Automated verification of termination certificates

In order to increase user confidence, many automated theorem provers provide certificates that can be independently verified. In this paper, we report on our progress in developing a standalone tool for checking the correctness of certificates for the termination of term rewrite systems, and formally proving its correctness in the proof assistant Coq. To this end, we use the extraction mechanism of Coq and the library on rewriting theory and termination called CoLoR

Polygraphs for termination of left-linear term rewriting systems

We present a methodology for proving termination of left-linear term rewriting systems (TRSs) by using Albert Burroni's polygraphs, a kind of rewriting systems on algebraic circuits. We translate the considered TRS into a polygraph of minimal size whose termination is proven with a polygraphic interpretation, then we get back the property on the TRS. We recall Yves Lafont's general translation of TRSs into polygraphs and known links between their termination properties. We give several conditions on the original TRS, including being a first-order functional program, that ensure that we can reduce the size of the polygraphic translation. We also prove sufficient conditions on the polygraphic interpretations of a minimal translation to imply termination of the original TRS. Examples are given to compare this method with usual polynomial interpretations.Comment: 15 page

Definitions by Rewriting in the Calculus of Constructions

The main novelty of this paper is to consider an extension of the Calculus of Constructions where predicates can be defined with a general form of rewrite rules. We prove the strong normalization of the reduction relation generated by the beta-rule and the user-defined rules under some general syntactic conditions including confluence. As examples, we show that two important systems satisfy these conditions: a sub-system of the Calculus of Inductive Constructions which is the basis of the proof assistant Coq, and the Natural Deduction Modulo a large class of equational theories.Comment: Best student paper (Kleene Award

Automated Termination Proofs for Logic Programs by Term Rewriting

There are two kinds of approaches for termination analysis of logic programs: "transformational" and "direct" ones. Direct approaches prove termination directly on the basis of the logic program. Transformational approaches transform a logic program into a term rewrite system (TRS) and then analyze termination of the resulting TRS instead. Thus, transformational approaches make all methods previously developed for TRSs available for logic programs as well. However, the applicability of most existing transformations is quite restricted, as they can only be used for certain subclasses of logic programs. (Most of them are restricted to well-moded programs.) In this paper we improve these transformations such that they become applicable for any definite logic program. To simulate the behavior of logic programs by TRSs, we slightly modify the notion of rewriting by permitting infinite terms. We show that our transformation results in TRSs which are indeed suitable for automated termination analysis. In contrast to most other methods for termination of logic programs, our technique is also sound for logic programming without occur check, which is typically used in practice. We implemented our approach in the termination prover AProVE and successfully evaluated it on a large collection of examples.Comment: 49 page

Inductive types in the Calculus of Algebraic Constructions

In a previous work, we proved that an important part of the Calculus of Inductive Constructions (CIC), the basis of the Coq proof assistant, can be seen as a Calculus of Algebraic Constructions (CAC), an extension of the Calculus of Constructions with functions and predicates defined by higher-order rewrite rules. In this paper, we prove that almost all CIC can be seen as a CAC, and that it can be further extended with non-strictly positive types and inductive-recursive types together with non-free constructors and pattern-matching on defined symbols.Comment: Journal version of TLCA'0

Termination of Narrowing with Dependency Pairs

In this work, we generalize the Dependency Pairs approach for automated proofs of termination to prove the termination of narrowing.We identify the phenomenon of echoing in infinite narrowing derivations and demonstrate that the new narrowing dependency pairs faithfully capture the shape of such derivations and provide a termination criterion.Iborra López, J. (2008). Termination of Narrowing with Dependency Pairs. http://hdl.handle.net/10251/13622Archivo delegad