Marcinkowski, J. and Pacholski, L. (1992). Undecidability of the Horn clause implication problem. In Proceedings of the 33rd Annual IEEE Symposium on Foundations of Computer Science (FOCS'92), pages 354--362. IEEE Computer Society Press.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....obtained by negating of clause B . Hence, the result of M. Schmidt Schau cited in Section 1 is an implication of his result that clause implication A ) B is decidable in case of A being a binary clause. If A contains four or more literals, the problem is undecidable. Marcinkowski and Pacholski [Mar92] have recently shown that clause implication is undecidable in case of L 1 R 11 ; R 12 = L 2 R 21; R 2k . The subject of [BHW92] and [Wur92] is cycle unification. Therein, a binary clause which can be applied to itself is called a cycle. In order to be able to control a cycle ....

J. Marcinkowski and L. Pacholski. Undecidability of the Hornclause implication problem. FOCS, 1992.

....[47] has shown that the two problems 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 ....

....by M. Schmidt Schau, who proved also it to be decidable. He had also shown that it becomes undecidable if A is a four literal clause [47] Later, J. Marcinkowski and L. Pacholski proved the three literal case (n = 2) to be undecidable as well. They proved this result for Horn clauses [36, 35]. Now let us consider that n = 2 and m = 1. Then the class of programs to be satisfied becomes : 8 B 1 : A A 1 ; A 2 : B : which is clearly close to the studied structure. We are optimistic that the results and or methods of the previous sections can help to establish the status ....

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Marcinkowski J., Pacholski L. "Undecidability of the Horn--Clause Implication Problem." FOCS 1992.

.... p(right 1 ) p(right 2 ) p(goal) where fact and goal are ground. The existence of solutions for this program is equivalent to the implication of clause A ) B. Where A is lef t right 1 ; right 2 and B is fact goal. The problem has been shown undecidable for the case where B is any in [15]. We hope our technique can solve this particular case. Acknowledgements : We would like to thank Prof. Jean Paul Delahaye, the basic idea of proof of Theorem 3.1 is due to him. ....

Marcinkowski J., Leszek Pacholski "Undecidability of the Horn--Clause Implication Problem" Proc. of the 33rd FOCS. 1992.

....studied. For instance, Implication of Horn clauses, denoted HC 1 ) HC 2 , also called generalized subsumption, is an important notion in learning theory and compilation. It was shown decidable if clause HC 1 is binary (two literal) 19] and has been shown recently undecidable if HC 1 is ternary [15]. More precisely, our paper is devoted to establish that all computation on Minsky machines can be expressed as the checking of consistency of the 4 formulas of the following form : 8x 1 Delta Delta Delta 8xm [A 1 (A 2 :A 3 ) A 4 ] that is 8 : A 1 : A 2 A 3 : A 4 : where, ....

....infer. The open problem concerning the total decoration property can be proved easily to be undecidable. Indeed for logic programs which contain only one binary clause, this property is equivalent to the non halting problem that we have shown to be undecidable. Another program scheme studied by [15] for implication of clause is the following one : 8 : G 1 : Delta Delta Delta Delta Delta Delta Gn : A A 1 ; A 2 : Gn 1 : where all G i are ground (without variables) and the Horn clause is now ternary. They proved, using a sophisticated technique ....

Marcinkowski J., Leszek Pacholski "Undecidability of the Horn--Clause Implication Problem" Proc. of the 33rd FOCS. 1992.

....and SLD resolution. In addition ILP has heavily utilized such theoretical results from computational logic as Lee s Subsumption Theorem [18] Gottlob s Lemma linking implication and subsumption [12] Marcinkowski and Pacholski s result on the undecidability of implication between de nite clauses [22], and many others. In addition to utilizing such theoretical results, ILP depends crucially on important advances in logic programming implementations. For example, many of the applications summarized in the next brief section were possible only because of fast deductive inference based on ....

J. Marcinkowski and L. Pacholski. Undecidability of the horn-clause implication problem. In Proceedings of the 33rd IEEE Annual Symposium on Foundations of Computer Science, pages 354-362. IEEE, 1992.

....general than D relative to background knowledge # ## is a set of clauses#, iffCg## logically implies D. Of these three orders, subsumption is the most tractable. In particular, subsumption is decidable, whereas logical implication is not decidable, not even for Horn clauses, as established by Marcinkowski and Pacholski #1992#. In turn, relative implication is harder than implication: both are undecidable, but proof procedures for implication need to take only derivations from fCg into account, whereas a proof procedure forrelative implication should check all derivations from fCg##. Within a generality order, there ....

Marcinkowski, J., & Pacholski, L. #1992#. Undecidability of the horn-clause implication problem. In Proceedings of the 33rdAnnual IEEE Symposium on Foundations of Computer Science, pp. 354#362 Pittsburg.

....clauses. For instance, the clause d above and the clause d 0 = p(f(f(f(X) p(X) have both c and the clause p(f(X) p(Y ) as least generalisations. Although Niblett claims that implication between Horn clauses is decidable, this has since been shown to be false by Marcinkowski and Pacholski [70]. Gottlob [42] also proves a number of properties concerning implication between clauses. Notably let c ; c Gamma be the positive and negative literals of c and d ; d Gamma be the same for d. Now if c j= d then c subsumes d and c Gamma subsumes d Gamma . 5.5.1. ....

J. Marcinkowski and L. Pacholski. Undecidability of the horn-clause implication problem. In Proceedings of the 33rd IEEE Annual Symposium on Foundations of Computer Science, pages 354--362. IEEE, 1992.

....clauses. In this setting, the word explain is interpreted as either subsume (denoted by ) or imply (denoted by j= In the latter case, note that the problem of whether or not a definite clause C implies another definite clause, called an implication problem, is undecidable in general [8]. On the other hand, if C is function free, then it is obvious that the implication problem is decidable. Flattening, which has been first introduced in the context of Inductive Logic Programming by Rouveirol [14] though similar ideas had already been used in other fields) is a 3 This work is ....

....propositional logic such as [1, 3] are directly applied to Inductive Logic Programming. On the other hand, if the converse holds, then the implication problem 5 j= D is decidable, because flat(5) and flat(D) are function free. However, it contradicts the undecidability of the implication problem [8, 15] or the satisfiability problem [5] In this paper, we show that the converse does not hold even if 5 = fCg, that is, there exist definite clauses C and D such that: C j= D but flat (C) 6j= flat(D) Furthermore, we investigate the conditions of C and D satisfying that C j= D if and only if flat ....

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Marcinkowski, J. and Pacholski, L.: Undecidability of the Horn-clause implication problem, Proc. 33rd Annual IEEE Symposium on Foundations of Computer Science, 354--362, 1992.

....p(X) and p(f(X) p(Y) as least generalisations. In [61] the existence and computability of a least generalisation under implication for any finite set of clauses that contains at least one non tautologous function free clause is proven. Since implication between Horn clauses is undecidable [53], different models of generality have been proposed. We here mainly discuss the generality models that are actually used in the overviewed special purpose techniques that are dedicated to the inductive synthesis of recursive logic programs, even though they are the weaker models. 2.3.1 ....

J. Marcinkowski and L. Pacholski. Undecidability of the Horn-clause implication problem. In Proc. of the 33rd IEEE Annual Symposium on Foundations of Computer Science, pp. 354--362. 1992.

....relative to background knowledge Sigma ( Sigma is a set of clauses) if fCg [ Sigma logically implies D. Of these three orders, subsumption is the most tractable. In particular, subsumption is decidable, whereas logical implication is not decidable, not even for Horn clauses, as established by Marcinkowski and Pacholski (1992). In turn, relative implication is harder than implication: both are undecidable, but proof procedures for implication need to take only derivations from fCg into account, whereas a proof procedure for relative implication should check all derivations from fCg [ Sigma. Within a generality order, ....

Marcinkowski, J., & Pacholski, L. (1992). Undecidability of the horn-clause implication problem. In Proceedings of the 33rd Annual IEEE Symposium on Foundations of Computer Science, pp. 354--362 Pittsburg.

....implication (LGIs) instead. Accordingly, we want to find out whether Plotkin s positive result on the existence of LGSs holds for LGIs as well. Most ILP researchers are inclined to believe that this question has a negative answer, due to the undecidability of logical implication between clauses [8]. If we restrict attention to Horn clauses (clauses with at most one positive literal) the question has indeed been answered negatively: there is no least Horn clause which implies both P (f 2 (x) P (x) and P (f 3 (x) P (x) 10] However, Muggleton and Page [12] have shown that the ....

J. Marcinkowski and L. Pacholski. Undecidability of the horn-clause implication problem. In Proceedings of the 33rd Annual IEEE Symposium on Foundations of Computer Science, pages 354--362, Pittsburg, 1992.

....by Schmidt Schauss (1988, page 294) Theorem 4 (Undecidability of implication between clauses) Let C and D be clauses. Then there exists no procedure to decide if C ) D. Niblett (1988) has claimed that implication between Horn clauses is decidable. This result has later been proved to be false (Marcinkowski Pacholski, 1992). The definition of a least general generalization under implication (LGGI) follows the definition of an LGG . Generalization of Clauses under Implication Definition A clause C is a generalization under implication of a set of clauses S = fD 1 ; D n g if and only if, for every 1 i n, ....

Marcinkowski, J., & Pacholski, L. (1992). Undecidability of the horn clause implication problem. In Proceedings of the Thirtythird Annual IEEE Symposium on Foundations of Computer Science, pp. 354--362 Pittsburg, Pennsylvania.

....The notion of the regular tree was inspired by an analysis of the properties of the prefix closure of the Thue closure of a set of strings as well as by the analysis of the derivation trees of inference rules defined by a Horn clause. The elements of the technique of Thue trees were announced in [42] where we proved undecidability of the Horn clause implication problem and where we also expresses our belief that this technique may have further applications. Here we present the proof of the Horn clause implication problem as well as other applications of the technique: to strengthen the result ....

....precise definitions) The strategy of proving undecidability results adapted in this paper is to reduce the problem of existence of a finite regular tree to a problem we consider. This research was partially supported by KBN grant 8 T11C 029 13. The results presented here have been announced in [42, 39, 40, 41]. Then, when a reduction was found, the result follows by our main technical result stating undecidability of the existence of finite regular trees. The latest result is obtained in Section 2 using variants of the theorem on E. Post [46] on the undecidability of finiteness of Thue closure, which ....

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J. Marcinkowski and L. Pacholski. Undecidability of the Horn clause implication problem. In Proceedings of 33rd Annual IEEE Symposium on the Foundations of Computer Science, pages 354--362, Los Alamitos, 1992.

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Marcinkowski, J. and Pacholski, L. (1992). Undecidability of the Horn clause implication problem. In Proceedings of the 33rd Annual IEEE Symposium on Foundations of Computer Science (FOCS'92), pages 354--362. IEEE Computer Society Press.

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Marcinkowski, J., L. Pacholski, Undecidability of the Horn clause implication problem. Proceedings of the 33 Annual IEEE Symposium on Foundations of Computer Science, Pittsburgh, 1992. 354-362.

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