Rewriting-based languages and systems

Many systems are at least partly or implicitly based on term rewriting. Examples are functional languages, computer algebra systems, and theorem provers. We give a brief survey of these systems, and provide a more in-depth comparison of the features of systems that use term rewriting as their primary execution mechanism. We give links to rewrite tools and projects as well as pointers to notions and techniques covered in the preceding chapters

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

A Lower Bound of the Number of Rewrite Rules Obtained by Homological Methods

It is well-known that some equational theories such as groups or boolean algebras can be defined by fewer equational axioms than the original axioms. However, it is not easy to determine if a given set of axioms is the smallest or not. Malbos and Mimram investigated a general method to find a lower bound of the cardinality of the set of equational axioms (or rewrite rules) that is equivalent to a given equational theory (or term rewriting systems), using homological algebra. Their method is an analog of Squier\u27s homology theory on string rewriting systems. In this paper, we develop the homology theory for term rewriting systems more and provide a better lower bound under a stronger notion of equivalence than their equivalence. The author also implemented a program to compute the lower bounds

Order-Sorted Equational Computation

The expressive power of many-sorted equational logic can be greatly enhanced by allowing for subsorts and multiple function declarations. In this paper we study some computational aspects of such a logic. We start with a self-contained introduction to order-sorted equational logic including initial algebra semantics and deduction rules. We then present a theory of order-sorted term rewriting and show that the key results for unsorted rewriting extend to sort decreasing rewriting. We continue with a review of order-sorted unification and prove the basic results. In the second part of the paper we study hierarchical order-sorted specifications with strict partial functions. We define the appropriate homomorphisms for strict algebras and show that every strict algebra is base isomorphic to a strict algebra with at most one error element. For strict specifications, we show that their categories of strict algebras have initial objects. We validate our approach to partial functions by proving that completely defined total functions can be defined as partial without changing the initial algebra semantics. Finally, we provide decidable sufficient criteria for the consistency and strictness of ground confluent rewriting systems

Homological Computations for Term Rewriting Systems

An important problem in universal algebra consists in finding presentations of algebraic theories by generators and relations, which are as small as possible. Exhibiting lower bounds on the number of those generators and relations for a given theory is a difficult task because it a priori requires considering all possible sets of generators for a theory and no general method exists. In this article, we explain how homological computations can provide such lower bounds, in a systematic way, and show how to actually compute those in the case where a presentation of the theory by a convergent rewriting system is known. We also introduce the notion of coherent presentation of a theory in order to consider finer homotopical invariants. In some aspects, this work generalizes, to term rewriting systems, Squier\u27s celebrated homological and homotopical invariants for string rewriting systems

Strong and weak operational termination of order-sorted rewrite theories

The final publication is available at Springer via http://dx.doi.org/10.1007/978-3-319-12904-4 10This paper presents several new results on conditional term rewriting within the general framework of order-sorted rewrite theories (OSRTs) which contains the more restricted framework of conditional term rewriting systems (CTRSs) as a special case. The results uncover some subtle issues about conditional termination. We first of all generalize a previous known result characterizing the operational termination of a CTRS by the quasi-decreasing ordering notion to a similar result for OSRTs. Second, we point out that the notions of *irreducible* term and of *normal form*, which coincide for unsorted rewriting are *totally different* for conditional rewriting and formally characterize that difference. We then define the notion of a *weakly operationally terminating* (or *weakly normalizing*) OSRT, give several evaluation mechanisms to compute normal forms in such theories, and investigate general conditions under which the rewriting-based operational semantics and the initial algebra semantics of a confluent OSRT coincide thanks to a notion of *canonical term algebra*. Finally, we investigate appropriate conditions and proof methods to ensure good executability properties of an OSRT for computing normal forms.Research partially supported by NSF grant CNS 13-19109. Salvador Lucas’ research was developed during a sabbatical year at the CS Dept. of the UIUC and was also partially supported by Spanish MECD grant PRX12/00214, MINECO project TIN2010-21062-C02-02, and GV grant BEST/2014/026 and project PROMETEO/2011/052.Lucas Alba, S.; Meseguer, J. (2014). Strong and weak operational termination of order-sorted rewrite theories. En Rewriting Logic and Its Applications. Springer Verlag (Germany). 178-194. https://doi.org/10.1007/978-3-319-12904-4_10S17819

On Causal Equivalence by Tracing in String Rewriting

We introduce proof terms for string rewrite systems and, using these, show that various notions of equivalence on reductions known from the literature can be viewed as different perspectives on the notion of causal equivalence. In particular, we show that permutation equivalence classes (as known from the lambda-calculus and term rewriting) are uniquely represented both by trace graphs (known from physics as causal graphs) and by so-called greedy multistep reductions (as known from algebra). We present effective maps from the former to the latter, topological multi-sorting TM, and vice versa, the proof term algebra [[ ]].Comment: In Proceedings TERMGRAPH 2022, arXiv:2303.1421