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Formalizing KnuthBendix orders and KnuthBendix completion
 In Proc. RTA ’13, volume 21 of LIPIcs
, 2013
"... We present extensions of our Isabelle Formalization of Rewriting that cover two historically related concepts: the KnuthBendix order and the KnuthBendix completion procedure. The former, besides being the first development of its kind in a proof assistant, is based on a generalized version of the ..."
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We present extensions of our Isabelle Formalization of Rewriting that cover two historically related concepts: the KnuthBendix order and the KnuthBendix completion procedure. The former, besides being the first development of its kind in a proof assistant, is based on a generalized version of the KnuthBendix order. We compare our version to variants from the literature and show all properties required to certify termination proofs of TRSs. The latter comprises the formalization of important facts that are related to completion, like Birkhoff’s theorem, the critical pair theorem, and a soundness proof of completion, showing that the strict encompassment condition is superfluous for finite runs. As a result, we are able to certify completion proofs.
A New and Formalized Proof of Abstract Completion?
"... Abstract. Completion is one of the most studied techniques in term rewriting. We present a new proof of the correctness of abstract completion that is based on peak decreasingness, a special case of decreasing diagrams. Peak decreasingness replaces Newman’s Lemma and allows us to avoid proof orders ..."
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Abstract. Completion is one of the most studied techniques in term rewriting. We present a new proof of the correctness of abstract completion that is based on peak decreasingness, a special case of decreasing diagrams. Peak decreasingness replaces Newman’s Lemma and allows us to avoid proof orders in the correctness proof of completion. As a result, our proof is simpler than the one presented in textbooks, which is confirmed by our Isabelle/HOL formalization. Furthermore, we show that critical pair criteria are easily incorporated in our setting. 1
Verified Computer Algebra in Acl2 (Gröbner Bases Computation)
"... In this paper, we present the formal verification of a Common Lisp implementation of Buchberger's algorithm for computing Gröbner bases of polynomial ideals. This work is carried out in the Acl2 system and shows how verified Computer Algebra can be achieved in an executable logic. ..."
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In this paper, we present the formal verification of a Common Lisp implementation of Buchberger's algorithm for computing Gröbner bases of polynomial ideals. This work is carried out in the Acl2 system and shows how verified Computer Algebra can be achieved in an executable logic.
Semantic Web Verification: Verifying Reasoning in the Logic ALC
, 2006
"... In the Semantic Web, knowledge is usually structured in the form of ontologies, using the Web Ontology Language (OWL), which is based in part on the Description Logics (DLs). DLs are a family of logical formalisms for representing and reasoning ..."
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In the Semantic Web, knowledge is usually structured in the form of ontologies, using the Web Ontology Language (OWL), which is based in part on the Description Logics (DLs). DLs are a family of logical formalisms for representing and reasoning
J.L. RUIZ–REINA, J.A. ALONSO, M.J. HIDALGO AND F.J. MARTÍN–MATEOS TERMINATION IN ACL2 USING MULTISET RELATIONS †
"... ABSTRACT: We present in this paper a case study of the use of the ACL2 system, describing an ACL2 formalization of multiset relations, and showing how multisets can be used to prove nontrivial termination properties. Every relation on a set A induces a relation on finite multisets over A; it can be ..."
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ABSTRACT: We present in this paper a case study of the use of the ACL2 system, describing an ACL2 formalization of multiset relations, and showing how multisets can be used to prove nontrivial termination properties. Every relation on a set A induces a relation on finite multisets over A; it can be shown that the multiset relation induced by a wellfounded relation is also wellfounded. We prove this property in the ACL2 logic, and use it by functional instantiation in order to provide wellfounded relations for the admissibility test of recursive functions. We also develope a macro defmul, to define wellfounded multiset relations in a convenient way. Finally, we present three examples illustrating how multisets are used to prove nontrivial termination properties in ACL2: a tailrecursive version of a general binary recursion scheme, a definition of McCarthy’s 91 function and a proof of Newman’s lemma for abstract reduction relations. These case studies show how nontrivial mathematical results can be stated and proved in the ACL2 logic, in spite of its apparent lack of expressiveness.
Formal Correctness of a Quadratic Unification Algorithm José–Luis Ruiz–Reina, Francisco–Jesús Martín–Mateos, José–Antonio
"... Abstract. We present a case study using ACL2 [5] to verify a nontrivial algorithm that uses efficient data structures. The algorithm receives as input two firstorder terms and it returns a most general unifier of these terms if they are unifiable, failure otherwise. The verified implementation sto ..."
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Abstract. We present a case study using ACL2 [5] to verify a nontrivial algorithm that uses efficient data structures. The algorithm receives as input two firstorder terms and it returns a most general unifier of these terms if they are unifiable, failure otherwise. The verified implementation stores terms as directed acyclic graphs by means of a pointer structure. Its time complexity is O(n 2) and its space complexity is O(n), and it can be executed in ACL2 at a speed comparable to a similar C implementation. We report the main issues encountered to achieve this formally verified implementation. 1.
The Milawa Rewriter and an ACL2 Proof of its Soundness
"... Abstract. Rewriting with lemmas is a central strategy in interactive theorem provers. We describe the Milawa rewriter, which makes use of assumptions, calculation, and conditional rewrite rules to simplify the terms of a firstorder logic. We explain how we have developed an ACL2 proof showing the r ..."
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Abstract. Rewriting with lemmas is a central strategy in interactive theorem provers. We describe the Milawa rewriter, which makes use of assumptions, calculation, and conditional rewrite rules to simplify the terms of a firstorder logic. We explain how we have developed an ACL2 proof showing the rewriter is sound, and how this proof can accommodate our rewriter’s many useful features such as freevariable matching, ancestors checking, syntatic restrictions, caching, and forcing.
Encapsulation for Practical Simplification Procedures
, 2003
"... ACL2 was used to prove properties of two simplification procedures. The procedures differ in complexity but solve the same programming problem that arises in the context of a resolution/paramodulation theorem proving system. Term rewriting is at the core of the two procedures, but details of the rew ..."
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ACL2 was used to prove properties of two simplification procedures. The procedures differ in complexity but solve the same programming problem that arises in the context of a resolution/paramodulation theorem proving system. Term rewriting is at the core of the two procedures, but details of the rewriting procedure itself are irrelevant. The ACL2 encapsulate construct was used to assert the existence of the rewriting function and to state some of its properties. Termination, irreducibility, and soundnessproperties were established for each procedure. The availability of the encapsulation mechanism in ACL2 is considered essential to rapid and efficient verification of this kind of algorithm.