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

A Call-by-Need Strategy for Higher-Order Functional-Logic Programming

We present an approach to truely higher-order functional-logic programming based on higher-order narrowing. Roughly speaking, we model a higherorder functional core language by higher-order rewriting and extend it by logic variables. For the integration of logic programs, conditional rules are supported. For solving goals in this framework, we present a complete calculus for higher-order conditional narrowing. We develop several refinements that utilize the determinism of functional programs. These refinements can be combined to a narrowing strategy which generalizes call-by-need as in functional programming, where the dedicated higher-order methods are only used for full higher-order goals. Furthermore, we propose an implementational model for this narrowing strategy which delays computations until needed

Two Decades of Maude

This paper is a tribute to José Meseguer, from the rest of us in the Maude team, reviewing the past, the present, and the future of the language and system with which we have been working for around two decades under his leadership. After reviewing the origins and the language's main features, we present the latest additions to the language and some features currently under development. This paper is not an introduction to Maude, and some familiarity with it and with rewriting logic are indeed assumed.Universidad de Málaga. Campus de Excelencia Internacional Andalucía Tech

Termination of rewriting strategies: a generic approach

We propose a generic termination proof method for rewriting under strategies, based on an explicit induction on the termination property. Rewriting trees on ground terms are modeled by proof trees, generated by alternatively applying narrowing and abstracting steps. The induction principle is applied through the abstraction mechanism, where terms are replaced by variables representing any of their normal forms. The induction ordering is not given a priori, but defined with ordering constraints, incrementally set during the proof. Abstraction constraints can be used to control the narrowing mechanism, well known to easily diverge. The generic method is then instantiated for the innermost, outermost and local strategies.Comment: 49 page

Guided Unfoldings for Finding Loops in Standard Term Rewriting

In this paper, we reconsider the unfolding-based technique that we have introduced previously for detecting loops in standard term rewriting. We improve it by guiding the unfolding process, using distinguished positions in the rewrite rules. This results in a depth-first computation of the unfoldings, whereas the original technique was breadth-first. We have implemented this new approach in our tool NTI and compared it to the previous one on a bunch of rewrite systems. The results we get are promising (better times, more successful proofs).Comment: Pre-proceedings paper presented at the 28th International Symposium on Logic-Based Program Synthesis and Transformation (LOPSTR 2018), Frankfurt am Main, Germany, 4-6 September 2018 (arXiv:1808.03326

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