E. Domenjoud, F. Klay, and C. Ringeissen. Combination techniques for non-disjoint equational theories. In A. Bundy, editor, 12th International Conference on Automated Deduction, 281, Nancy, France, 1994. Springer-Verlag.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....from scratch if a constraint solver is already available for each component domain. A lot of research has been done in recent years on domain combination (see, for instance [2, 3, 13, 1] although most of the efforts have been concentrated on unification problems and equational theories (see [1, 4, 5, 10, 14, 19], among others) The results of these investigations are still limited in scope and a deep understanding of many model and proof theoretic issues involved is still out of reach. In spite of that, we try to show the effectiveness of combination techniques by choosing one of the most general and ....

E. Domenjoud, F. Klay, and C. Ringeissen. Combination techniques for non- disjoint equational theories. In A. Bundy, editor, Proceedings l.th International Conference on Automated Deduction, Nancy (France}, volume 814 of Lecture Notes in Artificial Intelligence, pages 267-281. Springer-Verlag, 1994.

....which uses deterministic rules that may be applied in arbitrary order. In addition, we wanted to clarify the connection to Nelson and Oppen s combination method. An important open question is how far the combination result can be extended to theories sharing function symbols of larger arity. In [1], the problem of combining algorithm for the unification, matching, and word problem was investigated for theories haring constructors. For the word problem, this combination method presupposes the existence of a matching algorithm for certain restricted matching problems in the single theories. ....

....For the word problem, this combination method presupposes the existence of a matching algorithm for certain restricted matching problems in the single theories. For shared constants, it is easy to see that these restricted matching problems reduce to word problems, and thus the result in [1] also yields the modularity result of Theorem 5.9, even though this is not explicitly mentioned in the paper. However, the algorithm described in [1] is not rule based since it is a straightforward extension of the algorithms for the disjoint case, as described in [13, 9, 3] and thus shares the ....

[Article contains additional citation context not shown here]

E. Domenjoud, F. Klay, and C. Ringeissen. Combination techniques for non-disjoint equational theories. In A. Bundy, editor, Proceedings 12th International Conference on Automated Deduction, Nancy (France), volume 814 of Lecture Notes in Artificial Intelligence, pages 267-281. Springer-Verlag, 1994.

.... matching, and, e.g. 13, 4, 1] for unification) It is not hard to extend these results to theories sharing constant symbols [11, 7, 2] The only work we are aware of that presents a general combination approach for the union of equational theories having more than con stant symbols in common is [6], where the problem of combining algorithms for the unification, matching, and word problem is investigated for theories sharing so called constructors. In this paper, we restrict our attention to the word problem. The combination result we obtain improves on the corresponding result in [6] in ....

....is [6] where the problem of combining algorithms for the unification, matching, and word problem is investigated for theories sharing so called constructors. In this paper, we restrict our attention to the word problem. The combination result we obtain improves on the corresponding result in [6] in the following respects. Firsfly, we introduce a notion of constructors, modeled after the one introduced in [15] which is strictly more general than the one in [6] Whereas [6] does not allow for nontrivial identities between constructor terms, we only require the constructor theory to be ....

[Article contains additional citation context not shown here]

E. Domenjoud, F. Klay, and C. Ringeissen. Combination techniques for nondisjoint equational theories. In Proc. CADE-i, Springer LNAI 814, 1994.

....of unification with linear constant restrictions. Since then, the main open problem in the area was how to extend these results to the combination of theories having symbols in common. In general, the existence of shared symbols may lead to undecidability results for the union theory (see, e.g. [6, 5] for some examples) This means that a controlled form of sharing of symbols is necessary. For the word problem and for universally quantified formulae, a suitable notion of shared constructors has proved useful. In [5] Pigozzi s combination result for the word problem was extended to theories ....

.... amalgamation construction originally due to Schmidt Schau [11] As already mentioned in the introduction, the notion of constructors used here is taken from [5, 13] The only other work on combination methods for unification in the non disjoint case is due to Domenjoud, Ringeissen and Klay [6]. The main differences with our work are that (i) their notion of constructors is considerably more restrictive than ours; and (ii) they combine algorithms computing complete sets of unifiers, and so their method cannot be used to combine decision procedures. On the other hand, Domenjoud, ....

[Article contains additional citation context not shown here]

E. Domenjoud, F. Klay, and Ch. Ringeissen. Combination techniques for nondisjoint equational theories. In Proc. CADE-12, Springer LNCS 814, 1994.

....with disjoint signatures [12, 11, 13] Our results were achieved by assuming that the function symbols shared by the component theories were constructors in a appropriate sense. The notion of constructors presented in [4] was modeled after one first introduced in [14] and generalized that in [5]. Its formulation is based on the observation that some equational theories are such that the reducts of their free models to a subset # of their signature are themselves free. We would call constructors the symbols in #. The actual definition in [4] however, incorporated the restriction that the ....

E. Domenjoud, F. Klay, and Ch. Ringeissen. Combination techniques for nondisjoint equational theories. In A. Bundy, editor, Proc. of CADE-12, volume 814 of LNAI, pages 267--281. Springer-Verlag, 1994.

.... [GPT96] for a generalized approach) In recent years, considerable research has focused on both domain and solver combinations (see, for instance, BS95a, BS95b, KS96, NO79, Sho84] although most of the efforts have been concentrated on unification problems and equational theories ( BS92, Bou93, DKR94, Her86, KR92, SS89, Yel87] among others) The current results of these investigations are limited in scope, and a deep understanding of many model and proof theoretic issues involved is still out of reach. Despite that, in this paper we attempt to show the e#ectiveness of combination ....

....can still prove extremely di#cult. As a matter of fact, most combination problems have no possible solutions. It is relatively easy to find domains or theories such that a certain satisfiability problem is decidable in each of them and undecidable in their combination; for instance, see [DKR94, BT97] As a result, all the existing combination methods pose more or less serious restrictions on the classes of constraint domains and languages they can combine. Furthermore, essentially all of them require the component constraint languages to have pairwise disjoint signatures. A ....

E. Domenjoud, F. Klay, and C. Ringeissen. Combination techniques for non-disjoint equational theories. In A. Bundy, editor, Deduction (Nancy, France), volume 814 of Lecture Notes in Artificial Intelligence, pages 267--281, Berlin, 1994. Springer-Verlag.

....symbols that are constructors in a sense to be made 3i.e. equations of the form x t, where x is a variable occurring in the non variable term t. more precise later. The only previous work that presents a combination method for the word prob lem in the union of non disjoint theories is [9], where the problem of combining algorithms for the unification, matching, and word problem is investigated for theories sharing so called constructors. The combination method for the word prob lem described in [9] is not rule based since it is an extension of the algorithms for the disjoint ....

....method for the word prob lem in the union of non disjoint theories is [9] where the problem of combining algorithms for the unification, matching, and word problem is investigated for theories sharing so called constructors. The combination method for the word prob lem described in [9] is not rule based since it is an extension of the algorithms for the disjoint case, as described in [21, 23, 19, 13] We will show that the notion of a constructor introduced in [9] is a strict subcase of our notion, and that the combination result for the word problem presented in [9] can also ....

[Article contains additional citation context not shown here]

Eric Domenjoud, Francis Klay, and Christophe Ringeissen. Combination tech- niques for non-disjoint equational theories. In A. Bundy, editor, Proceedings of the 12th International Conference on Automated Deduction, Nancy (France), volume 814 of Lecture Notes in Atificial Intelligence, pages 267-281. SpringerVerlag, 1994. 57

....of uni cation with linear constant restrictions. Since then, the main open problem in the area was how to extend these results to the combination of theories having symbols in common. In general, the existence of shared symbols may lead to undecidability results for the union theory (see, e.g. [6, 5] for some examples) This means that a controlled form of sharing of symbols is necessary. For the word problem and for universally quanti ed formulae, a suitable notion of shared constructors has proved useful. In [5] Pigozzi s combination result for the word problem was extended to theories ....

.... amalgamation construction originally due to Schmidt Schau [11] As already mentioned in the introduction, the notion of constructors used here is taken from [5, 13] The only other work on combination methods for uni cation in the nondisjoint case is due to Domenjoud, Ringeissen and Klay [6]. The main di erences with our work are that (i) their notion of constructors is considerably more restrictive than ours; and (ii) they combine algorithms computing complete sets of uni ers, and so their method cannot be used to combine decision procedures. On the other hand, Domenjoud, ....

[Article contains additional citation context not shown here]

E. Domenjoud, F. Klay, and Ch. Ringeissen. Combination techniques for nondisjoint equational theories. In Proc. CADE-12, Springer LNCS 814, 1994.

....the context of logic and automated reasoning) This is not a restriction since the extension to the general 13 uni cation ( rst order case) is straightforward. A further improvement would be to consider the theories sharing non constant function symbols. Results in this direction can be found in [Baader99b, Domenjoud94]. This paper discussed only the general idea leaving out the details of implementation. For general data structures and algorithm in the syntactic uni cation, the interested reader can consult [Baader99a, Baader98] For A1 uni cation, Abdulrab92, Abdulrab93] gave the details of the ....

E. Domenjoud, F. Klay, C. Ringeissen, Combination Techniques for Non-Disjoint Equational Theories.

....uni cation with linear constant restrictions. Since then, the main open problem in the area has been how to extend these results to the combination of theories having symbols in common. In general, the existence of shared symbols may lead to undecidability results for the union theory (see, e.g. [DKR94, BT02] for some examples) This means that a controlled form of sharing of symbols is necessary. For the word problem and for universally quanti ed formulae, a suitable notion of shared constructors has turned out to be useful. In [BT02] Pigozzi s combination result for the word problem was extended to ....

.... amalgamation construction originally due to Schmidt Schau [SS89] As already mentioned in the introduction, the notion of constructors used here is taken from [BT02, TR02] The only other work on combining uni cation algorithms in the non disjoint case is due to Domenjoud, Ringeissen and Klay [DKR94]. The main di erences with our work are that (i) their notion of constructors is much more restrictive than ours (as shown in [BT02] and (ii) they combine algorithms computing complete sets of uni ers, and thus their method cannot be used for combining decision procedures. Our combined ....

[Article contains additional citation context not shown here]

Eric Domenjoud, Francis Klay, and Christophe Ringeissen. Combination techniques for non-disjoint equational theories. In A. Bundy, editor, Proceedings of the 12th International Conference on Automated Deduction, 36 Nancy (France), volume 814 of Lecture Notes in Articial Intelligence, pages 267-281. Springer-Verlag, 1994.

....this combination algorithm. All the combination results that will be presented in the following are restricted to the case of disjoint signatures. There are some approaches that try to weaken the disjointness assumption, but the theories to be combined must satisfy rather strong conditions [57, 26]. 4.2 A combination algorithm for E uni cation algorithms In the following, we consider equational theories E 1 and E 2 over the disjoint signatures 1 and 2 . We denote the union of the theories by E : E 1 [E 2 and 6 In retrospect, if one looks at the combination method for decision ....

....by Nelson Oppen, rst combination results for the non disjoint case have been obtained by Ch. Ringeissen and C. Tinelli [58, 71, 73] Similarly, the known combination methods for solving the word problem in the union of equational theories have been lifted to the case of non disjoint signatures in [26, 10 12]. Concerning the combination of uni cation algorithms for equational theories over non disjoint signatures, rst results have been presented in [26] Using the more abstract algebraic concepts that have been developed during the last years it should be possible to simplify and then generalize this ....

[Article contains additional citation context not shown here]

E. Domenjoud, F. Klay, and C. Ringeissen. Combination techniques for nondisjoint equational theories. In A. Bundy, editor, Proceedings of the 12th International Conference on Automated Deduction, volume 814 of Lecture Notes in Arti- cial Intelligence, pages 267-281, Nancy, France, 1994. Springer-Verlag.

....in the literature it is usually assumed that T 1 ; T 2 share a set of constructors (we prefer the terminology they are both constructible over T 0 ) There are various de nitions of constructors and depending on such de nitions there are variable strength results. Main papers on the subject are [5] and [3, 4] the second has a weaker de nition and consequently a stronger result. Our de nition is again weaker (see Section 10 for details) and, more important, it covers natural mathematical examples and does not make any strong assumption on T 0 (in [5] T 0 is assumed to be free, in [3, 4] to ....

....results. Main papers on the subject are [5] and [3, 4] the second has a weaker de nition and consequently a stronger result. Our de nition is again weaker (see Section 10 for details) and, more important, it covers natural mathematical examples and does not make any strong assumption on T 0 (in [5] T 0 is assumed to be free, in [3, 4] to be collapse free) 1 [5] and [3, 4] use quite di erent methods: in [3, 4] the combined decision algorithm is obtained through a refutation technique manipulating equations according to certain non deterministic rules. As such it has the advantage of being ....

[Article contains additional citation context not shown here]

E. Domenjoud, F. Klay, and C. Ringeissen. Combination techniques for nondisjoint equational theories. In Proc. CADE-12, volume 814 of Springer LNAI, pages 48-64, 1996.

....i = h(a i (x) h 0 (a i (y) i = 1; n) This amounts to decide the unifiability of h(ff) h 0 (fi) forbidding ff = and fi = This is therefore impossible. 2 Lemma 4.4 Forbidding oe in is necessary to decide unifiability. Proof : The following rewrite system comes from [12]. let R 2 = f f(a i (x) y) i (f(x; a i (y) j i = 1; n f 0 (a i (x) y) 0 i (f 0 (x; a i (y) j i = 1; n f( y) h(y) f 0 ( y) h(y) g where i and 0 i denote respectively OE(a i ) and OE 0 (a i ) The function f encodes OE, and the second ....

E. Domenjoud. Combination techniques for non-disjoint equational theories. In D. Kapur, editor, Proceedings 12th International Conference on Automated Deduction, Albany (NY, USA), LNCS, pages 267--281. Springer-Verlag, 1994.

.... Ringeissen from INRIA Lorraine has given optimal conditions for combining pattern matching algorithms modulo an equational theory satisfying appropriate properties [88] He and his colleagues have also given the first combination scheme for unification in non disjoint equational theories [36]. This question is important, since distinct theories often need to share constants originating from skolemization, and this is allowd by this schema. All these new algorithms are being made available in software systems developped independently at CIS (in collaboration with Aachen) and ....

E. Domenjoud, F. Klay, and Ch. Ringeissen. Combination techniques for non-disjoint equational theories. In Alan Bundy, editor, Proceedings 12th International Conference on Automated Deduction, Nancy (France), Lecture Notes in Artificial Intelligence, pages 267--281. Springer-Verlag, June/July 1994.

....as follows: given two unification algorithms in two (consistent) equational theories E 1 and E 2 , how to find a unification algorithm for E 1 [ E 2 . M. Schmidt Schau solved the general problem for disjoint function symbols sets [23] Some extensions of this result were considered: in [18] and [12], sharing of constants and constructors are respectively allowed. In [4] F. Baader K. Schulz have showed how to combine constraint solvers for two arbitrary Simply Combinable structures over disjoint signatures into a solver for their combined structure. In addition, many CLP dialects allow ....

E. Domenjoud, F. Klay, and Ch. Ringeissen. Combination techniques for nondisjoint equational theories. In Alan Bundy, editor, Proceedings 12th International Conference on Automated Deduction, Nancy (France), volume 814 of Lecture Notes in Artificial Intelligence, pages 267--281. Springer-Verlag, June/July 1994. 4

....satis able, according to some adopted notion of satis ability. 3 1. 1 Previous Work Most of the current work on the combination of constraints reasoners regards the combination of solvers for equational constraints, in particular, algorithms for E uni cation and related problems [BS95b, Bou93, DKR94, Her86, KR94a, KR94b, Rin92, SS89] In this context, the constraint language is restricted to quanti erfree formulae over a functional signature (no predicate symbols other than equality) each component constraint domain is axiomatized by an equational theory and the combined domain is ....

....properties it is possible to derive combination results for the word problem in the union of equational theories sharing constructors. 2 Outside the work on modular term rewriting, the rst combination results for the word problem in the union of non disjoint constraint theories were given in [DKR94] as a consequence of some combination techniques based on an adequate notion of (shared) constructors. The second of us used similar ideas later in [Rin96b] to extend the Nelson Oppen method to theories sharing constructors in a sense close to that of [DKR94] To our knowledge, the only new work ....

[Article contains additional citation context not shown here]

Eric Domenjoud, Francis Klay, and Christophe Ringeissen. Combination techniques for non-disjoint equational theories. In A. Bundy, editor, Proceedings of the 12th International Conference on Automated Deduction, Nancy (France), volume 814 of Lecture Notes in Articial Intelligence, pages 267-281. Springer-Verlag, 1994.

....equational theory E and its function symbols. It is a non disjoint combination of first order equational theories. Therefore, we cannot reuse the well known techniques developed for combining unification algorithms in the union of signature disjoint theories [BS96] and those developed in [DKR94] for some specific non disjoint unions of theories cannot be applied to this particular case. Thus, we are designing in this work a complete oeE unification procedure starting from the simple algorithm developed in [DHK95] This leads to additional transformation rules for dealing with E and with ....

E. Domenjoud, F. Klay, and C. Ringeissen. Combination techniques for non-disjoint equational theories. In A. Bundy, editor, Proceedings 12th International Conference on Automated Deduction, Nancy (France), volume 814 of Lecture Notes in Artificial Intelligence, pages 267--281. SpringerVerlag, June/July 1994.

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E. Domenjoud, F. Klay, and C. Ringeissen. Combination techniques for non-disjoint equational theories. In A. Bundy, editor, 12th International Conference on Automated Deduction, 281, Nancy, France, 1994. Springer-Verlag.

No context found.

E. Domenjoud, F. Klay, and C. Ringeissen. Combination techniques for nondisjoint equational theories. In Proc. CADE-12, volume 814 of Springer LNAI, pages 48--64, 1996.

No context found.

Eric Domenjoud, Francis Klay, and Christophe Ringeissen. Combination techniques for non-disjoint equational theories. In A. Bundy, editor, Proceedings of the 12th International Conference on Automated Deduction, Nancy (France), volume 814 of Lecture Notes in Artificial Intelligence, pages 267--281. SpringerVerlag, 1994.

No context found.

Eric Domenjoud, Francis Klay, and Christophe Ringeissen. Combination techniques for non-disjoint equational theories. In Alan Bundy, editor, Automated Deduction { CADE-12, volume 814 of Lecture Notes in Arti cial Intelligence, pages 267-281. Springer, 1994.

No context found.

E. Domenjoud, F. Klay, and C. Ringeissen. Combination techniques for nondisjoint equational theories. In A. Bundy, editor, Proceedings of the 12th International Conference on Automated Deduction, Nancy (France), volume 814 of Lecture Notes in Artificial Intelligence, pages 267--281. Springer-Verlag, 1994.

No context found.

Eric Domenjoud, Francis Klay, and Christophe Ringeissen. Combination techniques for non-disjoint equational theories. In Alan Bundy, editor, Automated Deduction { CADE-12, volume 814 of Lecture Notes in Arti cial Intelligence, pages 267-281. Springer, 1994.

No context found.

Eric Domenjoud, Francis Klay, and Christophe Ringeissen. Combination techniques for non-disjoint equational theories. In Alan Bundy, editor, Automated Deduction -- CADE-12, volume 814 of Lecture Notes in Computer Science, pages 267--281. Springer, 1994.

No context found.

DKR94. Eric Domenjoud, Francis Klay, and Christophe Ringeissen. Combination techniques for non-disjoint equational theories. In A. Bundy, editor, Proceedings of the 12th International Conference on Automated Deduction, Nancy (France), volume 814 of Lecture Notes in Arti- cial Intelligence, pages 267-281. Springer-Verlag, 1994.

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