A. Martelli, C. Moiso, and G.F. Rossi. Lazy unification algorithms for canonical rewrite systems. In H. At-Kaci and M. Nivat, editors, Resolution of Equations in Algebraic Structures, Vol. II, Rewriting Techniques, pages 245--274. Academic Press Press, 1989.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....means that for every solution to a given goal a solution that is at least as general is computed by the narrowing strategy. Since narrowing is a complicated operation, numerous calculi consisting of a small number of more elementary inference rules that simulate narrowing have been proposed (e.g. [8, 18, 23, 30, 17, 26, 10, 29, 22, 11]) Completeness issues for the lazy narrowing calculus lnc which is based on the calculus trans of Holldobler [18] have been extensively studied in [26] and [25] In [26] Middeldorp et al. prove that lnc is strongly complete whenever basic narrowing (Hullot [19] is complete. Strong ....

A. Martelli, C. Moiso, and G.F. Rossi. Lazy unification algorithms for canonical rewrite systems. In H. At-Kaci and M. Nivat, editors, Resolution of Equations in Algebraic Structures, Vol. II, Rewriting Techniques, pages 245--274. Academic Press Press, 1989.

....in the next step are underlined) 0 (s(0) s(z) 0 ) 0 (s(0 s(z) 0 ) 0 (s(s(z) 0 ) 0 0 0 (s(0) s(z) 0 ) 0 0 Obviously, the second lazy unification derivation should be preferred. There are many proposals for such lazy unification strategies. For instance, Martelli et al. [22] have proposed a lazy unification algorithm for confluent and terminating equational axioms. Due to the confluence requirement, equations are only applied in one direction. However, their method is not pure lazy since equations are applied to inner subterms in equations of the form x t where the ....

....have been proved to be complete w.r.t. domain based interpretations of rewrite rules [13, 23] Lazy unification is very similar to lazy narrowing but manipulates sets of equations rather than terms. It has been proved to be complete for canonical term rewriting systems w.r.t. standard semantics [6, 22]. From a practical point of view the most essential improvement of simple narrowing is normalizing narrowing [8] where the term is rewritten to its normal form before a narrowing step is applied. This optimization is important since it prefers deterministic computations: rewriting a term to ....

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A. Martelli, G.F. Rossi, and C. Moiso. Lazy Unification Algorithms for Canonical Rewrite Systems. In Hassan Ait-Kaci and Maurice Nivat, editors, Resolution of Equations in Algebraic Structures, Volume 2, Rewriting Techniques, chapter 8, pp. 245--274. Academic Press, New York, 1989.

....higher order narrowing [33] we show that LNT coincides with needed narrowing in the first order case. Note that steps of classic narrowing strategies (e.g. 14, 35] are defined as a variable instantiation followed by the reduction of some subterm, whereas more recent lazy narrowing strategies [11, 15, 20, 22] manipulate equation systems in the style of Martelli and Montanari s unification algorithm [19] and always reduce outermost function symbols instead of subterms. Thus, we present, for the first time, a needed narrowing calculus in the Martelli Montanari style and show its equivalence with the ....

A. Martelli, G.F. Rossi, and C. Moiso. Lazy unification algorithms for canonical rewrite systems. In Hassan Ait-Kaci and Maurice Nivat, editors, Resolution of Equations in Algebraic Structures, Volume 2, Rewriting Techniques, chapter 8, pages 245--274. Academic Press, New York, 1989.

.... s # 1 , s m = s # m , L 1 , L n , #) Mutate: f(s 1 , s m ) t, L 1 , L n , #) # (s 1 = l 1 , s m = l m , r = t, #) where f(l 1 , l m ) r is a renamed rule in R. 6 The above rules are a subset of the transformation rules used in [DS87, MMR89, Mit94] for semantic unification. They yield a complete forwarddecomposition calculus for solving goals of the form s = N , where N is in ground normal form, provided that the underlying TRS is convergent and either variable preserving or left linear (see [DMS92] In the next section, we present a ....

A. Martelli, C. Moiso, and G. F. Rossi. Lazy unification algorithms for canonical rewrite systems. In H. Ait-Kaci and M. Nivat, editors, Resolution of Equations in Algebraic Strucutures, pages 245--274. Academic Press, New York, 1989.

....are performed at the root only, where the unification of the left hand side of the rule with s again has to be done modulo R. Generalizing this to lazy higher order narrowing yields system LN, shown in Figure 1. It should be noted that our notion of lazy narrowing is also called lazy unification [ 7, 15 ] in the first order case. System LN essentially consists of the rules for higher order unification [ 29 ] plus the two narrowing rules. For instance, reconsider from Example 3.6 the R 0 matching problem x:H(f(x) x:h(g(x) f(x) where lazy narrowing yields fx:H 1 (f(x) x:g(x) x:H ....

A. Martelli, G. F. Rossi, and C. Moiso. Lazy unification algorithms for canonical rewrite systems. In H. Ait-Kaci and M. Nivat, editors, Resolution of Equations in Algebraic Structures, Vol. 2, Rewriting Techniques. Academic Press, 1989.

....: s m = s 0 m ; L 1 ; L n ; oe) Mutate: f(s 1 ; s m ) t; L 1 ; L n ; oe) s 1 = l 1 ; s m = l m ; r = t; oe) where f(l 1 ; l m ) r is a renamed rule in R. The above rules are a subset of the transformation rules which were used in [DS87, MMR89, Mit94] for semantic unification. They yield a complete forward decomposition calculus for solving goals of the form s = N , where N is in ground normal form, provided that the underlying TRS is convergent and either variable preserving or left linear [DMS92] In the next section, we will present a ....

A. Martelli, C. Moiso, and G.F. Rossi. Lazy unification algorithms for canonical rewrite systems. In H. Ait-Kaci and M. Nivat, editors, Resolution of Equations in Algebraic Strucutures, pages 245--274, New York, 1989. Academic Press.

....such as normalized and basic narrowing; see, for example, Fay, 1979; Hullot, 1980; Fribourg, 1985; Bosco et al. 1987; R ety, 1987; Nutt et al. 1989] Martelli and Montanari [1982] used transformations on systems of equations to describe syntactic unification. The method was later adapted in [Martelli et al. 1989] to provide a complete unification procedure for convergent rewrite systems. Furthermore, Gallier and Snyder have used transformations for describing equational and higher order unification [Gallier and Snyder, 1989; Snyder and Gallier, 1989] while Kirchner [1984] uses the technique for ....

....have used transformations for describing equational and higher order unification [Gallier and Snyder, 1989; Snyder and Gallier, 1989] while Kirchner [1984] uses the technique for unification in syntactic theories. Our method is a variant of narrowing, based on the top down approach outlined in [Martelli et al. 1989]. We achieve the effects of basic and normal narrowing in a lazy, topdown approach, and the introduction of directed goals (asymmetric goals, unlike the symmetric goals used, for example, by [Martelli et al. 1989] removes problems of generating extraneous reducible solutions. A number of ....

[Article contains additional citation context not shown here]

A. Martelli, G. F. Rossi and C. Moiso. Lazy unification algorithms for canonical rewrite systems. In H. Ait-Kaci and M. Nivat, editors, Resolution of Equations in Algebraic Structures, pages 245--274, Academic Press, New York, 1989.

.... Tenth Problem, shown to be undecidable in [ Matijasevic, 1970 ] cf. Bockmayr, 1987; Heilbrunner Holldobler, 1987 ] When a convergent R is available, a one way sort of paramodulation suffices, due to the existence of a rewrite proof for an arbitrary valid equation [ Dershowitz etal, 1987b; Martelli etal, 1989 ] The following set of rules, RU , restricts uses of equations to left hand sides of rules: Decompose: ff(s 1 ; s m ) f(t 1 ; t m )g [ P ; S) fs 1 t 1 ; s n t n g [ P ; S) Eliminate: fx sg [ P ; S) P oe; Soe [ fx = sg) if x 2 X , and x ....

A. Martelli, G. F. Rossi and C. Moiso, Lazy unification algorithms for canonical rewrite systems, in: Resolution of Equations in Algebraic Structures, H. Ait-Kaci, M. Nivat, ed., II: Rewriting Techniques, Academic Press, New York, pp. 245-274 (1989).

.... rule in R This forward decomposition calculus is a complete procedure for solving goals s = N , where N is in ground normal form, and the underlying TRS is canonical and variable preserving or left linear [DMS92] The above rules are a subset of the transformation rules which were used in [DS87, MMR89, Mit90, Mit94] for semantic unification. However, this approach does not explicitly exploit the fact that the right hand side of an initial goal has to be in ground normal form. In the next section we will present a calculus which takes more advantage of this since it starts with the result on the right hand ....

A. Martelli, C. Moiso, and G.F. Rossi. Lazy unification algorithms for canonical rewrite systems. In H. Ait-Kaci and M. Nivat, editors, Resolution of Equations in Algebraic Strucutures, pages 245--274, New York, 1989. Academic Press,.

....recursively enumerable. In other words, there exists a procedure that can find a unifier (match) whenever one exists. Such unification (matching) procedures have been studied extensively in the literature; see, for example [Fay, 1979; Hullot, 1980; Holldobler, 1987; Dershowitz and Sivakumar, 1987; Martelli et al. 1989; Gallier and Snyder, 1990; Dershowitz et al. 1990] and [Jouannaud and Kirchner, 1991] which is a survey of unification. If we restrict ourselves to convergent rewrite systems that are, additionally, either non erasing or left linear, then the non deterministic transformation rules of Table 2 ....

A. Martelli, G. F. Rossi, and C. Moiso. Lazy unification algorithms for canonical rewrite systems. In H. Ait-Kaci and M. Nivat, editors, Resolution of Equations in Algebraic Structures, volume 2: Rewriting Techniques, pages 245--274. Academic Press, New York, 1989.

....This means that we don t lose completeness when we restrict applications of the narrowing rule to a single equation in each goal. Since narrowing is not easily implemented, several authors studied calculi consisting of a small number of more elementary inference rules that simulate narrowing (e.g. [16, 8, 9, 14, 22, 6]) In this paper we are concerned with a subset (actually the specialization to confluent TRSs) of the calculus trans proposed by Holldobler [9] We call this calculus lazy narrowing calculus (lnc for short) Because the purpose of lnc is to simulate narrowing by more elementary inference rules, ....

....completeness of lnc like calculi is proved under the additional termination assumption. Without this assumption the completeness proof is significantly more involved. It is known that lnc like calculi generate many derivations which produce the same solutions (up to subsumption) Martelli et al. [16, 14] and Holldobler [9] among others, pointed out that many of these redundant derivations can be avoided by giving the variable elimination rule, one of the inference rules of lnc like calculi, precedence over the other inference rules. The problem whether this strategy is complete or not is called ....

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A. Martelli, C. Moiso, and G.F. Rossi, Lazy Unification Algorithms for Canonical Rewrite Systems, in: H. Ait-Kaci and M. Nivat, eds., Resolution of Equations in Algebraic Structures, Vol. II, Rewriting Techniques, (Academic Press, 1989) 245--274.

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