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22
Complexity and Expressive Power of Logic Programming
, 1997
"... This paper surveys various complexity results on different forms of logic programming. The main focus is on decidable forms of logic programming, in particular, propositional logic programming and datalog, but we also mention general logic programming with function symbols. Next to classical results ..."
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Cited by 299 (56 self)
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This paper surveys various complexity results on different forms of logic programming. The main focus is on decidable forms of logic programming, in particular, propositional logic programming and datalog, but we also mention general logic programming with function symbols. Next to classical results on plain logic programming (pure Horn clause programs), more recent results on various important extensions of logic programming are surveyed. These include logic programming with different forms of negation, disjunctive logic programming, logic programming with equality, and constraint logic programming. The complexity of the unification problem is also addressed.
Deconstructing Shostak
, 2002
"... Decision procedures for equality in a combination of theories are at the core of a number of verification systems. Shostak's decision procedure for equality in the combination of solvable and canonizable theories has been around for nearly two decades. Variations of this decision procedure have ..."
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Cited by 35 (3 self)
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Decision procedures for equality in a combination of theories are at the core of a number of verification systems. Shostak's decision procedure for equality in the combination of solvable and canonizable theories has been around for nearly two decades. Variations of this decision procedure have been implemented in a number of systems including STP, Ehdm, PVS, STeP, and SVC. The algorithm is quite subtle and a correctness argument for it has remained elusive. Shostak's algorithm and all previously published variants of it yield incomplete decision procedures. We describe a variant of Shostak's algorithm along with proofs of termination, soundness, and completeness.
An Eunification algorithm for analyzing protocols that use modular exponentiation
, 2003
"... Modular multiplication and exponentiation are common operations in modern cryptography. Uni cation problems with respect to some equational theories that these operations satisfy are investigated. Two dierent but related equational theories are analyzed. A uni cation algorithm is given for one of ..."
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Cited by 20 (0 self)
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Modular multiplication and exponentiation are common operations in modern cryptography. Uni cation problems with respect to some equational theories that these operations satisfy are investigated. Two dierent but related equational theories are analyzed. A uni cation algorithm is given for one of the theories which relies on solving syzygies over multivariate integral polynomials with noncommuting indeterminates. For the other theory, in which the distributivity property of exponentiation over multiplication is assumed, the uni ability problem is shown to be undecidable by adapting a construction developed by one of the authors to reduce Hilbert's 10th problem to the solvability problem for linear equations over semirings. A new algorithm for computing strong Grobner bases of right ideals over the polynomial semiring Z<X 1 ; : : : ; Xn> is proposed; unlike earlier algorithms proposed by Baader as well as by Madlener and Reinert which work only for right admissible term orderings with the boundedness property, this algorithm works for any right admissible term ordering. The algorithms for some of these uni cation problems are expected to be integrated into Research supported in part by the NSF grant nos. CCR0098114 and CDA9503064, the ONR grant no. N000140110429, and a grant from the Computer Science Research Institute at Sandia National Labs.
From Set to Hyperset Unification
, 1999
"... In this paper we show how to extend a set unification algorithm  i.e., an extended unification algorithm incorporating the axioms of a simple theory of sets  to hyperset unification, that is to sets in which, roughly speaking, membership can form cycles. This is obtained by enlarging the domain ..."
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Cited by 9 (8 self)
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In this paper we show how to extend a set unification algorithm  i.e., an extended unification algorithm incorporating the axioms of a simple theory of sets  to hyperset unification, that is to sets in which, roughly speaking, membership can form cycles. This is obtained by enlarging the domain from that of terms (hence, trees) to that of graphs involving free as well as interpreted function symbols (namely, the set element insertion and the empty set), which can be regarded as a convenient denotation of hypersets. We present a hyperset unification algorithm which (nondeterministically) computes, for each given unification problem, a finite collection of systems of equations in solvable form whose solutions represent a complete set of solutions for the given unification problem. The crucial issue of termination of the algorithm is addressed and solved by the addition of simple nonmembership constraints. Finally, the hyperset unification problem dealt with is proved to be NPcomp...
Set Unification
, 2001
"... The goal of this paper is to provide a uniform overview of the unification problem in algebras capable of describing sets. The problem has been tackled, directly and indirectly, by many researchers and it can find important applications in various research areas  e.g., deductive databases, theorem ..."
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Cited by 8 (4 self)
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The goal of this paper is to provide a uniform overview of the unification problem in algebras capable of describing sets. The problem has been tackled, directly and indirectly, by many researchers and it can find important applications in various research areas  e.g., deductive databases, theorem proving, static analysis, rapid software prototyping. The problem has been explored in depth, but the various solutions proposed are spread across a large literature, and some of the approaches have been ignored and/or rediscovered by different researchers. In this
On the Implementation of a RuleBased Programming System and Some of its Applications
 Proc. of the 4th Int. Workshop on the Implementation of Logics (WIL'03
, 2003
"... We describe a rulebased programming system where rules specify nondeterministic computations. The system is called FunLog and has constructs for defining elementary rules, and to build up complex rules from simpler ones via operations akin to the standard operations from abstract rewriting. The ..."
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Cited by 8 (6 self)
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We describe a rulebased programming system where rules specify nondeterministic computations. The system is called FunLog and has constructs for defining elementary rules, and to build up complex rules from simpler ones via operations akin to the standard operations from abstract rewriting. The system has been implemented in Mathematica and is, in particular, useful to program procedures which can be encoded as sequences of rule applications which follow a certain reduction strategy. In particular, the procedures for unification with sequence variables in free, flat, and restricted flat theories can be specified via a set of inference rules which should be applied in accordance with a certain strategy. We illustrate how these unification procedures can be expressed in our framework.
Optimisation Techniques for Combining Constraint Solvers
 IN MAARTEN DE RIJKE AND
, 1998
"... In recent years, techniques that had been developed for the combination of unification algorithms for equational theories were extended to combining constraint solvers. These techniques inherited an old deficit that was already present in the combination of equational theories which makes them rathe ..."
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Cited by 5 (2 self)
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In recent years, techniques that had been developed for the combination of unification algorithms for equational theories were extended to combining constraint solvers. These techniques inherited an old deficit that was already present in the combination of equational theories which makes them rather unsuitable for practical use: The underlying combination algorithms are highly nondeterministic. This paper is concerned with the practical problem of how to optimise the combination method of Baader and Schulz. We present an optimisation method, called the deductive method, which uses specific algorithms for the components to reach certain decisions deterministically. We also give a strategy how to select an order of nondeterministic decisions. Run time tests of our implementation indicate that the optimised combination method yields combined decision procedures that are efficient enough to be used in practice.
Combining Constraint Solving
, 2001
"... this paper. On the one hand, dening a semantics for the combined system may depend on methods and results from formal logic and universal algebra. On the other hand, an ecient combination of the actual constraint solvers often requires the possibility of communication and cooperation between the sol ..."
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Cited by 5 (0 self)
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this paper. On the one hand, dening a semantics for the combined system may depend on methods and results from formal logic and universal algebra. On the other hand, an ecient combination of the actual constraint solvers often requires the possibility of communication and cooperation between the solvers.
Set Unification Revisited
, 2001
"... The goal of this paper is to provide a uniform overview of the unification problem in the algebras capable of describing sets. The problem has been tackled, directly and indirectly, by many researchers and finds important applications in various research areas  e.g., deductive databases, theorem p ..."
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Cited by 4 (4 self)
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The goal of this paper is to provide a uniform overview of the unification problem in the algebras capable of describing sets. The problem has been tackled, directly and indirectly, by many researchers and finds important applications in various research areas  e.g., deductive databases, theorem proving, static analysis, rapid software prototyping. The problem has been explored in depth, but the various solutions proposed are spread across a large literature, and some of the approaches have been ignored and/or rediscovered by different researchers. In this paper we provide a uniform presentation of unification of sets, starting with the simpler instances of the problem  e.g., matching of completely ground terms  and proceeding to progressively more complex problems  e.g., unification between general ACI terms. The algorithms presented are partly drawn from the literature  and properly revisited and analyzed  and partly novel proposals by the authors. The major contribution of this work is to provide the first uniform presentation of the problem, covering all its different instances, surveying at the same time the different solutions developed.