Termination orderings for associative-commutative rewriting systems

In this paper we describe a new class of orderings—associative path orderings—for proving termination of associative-commutative term rewriting systems .These orderings are based on the concept of simplification orderings and extend the well-known recursive path orderings to E - congruence classes, where E is an equational theory consisting of associativity and commutativity axioms. Associative path orderings are applicable to term rewriting systems for which a precedence ordering on the set of operator symbols can be defined that satisfies a certain condition,the associative path condition. The precedence ordering can often be derived from the structure of the reduction rules. We include termination proofs for various term rewriting systems (for rings,boolean algebra,etc.) and, in addition, point out ways to handle situations where the associative path condition is too restrictive

Constrained completion: Theory, implementation, and results

The Knuth-Bendix completion procedure produces complete sets of reductions but can not handle certain rewrite rules such as commutativity. In order to handle such theories, completion procedure were created to find complete sets of reductions modulo an equational theory. The major problem with this method is that it requires a specialized unification algorithm for the equational theory. Although this method works well when such an algorithm exists, these algorithms are not always available and thus alternative methods are needed to attack problems. A way of doing this is to use a completion procedure which finds complete sets of constrained reductions. This type of completion procedure neither requires specialized unification algorithms nor will it fail due to unorientable identities. We present a look at complete sets of reductions with constraints, developed by Gerald Peterson, and the implementation of such a completion procedure for use with HIPER - a fast completion system. The completion procedure code is given and shown correct along with the various support procedures which are needed by the constrained system. These support procedures include a procedure to find constraints using the lexicographic path ordering and a normal form procedure for constraints. The procedure has been implemented for use under the fast HIPER system, developed by Jim Christian, and thus is quick. We apply this new system, HIPER- extension, to attack a variety of word problems. Implementation alternatives are discussed, developed, and compared with each other as well as with the HIPER system. Finally, we look at the problem of finding a complete set of reductions for a ternary boolean algebra. Given are alternatives to attacking this problem and the already known solution along with its run in the HIPER-extension system --Abstract, page iii

Normalized Rewriting: an Alternative to Rewriting modulo a Set of Equations

AbstractIn the first part of this paper, we introducenormalized rewriting, a new rewrite relation. It generalizes former notions of rewriting modulo a set of equationsE, dropping some conditions onE. For example,Ecan now be the theory of identity, idempotence, the theory of Abelian groups or the theory of commutative rings. We give a new completion algorithm for normalized rewriting. It contains as an instance the usual AC completion algorithm, but also the well-known Buchberger algorithm for computing Gröbner bases of polynomial ideals. In the second part, we investigate the particular case of completion of ground equations. In this case we prove by a uniform method that completion moduloEterminates, for some interesting theoriesE. As a consequence, we obtain the decidability of the word problem for some classes of equational theories, including the AC-ground case (a result known since 1991), the ACUI-ground case (a new result to our knowledge), and the cases of ground equations modulo the theory of Abelian groups and commutative rings, which is already known when the signature contains only constants, but is new otherwise. Finally, we give implementation results which show the efficiency of normalized completion with respect to completion modulo AC

REST: Integrating Term Rewriting with Program Verification (Extended Version)

We introduce REST, a novel term rewriting technique for theorem proving that uses online termination checking and can be integrated with existing program verifiers. REST enables flexible but terminating term rewriting for theorem proving by: (1) exploiting newly-introduced term orderings that are more permissive than standard rewrite simplification orderings; (2) dynamically and iteratively selecting orderings based on the path of rewrites taken so far; and (3) integrating external oracles that allow steps that cannot be justified with rewrite rules. Our REST approach is designed around an easily implementable core algorithm, parameterizable by choices of term orderings and their implementations; in this way our approach can be easily integrated into existing tools. We implemented REST as a Haskell library and incorporated it into Liquid Haskell's evaluation strategy, extending Liquid Haskell with rewriting rules. We evaluated our REST implementation by comparing it against both existing rewriting techniques and E-matching and by showing that it can be used to supplant manual lemma application in many existing Liquid Haskell proofs

AC-KBO Revisited

Equational theories that contain axioms expressing associativity and commutativity (AC) of certain operators are ubiquitous. Theorem proving methods in such theories rely on well-founded orders that are compatible with the AC axioms. In this paper we consider various definitions of AC-compatible Knuth-Bendix orders. The orders of Steinbach and of Korovin and Voronkov are revisited. The former is enhanced to a more powerful version, and we modify the latter to amend its lack of monotonicity on non-ground terms. We further present new complexity results. An extension reflecting the recent proposal of subterm coefficients in standard Knuth-Bendix orders is also given. The various orders are compared on problems in termination and completion.Comment: 31 pages, To appear in Theory and Practice of Logic Programming (TPLP) special issue for the 12th International Symposium on Functional and Logic Programming (FLOPS 2014

Well-founded Path Orderings for Drags

International audienceThe definition herein of the Graph Path Ordering (GPO) on certain graph expressions is inspired by that of the Recursive Path Ordering (RPO), and enjoys all those properties that have made RPO popular, in particular, well-foundedness and monotonicity on variable-free terms.We are indeed interested in a generalization of algebraic expressions called operadic expressions, which are finite graphs each vertex of which is labelled by a function symbol, the arity of which governs the number of vertices it relates to in the graph. These graphs are seen here as terms with sharing and back-arrows. Operadic expressions arethemselves multiplied (an associative operation) to form monomials, which are in turn summed up (an associative commutative operation) to form polynomials. Operadic expressions and their polynomials occur in algebraic topology, and in various areas of computer science, notably concurrency and type theory. Rewriting basic operadic expressions isvery much like rewriting algebraic expressions, while rewriting their monomials and polynomials is very much like the Groebner basis theory. GPO provides an initial building block for computing with operadic expressions and their polynomials

Ordering constraints on trees

We survey recent results about ordering constraints on trees and discuss their applications. Our main interest lies in the family of recursive path orderings which enjoy the properties of being total, well-founded and compatible with the tree constructors. The paper includes some new results, in particular the undecidability of the theory of lexicographic path orderings in case of a non-unary signature

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