Manfred Schmidt-Schau and Klaus U. Schulz. On the exponent of periodicity of minimal solutions of context equations. In Proceedings of the 9th Int. Conf. on Rewriting Techniques and Applications, volume 1379 of Lecture Notes in Computer Science, pages 61--75, 1998.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... constraint programming language Oz [Smo95] Context uni cation (CU) was introduced in rewriting and uni cation theory [Com92,SS94] CU can be considered as second order linear uni cation [L ev96] which is a restriction of higher order uni cation, 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 ....

Manfred Schmidt-Schau and Klaus Schulz. On the exponent of periodicity of minimal solutions of context equations. In International Conference on Rewriting Techniques and Applications, volume 1379 of LNCS, 1998.

....applying the reduction to the results of [LV00] proves that second order uni cation is undecidable for one binary function symbol and one second order variable occurring four times. Moreover, the same reduction is applicable to context uni cation [Com98] for which decidability is still unknown [Com98,LV01,SS96,Lev96,SS98,SSS98,SSS99], and it allows concentrating the e orts in a very simple signature. We also think that currying could help to simplify the signature used in higher order matching, and this could help to prove its decidability (or undecidability) If we currify function applications in a second order [or ....

Manfred Schmidt-Schau and Klaus U. Schulz. On the exponent of periodicity of minimal solutions of context equations. In Proceedings of the 9th Int. LNCS, pages 61-75, Tsukuba, Japan, 1998.

....choice of p; q; r k satisfying the bound. The converse reduction does not seem easy to find. The bounded second order unification problem has recently been proved decidable [SS99b] The relationship between context unification and word unification [Mak77] was originally suggested in [Lev96] In [SSS98] it is proved that the exponent of periodicity lemma also holds for context unification. We can easily reduce word unification to context unification by encoding any word unification problem, like F a G = G a F , as a monadic context unification problem F (a(G(b) G(a(F (b) where b ....

M. Schmidt-Schau and K. U. Schulz. On the exponent of periodicity of minimal solutions of context equations. In Proceedings of the 9th Int. Conf. on Rewriting Techniques and Applications (RTA'98), volume 1379 of LNCS, pages 61--75, Tsukuba, Japan, 1998.

.... constraint programming language Oz [Smo95] Context unification (CU) was introduced in rewriting and unification theory [Com92,SS94] CU can be considered as second order linear unification [L ev96] which is a restriction of higher order unification, or as an extension of string unification [SSS98]. The decidability question for CU is a prominent open problem [RTA98] A decidable fragment of CU called stratified unification has been used to show the decidability of distributive unification [SS97] and for solving onestep rewriting constraints [NPR97a,NTT99] It is shown in [SSS99] that ....

Manfred Schmidt-Schau and Klaus Schulz. On the exponent of periodicity of minimal solutions of context equations. In International Conference on Rewriting Techniques and Applications, volume 1379 of LNCS, 1998.

....uni cation, which is known to be undecidable. Similarly, the problem of disambiguating arbitrary CLLS descriptions (that is, of enumerating all most general solved forms of a constraint) is very complex. It was shown in Niehren and Koller (1998) that CLLS is equivalent to context uni cation (Schmidt Schau and Schulz, 1998; Niehren et al. 1997a) There are semi decision procedures 0.tex; 4 02 2000; 14:18; p.26 The Constraint Language for Lambda Structures 27 for context uni cation, but its decidability is a prominent open problem in theoretical computer science (RTA, 1998) For the sublanguage of dominance ....

Schmidt-Schau M. and K. Schulz: 1998, `On the Exponent of Periodicity of Minimal Solutions of Context Equations'. In: T. Nipkow (ed.): International Conference on Rewriting Techniques and Applications. to appear.

....required to be linear, i.e. of the form x:t(x) where t(x) contains exactly one occurrence of x. Context unification is useful in different areas of Computer Science: term rewriting, theorem proving, equational unification, constraint solving, computational linguistics, software engineering [11, 9, 12]. CUP is stated as follows: Given a pair of terms t, t 0 built as usual from symbols of a signature Sigma , firstorder variables w, and unary function variables F , does there exist an assignment of terms to w and linear second order functions to F such that (t) t 0 ) Thus, CUP is a ....

....to F such that (t) t 0 ) Thus, CUP is a decidability problem for the existentially quantified equations (9 equational theory) of the form 9 F 9 w t = t 0 ; 1) where the quantified context variables F range over linear functions. Currently the decidability of CUP is an open problem [11, 9, 12]. Most researchers conjecture and hope that CUP is decidable. All the above papers provide some approximations: either prove decidability of particular cases, or settle undecidability of some generalizations, or provide technical results towards decidability of CUP. Presumably, CUP is very hard ....

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M. Schmidt-Schau and K. U. Schulz. On the exponent of periodicity of minimal solutions of context equations. In Rewriting Techniques and Applications'98, volume 1379 of Lect. Notes Comput. Sci., pages 61--75. Springer-Verlag, 1998.

....first aspect possible for larger examples. The investigation of the computational aspects of context unification is an active field of research in theoretical computer science that has spawned several sound and complete algorithms for context unification and various related problems [L evy, 1996, Schmidt Schau Schulz, 1998, Niehren et al. 1997b] Unfortunately, each of these algorithms has drawbacks that prevent its use for the linguistic application, and we have to come up with an algorithm of our own that is better suited for this domain before we can evaluate context unification fairly. In the next section, we ....

....of them is sufficient for our application. For example, there is a decision procedure for stratified context constraints context constraints where every occurrence of a variable must have the same second order prefix, i.e. the context variables visited to reach the variable must be the same [Schmidt Schau Schulz, 1998, Hohl, 1997] This fragment is too small for the linguistic application; consider the following example: 14.7) Every student read a book. Every professor who hadn t read every paper did too. 14.8) C s = X:C 1 (every student lam x (C 2 (X) C s = X:C 3 (a book lam y (C 4 (X) X s = C s ....

Manfred Schmidt-Schauß and Klaus Schulz. On the exponent of periodicity of minimal solutions of context equations. In Tobias Nipkow, editor, 8th International Conference on Rewriting Techniques and Applications, Lecture Notes in Computer Science, Rutgers University, NJ, USA, 1998. Springer-Verlag. To appear.

.... appears as a subproblem in constraint solving with membership constraints [1] distributive unification [21] and completion of birewriting systems [13] The decidability of context unification is a difficult open problem, there has been some progress though, towards a decidability result [22]. Recently, a natural reduction from simultaneous rigid E unification was found to secondorder unification [15, 26] complementing the previously known converse reduction [4] and implying that the problems are in principal the same. Due to the fundamental role of simultaneous rigid E unification ....

....problem for second order languages L such that jF L j = 1. 2.2 Context Unification and Linear SOU Linear SOU [14] is SOU where substitutions are required to map second order variables of arity n to closed terms with exactly one occurrence of each bound variable z i for i n. Context Unification[1, 20, 22] is Linear SOU for languages of degree 1. 2.3 Simultaneous Rigid E Unification Let L be a first order language. A rigid equation in L is a pair (E; e) where E is a system of equations in L and e is an equation in L. Simultaneous Rigid E Unification [9] or SREU, for L is the following decision ....

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M. Schmidt-Schauß and Klaus U. Schulz. On the exponent of periodicity of minimal solutions of context equations. In T. Nipkow, editor, Rewriting Techniques and Applications, volume 1379 of Lecture Notes in Computer Science, pages 61--75. Springer Verlag, 1998.

.... concurrent constraint programming language Oz [Smo95] Context uni cation (CU) was introduced in rewriting and uni cation theory [Com92,SS94] CU can be considered as second order linear uni cation [L ev96] which is a restriction of higher order uni cation, or as an extension of string uni cation [SSS98]. 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 ....

Manfred Schmidt-Schau and Klaus Schulz. On the exponent of periodicity of minimal solutions of context equations. In International Conference on Rewriting Techniques and Applications, volume 1379 of LNCS, 1998.

No context found.

Manfred Schmidt-Schau and Klaus U. Schulz. On the exponent of periodicity of minimal solutions of context equations. In Proceedings of the 9th Int. Conf. on Rewriting Techniques and Applications, volume 1379 of Lecture Notes in Computer Science, pages 61--75, 1998.

....higher order uni cation where we start with given bounds for the variables in the problem and increase the bounds as long as we have an unsolvable problem. The proof technique uses a lemma on an upper bound for the exponent of periodicity for a minimal uni er for context uni cation from [SSS98] which is a generalization of a lemma that appeared in the decidability proof of word uni cation by Makanin [Mak77] An improvement of the latter result was given in [KP96] This link to word uni cation ( Mak77,Sch90,Sch93,Gut98,Pla99] is not accidental. The relationship between word uni ....

....be an elementary constant occurring in the codomain of that does not occur in S. Part 2 shows that is a target type in subt(S) Hence, as in Part 1 of the proof a can be replaced by an elementary constant b 2 0 . ut 4. 2 The Exponent of Periodicity The exponent of periodicity (see also [SSS98]) of a uni er of (S; b) is the maximal number n such that for some variable x occurring in S the image (x) contains a subterm of the form C [t] where C is a nontrivial ground rst order context. De nition 4.2. Let t be a ground term in normal form. Assume that we color in t the ....

[Article contains additional citation context not shown here]

Manfred Schmidt-Schau and Klaus U. Schulz. On the exponent of periodicity of minimal solutions of context equations. In Proceedings of the 9th Int. Conf. on Rewriting Techniques and Applications, volume 1379 of Lecture Notes in Computer Science, pages 61-75, 1998.

....variables represent terms with exactly one hole in contrast to a term with an arbitrary number of (equally named) holes in the general secondorder case. The name contexts was coined in [Com93] Currently, it is not known whether context unification is decidable. It is known that it is NP hard (cf. [SSS98]) and that satisfiability of formulas in a logical theory of context unification is undecidable [NPR97a,Vor98] There are some decidable fragments: i. If for every context variable X, all occurrences of X have the same argument [Com98a,Com98b] ii. If the number of occurrences of every ....

....introducing integer exponents for ground contexts, the algorithm SCU uses an n fold copy. This again simplifies the description of the algorithm sacrificing efficiency. SCU makes use of a lemma on the exponent of periodicity of a minimal solution of context unification problems, proved in [SSS98], which is a generalization of a similar result for string unification [Mak77,KP96] An experimental implementation of stratified context unification (with exponents) was done in [Hoh97] The following result is proved in this paper: Theorem: Stratified context unification is decidable. A ....

[Article contains additional citation context not shown here]

Manfred Schmidt-Schau and Klaus U. Schulz. On the exponent of periodicity of minimal solutions of context equations. In Proceedings of the 9th Int. Conf. on Rewriting Techniques and Applications, volume 1379 of Lecture Notes in Computer Science, pages 61--75, 1998.

....for nding corresponding positions . We give two criteria, one that approximates the problem by linearizing the variables, and another that derives a system of linear equations from the number of occurrences of function symbols. General context matching was previously known to be NP complete [21]. Context matching is a restricted form of linear higher order matching, which was shown to be in NP and hence NP complete by de Groote [9] Here linear means that only solutions where all functions are linear, i.e. contain each of their bound variable exactly once, are considered. A context may ....

....landscape of the secondorder matching problem with respect to several restrictions, i.e. number of second order variables, number of occurrences of variables, ground, function free, but not strati cation. Our interest for the strati ed fragment has been motivated by the results on its decidability [21, 20, 19]. Other decidable fragments are the varity 2 fragment [16] 1 and the two context variable fragment [22] whose corresponding matching problems we also consider. 2 Preliminaries We assume a xed in nite set X of (individual) variables, a xed in nite set C of context variables, and a xed set ....

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Manfred Schmidt-Schau and Klaus U. Schulz. On the exponent of periodicity of minimal solutions of context equations. In Proc. 9th Int. Conf. on Rewriting Techniques and Applications, LNCS 1379, pages 61-75, Tsukuba, Japan, 1998. Springer.

....ordinary higher order uni cation where we start with given bounds for the variables in the problem and increase the bounds as long as we have an unsolvable problem. The proof technique uses a lemma on an upper bound for the exponent of periodicity for a minimal uni er for context uni cation from [SSS98] which is a generalization of a lemma that appeared in the decidability proof of word uni cation by Makanin [Mak77] An improvement of the latter result was given in [KP96] This link to word uni cation ( Mak77,Sch90,Sch93,Gut98,Pla99] is not accidental. The relationship between word uni cation ....

....an elementary constant occurring in the codomain of that does not occur in S. Part 2 shows that is a target type in subt(S) Hence, as in Part 1 of the proof a can be replaced by an elementary constant b 2 0 . ut 4. 2 The Exponent of Periodicity The exponent of periodicity (see also [SSS98]) of a uni er of (S; b) is the maximal number n such that for some variable x occurring in S the image (x) contains a subterm of the form C n [t] where C is a nontrivial ground rst order context. De nition 4.2. Let t be a ground term in normal form. Assume that we color in t the ....

[Article contains additional citation context not shown here]

Manfred Schmidt-Schau and Klaus U. Schulz. On the exponent of periodicity of minimal solutions of context equations. In Proceedings of the 9th Int. Conf. on Rewriting Techniques and Applications, volume 1379 of Lecture Notes in Computer Science, pages 61-75, 1998.

....it is meant as a meta notation of a term, not as explicit syntax. For integers we use i mod n, which is the unique number j 2 [1: n] with i j j(mod n) An equation system is a set of equations s : t, also called unification problem. A ground substitution oe has exponent of periodicity n ([Mak77,SSS98]) iff i) for every X , if there are ground contexts A; B; C with B nontrivial, such that oe(X ) AB m C, then m n; ii) there is some X , such that oe(X ) AB n C, for appropriate ground contexts A; B; C where B is nontrivial. The following lemma is a generalization of [KP96] Lemma 2.1. ....

....iff i) for every X , if there are ground contexts A; B; C with B nontrivial, such that oe(X ) AB m C, then m n; ii) there is some X , such that oe(X ) AB n C, for appropriate ground contexts A; B; C where B is nontrivial. The following lemma is a generalization of [KP96] Lemma 2.1. [SSS98]) There is a constant c, such that for every unifiable context unification problem Gamma its exponent of periodicity is at most 2 cd , where d is the size of Gamma . Definition 2.2. We define SO prefixes as words in V 2 . An SO prefix of a position p in a term t is the word consisting of ....

[Article contains additional citation context not shown here]

Manfred Schmidt-Schau and Klaus U. Schulz. On the exponent of periodicity of minimal solutions of context equations. In Proceedings of the 9th Int. Conf. on Rewriting Techniques and Applications, volume 1379 of Lecture Notes in Computer Science, pages 61--75, 1998.

....It also improves and generalizes the algorithm in [SS98] where the signature is restricted to containing only one non constant function symbol. SCU makes use of a lemma for string unification [Mak77, KP96] on the exponent of periodicity of a minimal solution extended to context unification in [SSS98]. Instead of syntactically introducing integer exponents for ground contexts, the algorithm uses an n fold copy. An experimental implementation of stratified context unification (with exponents) was done in [Hoh97] The following result is proved in this paper: Theorem: stratified context ....

....n , where C[ Delta] is a context and n is an integer. This is defined as C[ Delta] 1 : C[ Delta] C[ Delta] n 1 : C[C[ Delta] n ] If we use this notation in a term, it is meant as a meta notation of a term, not as explicit syntax. A ground substitution oe has exponent of periodicity n ([Mak77, SSS98]) iff there are ground contexts A; B; C, such that oe(X) AB n C, but this is false for n 1. Lemma 1. SSS98] There is a constant c, such that for every unifiable context unification problem Gamma its exponent of periodicity is at most 2 cd , where d is the size of Gamma . Definition ....

[Article contains additional citation context not shown here]

Manfred Schmidt-Schau and Klaus U. Schulz. On the exponent of periodicity of minimal solutions of context equations. In Proceedings of the 9th Int. Conf. on Rewriting Techniques and Applications, volume 1379 of Lecture Notes in Computer Science, pages 61--75, 1998.

....the problem of solvability of word equations known to be decidable (Makanin 1977) This reduction augmented with linear integer constraint solving provides a decision procedure in the case when resulting word equations have finite minimal complete sets of unifiers. A recent breakthrough result of (Schmidt Schau Schulz 1997) on the upper bound on the exponent of periodicity of solutions to context equations suggests that the ultimate decidability of context unification may possibly be obtained by using this reduction and a termination argument similar to Makanin s. Keywords Context unification, solvability of ....

....(Goldfarb 1981, Farmer 1991) CUP is almost SOU, with only unary function variables allowed and solutions required to be linear, i.e. of the form x:t(x) where t(x) contains exactly one occurrence of x. CUP is claimed open in (Schmidt Schau 1994, Levy 1996, Niehren, Pinkal Ruhrberg 1997, Schmidt Schau Schulz 1997). Most researchers conjecture and hope that CUP is decidable. All the above papers provide some approximations: either prove decidability of particular cases, or settle undecidability of some generalizations, or provide technical results towards decidability of CUP. The positive solution would be ....

[Article contains additional citation context not shown here]

Schmidt-Schau, M. & Schulz, K. U. (1997), On the exponent of periodicity of minimal solutions of context equations, in `Rewriting Techniques and Applications'98', Lect. Notes Comput. Sci., Springer-Verlag. To appear.

....and the equations may contain an arbitrary number of individual variables as well. The result holds under the assumption that the first order background signature is finite. The paper provides a partial solution to the Context Unification Problem that has recently attracted considerable attention [3, 4, 13, 29, 18, 19, 24]: Is solvability of arbitrary context equations decidable The interest in this problem relies on its close connection to several other well studied decision problems. Context unification represents a natural variant of second order unification, which is known to be undecidable ( 8, 6, 14] ....

....In [21] there is an algorithm and a sketch of a proof showing that solvability of so called stratified context unification problems is decidable; a complete proof for a restricted signature is published in [22] Stratification imposes strong restrictions on the nesting of context variables. In [24] the authors have given an upper bound on the so called exponent of periodicity of a minimal solution of a context equation. In the case of word equations a similar bound was a key ingredient of Makanin s decidability result. J. Niehren, M. Pinkal and P. Ruhrberg [18] showed that context ....

M. Schmidt-Schau and K. U. Schulz. On the exponent of periodicity of minimal solutions of context equations. In Rewriting Techniques and Applications, Proc. RTA'98, volume 1379 of LNCS, pages 61--75. Springer-Verlag, 1998.

No context found.

M. Schmidt-Schau and K.U. Schulz. On the exponent of periodicity of minimal solutions of context equations. In 9th Int. Conf. on Rewriting Techniques and Applications, RTA'98, volume 1379 of LNCS, pages 61--75, Tsukuba, Japan, 1998.

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