Jean-[ Hullot. Canonical forms and unification. In W. Bibel and R. Kowalski, editors, Proceedings of 5th ConferenceonAutomatedDeduction, pages 318--  Home/Search   Document Not in Database   Summary   Related Articles   Check

....definition of term graph narrowing was introduced in [46] It extends the definition in [45] in that the latter corresponds to the special case where all nodes in the set U represent the same term. The paper [46] also studies basic term graph narrowing, an analogue to basic term based narrowing [55]. Roughly speaking, this strategy forbids narrowing steps at nodes that have been created by the substitutions of previous steps. It turns out that minimally collapsing basic narrowing is complete if ) coll is innermost normalizing and confluent, and that Theorem 8.8 holds for maximally collapsing ....

Jean-Marie Hullot. Canonical forms and unification. In Proc. 5th International Conference on Automated Deduction, volume 87 of Lecture Notes in Computer Science, pages 318--334. Springer-Verlag, 1980.

....already type checked. The types of logical variables are restricted to first order types thus ensuring first order unification. 3 An Informal Introduction to Narrowing Narrowing has been introduced as a general E unification algorithm used to prove existentially quantified equational formulas [Fay79, Hul80] in equational logic. Later, a restricted algorithm has been independently proposed by [Red85, Red86] and by [DFP86] although the latter does not use the term narrowing) as the operational model for evaluating functional expressions containing free variables. It can be understood as functional ....

Jean-Marie Hullot. Canonical forms and unification. In 5th Conf. on Automated Deduction. LNCS 87, 1980.

....narrowing was introduced in [44] It extends the definition in [46] in that the latter corresponds to the special case where all nodes in the set U represent the same term. The paper [44] also 1.9. FURTHER TOPICS 53 studies basic term graph narrowing, an analogue to basic term based narrowing [55]. Roughly speaking, this strategy forbids narrowing steps at nodes that have been created by the substitutions of previous steps. It turns out that minimally collapsing basic narrowing is complete if ) coll is innermost normalizing and confluent, and that Theorem 1.8.8 holds for maximally ....

Jean-Marie Hullot. Canonical forms and unification. In Proc. 5th International Conference on Automated Deduction, volume 87 of Lecture Notes in Computer Science, pages 318--334. Springer-Verlag, 1980.

....of attempts to integrate equality into resolution based theorem provers [80] Narrowing was introduced by Slagle [86] Later, narrowing was used as a basis for semantic unification (i.e. unification modulo a set of rules) algorithms. Basic narrowing appeared for the first time in Hullot s work [57]. The completeness result for innermost narrowing in the context of canonical term rewriting systems is originally due to Fribourg [22] Holldobler s thesis [54] gives a systematic presentation of the current state of art of this field including also interesting historical references. Our ....

Jean-Marie Hullot. Canonical forms and unification. In Wolfgang Bibel and Robert Kowalski, editors, Proceedings, 5th Conference on Automated Deduction, pages 318-- 334. Springer, 1980. Lecture Notes in Computer Science, Volume 87.

....of ELP use a restricted form of paramodulation. The best known are narrowing and its refinements. Narrowing was introduced by Slagle [35] and later used as a basis for semantic unification algorithms (i.e. for unification modulo a set of rules [12] Basic narrowing was introduced by Hullot [25]. The completeness of innermost narrowing for canonical term rewriting systems is due to Fribourg [13] Holldobler s thesis [22] gives a systematic account of this field, with interesting historical references. 4.2.2 Resolution as Restricted Paramodulation. Resolution can be regarded as ....

Jean-Marie Hullot. Canonical forms and unification. In Wolfgang Bibel and Robert Kowalski, editors, Proceedings, 5th Conference on Automated Deduction, pages 318--334. Springer, 1980. Lecture Notes in Computer Science, Volume 87.

....term is still firstorder, as developed in [ 14 ] 1 Examples from other areas, e.g. formalizing logics, can be found in [ 21 ] The structure of the work is as follows. The first approach we consider is the general notion of narrowing, for which many refinements exist, e.g. basic narrowing [ 9 ] . For this, Section 3 presents an abstract view of higher order narrowing, where a problem with locally bound variables in the solutions becomes apparent. We show in Section 3.1 that the first order notion of narrowing can be lifted to higher order patterns and argue that it is problematic when ....

Jean-Marie Hullot. Canonical forms and unification. In W. Bibel and R. Kowalski, editors, Proceedings of 5th Conference on Automated Deduction, pages 318-- 334. Springer Verlag, LNCS, 1980.

....describing groups (see Example 12) Note that unification of external objects does not pose any problems since in our setting they can be treated as constants. 6. 4 Generic Equation Solving If there is a canonical term rewriting system that is equivalent to a set of equations E , narrowing [Hul80] provides a semi decision procedure for unification problems modulo E . Thus we can use narrowing based on a canonical simplification relation as an approach to a generic procedure to solve equations in the built in domain. Again decomposition freedom turns out to be important because with a ....

Jean-Marie Hullot. Canonical forms and unification. In Proc. Fifth International Conference on Automated Deduction (LNCS 87), pages 318--334. Springer-Verlag, 1980.

....applications of the above results, we mention some termination criteria for narrowing. Their combination using the above results is straightforward. If all right hand sides are (either constructor terms or) ground terms, then semantic unification is decidable, i.e. basic narrowing terminates [13]. Let R be a convergent rewrite system in which every left hand side is of the form f(t 1 ; t n ) such that each t i is either a variable or a ground term. Then narrowing terminates [6] 5.2 Modular Narrowing Strategies We now apply the results of the preceding sections to optimize ....

....n 1 ; R2 1 : R2 k n k ; where = k : 1 , and n 1 is R 1 reducible THEN stop (prune derivation) Since the used narrowing strategies are free to compute normal forms, any complete strategies can be integrated with the above optimizations. For instance, basic narrowing [13], needed narrowing [1] and LSE narrowing [4] are possible candidates. Example 3. Assume R 1 = fx 0 x; x s(y) s(x y) x ( Gammay) Gamma( Gammax y)g R 2 = f Gamma0 0; Gamma Gamma x x; s( Gammas(x) Gammaxg; Geser [9] showed that R 1 commutes over R 2 . Furthermore, it ....

Jean-Marie Hullot. Canonical forms and unification. In W. Bibel and R. Kowalski, editors, Proceedings of 5th Conference on Automated Deduction, pages 318--334. Springer Verlag, LNCS, 1980.

....model of the language is enhanced to be able to support querying a functional program in terms of constraints. Following this approach, a non ground expression may be invoked when solving a querying constraint. Therefore, the most fundamental extension to the reduction model is to use narrowing [Hul80] to compute non ground expressions. Any non ground expression stands for a set of values corresponding to all possible instantiations of its free variables. That is E [ e(x i ) j = fE [ e] jj x i g where j x i stands for an environment for free variables x i , j is the global environment ....

....ff k fi k is renamed so that #(ff k ) #(e) We call the substitution ae the output substitution and the substitution oe the input substitution. In the following presentation, we write e ; ae] e 0 for a narrowing step whenever the occurrence and the applied rule is not of interest. In [Hul80], narrowing was proved to be complete for E unification when the equational theory is defined in terms of a canonical (i.e. confluent and terminating) term rewriting system. The fundamental lemma for proving completeness is the so called lifting lemma due to Hullot [Hul80] Lemma 3.2.1 (Lifting ....

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Jean-Marie Hullot. Canonical forms and unification. In 5th Conf. on Automated Deduction. LNCS 87, 1980.

.... y) max(x; 0) x max(0; x) x max(s(x) s(y) s(max(x; y) sort(nil) nil sort(cons(x; y) insert(x; sort(y) insert(x; nil) cons(x; nil) insert(x; cons(y; z) cons(min(x; y) insert(max(x; y) z) 5 Related Results Some results similar to those given here have been reported in [Hul80, KN87], where they are interested in the more general problem of semantic unification. Hullot [Hul80] shows that the narrowing procedure terminates when all right hand sides are either variables or ground terms. Furthermore, it has been demonstrated by Kapur and Narendran [KN87] that if each right hand ....

.... insert(x; sort(y) insert(x; nil) cons(x; nil) insert(x; cons(y; z) cons(min(x; y) insert(max(x; y) z) 5 Related Results Some results similar to those given here have been reported in [Hul80, KN87] where they are interested in the more general problem of semantic unification. Hullot [Hul80] shows that the narrowing procedure terminates when all right hand sides are either variables or ground terms. Furthermore, it has been demonstrated by Kapur and Narendran [KN87] that if each right hand side is either ground or a subterm of the left hand side, the unification problem for the ....

Jean-Marie Hullot. Canonical forms and unification. In R. Kowalski, editor, Proceedings of the Fifth International Conference on Automated Deduction, pages 318--334, Les Arcs, France, July 1980. Vol. 87 of Lecture Notes in Computer Science, Springer, Berlin.

....several systems such as Elf [ 27 ] and Isabelle 3 have already resorted to higherorder patterns, where unification behaves much like the first order case. There is an interesting variety of applications where linearity is a common and sometimes also useful restriction. For instance, narrowing [ 18 ] is a general method to solve equations modulo a theory given by a term rewrite system. Then we can define a second order version of narrowing with decidable unification as long as the left hand sides of the used rules are linear patterns. This is in fact a common restriction for constructor based ....

Jean-Marie Hullot. Canonical forms and unification. In W. Bibel and R. Kowalski, editors, Proceedings of 5th Conference on Automated Deduction, pages 318--334. Springer Verlag, LNCS, 1980.

.... have been used for doing data type induction [Mus80] interpreting equational logic programs [GM86] proving theorems in first order theories [HD83] debugging specifications [GGH90] proving equivalence of algebras [Mar86] and automatically generating equational unification algorithms [Hul80] The original completion procedure was discovered by Knuth and Bendix [KB70] and has since been studied, modified, and extended. See [Buc85] for a historical survey of completion procedures, with more than 200 references that include algorithms, applications, and implementations. A careful ....

....problems can be further generalized to allow new function symbols in the rewriting systems; for example, some extension of completion allow function symbols in R that are not in E, and require only that R be a conservative extension of E . Other variations on the completion includes narrowing [Hul80] and completion modulo equations [Hue80, PS81] Although these problems fall in the class addressed by our approach, we consider only traditional completion in this thesis. All these variations on completion are unsolvable, and in fact the problem of determining whether a set of rules is ....

Jean-Marie Hullot. Canonical forms and unification. In W. Bibel and Robert A. Kowalski, editors, Proceedings of the Fifth Conference on Automated Deduction, pages 318--334. LNCS 87, Springer-Verlag, July 1980.

....form. Definition27 Narrowing. A term s narrows to a term t, symbolized s ; t, if t = s [r ] for some non variable subterm s 0 of s, renamed rule l r in R, and most general unifier of s 0 and l. By ; we denote the reflexive transitive closure of this narrowing relation. Theorem28 [48]. If R is a rewrite system, oe is an irreducible substitution (that is, xoe is irreducible for all variables x) and soe t, then there exists a term u such that s ; u and u is at least as general as t. A term u is at least as general as a term t if there is a substitution such that t = ....

Jean-Marie Hullot. Canonical forms and unification. In R. Kowalski, editor, Proceedings of the Fifth International Conference on Automated Deduction (Les Arcs, France), volume 87 of Lecture Notes in Computer Science, pages 318--334, Berlin, July 1980. Springer-Verlag.

.... an atom or equation, a computational model must verify Gamma j= 9x 1 ; x n A 1 ; A n by computing an answer substitution such that Gamma j= 8( A 1 : A n ) Such models integrate SLD resolution with some form of equational deduction such as paramodulation [ or narrowing [Hul80]. A complete computational model was proposed recently by Snyder et al. Sny90] as a goal directed inference system. Systems which aim to support the full power of Horn clause logic with equality include Eqlog [GM84] which exploits fully the order sorted variation of the logic, SLOG [Fri85] in ....

....E unification procedure which solves equations over the equality defined by the equational subprogram. Another way to restrict the computational explosiveness of general equational deduction is to use equational clauses as directed rewrite rules. A full discussion may be found in [DO88] Narrowing [Hul80] (resp. conditional narrowing [DO88] is employed to solve equations in a rewriting system (resp. conditional rewriting system) Many languages have been developed along this line, e.g. RITE [DP86] K Leaf [EGP86] They represent enhanced Prolog systems in which a rewrite relation is defined ....

Jean-Marie Hullot. Canonical forms and unification. In 5th Conf. on Automated Deduction. LNCS 87, 1980.

.... mechanisms for generating solutions, in that a solution at least as general as any that satisfies the query can always be found (solutions that are provably equal are considered to be the same) More specifically, with ground confluence, any irreducible solution to a goal can be found by narrowing [21, 25]. An irreducible solution assigns normal forms to each variable. Limiting one to irreducible solutions is justifiable in a functional setting, since the values one is looking for are always constructor terms. The orthogonal approach to equational programming leads to a lazy, outermost ....

Jean-Marie Hullot. Canonical forms and unification. In R. Kowalski, editor, Proceedings of the Fifth International Conference on Automated Deduction (Les Arcs, France), volume 87 of Lecture Notes in Computer Science, pages 318--334, Berlin, July 1980. Springer-Verlag.

....and hence is complete for equational deduction. Based on this completeness we establish the completeness of term graph narrowing for solving equations, or E unification, over convergent term graph rewriting systems. We thus obtain an analogue to Hullot s completeness result for term narrowing [3]. Corradini and Wolz [1] consider narrowing on term graphs with multiple roots, so called jungles. In their approach, unification and rewriting is based on jungle pushouts instead on substitutions, and rewriting does neither include garbage collection nor collapsing. The results in [1] aim at ....

....terms need not be equivalent if collapsing is not included in )R . 6 Narrowing The goal of the following is to solve equations by transformations on term graphs. To this end we define term graph narrowing and show a completeness result which corresponds to Hullot s result for term narrowing [3,5]. An equation s = t is a pair of terms s and t. We are interested in solutions to such equations modulo the equational theory induced by a set of equations E. We assume that E is given in form of a term rewriting system R, that is, E = fl = r j l r 2 Rg. Then s = t has a solution (or s and t ....

Jean-Marie Hullot. Canonical forms and unification. In Proc. 5th International Conference on Automated Deduction, pages 318--334. Springer Lecture Notes in Computer Science 87, 1980. 3 A term or a term graph is a normal form if it does not initiate any rewrite step. Habel and Plump

....procedure which solves equations over the equality defined by the equational subprogram. Another way to restrict the computational explosiveness of general equational deduction is to use equational clauses as directed rewrite rules. A full discussion may be found in [DO88] Narrowing [Hul80] (resp. conditional narrowing [DO88] is employed to solve equations in a rewriting system (resp. conditional rewriting system) Many languages have been developed along this line, e.g. RITE [DP86] K Leaf [EGP86] They represent enhanced Prolog systems in which a rewrite relation is defined ....

Jean-Marie Hullot. Canonical forms and unification. In 5th Conf. on Automated Deduction. LNCS 87, 1980.

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Jean-[ Hullot. Canonical forms and unification. In W. Bibel and R. Kowalski, editors, Proceedings of 5th ConferenceonAutomatedDeduction, pages 318--

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