J-P. Jouannaud and M. Okada. Satisfiability of systems of ordinal notations with the subterm property is decidable. In 18th International Colloquium Automata, Languages and Programming (ICALP), LNCS 510, pages 455--468, Madrid, Spain, July 16--20 1991. Springer-Verlag.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....the Spanish Cicyt project HEMOSS, ref. TIC98 0949C02 01. that can be used for deduction with other clauses containing arbitrary new (e.g. Skolem) symbols. The satisfiability problem for ordering constraints was first shown decidable for fixed signatures when is a total LPO [1] or a total RPO [7]. For extended signatures, decidability was shown for LPO in [17] and for RPO in [14] Regarding complexity, NP algorithms for LPO (fixed and extended signatures) and RPO (extended ones) were given in [14] More recently, an NP algorithm has been given as well for RPO under fixed signatures in ....

....correspond to the natural number fragment, can then be dealt with independently. This problem is solved for the case f 2 lex again by a transitivity closure, but now over the natural number ordering. For the case where f is not unary and has multiset status, we rely on the existing methods of [7, 13] for dediding the satisfiability of multiset constraints on natural numbers. An improvement with respect to the earlier short version of this work [15] is that here we introduce from the beginning the additional predicate in the constraint language, which leads to a better performance, since ....

[Article contains additional citation context not shown here]

J-P. Jouannaud and M. Okada. Satisfiability of systems of ordinal notations with the subterm property is decidable. In 18th International Colloquium Automata, Languages and Programming (ICALP), LNCS 510, pages 455--468, Madrid, Spain, July 16--20 1991. Springer-Verlag.

....recall that lexicographic path ordering (lpo) and the recursive path ordering and many other orderings such as [13, 10] coincide in the unary case. Among the positive results it is known that the existential theory of total lpo is decidable [3, 17] The same result holds for the case of total rpo [8, 15]. The proof technique we use for our decidability result might be interesting by itself. It relies on encoding of words as trees and then on building a tree automaton to recognize the rpo relation. Key words: Recursive path ordering, first order theory, ground reducibility, tree automata, ordered ....

J.-P. Jouannaud and M. Okada. Satisfiability of systems of ordinal notations with the subterm ordering is decidable. In 18th International Colloquium on Automata, Languages and Programming (ICALP), volume 510 of Lecture Notes in Computer Science, pages 455-468, Madrid, Spain, July 1991. Springer-Verlag.

....new (e.g. Skolem) symbols, but it is less restrictive and hence less powerful for refutational theorem proving. The satisfiability problem for ordering constraints was first shown decidable for the well known recursive path orderings (RPO) introduced by N. Dershowitz [Der82] for fixed signatures [Com90,JO91] and extended ones [NR95,Nie93] NP algorithms (fixed and extended signatures) were given in [Nie93,NRV99] For the Knuth Bendix ordering (KBO) this result has only recently been obtained (for fixed signatures) in [KV00] Ordered strategies and ordering constraint inheritance can be used without ....

J-P. Jouannaud and M. Okada. Satisfiability of systems of ordinal notations with the subterm property is decidable. In 18th International Colloquium Automata, Languages and Programming (ICALP), LNCS 510, pages 455--468, Madrid, Spain, July 16--20 1991. Springer-Verlag.

....of saturated sets of ordering constrained clauses that can be used for deduction with other clauses containing arbitrary new (e.g. Skolem) symbols. The satisfiability problem for ordering constraints was first shown decidable for fixed signatures when is a total LPO [Com90] or a total RPO [JO91]. For extended signatures, decidability was shown for LPO in [NR95] and for RPO in [Nie93] Regarding complexity, NP algorithms for LPO (fixed and extended signatures) and RPO (extended ones) were given in [Nie93] More recently, an NP algorithm has been given as well for RPO under fixed ....

....correspond to the natural number fragment, can then be dealt with independently. This problem is solved for the case f 2 lex again by a transitivity closure, but now over the natural number ordering. For the case where f is not unary and has multiset status, we rely on the existing methods of [JO91,NRV99] for dediding the satisfiability of multiset constraints on natural numbers. An improvement with respect to the earlier short version of this work [NR99] is that here we introduce from the beginning the additional predicate in the constraint language, which leads to a better performance, since ....

[Article contains additional citation context not shown here]

J-P. Jouannaud and M. Okada. Satisfiability of systems of ordinal notations with the subterm property is decidable. In 18th International Colloquium Automata, Languages and Programming (ICALP), LNCS 510, pages 455-- 468, Madrid, Spain, July 16--20 1991. Springer-Verlag.

.... Delta Delta C 0 ) then it is regarded as a tree hAn ; A 1 ; A 0 i labeled by on the root. Note that for the nodes labeled by or Theta there are no orders on immediate successors, while for the nodes labeled by there are orders. Following the recursive path ordering with status [24, 20], we define a well order on the class of algebras incorporating both multiset path ordering and lexicographic path ordering depending on the labels on the nodes. The precedence order of the labels is ; OE 1 OE OE Theta OE . In the following definition, the term subalgebra is used to denote the ....

J.-P. Jouannaud and M. Okada, Satisfiability of systems of ordinal notations with the subterm property is decidable, preprint.

....recursive path order or a Knuth Bendix order, where compatible with an order means that l r for all rules l r. For recursive path order this is can be derived directly from the definition; for Knuth Bendix order a decision procedure has been described in Dick et al. 1990) In Comon (1990) Jouannaud and Okada (1991) it has been proved that it is also decidable whether l oe r oe for all rules l r and all ground substitutions oe, where is any recursive path order. The following implications hold for finite signatures: compatible with a recursive path order or a Knuth Bendix order total ....

J.-P. Jouannaud and M. Okada. Satisfiability of systems of ordinal notations with the subterm property is decidable. In Proceedings of the 18th International Colloquium on Automata, Languages and Programming (ICALP91), volume 510 of Lecture Notes in Computer Science, pages 455--468. Springer, 1991.

.... are built over a given signature F) and extended signature semantics (new symbols are allowed to appear in solutions) Nieuwenhuis and Rubio 1992b] The satisfiability problem for ordering constraints was first shown decidable for fixed signatures when is a total LPO [Comon 1990] or a total RPO [Jouannaud and Okada 1991]. For extended signatures, decidability was shown for LPO in [Nieuwenhuis and Rubio 1995] and for RPO in [Nieuwenhuis 1993] Regarding complexity, NP algorithms for LPO (fixed and extended signatures) and RPO (extended ones) were given in [Nieuwenhuis 1993] Recently, an NP algorithm has been ....

Jouannaud J.-P. and Okada M. [1991], Satisfiability of systems of ordinal notations with the subterm property is decidable, in `18th International Colloquium Automata, Languages and Programming (ICALP)', LNCS 510, Springer-Verlag, Madrid, Spain, pp. 455--468.

....partially supported by the ESPRIT Basic Research WG CCL II. F) and extended signature semantics (new symbols are allowed to appear in solutions) NR95] The satisfiability problem for ordering constraints was first shown decidable for fixed signatures when is a total LPO [Com90] or a total RPOS [JO91]. For extended signatures, decidability was shown for LPO in [NR95] and for RPO in [Nie93] Regarding complexity, NP algorithms for LPO (fixed and extended signatures) and RPO (extended ones) were given in [Nie93] Very recently, an NP algorithm has been given as well for RPO under fixed ....

....that is, constraints built only over f , 0, and possibly other constants smaller than f , and with solutions over this same signature. If f 2 lex this is a simple problem over the natural numbers, but for f 2 mul this seems not the case. Hence for the moment we propose to use the algorithm of [JO91] or the NP one of [NRV98] for SN , which is normally a minor part of S. The Lexicographic Case Let (F ; F ) be such that f F a 1 F : F an F 0, where the smallest non constant symbol f is in lex (the case with at most one constant below f has been treated already in Subsection 6.2) Then a ....

J-P. Jouannaud and M. Okada. Satisfiability of systems of ordinal notations with the subterm property is decidable. In Proc 18th ICALP, LNCS 510, Madrid, Spain, July 16--20 1991. Springer-Verlag.

....on symbolic constraint solving, an essential ingredient for deduction with clauses with constraints. The problem of deciding satisfiability of ordering constraints over the Lexicographic Path Ordering (LPO) is solved in [227] this is extended to the Recursive Path Ordering with Status (RPOS) in [521]. The practical feasibility of these algorithms, and some simpler subcases of the problems are addressed in [721] and in [717] NP completeness of several cases is shown. For ordering constraint solving in the case where axioms for associativity and commutativity (AC) are built in into the logic, ....

J.-P. Jouannaud and M. Okada. Satisfiability of systems of ordinal notations with the subterm property is decidable. In Automata, Languages and Programming, 18th International Colloquium, Madrid, Spain, July 1991. Springer LNCS 510. \Phi.

....total on ground terms. A typical example of such an ordering is the lexicographic path ordering extending a total precedence, whose existential fragment has been shown decidable [11] it is actually NP complete [38] This result has been extended to other total recursive path (quasi )orderings [31]. The decidability of the full first order theory of these orderings is an open question (problem 24 in [21] ffl The theory of partial recursive path orderings appears to be even more difficult; the Sigma 4 fragment has been shown undecidable [44] The decidability of the existential fragment ....

J.-P. Jouannaud and M. Okada. Satisfiability of systems of ordinal notations with the subterm property is decidable. In Proc. 18th Int. Coll. on Automata, Languages and Programming, Madrid, LNCS 510, 1991.

....freely interpreted and is interpreted as a wellfounded monotonic term ordering which is total on ground terms. Typical examples of such orderings are the total recursive path orderings (RPO) introduced by N. Dershowitz [6] The existential fragment of the theory of each RPO was shown decidable [4, 8] and actually NP complete [13] Ordered strategies and ordering constraints inheritance can be used in presence of AC symbols without loosing completeness [15, 19] This requires first an ordering which, in addition to former requirements, is also AC compatible. Then we need a constraint solving ....

....occur in the original constraint, like in the example above. We believe, however, that the following theorem also holds for the other cases, if one first detects and eliminates constraints that are unsatisfiable because of the lack of room between two terms, like it is done for RPO constraints in [5, 8, 13] (for instance in the previous example there is no value for x between two successor terms, like in f(0; h(0) x f(0; 0) Theorem 7.2 Let F be a precedence with smallest non constant and constant h and 0 resp. such that h = 2 AC and there is no g 6= 0 with h F g, and suppose C is a ....

J.-P. Jouannaud and M. Okada. Satisfiability of systems of ordinal notations with the subterm property is decidable. In Automata, Languages and Programming, 18th International Colloquium, LNCS 510, Madrid, Spain, July 16--20 1991. Springer-Verlag.

....this area was the proof by H. Comon that the Lexicographic Path Ordering (LPO) constraint solving problem is decidable [1] Subsequently R. Nieuwenhuis [14] showed that the problem is NP complete. In the case of RPO, the decidability of constraint solving was proved by J.P. Jouannaud and M. Okada [8]. In fact, they gave an algorithm for a general ordering combining both RPO and LPO. The RPO case is also considered by [10] where a polynomial bound on the depths of the terms in a solution is derived. This is the only upper bound that has been given for RPO constraint solving so far, as far as ....

....This is the only upper bound that has been given for RPO constraint solving so far, as far as we know; however, it does not help us show that the solvability problem is in NP. In this paper we show it to be in NP for the first time, and therefore NP complete since it is NP hard by [14] Unlike [8] our algorithm does not use any predecessor function. Such a function creates big terms in the solving process and it is unlikely that the algorithm in [8] could be modified into a NP one. Note also that the proof that LPO constraint solving is NP cannot be adapted for RPO since it relies heavily ....

[Article contains additional citation context not shown here]

J.-P. Jouannaud and M. Okada. Satisfiability of systems of ordinal notations with the subterm ordering is decidable. In 18th International Colloquium on Automata, Languages and Programming (ICALP), volume 510 of Lecture Notes in Computer Science, pages 455-468, Madrid, Spain, July 1991. Springer-Verlag.

....total on ground terms as permuting the direct subterms of a function symbol whose status is multiset leads to incomparable terms. However, modulo such permutations, the (quasi )ordering is total. With such an extension to a total quasi ordering, constraint solving is still possible: Theorem 6 ([13]) The existential fragment of the theory of a total recursive path (quasi )ordering with status is decidable. Actually, as above, the fragment is NP complete. Satisfiability over an extended signature is NP complete as well [18] 4.3 Partial recursive path orderings Although less interesting ....

J.-P. Jouannaud and M. Okada. Satisfiability of systems of ordinal notations with the subterm property is decidable. In Proc. 18th Int. Coll. on Automata, Languages and Programming, Madrid, Lecture Notes in Computer Science, vol. 510, pages 455--468, 1991. Springer-Verlag.

....total lexicographic path ordering has been investigated by H. Comon and its existential fragment has been shown decidable [2] This frag2 ment is actually NP complete, as shown by R. Nieuwenhuis [12] The existential fragment of the theory of any total recursive path ordering is actually decidable [9]. On the other side, R. Treinen has shown that the full first order theory (actually the 9 8 9 8 fragment) of the theory of a partial recursive path ordering is undecidable [15] This leaves as open questions the existential fragment of a partial recursive path ordering on the one ....

J.-P. Jouannaud and M. Okada. Satisfiability of systems of ordinal notations with the subterm property is decidable. In J. L. Albert, B. Monien, and M. R. Artalejo, editors, 18th International Colloquium on Automata, Languages and Programming, Lecture Notes in Computer Science, vol. 510, pages 455--468, Madrid, Spain, 1991. Springer-Verlag.

.... semantics (solutions are built over a given signature F) and extended signature semantics (new symbols are allowed to appear in solutions) NR95] The satisfiability problem for ordering constraints was first shown decidable for fixed signatures when is a total LPO [Com90] or a total RPOS [JO91]. For extended signatures, decidability was shown for LPO in [NR95] and for RPO in [Nie93] Regarding complexity, NP algorithms for LPO (fixed and extended signatures) and RPO (extended ones) were given in [Nie93] Very recently, an NP algorithm has been given as well for RPO under fixed ....

....that is, constraints built only over f , 0, and possibly other constants smaller than f , and with solutions over this same signature. If f 2 lex this is a simple problem over the natural numbers, but for f 2 mul this seems not the case. Hence for the moment we propose to use the algorithm of [JO91] or the NP one of [NRV98] for SN , which is normally a minor part of S. 6.4.2 The lexicographic case Let (F ; F ) be such that f F a 1 F : F a n F 0, where the smallest non constant symbol f is in lex (the case with at most one constant below f has been treated already in Subsection ....

J-P. Jouannaud and M. Okada. Satisfiability of systems of ordinal notations with the subterm property is decidable. In Automata, Languages and Programming, 18th International Colloquium, LNCS 510, Madrid, Spain, July 16--20 1991. Springer-Verlag.

....symbol greater than f ; if f is not unary then is f(0; 0; t; 0) where t is the second smallest ground term. Note that always f(0; 0; t) is the k th successor term of a ground term t if t is in N , and otherwise f(0; 0; t) is the first successor of t. cf. Com90] or [JO91] for more details) Here we consider simple systems t 1 # : # t p Gamma1 t p 1 # : # t n Gamma1 t n where t n is the smallest constant 0. Now t p 1 # : # t n is called the natural part. and t 1 # : # t p Gamma1 is called the non natural part of the system. ....

J-P. Jouannaud and M. Okada. Satisfiability of systems of ordinal notations with the subterm property is decidable. In Automata, Languages and Programming, 18th International Colloquium, LNCS 510, Madrid, Spain, July 16--20 1991. Springer-Verlag.

.... which have turned out quite useful in the AC case, as it was already pointed out in [KKR90] Furthermore, we believe that a simple ordering like the one defined here can also be a first step towards a decision procedure for the satisfiability of AC ordering constraints, like done in [Com90] and [JO91] for LPO and RPO. Let us now first give some intuition about the definition of the ordering. Let rpo be a recursive path ordering (with status) generated by a total precedence F , in which AC symbols, denoted by FAC , have multiset status and where all other symbols have lexicographical ....

J-P. Jouannaud and M. Okada. Satisfiability of systems of ordinal notations with the subterm property is decidable. In Automata, Languages and Programming, 18th International Colloquium, LNCS 510, Madrid, Spain, July 16--20 1991. Springer-Verlag.

....not total on ground terms as permuting the direct subterms of a function symbol whose status is multiset leads to incomparable terms. However, modulo such permutations, the (quasi )ordering is total. With such an extension to a total quasi ordering, constraint solving is still possible: Theorem6 [13]. The existential fragment of the theory of a total recursive path (quasi )ordering with status is decidable. Actually, as above, the fragment is NP complete. Satisfiability over an extended signature is NP complete as well [18] 4.3 Partial recursive path orderings Although less interesting ....

J.-P. Jouannaud and M. Okada. Satisfiability of systems of ordinal notations with the subterm property is decidable. In Proc. 18th Int. Coll. on Automata, Languages and Programming, Madrid, Lecture Notes in Computer Science, vol. 510, pages 455--468, 1991. Springer-Verlag.

.... The first method for deciding the satisfiability under a given signature of ordering constraints was defined by Comon (1990) for the case in which is interpreted as the lexicographic path ordering (LPO) Here we cannot use its extension to the recursive path ordering with status (RPOS) given by Jouannaud and Okada (1991) , since RPO is not total on ground terms. However, since these methods are quite inefficient in practice, we have developed a new faster algorithm for the LPO case that can be used to decide satisfiability under a given signature and under extended signatures, which is, as we will see in the ....

Jouannaud, J.P., Okada, M. (1991). Satisfiability of systems of ordinal notations with the subterm property is decidable. In Automata, Languages and Programming, 18th International Colloquium, LNCS 510, Madrid, Spain. Springer-Verlag.

....in first order theorem proving involving This work is partly supported by the GDR Programmation du CNRS , the ESPRIT Working Group CCL and the ESPRIT Basic Research Action TYPES . AC operators [28,29] Solving ordering constraints is also known to be easier when using total orderings [6, 13], hence having simply defined total AC orderings is certainly a good basis for use in AC ordering constraints. Defining a total AC compatible reduction ordering was a very difficult problem and has been solved only very recently (in 1991) by Narendran and Rusinowitch [24] Unfortunately their ....

J.-P. Jouannaud and M. Okada. Satisfiability of systems of ordinal notations with the subterm property is decidable. In Proc. 18th Int. Coll. on Automata, Languages and Programming, Madrid, LNCS 510, 1991.

....an ordering and extend it to constrained terms, defining OE : s OE : t iff, for every solution oe of OE, soe toe. However, we do not know yet whether such conditions are decidable. For inequations alone this has been proved recently decidable when is a recursive path ordering (Comon, 1990; Jouannaud Okada, 1991). Completion of Rewrite Systems with Membership Constraints. Part I: Deduction Rules 19 Delete E [ fOE : s = tg; R E;R If OE s 6= t is not satisfiable. Simplify E [ fOE : s = tg; R E [ fOE : s 0 = tg; R If OE : s R OE : s 0 Compose E;R [ fOE : l rg E;R [ fOE : l r 0 g If OE ....

Jouannaud, Jean-Pierre, & Okada, Mitsuhiro. 1991. Satisfiability of systems of ordinal notations with the subterm property is decidable. In: (Albert et al., 1991).

.... of decidability of the theory of a total simplification ordering has been posed in Comon (1988) The decidability of the existential fragment of a total lexicographic path ordering (lpo for short) is shown in Comon (1990) the analogous result for a total recursive path ordering has been given in Jouannaud Okada (1991). We prove in Example (C) the undecidability of the Sigma 4 fragment of a partial lpo. Unfortunately there still remain two big gaps between these results (see Section 5) The undecidability of the Sigma 2 fragment of complete number theory (Example (G) is of course by no means a new result; it ....

Jouannaud, J.-P., Okada, M. (1991). Satisfiability of systems of ordinal notation with the subterm property is decidable. In Albert, J. L., Monien, B., Artalejo, M. R., editors, 18th International Colloquium on Automata, Languages and Programming, Lecture Notes in Computer Science, vol.

....i . On the other hand, it would be very interesting to solve the decidability of the existential fragment of the first order theory of embedding, since this would open the way to the decidability of the same fragment for the recursive path ordering 3 , which is known for a total precedence only [3, 17]. ....

J.-P. Jouannaud and M. Okada. Satisfiability of systems of ordinal notations with the subterm property is decidable. In Proc. 18th Int. Coll. on Automata, Languages and Programming, Madrid, LNCS 510, 1991.

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