José-Antonio Alonso

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 first-order, 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 Newmans lemma (for abstract reductions) and Knuth–Bendix critical pair theorem (for term rewriting).

Formal Correctness of a Quadratic Unification Algorithm

We present a case study using ACL2 to verify a nontrivial 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

A Formal Proof of Dickson's Lemma in ACL2

O(n^2)

Formalizing Rewriting in the ACL2 Theorem Prover

and its space complexity

A Formally Verified Prover for the \mathcal{ALC\,} Description Logic

O(n)

Proof pearl: a formal proof of Higman's Lemma in ACL2

, 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.

Rete algorithm applied to robotic soccer

Dickson’s Lemma is the main result needed to prove the termination of Buchberger’s algorithm for computing Grobner basis of polynomial ideals. In this case study, we present a formal proof of Dickson’s Lemma using the ACL2 system. Due to the limited expressiveness of the ACL2 logic, the classical non-constructive proof of this result cannot be done in ACL2. Instead, we formalize a proof where the termination argument is justified by the multiset extension of a well-founded relation.

Constructing Formally Verified Reasoners for the ALC Description Logic

We present an application of the ACL2 theorem prover to formalize and reason about rewrite systems theory. This can be seen as a first approach to apply formal methods, using ACL2, to the design of symbolic computation systems, since the notion of rewriting or simplification is ubiquitous in such systems. We concentrate here on formalization and representation aspects of abstract reduction and term rewriting systems, using the first-order, quantifier-free ACL2 logic based on Common Lisp.

Verification in ACL2 of a generic framework to synthesize SAT-provers

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 \(\mathcal{ALC\,}\)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 \(\mathcal{ALC\,}\), 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 \(\mathcal{ALC\,}\) prover.

Verifying an Applicative ATP Using Multiset Relations

In this paper we present a formalization and proof of Higmans Lemma in ACL2. We formalize the constructive proof described in [10] where the result is proved using a termination argument justified by the multiset extension of a well-founded relation. To our knowledge, this is the first mechanization of this proof.