127 research outputs found

    Formal proofs about rewriting using ACL2

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    We present an application of the ACL2 theorem prover to reason about rewrite systems theory. We describe the formalization and representation aspects of our work using the firstorder, quantifier-free logic of ACL2 and we sketch some of the main points of the proof effort. First, we present a formalization of abstract reduction systems and then we show how this abstraction can be instantiated to establish results about term rewriting. The main theorems we mechanically proved are Newman’s lemma (for abstract reductions) and Knuth–Bendix critical pair theorem (for term rewriting).Ministerio de Educación y Ciencia TIC2000-1368-CO3-0

    Encapsulation for Practical Simplification Procedures

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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 soundness properties 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.Comment: 6 page

    A Theory About First-Order Terms in ACL2

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    We describe the development in ACL2 of a library of results about first-order terms. In particular, we present the formalization of some of the main properties of the complete lattice of first-order terms with respect to the subsumption relation. As a byproduct, verified executable implementations are obtained for some basic operations on firstorder terms, including matching, renaming, unification and anti-unification. This work can be seen as a basis for further studies about the formal properties of automated reasoning and symbolic computation systems.Ministerio de Ciencia y Tecnología TIC2000-1368-CO3-0

    Verifying the bridge between simplicial topology and algebra: the Eilenberg–Zilber algorithm

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    The Eilenberg–Zilber algorithm is one of the central components of the computer algebra system called Kenzo, devoted to computing in Algebraic Topology. In this article we report on a complete formal proof of the underlying Eilenberg–Zilber theorem, using the ACL2 theorem prover. As our formalization is executable, we are able to compare the results of the certified programme with those of Kenzo on some universal examples. Since the results coincide, the reliability of Kenzo is reinforced. This is a new step in our long-term project towards certified programming for Algebraic Topology.Ministerio de Ciencia e Innovación MTM2009-13842European Union’s 7th Framework Programme [243847] (ForMath)

    Certified Symbolic Manipulation: Bivariate Simplicial Polynomials

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    Certified symbolic manipulation is an emerging new field where programs are accompanied by certificates that, suitably interpreted, ensure the correctness of the algorithms. In this paper, we focus on algebraic algorithms implemented in the proof assistant ACL2, which allows us to verify correctness in the same programming environment. The case study is that of bivariate simplicial polynomials, a data structure used to help the proof of properties in Simplicial Topology. Simplicial polynomials can be computationally interpreted in two ways. As symbolic expressions, they can be handled algorithmically, increasing the automation in ACL2 proofs. As representations of functional operators, they help proving properties of categorical morphisms. As an application of this second view, we present the definition in ACL2 of some morphisms involved in the Eilenberg-Zilber reduction, a central part of the Kenzo computer algebra system. We have proved the ACL2 implementations are correct and tested that they get the same results as Kenzo does.Ministerio de Ciencia e Innovación MTM2009-13842Unión Europea nr. 243847 (ForMath
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