P. Devienne. Weighted graphs: A tool for studying the halting problem and time complexity in term rewriting systems and logic programming. Journal of Theoretical Computer Science, 75:157--215, 1990.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....to be complete. Unfortunately, answers to the generalized goal need not to be answers to the initial goal. M. Schmidt Schau [16] has shown that cycle unification is decidable provided that the goal and the fact are ground, i.e. they do not contain variable occurrences. Independently, P. Devienne [7] has given a more general result for cycle unification problems with linear goals and facts, i.e. each variable occurs at most once in the goal and the fact. He uses essentially the same ideas as Schmidt Schau, but a very special technique based on directed weighted graphs. Devienne s results were ....

....are decidable and finitary, and have specified a minimal and complete unification algorithm for these classes. Table 1 gives an overview of these results as well as of previous work. In each row 13 Class Decidability Type Algorithm References C open infinitary open C l decidable infinitary open [7] C g decidable unitary yes [16] Cm open infinitary open C nr decidable finitary yes this paper Cu decidable finitary yes [22] Table 1: Properties of cycle unification classes. C C l C g Cm C nr Cu Figure 2: The relation between the classes C , C l , C g , Cm , C nr , and Cu . we state ....

P. Devienne. Weighted graphs: A tool for studying the halting problem and time complexity in term rewriting systems and logic programming. Journal of Theoretical Computer Science, 75:157--215, 1990.

....be complete. Unfortunately, answers to the generalized goal need not to be answers to the initial goal. M. Schmidt Schau [SS88] has shown that cycle unification is decidable provided that the goal and the fact are ground, i.e. they do not contain variable occurrences. Independently, P. Devienne [Dev90] has given a more general result for cycle unification problems with linear goals and facts, i.e. each variable occurs at most once in the goal and the fact. He uses essentially the same ideas as Schmidt Schau, but a very special technique based on directed weighted graphs. Devienne s results ....

....classes are related as shown in Figure 10. Our most general result concerns the class of unifying cycles. For this class we have shown that we only have to consider finitely many iterations 28 Class Decidability Type Algorithm References C open infinitary open C l decidable infinitary open [Dev90] C g decidable unitary yes [SS88] Cm open infinitary open C nrm decidable finitary yes [BHW91] C u decidable finitary yes in this paper Table 1: Properties of cycle unification classes C C l C g Cm C nrm C u Figure 10: The relation between the classes C , C l , C g , Cm , C nrm , and C ....

P. Devienne. Weighted graphs: A tool for studying the halting problem and time complexity in term rewriting systems and logic programming. Journal of Theoretical Computer Science, 75:157-- 215, 1990.

....as the fact are ground, they define upper bounds on the depth of the terms occurring in the binary rule. After a finite sequence of self application steps, any further self application either leaves the rule invariant or increases the depth of the terms occurring in it. Independently, P. Devienne [Dev90] has given a more general result for programs with linear goals and facts, i.e. each variable occurs at most once in the goal and the fact. He uses essentially the same ideas as Schmidt Schau, but a specialized technique based on directed weighted graphs. Both do not A Horn clause is a ....

....non terminating queries [SGG86] As a first step research in the special case of binary clauses was started. Much effort was devoted to control the self applicability of binary clauses in order to detect non terminating queries and to speed up the computation in terminating cases; cf. for instance [Dev90], DVB90] or [UvG88] 2 Reduction For basic notions such as substitution, unification, SLD resolution from logic programming we refer to [Llo87] The undecidability of the satisfiability of the smallest binary program is shown by reducing the Post Correspondence Problem to it. First, we recall ....

P. Devienne. Weighted graphs: A tool for studying the halting problem and time complexity in term rewriting systems and logic programming. Journal of Theoretical Computer Science, 75:157-- 215, 1990.

....termination proof process [Nai83, UVG88, APP 89, BS89b, DSVB90, Plu90a, Plu90b, Sag91, SVG91] Another approach concerns the characterization of terminating logic programs. It aims at the treatment of negation as finite failure or the better understanding of decidability issues [AP90, AB91, Dev90] These rather theoretical works usually provides manually verifiable criteria for termination. Undecidability of the halting problem, a classical result of theoretical computer science, states that it is undecidable to determine whether or not any program terminates. The proposed method of this ....

P. Devienne. Weighted graphs: a tool for studying the halting problem and time complexity in term rewriting systems and logic programming. Theoretical Computer Science, 75(2):157--215, 1990.

....are decidable when goal and fact are ground . This result is a corollary of his work on the implication of clauses, or equivalently on the decision problem of clause sets consisting of one clause and some ground units (one literal clause) see also [36] M. Dauchet, P. Devienne and P. Leb egue [11, 17] studied the linear case and proved it decidable as well. They used a new technic based on weighted directed graph (an extension of the directed graphs) W. Bibel, S. Holldobler and J. Wurtz [2] considered the emptiness problem and proved it decidable for some particular cases . They denoted ....

....the Conway relation j g , associated with Sigma, is so. 6. THE HALTING PROBLEM In this section, we will provide the answer to the first problem : does the resolution of a binary recursive Horn clause when given a goal halt While it has been established decidable in the ground [47] and linear [17] goal case, we will establish here the undecidability in the general case [19] Already a right linear rule is sufficient. In order to complete the answer, we will show the decidability if the rule is left linear. 17 6.1. The General Case Theorem 6.1. The halting problem, according to ....

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Devienne P. "Weighted graphs -- tool for studying the halting problem and time complexity in term rewriting systems and logic programming." Journal of Theoretical 75, pp. 157--215. 1990.

....: finite or infinite computation ; null, finite or infinite number of solutions. The computational power of this class is also interesting. M. Schmidt Schauss [17] has shown that the two problems are decidable when goal and fact are ground 1 . M. Dauchet, P. Devienne and P. Leb egue [4] [6] studied the linear 2 case and proved it decidable as well. W. Bibel, S. Holldobler and J. Wurtz [2] have considered the emptiness problem and have proved it decidable for some particular cases (see also [16, 18] In [7] we have proved the halting problem to be undecidable in the general case. ....

....program depending on the linearity of the terms goal, left, right and fact. We prove that the halting problem becomes decidable as soon goal or left are linear. The emptiness problem remains undecidable in the linear Horn clause case. The proof in the first case is based on the weighted graphs [4, 5, 12, 6]. In the second case, we use the same method as for Theorem 5.1, we simply transform any append like program in an equivalent one by linearizing the Horn clause. We are not going to give the detailed proofs here, they will appear soon in a extended report and can be actually communicated to all ....

[Article contains additional citation context not shown here]

Devienne P. "Weighted graphs -- tool for studying the halting problem and time complexity in term rewriting systems and logic programming. " Journal of Theoretical Computer Science, n o 75, pp. 157--215. 1990.

....for which no decision algorithm, given a goal, always decides in a finite number of steps whether or not the resolution using this clause is finite. The halting problem of derivations w.r.t. one binary Horn clause had been shown decidable if the goal is ground [SS88] or if the goal is linear [Dev88, Dev90, DLD90]. The undecidability in the non linear case is an unexpected extension. The proof of the main result is based on the unpredictable iterations of periodically linear functions defined by J.H. Conway within number theory. Let us note that these new undecidability results are proved w.r.t. any type ....

Devienne P. "Weighted graphs -- tool for studying the halting problem and time complexity in term rewriting systems and logic programming." Journal of Theoretical Computer Science, n o 75, pp. 157--215. 1990.

....for which no decision algorithm, given a goal, always decides in a finite number of steps whether or not the resolution using this clause is finite. The halting problem of derivations w.r.t. one binary Horn clause had been shown decidable if the goal is ground [SS88] or if the goal is linear [Dev88, Dev90, DLD90]. The undecidability in the non linear case is an unexpected extension. The proof of the main result is based on the unpredictable iterations of periodically linear functions defined by J.H. Conway within number theory. Let us note that these new undecidability results are proved w.r.t. any type ....

Devienne P. "Weighted graphs -- tool for studying the halting problem and time complexity in term rewriting systems and logic programming (extended abstract)." Fifth Generation Computer Systems 88, Tokyo, Japan. 1988.

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