Joachim Niehren, Manfred Pinkal, and Peter Ruhrberg. On equality up-to constraints over nite trees, context uni cation and one-step rewriting. In William McClune, editor, 14th International Conference on Automated Deduction, volume 1249 of Lecture Notes in Arti cial Intelligence, pages 34-48, Townsville, Australia, July 1997. Springer-Verlag.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....be constant functions, which means that equations where both sides start with variables are always trivially solvable. This e ect is heavily used in the decidability results for bounded second order higher order uni cation given in [SS99a,SS01] and in the present paper. In context uni cation [NPR97,Vor98,NTT00,SS99b,SSS00,LV00b], solution values of context variables are tree functions with one argument that has exactly one occurrence position. Hence solving a context equation leads to a complete structural alignment of both sides as trees. As we explain in Section 10.3, bounded) second order uni cation can be ....

Joachim Niehren, Manfred Pinkal, and Peter Ruhrberg. On equality upto constraints over nite trees, context uni cation, and one-step rewriting. In Proceedings of the International Conference on Automated Deduction, volume 1249 of Lecture Notes in Computer Science, pages 34-48, 1997.

....may be constant functions, which means that equations where both sides start with variables are always trivially solvable. This e ect is heavily used in the decidability results for bounded second order higher order uni cation given in [SS99a,SS01] and in the present paper. In context uni cation [NPR97,Vor98,NTT00,SS99b,SSS00,LV00b], solution values of context variables are tree functions with one argument that has exactly one occurrence position. Hence solving a context equation leads to a complete structural alignment of both sides as trees. As we explain in Section 10.3, bounded) second order uni cation can be ....

Joachim Niehren, Manfred Pinkal, and Peter Ruhrberg. On equality upto constraints over nite trees, context unication, and one-step rewriting. In Proceedings of the International Conference on Automated Deduction, volume 1249 of Lecture Notes in Computer Science, pages 34-48, 1997.

.... be shown even for very simple classes of rewrite systems [19,20,10,18] The question of decidability of the purely existential fragment of positive and negative rewrite constraints remains open, even though some cases for restricted classes of rewrite systems are solved [2,8] It has been shown in [12] that satis ability of RC can be expressed as satis ability of strati ed CU and hence is decidable. However, it was not known whether strati ed CU really is more dicult than solving RC. In this paper, we propose a minor extension of RC and show that it is in fact equivalent to strati ed CU, ....

....as CU Equations Our main result is Theorem 3 For every signature, there is a linear time, satis ability preserving translation which maps a strati ed system of CU equations to a rewrite constraint, and vice versa. 3 Rewrite Constraints as Strati ed CU Equations It was already shown in [12] that rewrite constraints of the form x y by t t can be translated into a strati ed system of CU equations. This translation is extended in Figure 3 to the slightly more general rewrite constraints that we consider in this article. The correctness of the translation of C D by rule (U3) ....

J. Niehren, M. Pinkal, and P. Ruhrberg. On equality up-to constraints over nite trees, context uni cation and one-step rewriting. In 14th Int. Conference on Automated Deduction, volume 1249 of LNAI, pages 34-48, 1997.

.... or as an extension of string uni cation [SSS98] The decidability question for CU is a prominent open problem [RTA98] A decidable fragment of CU called strati ed uni cation has been used to show the decidability of distributive uni cation [SS97] and for solving onestep rewriting constraints [NPR97a,NTT99]. It is shown in [SSS99] that context uni cation with two context variables each of which may occur an arbitrary number of times is decidable. The proof is by reduction to string uni cation, which is decidable according to Makanin s famous result [Mak77,Sch93] Tree Descriptions in Semantic ....

....string uni cation, which is decidable according to Makanin s famous result [Mak77,Sch93] Tree Descriptions in Semantic Underspeci cation. Recently, tree descriptions based on dominance constraints and context uni cation have been proposed for the same application to natural language semantics [ENRX98,NPR97a,Kol99]. There, the goal was to nd a uniform language providing underspeci ed representations for the semantics of scope, parallelism, anaphora, and their interactions (for a survey of semantic underspeci cation, see e.g. vP96] The common characteristic of both approaches is that they view the ....

[Article contains additional citation context not shown here]

Joachim Niehren, Manfred Pinkal, and Peter Ruhrberg. On equality up-to constraints over nite trees, context uni cation and one-step rewriting. In Proc. of the CADE, volume 1249 of LNCS, pages 34-48, 1997.

....of second order linear uni cation [7, 12] or as an extension of string uni cation. The decidability of CU is still open. A decidable fragment of CU called strati ed uni cation has been used to show the decidability of distributive uni cation [15] and for solving one step rewriting constraints [10]. In natural language semantics, CU has been applied in an underspeci ed treatment of scope, parallelism, and their interaction [11] The treatment of parallelism in this approach is related to analyses via higher order uni cation [3, 5] The constraint language for lambda structures (CLLS) ....

....of dominance and parallelism constraints as context constraints. We then lift this encoding to the rstorder theories of CLLS 0 and context constraints and prove its correctness. We also show that context uni cation can be encoded into CLLS 0 . This result is non trivial but mainly follows from [10]. ....

J. Niehren, M. Pinkal, and P. Ruhrberg. On equality up-to constraints over nite trees, context uni cation and one-step rewriting. In Proc. of the CADE, volume 1249 of LNCS, pages 34-48, 1997.

.... be shown even for very simple classes of rewrite systems [19,20,10,18] The question of decidability of the purely existential fragment of positive and negative rewrite constraints remains open, even though some cases for restricted classes of rewrite systems are solved [2,8] It has been shown in [12] that satis ability of RC can be expressed as satis ability of strati ed CU and hence is decidable. However, it was not known whether strati ed CU really is more dicult than solving RC. In this paper, we propose a minor extension of RC and show that it is in fact equivalent to strati ed CU, with ....

....Fig. 3. Rewrite Constraints as CU Equations Theorem 3 For every signature, there is a linear time, satis ability preserving translation which maps a strati ed system of CU equations to a rewrite constraint, and vice versa. 3 Rewrite Constraints as Strati ed CU Equations It was already shown in [12] that rewrite constraints of the form x y by t t 0 can be translated into a strati ed system of CU equations. This translation is extended in Figure 3 to the slightly more general rewrite constraints that we consider in this article. The correctness of the translation of C D by rule (U3) ....

J. Niehren, M. Pinkal, and P. Ruhrberg. On equality up-to constraints over nite trees, context unication and one-step rewriting. In 14th Int. Conference on Automated Deduction, volume 1249 of LNAI, pages 34-48, 1997.

.... The decidability question for CU is still open, even though decidability is claimed in an as yet unpublished paper [LV99] A decidable fragment of CU called strati ed uni cation has been used to show the decidability of distributive uni cation [SS97] and for solving one step rewriting constraints [NPR97a]. It is shown in [SSS99] that context uni cation with only two context variables each of which may occur an arbitrary number of times is decidable. The proof is by reduction to string uni cation, which is decidable according to Makanin s famous result [Mak77,Sch93] Tree Descriptions in ....

....encodings of CLLS 0 into CU and vice versa. The most interesting (and non obvious) part of the construction is to show how to encode dominance constraints in context uni cation. Once we know how to do that, the rest of this direction is easy. The inverse encoding can be deduced from a result in [NPR97a]. Plan of the Paper. In Section 2 we introduce dominance constraints and context uni cation informally and discuss their relationship at an example. We also show there why it is nontrivial to encode dominance constraints in context uni cation. In Section 3, we de ne dominance and parallelism ....

[Article contains additional citation context not shown here]

Joachim Niehren, Manfred Pinkal, and Peter Ruhrberg. On equality up-to constraints over nite trees, context unication and one-step rewriting. In Proc. of the CADE, volume 1249 of LNCS, pages 34-48, 1997.

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Joachim Niehren, Manfred Pinkal, and Peter Ruhrberg. On equality up-to constraints over nite trees, context uni cation and one-step rewriting. In William McClune, editor, 14th International Conference on Automated Deduction, volume 1249 of Lecture Notes in Arti cial Intelligence, pages 34-48, Townsville, Australia, July 1997. Springer-Verlag.

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