Paul Chew. Unique normal forms in term rewriting systems with repeated variables. In Proceedings of the Thirteenth Annual Symposium on Theory of Computing, pages 7-18. ACM, 1981.  Home/Search   Document Not in Database   Summary   Related Articles   Check

...., b t 2 being normal forms of t 1 and t 2 , respectively. In case b t 1 6= b t 2 , g( b t 1 ; b t 2 ) is already in normal form, otherwise it reduces in one step to the normal form a. For (ii) we observe that R is strongly non overlapping, hence it has unique normal forms (UN) by Chew s Theorem ([Che81], KV89] Using (i) and the basic fact UN WN = CR, this yields confluence of R. For proving (iii) it suffices to show that f(a; a) a does not hold (because R a is obviously true) This is equivalent to show f(a; a) 6 R 0 a where = fb a; g(a; f(x; a) f(x; x) g(x; x) ag since ....

Paul Chew. Unique normal forms in term rewriting systems with repeated variables. In Proc. 13th Annual Symp. of Theory of Computing, pages 7--18, 1981.

...., b t 2 being normal forms of t 1 and t 2 , respectively. In case b t 1 6= b t 2 , g( b t 1 ; b t 2 ) is already in normal form, otherwise it reduces in one step to the normal form a. For (ii) we observe that R is strongly non overlapping, hence it has unique normal forms (UN) by Chew s Theorem ([Che81], KV89] Using (i) and the basic fact UN WN = CR, this yields confluence of R. For proving (iii) it suffices to show that f(a; a) R# a does not hold (because f(a; a) R a is obviously true) This is equivalent to show f(a; a) 6 R 0 a where R 0 = fb a; g(a; f(x; a) f(x; x) ....

Paul Chew. Unique normal forms in term rewriting systems with repeated variables. In Proc. 13th Annual Symp. of Theory of Computing, pages 7--18, 1981.

....the rewriting rules. Secondly, although the general method is essentially proof theoretic, our new application uses a lemma that depends on a modeltheoretic argument, using the graph model Pw. The two applications of our method that were mentioned above, also follow from a theorem stated in Chew [1981], establishing uniqueness of normal forms for a wider class of non leftlinear TRSs. The proof offered by Chew for his theorem seems inconclusive, though. After this became apparent, there has been a renewed interest in finding a complete and convincing proof, notably by Mano and Ogawa [1997] ....

....has unique normal forms. PROOF. i) The Curch Rosser property for CL pc L follows by Proposition 4.4 from 4.3 and 4.1. ii) Since we have confluence for the linearization CL pc L , Theorem 3.8 can be applied. 11 5. Chew s Theorem We will now give a brief account of a theorem stated in Chew [1981], giving sufficient conditions for a TRS to have unique normal forms. In the light of the present paper, Chew s theorem can be viewed as a generalization of Theorem 3.9: the condition of strong non ambiguity is relaxed to allow overlap at the root between the lefthand sides of rules, but only when ....

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CHEW, P. (1981), Unique normal forms in term rewriting systems with repeated variables. In: 13th Annual ACM Symposium on the theory of Computing, p. 7-18.

.... R 2 is not CR[3] R 1 = 8 : d(x; x) 0 d(x; f (x) 1 2 f(2) 9 = R 2 = 8 : d(x; x) 0 f (x) d(x; f(x) 1 f(1) 9 = Chew and Klop have shown the sufficient condition for the uniquely normalizing (UN) property instead of CR that is, a strongly nonoverlapping TRS is UN[3, 7]. A TRS is said to be strongly nonoverlapping if its linearization (i.e. the renaming of repeated variables with fresh individual variables) is nonoverlapping. Chew also states, more generally that a compatible TRS is UN[7] In the compatible case, however, Chew s proof is hard to recognize, and ....

....property instead of CR that is, a strongly nonoverlapping TRS is UN[3, 7] A TRS is said to be strongly nonoverlapping if its linearization (i.e. the renaming of repeated variables with fresh individual variables) is nonoverlapping. Chew also states, more generally that a compatible TRS is UN[7]. In the compatible case, however, Chew s proof is hard to recognize, and its journal version has not yet been published[4] Several trials to get a new proof have been reported[3, 15] and they show partial answers. Their main technique is to first transform a nonlinear TRS to a linear TRS ....

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P.Chew, "Unique Normal Forms in Term Rewriting Systems with Repeated Variables",Proc. the 13th ACM Sympo. on Theory of Computing, pp.7-18 (1981)

....modulo an associated equational logic E. In section 4, a decidable condition for E nonoverlapping property is proposed. The statement is, an nonoverlapping TRS is E nonoverlapping. Thus, an nonoverlapping TRS is proved to be UC. This result is also compared with classical results found in [4, 10]. 2 Reduction systems 2.1 Abstract reduction systems A reduction system is a structure R = hA; i consisting of an object set A and any binary relation on A (i.e. A2 A) called a reduction relation. A reduction (starting with x 0 ) in R is a finite or an infinite sequence x 0 x 1 x 2 ....

....TRS R is UC. The assumption nonoverlapping is weaker than strongly nonoverlapping , and the result UC is stronger than UN . Thus, corollary 1 is a simple but more powerful result than the following classical theorem. Figure 3: Relation among nonoverlapping properties. Theorem [4] A TRS R is UN if the following conditions are met : ffl R is strongly nonoverlapping. ffl R is compatible. In fact, the theorem above shows that Example 4 is UN . Further, theorem 2 shows that the example is UC, though it is not CR (See Figure 4) Example 4 R 4 def = 8 : d(x; x) ....

Chew,P., "Unique Normal Forms in Term Rewriting Systems with Repeated Variables", Proc. 13th ACM STOC , pp.7-18 (1981)

....of the left in normal form) that is not length decreasing Problem 58 (M. Oyamaguchi) Is any strongly non overlapping right linear termrewriting system confluent ( Strong in the sense that left hand sides are non overlapping even when the occurrences of variables have been renamed apart [ 21 ] . On the one hand, strongly non overlapping systems need not be confluent [ 46 ] on the other hand, strongly non overlapping right ground systems are [ 88 ] A partial positive solution is given in [ 83; 99 ] namely, any strongly non overlapping right linear term rewriting system is ....

P. Chew. Unique normal forms in term rewriting systems with repeated variables. In Proceedings of the Thirteenth Annual Symposium on Theory of Computing, pages 7--18. ACM, 1981.

....theorem, which states that normal forms are unique up to conversion in compatible term rewriting systems. 1 Introduction A term rewriting system (TRS) R is compatible if for each pair of rules in R, there exist appropriate linearizations and they are almost non overlapping. Chew s theorem [Che81] states that the unique normal form property (UN) holds in a compatible TRS, i.e. normal forms are unique up to conversion. The theorem is important since compatibility is a syntactic condition and the class partly contains nonleft linear non terminating TRSs. However, there is a general feeling ....

....(UN) holds in a compatible TRS, i.e. normal forms are unique up to conversion. The theorem is important since compatibility is a syntactic condition and the class partly contains nonleft linear non terminating TRSs. However, there is a general feeling of doubt about the original proof in [Che81]. In fact, there is a gap in the proof of a key lemma 1 . There have been several attempts at a new proof, and partial answers have been obtained [dV90, Oga92, TO94] De Vrijer showed that UN of a TRS R can be reduced to the Church Rosser property (CR) of its conditional linearization, R L ....

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P. Chew. Unique normal forms in term rewriting systems with repeated variables. In the 13th STOC, pp. 7--18, 1981.

....a non overlapping left linear pattern HORS is shown to satisfy the conditions (see also [vOvR94] vR96] It is well known that a non overlapping left linear TRS has the Church Rosser property. Without left linearity and with a slight modification of the non overlap requirement, some results [Che81, dV90, TO94, MO95] have concluded the unique normal form property of TRSs. The unique normal form property is a sufficient condition for consistency of the system [KdV89] and weaker than the Church Rosser property. Let us briefly introduce the methodology in [dV90] see also [KdV90] The following theorem ....

.... i: 9 = which is called Parallel Conditional [KdV90] Though this system has only trivial overlap, any CCL of this system has non trivial overlap. It would be solved either by a model theoretic approach [dV90, KdV90] or by extending the notion of compatibility of TRSs [Che81, MO95]. The former was also used to conclude the unique normal form property of untyped calculus with Surjective Pairing in [KdV89] whereas the latter would give a decidable condition. Acknowledgements We would like to thank Vincent van Oostrom for his helpful comments, and the members of the TRS ....

P. Chew. Unique normal forms in term rewriting systems with repeated variables. In the 13th STOC, pp. 7--18, 1981.

....nite terms in logic programming. In International conference on cation with in cation without occur check [MR84] have unique normal forms This conjecture was originally proposed in [OO89] with an incomplete proof, as an extension of the result on strongly nonoverlapping systems [Klo80][Che81]. Related results appear in [OO93] TO94] MO94] but the original conjecture is still open. This is related to Problem 58. This problem is also related with modularity of con uence of systems sharing constructors, see [Ohl94] Remark: The answer is yes if the system is also nonduplicating [Ver96] ....

.... The answer is yes if the system is also nonduplicating [Ver96] A direct technique is given in [Ver96] The nonduplicating condition can be relaxed under a certain technical condition [Ver96] Some extensions to handle root overlaps are given in [Ver97] and a restricted version of the result in [Che81] is also proved using the direct technique in [Ver97] ....

Paul Chew. Unique normal forms in term rewriting systems with repeated variables. In Proceedings of the Thirteenth Annual Symposium on Theory of Computing, pages 7-18. ACM, 1981.

....of the left in normal form) which is not length decreasing Problem 58 (M. Oyamaguchi) Is any strongly non overlapping right linear term rewriting system confluent ( Strong in the sense that left hand sides are nonoverlapping even when the occurrences of variables have been renamed apart [ Chew, 1981 ] On the one hand, strongly non overlapping systems need not be confluent [ Huet, 1980 ] on the other hand, strongly non overlapping right ground systems are [ Oyamaguchi and Ohta, 1993 ] Problem 59 (M. Kurihara, M. Krishna Rao) One of the earliest results established on modularity of ....

Paul Chew. Unique normal forms in term rewriting systems with repeated variables. In Proceedings of the Thirteenth Annual Symposium on Theory of Computing, pages 7--18. ACM, 1981.

....systems of equations. There are also many natural equations, such as equal(x; x) true, that violate the left linearity constraint. Such violations seem to be less common, and less crucial in practice, and the theoretical results needed to deal with them appear to be quite difficult. Paul Chew [Che81] proved that nonoverlapping, but not necessarily left linear, equations produce unique normal forms, but they do not always have the Church Rosser property. Uniqueness of normal forms, without the Church Rosser property, is not enough to guarantee completeness of a term rewriting implementation, ....

L. P. Chew. Unique normal forms in term rewriting systems with repeated variables. In 13th Annual ACM Symposium on Theory of Computing, pages 7--18, 1981.

....sequences are allowed. Chew showed that uniqueness of normal forms still holds in a more liberal set of systems without restriction 3 (i.e. repeated variables are allowed) but where restriction 4 must hold for the related system in which left hand side variables are renamed to be distinct [Che81] (i.e. f(g(x; x) x) is defined to overlap with g(a; b) in spite of the fact that the a and b are distinct and cannot be substituted for the same variable x) The Church Rosser property does not always hold in this case, and I do not know any efficient, complete techniques for implementing ....

L. P. Chew. Unique normal forms in term rewriting systems with repeated variables. In 13th Annual ACM Symposium on Theory of Computing, pages 7--18, 1981.

....pair. In the above examples, R 1 is not strongly non overlapping and R 2 is not simple right linear. With neither linearity nor termination, it seems difficult to establish CR, but there are some results concluding UN itself instead of CR. Chew showed that a strongly non overlapping TRS is UN [4]. For example, R 2 above is strongly non overlapping, and therefore UN. As an extension, he also stated that UN holds for a compatible TRS, where a TRS is compatible if, for each pair of rules, appropriate linearizations of the rules exist that are almost non overlapping. However, there is a ....

....and therefore UN. As an extension, he also stated that UN holds for a compatible TRS, where a TRS is compatible if, for each pair of rules, appropriate linearizations of the rules exist that are almost non overlapping. However, there is a general feeling of doubt about the original proof in [4]. In fact, there is a gap in the proof of a key lemma (see Appendix for details) There have been several attempts at a new proof of Chew s theorem, and partial answers have been obtained [35,21,31] De Vrijer refined Chew s methodology in terms of conditional linearization [35] The ....

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P. Chew. Unique normal forms in term rewriting systems with repeated variables. In Proc. the 13th ACM STOC, pages 7--18, 1981.

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Paul Chew. Unique normal forms in term rewriting systems with repeated variables. In Proceedings of the Thirteenth Annual Symposium on Theory of Computing, pages 7-18. ACM, 1981.

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P. Chew. Unique normal forms in term rewriting systems with repeated variables. In Proceedings of the Thirteenth Annual SymposiumonTheory of Computing, pages 7#18. ACM, 1981.

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