A Finite Representation of the Narrowing Space

Abstract

Abstract. Narrowing basically extends rewriting by allowing free variables in terms and by replacing matching with unification. As a consequence, the search space of narrowing becomes usually infinite, as in logic programming. In this paper, we introduce the use of some operators that allow one to always produce a finite graph that still represents all the narrowing derivations. Furthermore, we introduce a novel, compact equational representation of the (possibly infinite) answers computed by narrowing for a given initial term. Both the finite graphs and the equational representation of the computed answers might be useful in a number of areas, like program comprehension, static analysis, program transformation, etc.

A framework for computing finite SLD trees

“…We follow a similar construction as the one in [26] for rewrite systems. First, our equational notation considers the following three operators:…”

“…Finally, some of the ideas in this paper can also be found in our previous work [37,26], where rewrite systems are considered instead. However, besides the paradigm shift, which is not trivial at all, 9 in this paper we also introduce a new, parametric notion of closed SLD tree (i.e., the notion of extended SLD tree that includes all possible closed SLD trees, which are then built using particular strategies); moreover, we also prove some new properties for our closed SLD trees and introduce novel specific strategies with different levels of abstraction for their construction.…”

On Transforming Narrowing Trees into Regular Tree Grammars Generating Ranges of Substitutions

“…From this viewpoint, for a pair of terms, the enumeration of substitutions obtained by narrowing would be useful in analyzing rewriting that starts with instances of the pair. To analyze sets of substitutions derived by innermost narrowing, narrowing trees [23] are useful. For example, infeasibility of conditional critical pairs of some normal 1-CTRS can be proved by using the grammar representation of a narrowing tree [21].…”

“…, u n . A narrowing tree [23,21] of R with the root pair s t is a finite representation that defines the set of substitutions θ such that the pair s t narrows to a particular ground term u consisting of a special binary symbol & and a special constant by innermost narrowing i R with a substitution θ (i.e., (s t) i * θ ,R u and thus θ s c → * R θt). Note that is considered a binary symbol, (x x) → is assumed to be implicitly included in R, and c → R denotes the constructor-based rewriting step which applies rewrite rules to basic terms.…”

“…Let us now consider to prove infeasibility of x < y true, y < x true. A narrowing tree for x < y true & y < x true can be represented by the following grammar representation [23,21] that can be considered an RTG (see Section 4):…”