Lazy narrowing: Strong completeness and eager variable elimination (extended abstract)

Abstract

Narrowing is an important method for solving unification problems in equational theories that are presented by confluent term rewriting systems. Because narrowing is a rather complicated operation, several authors studied calculi in which narrowing is replaced by more simple inference rules. This paper is concerned with one such calculus. Contrary to what has been stated in the literature, we show that the calculus lacks strong completeness, so selection functions to cut down the search space are not applicable. \u prove completeness of the calculus and we establish an interesting connection between its strong completeness and the completeness of basic narrowing. We also address the eager variable elimination problem. It is known that many redundant derivations can be avoided if the variable elimination rule, one of the inference rules of our calculus, is given precedence over the other inference rules. We prove the completeness of a restricted variant of eager variable elimination in the case of orthogonal term rewriting systems.

A rewriting logic for declarative programming

“…The rest of the rules are completely deterministic (modulo the symmetry of ~) and, what is more important, if several H--rules are applicable to a given goal, a don't care choice among them can be done, as a consequence of the Progress Lemma 21 below. This kind of strong completeness doesn't hold in general for other lazy narrowing calculi, as shown in [21].…”

“…those in [5,20,1,11,21], completeness of our calculus holds without any confluence or non-ambiguity hypothesis, even for conditional rewrite systems with extra variables in the conditions, which may cause incompleteness of narrowing w.r.t, the semantics of equational logic [19]. To gain this generality, we restrict ourselves to left-linear, constructor-based rewrite rules (which are expressive enough for programming), and we replace equational logic by a rewriting logic.…”

Análisis semántico y transformación de programas lógico-funcionales.

“…La completitud fuerte significa completitud independientemente de la función de selección de ecuaciones empleada [Höl89,OMI95], i.e. no se pierde completitud cuando se restringen las aplicaciones de la regla de narrowing a una sola ecuación en cada paso (las ecuaciones del objetivo se pueden seleccionar de forma indeterminista dont't care).…”

“…Enunciamos, en primer lugar, una serie de definiciones y lemas previos. La siguiente definición introduce una noción de complejidad asociada con una derivación, similar a la que puede encontrarse en [OMI95].…”

“…El resultado se prueba, de forma inmediata, siguiendo las líneas de la demostración de la completitud fuerte del narrowing ordinario respecto a respuestas normalizadas [OMI95], explotando el hecho de que el término asociado a la variable nueva x en θ no se reduce por narrowing en D .…”