Improving Rewriting Induction Approach for Proving Ground Confluence

In (Aoto&Toyama, FSCD 2016), a method to prove ground confluence of many-sorted term rewriting systems based on rewriting induction is given. In this paper, we give several methods that add wider flexibility to the rewriting induction approach for proving ground confluence. Firstly, we give a method to deal with the case in which suitable rules are not presented in the input system. Our idea is to construct additional rewrite rules that supplement or replace existing rules in order to obtain a set of rules that is adequate for applying rewriting induction. Secondly, we give a method to deal with non-orientable constructor rules. This is accomplished by extending the inference system of rewriting induction and giving a sufficient criterion for the correctness of the system. Thirdly, we give a method to deal with disproving ground confluence. The presented methods are implemented in our ground confluence prover AGCP and experiments are reported. Our experiments reveal the presented methods are effective to deal with problems for which state-of-the-art ground confluence provers can not handle

Ground Confluence Prover based on Rewriting Induction

Ground confluence of term rewriting systems guarantees that all ground terms are confluent. Recently, interests in proving confluence of term rewriting systems automatically has grown, and confluence provers have been developed. But they mainly focus on confluence and not ground confluence. In fact, little interest has been paid to developing tools for proving ground confluence automatically. We report an implementation of a ground confluence prover based on rewriting induction, which is a method originally developed for proving inductive theorems

Termination of rewrite relations on λ\lambda-terms based on Girard's notion of reducibility

In this paper, we show how to extend the notion of reducibility introduced by Girard for proving the termination of $\beta$-reduction in the polymorphic $\lambda$-calculus, to prove the termination of various kinds of rewrite relations on $\lambda$-terms, including rewriting modulo some equational theory and rewriting with matching modulo $\beta$$\eta$, by using the notion of computability closure. This provides a powerful termination criterion for various higher-order rewriting frameworks, including Klop's Combinatory Reductions Systems with simple types and Nipkow's Higher-order Rewrite Systems

Inductive Theorem Proving Using Refined Unfailing Completion Techniques

We present a brief overview on completion based inductive theorem proving techniques, point out the key concepts for the underlying "proof by consistency" - paradigm and isolate an abstract description of what is necessary for an algorithmic realization of such methods. In particular, we give several versions of proof orderings, which - under certain conditions - are well-suited for that purpose. Together with corresponding notions of (positive and negative) covering sets we get abstract "positive" and "negative" characterizations of inductive validity. As a consequence we can generalize known criteria for inductive validity, even for the cases where some of the conjectures may not be orientable or where the base system is terminating but not necessarily ground confluent. Furthermore we consider several refinements and optimizations of completion based inductive theorem proving techniques. In particular, sufficient criteria for being a covering set including restrictions of critical pairs (and the usage of non-equational inductive knowledge) are discussed. Moreover a couple of lemma generation methods are briefly summarized and classified. A new techniques of save generalization is particularly interesting, since it provides means for syntactic generalizations, i.e. simplifications, of conjectures without loosing semantic equivalence. Finally we present the main features and characteristics of UNICOM, an inductive theorem prover with refined unfailing completion techniques and built on top of TRSPEC, a term rewriting based system for investigating algebraic specifications

Complete Sets of Transformations for General \u3cem\u3eE\u3c/em\u3e-Unification

This paper is concerned with E-unification in arbitrary equational theories. We extend the method of transformations on systems of terms, developed by Martelli-Montanari for standard unification, to E-unification by giving two sets of transformations, BT and T, which are proved to be sound and complete in the sense that a complete set of E-unifiers for any equational theory E can be enumerated by either of these sets. The set T is an improvement of BT, in that many E-unifiers produced by BT will be weeded out by T. In addition, we show that a generalization of surreduction (also called narrowing) combined with the computation of critical pairs is complete. A new representation of equational proofs as certain kinds of trees is used to prove the completeness of the set BT in a rather direct fashion that parallels the completeness of the transformations in the case of (standard) unification. The completeness of T and the generalization of surreduction is proved by a method inspired by the concept of unfailing completion, using an abstract (and simpler) notion of the completion of a set of equations

Jack polynomials and orientability generating series of maps

We study Jack characters, which are the coefficients of the power-sum expansion of Jack symmetric functions with a suitable normalization. These quantities have been introduced by Lassalle who formulated some challenging conjectures about them. We conjecture existence of a weight on non-oriented maps (i.e., graphs drawn on non-oriented surfaces) which allows to express any given Jack character as a weighted sum of some simple functions indexed by maps. We provide a candidate for this weight which gives a positive answer to our conjecture in some, but unfortunately not all, cases. In particular, it gives a positive answer for Jack characters specialized on Young diagrams of rectangular shape. This candidate weight attempts to measure, in a sense, the non-orientability of a given map.Comment: v2: change of title, substantial changes of the content v3: substantial changes in the presentatio

Equational logic and rewriting

International audienceIn this survey, we do not address higher order logics nor type theory, but rather restrict to first-order concepts. We focus on equational logic and its relation to rewriting logic and we consider their impact on automated deduction and programming languages