Term rewriting systems from Church-Rosser to Knuth-Bendix and beyond

Term rewriting systems are important for computability theory of abstract data types, for automatic theorem proving, and for the foundations of functional programming. In this short survey we present, starting from first principles, several of the basic notions and facts in the area of term rewriting. Our treatment, which often will be informal, covers abstract rewriting, Combinatory Logic, orthogonal systems, strategies, critical pair completion, and some extended rewriting formats

A rationale for conditional equational programming

AbstractConditional equations provide a paradigm of computation that combines the clean syntax and semantics of LISP-like functional programming with Prolog-like logic programming in a uniform manner. For functional programming, equations are used as rules for left-to-right rewriting; for logic programming, the same rules are used for conditional narrowing. Together, rewriting and narrowing provide increased expressive power. We discuss some aspects of the theory of conditional rewriting, and the reasons underlying certain choices in designing a language based on them. The most important correctness property a conditional rewriting program may posses is ground confluence; this ensures that at most one value can be computed from any given (variable-free) input term. We give criteria for confluence. Reasonable conditions for ensuring the completeness of narrowing as an operational mechanism for solving goals are provided; these results are then extended to handle rewriting with existentially quantified conditions and built-in predicates. Some termination issues are also considered, including the case of rewriting with higher-order terms

Constrained narrowing for conditional equational theories modulo axioms

For an unconditional equational theory (Sigma, E) whose oriented equations (E) over arrow are confluent and terminating, narrowing provides an E-unification algorithm. This has been generalized by various authors in two directions: (i) by considering unconditional equational theories (Sigma, E boolean OR B) where the (E) over arrow are confluent, terminating and coherent modulo axioms B, and (ii) by considering conditional equational theories. Narrowing for a conditional theory (Sigma, E boolean OR B) has also been studied, but much less and with various restrictions. In this paper we extend these prior results by allowing conditional equations with extra variables in their conditions, provided the corresponding rewrite rules (E) over arrow are confluent, strictly coherent, operationally terminating modulo B and satisfy a natural determinism condition allowing incremental computation of matching substitutions for their extra variables. We also generalize the type structure of the types and operations in Sigma to be order-sorted. The narrowing method we propose, called constrained narrowing, treats conditions as constraints whose solution is postponed. This can greatly reduce the search space of narrowing and allows notions such as constrained variant and constrained unifier that can cover symbolically possibly infinite sets of actual variants and unifiers. It also supports a hierarchical method of solving constraints. We give an inference system for hierarchical constrained narrowing modulo B and prove its soundness and completeness. (C) 2015 Elsevier B.V. All rights reserved.We thank the anonymous referees for their constructive criticism and their very detailed and helpful suggestions for improving an earlier version of this work. We also thank Luis Aguirre for kindly giving us additional suggestions to improve the text. This work has been partially supported by NSF Grant CNS 13-19109 and by the EU (FEDER) and the Spanish MINECO under grant TIN 2013-45732-C4-1-P, and by Generalitat Valenciana PROMETEOII/2015/013.Cholewa, A.; Escobar Román, S.; Meseguer, J. (2015). Constrained narrowing for conditional equational theories modulo axioms. Science of Computer Programming. 112:24-57. https://doi.org/10.1016/j.scico.2015.06.001S245711

Nondeterminism in algebraic specifications and algebraic programs

"Nondeterminism in Algebraic Specifications and Algebraic Programs" presents a mathematical theory for the integration of three concepts: non-determinism, axiomatic specification and term rewriting. For non-deterministic programs, an algebraic specification language is provided which admits the application of automated tools based on term rewriting techniques. This general framework is used to explore connections between logic programming and algebraic programming. Examples from various areas of computer science are given, including results of computer experiments with a prototypical implementation. This book should be of interest to readers working within several fields of theoretical computer science, from algebraic specification theory to formal descriptions of distributed systems

Higher Order Unification via Explicit Substitutions

AbstractHigher order unification is equational unification for βη-conversion. But it is not first order equational unification, as substitution has to avoid capture. Thus, the methods for equational unification (such as narrowing) built upon grafting (i.e., substitution without renaming) cannot be used for higher order unification, which needs specific algorithms. Our goal in this paper is to reduce higher order unification to first order equational unification in a suitable theory. This is achieved by replacing substitution by grafting, but this replacement is not straightforward as it raises two major problems. First, some unification problems have solutions with grafting but no solution with substitution. Then equational unification algorithms rest upon the fact that grafting and reduction commute. But grafting and βη-reduction do not commute in λ-calculus and reducing an equation may change the set of its solutions. This difficulty comes from the interaction between the substitutions initiated by βη-reduction and the ones initiated by the unification process. Two kinds of variables are involved: those of βη-conversion and those of unification. So, we need to set up a calculus which distinguishes between these two kinds of variables and such that reduction and grafting commute. For this purpose, the application of a substitution of a reduction variable to a unification one must be delayed until this variable is instantiated. Such a separation and delay are provided by a calculus of explicit substitutions. Unification in such a calculus can be performed by well-known algorithms such as narrowing, but we present a specialised algorithm for greater efficiency. At last we show how to relate unification in λ-calculus and in a calculus with explicit substitutions. Thus, we come up with a new higher order unification algorithm which eliminates some burdens of the previous algorithms, in particular the functional handling of scopes. Huet's algorithm can be seen as a specific strategy for our algorithm, since each of its steps can be decomposed into elementary ones, leading to a more atomic description of the unification process. Also, solved forms in λ-calculus can easily be computed from solved forms in λσ-calculus

Twenty years of rewriting logic

AbstractRewriting logic is a simple computational logic that can naturally express both concurrent computation and logical deduction with great generality. This paper provides a gentle, intuitive introduction to its main ideas, as well as a survey of the work that many researchers have carried out over the last twenty years in advancing: (i) its foundations; (ii) its semantic framework and logical framework uses; (iii) its language implementations and its formal tools; and (iv) its many applications to automated deduction, software and hardware specification and verification, security, real-time and cyber-physical systems, probabilistic systems, bioinformatics and chemical systems