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Enhancing the Nuprl Proof Development System and Applying it to Computational Abstract Algebra
, 1995
"... This thesis describes substantial enhancements that were made to the software tools in the Nuprl system that are used to interactively guide the production of formal proofs. Over 20,000 lines of code were written for these tools. Also, a corpus of formal mathematics was created that consists of rou ..."
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Cited by 47 (4 self)
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This thesis describes substantial enhancements that were made to the software tools in the Nuprl system that are used to interactively guide the production of formal proofs. Over 20,000 lines of code were written for these tools. Also, a corpus of formal mathematics was created that consists of roughly 500 definitions and 1300 theorems. Much of this material is of a foundational nature and supports all current work in Nuprl. This thesis concentrates on describing the half of this corpus that is concerned with abstract algebra and that covers topics central to the mathematics of the co...
Rewrite Techniques for Transitive Relations
 IN PROC., 9TH IEEE SYMPOSIUM ON LOGIC IN COMPUTER SCIENCE
, 1994
"... We propose inference systems for dealing with transitive relations in the context of resolutiontype theorem proving. These inference mechanisms are based on standard techniques from term rewriting and represent a refinement of chaining methods. We establish their refutational completeness and al ..."
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Cited by 38 (6 self)
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We propose inference systems for dealing with transitive relations in the context of resolutiontype theorem proving. These inference mechanisms are based on standard techniques from term rewriting and represent a refinement of chaining methods. We establish their refutational completeness and also prove their compatibility with the usual simplification techniques used in rewritebased theorem provers. A key to the practicality of chaining techniques is the extent to which socalled variable chainings can be restricted. We demonstrate that rewrite techniques considerably restrict variable chaining, though we also show that they cannot be completely avoided for transitive relations in general. If the given relation satisfies additional properties, such as symmetry, further restrictions are possible. In particular, we discuss (partial) equivalence relations and congruence relations.
Ordered Chaining Calculi for FirstOrder Theories of Transitive Relations
 Journal of the ACM
, 1998
"... this paper have been presented at the 12th International Conference on Automated Deduction (Nancy, France, June/July 1994) and the 9th IEEE Symposium on Logic in Computer Science (Paris, France, July 1994). ..."
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Cited by 33 (4 self)
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this paper have been presented at the 12th International Conference on Automated Deduction (Nancy, France, June/July 1994) and the 9th IEEE Symposium on Logic in Computer Science (Paris, France, July 1994).
Birewrite systems
, 1996
"... In this article we propose an extension of term rewriting techniques to automate the deduction in monotone preorder theories. To prove an inclusion a ⊆ b from a given set I of them, we generate from I, using a completion procedure, a birewrite system 〈R⊆, R⊇〉, that is, a pair of rewrite relations ..."
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Cited by 30 (8 self)
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In this article we propose an extension of term rewriting techniques to automate the deduction in monotone preorder theories. To prove an inclusion a ⊆ b from a given set I of them, we generate from I, using a completion procedure, a birewrite system 〈R⊆, R⊇〉, that is, a pair of rewrite relations −−− → R ⊆ and −−− → R ⊇ , and seek a common term c such that a −−−→ R ⊆ c and b −−−→
Ordered Semantic HyperLinking
, 1994
"... We propose a method for combining the clause linking theorem proving method with theorem proving methods based on orderings. This may be useful for incorporating termrewriting based approaches into clause linking. In this way, some of the propositional inefficiencies of orderingbased approaches ..."
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Cited by 29 (2 self)
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We propose a method for combining the clause linking theorem proving method with theorem proving methods based on orderings. This may be useful for incorporating termrewriting based approaches into clause linking. In this way, some of the propositional inefficiencies of orderingbased approaches may be overcome, while at the same time incorporating the advantages of ordering methods into clause linking. The combination also provides a natural way to combine resolution on nonground clauses, with the clause linking method, which is essentially a ground method. We describe the method, prove completeness, and show that the enumeration part of clause linking with semantics can be reduced to polynomial time in certain cases. We analyze the complexity of the proposed method, and also give some plausibility arguments concerning its expected performance.
The Saturate System
, 1998
"... The Saturate system is an experimental theorem prover for firstorder logic, primarily based on saturation. Saturate uses techniques of ordered chaining for arbitrary transitive relations, including orderings, equivalence relations and congruences, and integrates CNF transformation lazily into satur ..."
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Cited by 21 (11 self)
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The Saturate system is an experimental theorem prover for firstorder logic, primarily based on saturation. Saturate uses techniques of ordered chaining for arbitrary transitive relations, including orderings, equivalence relations and congruences, and integrates CNF transformation lazily into saturation.
Herbrand’s theorem for prenex Gödel logic and its consequences for theorem proving
 IN LOGIC FOR PROGRAMMING AND AUTOMATED REASONING LPAR’2001, 201–216. LNAI 2250
, 2001
"... Herbrand’s Theorem £¥ ¤ ¦ for, i.e., Gödel logic enriched by the projection § operator is proved. As a consequence we obtain a “chain normal form” and a translation of £ ¤ ¦ prenex into (order) clause logic, referring to the classical theory of dense total orders with endpoints. A chaining calc ..."
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Cited by 16 (11 self)
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Herbrand’s Theorem £¥ ¤ ¦ for, i.e., Gödel logic enriched by the projection § operator is proved. As a consequence we obtain a “chain normal form” and a translation of £ ¤ ¦ prenex into (order) clause logic, referring to the classical theory of dense total orders with endpoints. A chaining calculus provides a basis for efficient theorem proving.
Buchberger's algorithm: A constraintbased completion procedure
, 1994
"... We present an extended completion procedure with builtin theories defined by a collection of associativity and commutativity axioms and additional ground equations, and reformulate Buchberger's algorithm for constructing Gröbner bases for polynomial ideals in this formalism. The presentation o ..."
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Cited by 15 (2 self)
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We present an extended completion procedure with builtin theories defined by a collection of associativity and commutativity axioms and additional ground equations, and reformulate Buchberger's algorithm for constructing Gröbner bases for polynomial ideals in this formalism. The presentation of completion is at an abstract level, by transition rules, with a suitable notion of fairness used to characterize a wide class of correct completion procedures, among them Buchberger's original algorithm for polynomial rings over a field.
Theorem Proving in Cancellative Abelian Monoids
, 1996
"... We describe a refined superposition calculus for cancellative abelian monoids. They encompass not only abelian groups, but also such ubiquitous structures as the natural numbers or multisets. Both the AC axioms and the cancellation law are difficult for a general purpose superposition theorem prover ..."
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Cited by 9 (1 self)
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We describe a refined superposition calculus for cancellative abelian monoids. They encompass not only abelian groups, but also such ubiquitous structures as the natural numbers or multisets. Both the AC axioms and the cancellation law are difficult for a general purpose superposition theorem prover, as they create many variants of clauses which contain sums. Our calculus requires neither explicit inferences with the theory clauses for cancellative abelian monoids nor extended equations or clauses. Improved ordering constraints allow us to restrict to inferences that involve the maximal term of the maximal sum in the maximal literal. Furthermore, the search space is reduced drastically by certain variable elimination techniques.
Combining Algebra and Universal Algebra in FirstOrder Theorem Proving: The Case of Commutative Rings
 In Proc. 10th Workshop on Specification of Abstract Data Types
, 1995
"... . We present a general approach for integrating certain mathematical structures in firstorder equational theorem provers. More specifically, we consider theorem proving problems specified by sets of firstorder clauses that contain the axioms of a commutative ring with a unit element. Associativeco ..."
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Cited by 7 (4 self)
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. We present a general approach for integrating certain mathematical structures in firstorder equational theorem provers. More specifically, we consider theorem proving problems specified by sets of firstorder clauses that contain the axioms of a commutative ring with a unit element. Associativecommutative superposition forms the deductive core of our method, while a convergent rewrite system for commutative rings provides a starting point for more specialized inferences tailored to the given class of formulas. We adopt ideas from the Grobner basis method to show that many inferences of the superposition calculus are redundant. This result is obtained by the judicious application of the simplification techniques afforded by convergent rewriting and by a process called symmetrization that embeds inferences between single clauses and ring axioms. 1 Introduction 1.1 Motivation Specifications of programs include both symbols with their usual mathematical meaning as well as additional f...