Verification in ACL2 of a Generic Framework to Synthesize SAT–Provers

We present in this paper an application of the ACL2 system to reason about propositional satisfiability provers. For that purpose, we present a framework where we define a generic transformation based SAT–prover, and we show how this generic framework can be formalized in the ACL2 logic, making a formal proof of its termination, soundness and completeness. This generic framework can be instantiated to obtain a number of verified and executable SAT–provers in ACL2, and this can be done in an automatized way. Three case studies are considered: semantic tableaux, sequent and Davis–Putnam methods.Ministerio de Ciencia y Tecnología TIC2000-1368-C03-0

Formal Verification of Molecular Computational Models in ACL2: A Case Study

Theorem proving is a classical AI problem with a broad range of applications. Since its complexity is exponential in the size of the problem, many methods to parallelize the process has been proposed. One of these approaches is based on the massive parallelism of molecular reactions. ACL2 is an automated theorem prover especially adequate for algorithm verification. In this paper we present an ACL2 formalization of a molecular computational model: Adleman’s restricted model. As an application of this model, an implementation of Lipton’s experiment solving SAT is described. We use ACL2 to make a formal proof of the completeness and soundness properties of this implementation.Ministerio de Ciencia y Tecnología TIC2000-1368-C03-0

Formal verification of a generic framework to synthesize SAT-provers

We present in this paper an application of the ACL2 system to generate and reason about propositional satis ability provers. For that purpose, we develop a framework where we de ne a generic SAT-prover based on transformation rules, and we formalize this generic framework in the ACL2 logic, carrying out a formal proof of its termination, soundness and completeness. This generic framework can be instantiated to obtain a number of veri ed and executable SAT-provers in ACL2, and this can be done in an automated way. Three instantiations of the generic framework are considered: semantic tableaux, sequent and Davis-Putnam-Logeman-Loveland methods.Ministerio de Ciencia y Tecnología TIC2000-1368-C03-0

Formal Correctness of a Quadratic Unification Algorithm

We present a case study using ACL2 [5] to verify a non-trivial algorithm that uses efficient data structures. The algorithm receives as input two first-order 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(n2) 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

Constructing Formally Verified Reasoners for the ALC Description Logic

Description Logics are a family of logics used to represent and reason about conceptual and terminological knowledge. Recently, its importance has been increased since they are used as a basis for the Ontology Web Language (OWL) used for the Semantic Web. In previous work, we have developed in PVS a generic framework for reasoning in the ALC description logic, proving its termination, soundness and completeness. In this paper we present the construction, from the generic framework, of a formally verified generic tableau– based algorithm for checking satisfiability of ALC –concepts. We do it using a methodology of refinements to transfer the properties from the framework to the algorithm. We also obtain some verified reasoners from the algorithm by a process of instantiation.Ministerio de Educación y Ciencia TIN2004–0388

Proof Pearl: a Formal Proof of Higman’s Lemma in ACL2

Higman’s lemma is an important result in infinitary combinatorics, which has been formalized in several theorem provers. In this paper we present a formalization and proof of Higman’s Lemma in the ACL2 theorem prover. Our formalization is based on a proof by Murthy and Russell, where the key termination argument is justified by the multiset relation induced by a well-founded relation. To our knowledge, this is the first mechanization of this proof.Ministerio de Ciencia e Innovación MTM2009-13842-C02-0

Formal proofs about rewriting using ACL2

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

A Theory About First-Order Terms in ACL2

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

Extending Attribute Exploration by Means of Boolean Derivatives

We present a translation of problems of Formal Context Analysis into ideals problems in F2[x] through the Boolean derivatives. The Boolean derivatives are introduced as a kind of operators on propositional formulas which provide a complete calculus. They are useful to refine stem basis as well as for extending attribute exploration

A Formally Verified Prover for the ALC Description Logic

The Ontology Web Language (OWL) is a language used for the Semantic Web. OWL is based on Description Logics (DLs), a family of logical formalisms for representing and reasoning about conceptual and terminological knowledge. Among these, the logic ALC is a ground DL used in many practical cases. Moreover, the Semantic Web appears as a new field for the application of formal methods, that could be used to increase its reliability. A starting point could be the formal verification of satisfiability provers for DLs. In this paper, we present the PVS specification of a prover for ALC , as well as the proofs of its termination, soundness and completeness. We also present the formalization of the well–foundedness of the multiset relation induced by a well–founded relation. This result has been used to prove the termination and the completeness of the ALC prover.Ministerio de Educación y Ciencia TIN2004–0388