...
Predicative Recursion and Computational Complexity
, 1992
"... The purpose of this thesis is to give a "foundational" characterization of some common complexity classes. Such a characterization is distinguished by the fact that no explicit resource bounds are used. For example, we characterize the polynomial time computable functions without making any direct r ..."
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Cited by 45 (3 self)
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The purpose of this thesis is to give a "foundational" characterization of some common complexity classes. Such a characterization is distinguished by the fact that no explicit resource bounds are used. For example, we characterize the polynomial time computable functions without making any direct reference to polynomials, time, or even computation. Complexity classes characterized in this way include polynomial time, the functional polytime hierarchy, the logspace decidable problems, and NC. After developing these "resource free" definitions, we apply them to redeveloping the feasible logical system of Cook and Urquhart, and show how this firstorder system relates to the secondorder system of Leivant. The connection is an interesting one since the systems were defined independently and have what appear to be very different rules for the principle of induction. Furthermore it is interesting to see, albeit in a very specific context, how to retract a second order statement, ("inducti...
Feasible Computation through Model Theory
, 1993
"... The computational complexity of a problem is usually defined in terms of the resources required on some machine model of computation. An alternative view looks at the complexity of describing the problem (seen as a collection of relational structures) in a logic, measuring logical resources such as ..."
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Cited by 36 (7 self)
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The computational complexity of a problem is usually defined in terms of the resources required on some machine model of computation. An alternative view looks at the complexity of describing the problem (seen as a collection of relational structures) in a logic, measuring logical resources such as the number of variables, quantifiers, operators, etc. A close correspondence has been observed between these two, with many natural logics corresponding exactly to independently defined complexity classes. For the complexity classes that are generally identified with feasible computation, such characterizations require the presence of a linear order on the domain of every structure, in which case the class PTIME is characterized by an extension of firstorder logic by means of an inductive operator. No logical characterization of feasible computation is known for unordered structures. We approach this question from two directions. On the one hand, we seek to accurately characterize the expre...
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 ..."
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Cited by 36 (5 self)
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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...
A Fixpoint Approach to SecondOrder Quantifier Elimination with Applications to Correspondence Theory
, 1995
"... This paper is about automated techniques for (modal logic) correspondence theory. The theory we deal with concerns the problem of finding fixpoint characterizations of modal axiom schemata. Given a modal schema and a semantics based method of translating modal formulae into classical ones, we try to ..."
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Cited by 27 (7 self)
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This paper is about automated techniques for (modal logic) correspondence theory. The theory we deal with concerns the problem of finding fixpoint characterizations of modal axiom schemata. Given a modal schema and a semantics based method of translating modal formulae into classical ones, we try to derive automatically a fixpoint formula characterizing precisely the class of frames validating this schema. The technique we consider can, in many cases, be easily applied without any computer support. Although we mainly concentrate on Kripke semantics, our fixpoint approach is much more general, as it is based on the elimination of secondorder quantifiers from formulae. Thus it can be applied in secondorder theorem proving as well. We show some application examples for the method which may serve as new, automated proofs of the respective correspondences.
Descriptive Complexity Theory over the Real Numbers
 LECTURES IN APPLIED MATHEMATICS
, 1996
"... We present a logical approach to complexity over the real numbers with respect to the model of Blum, Shub and Smale. The logics under consideration are interpreted over a special class of twosorted structures, called Rstructures: They consist of a finite structure together with the ordered field ..."
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Cited by 24 (8 self)
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We present a logical approach to complexity over the real numbers with respect to the model of Blum, Shub and Smale. The logics under consideration are interpreted over a special class of twosorted structures, called Rstructures: They consist of a finite structure together with the ordered field of reals and a finite set of functions from the finite structure into R. They are a special case of the metafinite structures introduced recently by Grädel and Gurevich. We argue that Rstructures provide the right class of structures to develop a descriptive complexity theory over R. We substantiate this claim by a number of results that relate logical definability on Rstructures with complexity of computations of BSSmachines.
On Winning Ehrenfeucht Games and Monadic NP
 Annals of Pure and Applied Logic
, 1996
"... Inexpressibility results in Finite Model Theory are often proved by showing that Duplicator, one of the two players of an Ehrenfeucht game, has a winning strategy on certain structures. In this article a new method is introduced that allows, under certain conditions, the extension of a winning strat ..."
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Cited by 21 (3 self)
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Inexpressibility results in Finite Model Theory are often proved by showing that Duplicator, one of the two players of an Ehrenfeucht game, has a winning strategy on certain structures. In this article a new method is introduced that allows, under certain conditions, the extension of a winning strategy of Duplicator on some small parts of two finite structures to a global winning strategy. As applications of this technique it is shown that (*) Graph Connectivity is not expressible in existential monadic secondorder logic (MonNP), even in the presence of a builtin linear order, (*) Graph Connectivity is not expressible in MonNP even in the presence of arbitrary builtin relations of degree n^o(1), and (*) the presence of a builtin linear order gives MonNP more expressive power than the presence of a builtin successor relation.
The Closure of Monadic NP
 Journal of Computer and System Sciences
, 1997
"... It is a wellknown result of Fagin that the complexity class NP coincides with the class of ..."
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Cited by 21 (0 self)
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It is a wellknown result of Fagin that the complexity class NP coincides with the class of