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Soft typing for ordered resolution
 IN `PROCEEDINGS OF THE 14TH INTERNATIONAL CONFERENCE ON AUTOMATED DEDUCTION, CADE14
"... We propose a variant of ordered resolution with semantic restrictions based on interpretations which are identi ed by the given atom ordering and selection function. Techniques for effectively approximating validity (satisfiability) in these interpretations are presented. They are related to methods ..."
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Cited by 17 (6 self)
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We propose a variant of ordered resolution with semantic restrictions based on interpretations which are identi ed by the given atom ordering and selection function. Techniques for effectively approximating validity (satisfiability) in these interpretations are presented. They are related to methods of soft typing for programming languages. The framework is shown to be strictly more general than certain previously introduced approaches. Implementation of some of our techniques in the Spass prover has lead to encouraging experimental results.
Unification in extensions of shallow equational theories
 REWRITING TECHNIQUES AND APPLICATIONS, 9TH INTERNATIONAL CONFERENCE, RTA98', VOL. 1379 OF LNCS
, 1998
"... We show that unification in certain extensions of shallow equational theories is decidable. Our extensions generalize the known classes of shallow or standard equational theories. In order to prove decidability of unification in the extensions, a class of Horn clause sets called sorted shallow equa ..."
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Cited by 10 (2 self)
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We show that unification in certain extensions of shallow equational theories is decidable. Our extensions generalize the known classes of shallow or standard equational theories. In order to prove decidability of unification in the extensions, a class of Horn clause sets called sorted shallow equational theories is introduced. This class is a natural extension of tree automata with equality constraints between brother subterms as well as shallow sort theories. We show that saturation under sorted superposition is effective on sorted shallow equational theories. So called semilinear equational theories can be e ectively transformed into equivalent sorted shallow equational theories and generalize the classes of shallow and standard equational theories.
Basic Syntactic Mutation
"... We give a set of inference rules for Eunification, similar to the inference rules for Syntactic Mutation. If the E is finitely saturated by paramodulation, then we can block certain terms from further inferences. Therefore, ..."
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Cited by 5 (1 self)
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We give a set of inference rules for Eunification, similar to the inference rules for Syntactic Mutation. If the E is finitely saturated by paramodulation, then we can block certain terms from further inferences. Therefore,
From Search to Computation: Redundancy Criteria and Simplification at Work
"... The concept of redundancy and simplification has been an ongoing theme in Harald Ganzinger’s work from his first contributions to equational completion to the various variants of the superposition calculus. When executed by a theorem prover, the inference rules of these calculi usually generate a t ..."
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The concept of redundancy and simplification has been an ongoing theme in Harald Ganzinger’s work from his first contributions to equational completion to the various variants of the superposition calculus. When executed by a theorem prover, the inference rules of these calculi usually generate a tremendously growing search space. The redundancy and simplification concept is indispensable for cutting down this search space to a manageable size. For a number of subclasses of firstorder logic appropriate redundancy and simplification concepts even turn the superposition calculus into a decision procedure. Hence, the key to successfully applying firstorder theorem proving to a problem domain is to find those simplifications and redundancy criteria that fit this domain and can be effectively implemented. We present Harald Ganzinger’s work in the light of the simplification and redundancy techniques that have been developed for concrete problem areas. This includes a variant of contextual rewriting to decide a subclass of Euclidean geometry, ordered chaining techniques for ChurchRosser and priority queue proofs, contextual rewriting and historydependent complexities for the completion of conditional rewrite systems, rewriting with equivalences for theorem proving in set theory, soft typing for the exploration of sort information in the context of equations, and constraint inheritance for automated complexity analysis.
From Search to Computation: Redundancy Criteria and Simplification at Work
"... Abstract. The concept of redundancy and simplification has been an ongoing theme in Harald Ganzinger’s work from his first contributions to equational completion to the various variants of the superposition calculus. When executed by a theorem prover, the inference rules of logic calculi usually gen ..."
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Abstract. The concept of redundancy and simplification has been an ongoing theme in Harald Ganzinger’s work from his first contributions to equational completion to the various variants of the superposition calculus. When executed by a theorem prover, the inference rules of logic calculi usually generate a tremendously huge search space. The redundancy and simplification concept is indispensable for cutting down this search space to a manageable size. For a number of subclasses of firstorder logic appropriate redundancy and simplification concepts even turn the superposition calculus into a decision procedure. Hence, the key to successfully applying firstorder theorem proving to a problem domain is to find those simplifications and redundancy criteria that fit this domain and can be effectively implemented. We present Harald Ganzinger’s work in the light of the simplification and redundancy techniques that have been developed for concrete problem areas. This includes a variant of contextual rewriting to decide a subclass of Euclidean geometry, ordered chaining techniques for ChurchRosser and priority queue proofs, contextual rewriting and historydependent complexities for the completion of conditional rewrite systems, rewriting with equivalences for theorem proving in set theory, soft typing for the exploration of sort information in the context of equations, and constraint inheritance for automated complexity analysis. 1
Complexity of Linear Standard Theories
, 2001
"... We give an algorithm for deciding Eunification problems for linear standard equational theories (linear equations with all shared variables at a depth less than two) and varity 1 goals (linear equations with no shared variables). We show that the algorithm halts in quadratic time for the nonun ..."
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We give an algorithm for deciding Eunification problems for linear standard equational theories (linear equations with all shared variables at a depth less than two) and varity 1 goals (linear equations with no shared variables). We show that the algorithm halts in quadratic time for the nonuniform Eunification problem, and linear time if the equational theory is varity 1. The algorithm is still polynomial for the uniform problem. The size of the complete set of unifiers is exponential, but membership in that set can be determined in polynomial time. For any goal (not just varity 1) we give a NEXPTIME algorithm.