Aart Middeldorp and Yoshihito Toyama. Completeness of combinations of constructor systems. In Book [Boo91], pages 174{ 187.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....F . If for all rewrite rules l r of R one has l rpo r, then the TRS (F ; R) is terminating. Constructor systems are a subclass of TRSs. It turns out that the transformations of well moded logic programs as presented in Section 2 always yields a constructor system. 1.19. Definition (cf. [MT91], Gra93] A constructor system (CS for short) is a TRS (F ; R) with the property that F can be partitioned into disjoint sets D and C such that every left hand side F (t 1 ; t n ) of a rewrite rule of R satisfies F 2 D and t 1 ; t n 2 T (C; V) Function symbols in D are called ....

Aart Middeldorp and Yoshihito Toyama. Completeness of combinations of constructor systems. Proceedings of RTA-91, Lecture Notes in Computer Science(488):188--199, April 1991. 26

....The usual notions of unions can be expressed in terms of modules in a straightforward way. We will say that [F 1 j R 1 ] extends [F 0 j R 0 ] regardless of [F 2 j R 2 ] if F 1 F 2 = F 0 j R 0 ] F 1 j R 1 ] and [F 0 j R 0 ] F 2 j R 2 ] such extension is indeed a union of composable TRS [9, 12, 15]. We will talk about the disjoint union R 1 [ R 2 if [F 1 j R 1 ] and [F 2 j R 2 ] extend [ j ; The union will be constructor sharing if [F 1 j R 1 ] and [F 2 j R 2 ] extend [F 0 j ; A property P is said modular for a specific kind of union if R 1 and R 2 having property P implies that the ....

A. Middeldorp and Y. Toyama. Completeness of combinations of constructor systems. In Proc. 4th Rewriting Techniques and Applications, LNCS 488, Como, Italy, 1991.

....of selection invariance In this section, we give a straightforward proof for one of the main results of [9] using the selection invariance result of the previous section. In [9] modularity of completeness is established for a class of hierarchical combinations generalizing the main result of [15]. The main step in proving that result is the proof of strong innermost normalization of R 0 #R 1 when R 0 and R 1 are complete. From this and Theorem 3 follows the modularity of completeness for the following class of hierarchical combinations. 3 The proof given in [9] is somewhat contrived ....

A. Middeldorp, Y. Toyama, Completeness of combinations of constructor systems, J. Symbolic Comput. 15 (1993) 331--348.

....has been done on modular properties of term rewrite systems, some of which is summarized in Theorem 2.3. Unfortunately, applicative systems are never disjoint, as they all contain the symbol Ap. Thus none of these modularity theorems apply to unions of applicative systems. Some authors (e.g. Middeldorp and Toyama, 1991; Krishna Rao, 1993) have extended the notion to allow certain types of shared symbols, but unions of applicative systems still fall outside the scope of known results. Definition 8.1. Two applicative systems are applicatively disjoint if their only common function symbol is Ap. A property P of ....

Middeldorp, A. and Toyama, Y. (1991). Completeness of combinations of constructor systems. J.

.... R 2 , where the operators defined by R 2 do not occur in R 1 , and the operators defined by R 1 are not defined by R 2 . Suppose that R 1 is SN. Then R 1 R 2 is SN i# SN (R 1 ) R 2 is SN. 2 Proof. Since both systems are confluent, this is a special case of Theorem 3. 5 of Middeldorp and Toyama [MT91]. 2 Suppose we have an SN orthogonal constructor TRS, and augment it by rules defining a new operator F . Then Theorem 7.3 implies that we can prove SN of the new system by proving SN for just the rules for F over terms in normal form w.r.t. the rules of the old system, with those rules being ....

A. Middeldorp and Y. Toyama. Completeness of combinations of constructor systems. In 4th Int. Conf. on Rewriting Techniques and Applications, pages 188--199, 1991. Lecture Notes in Computer Science, vol. 488.

....to separate the distinct atoms in the body of a clause. In Section 5 some examples are given to clarify how this algorithm transforms logic programs into TRSs. It turns out that well moded logic programs transform into TRSs of a special form, so called constructor systems. 3.1. Definition (cf. [MT91], Gra92] A constructor system (CS for short) is a TRS (F ; R) with the property that F can be partitioned into disjoint sets D and C such that every left hand side F (t 1 ; t n ) of a rewrite rule of R satisfies F 2 D and t 1 ; t n 2 T (C; V) Function symbols in D are called ....

Aart Middeldorp and Yoshihito Toyama. Completeness of combinations of constructor systems. Proceedings of RTA-91, LNCS(488):188--199, April 1991.

.... There has been considerable work to show modularity for properties of term rewrite systems (TRS) such as confluence and termination, e.g. 16, 24, 10] Most of the results on modularity concern the union of term rewrite systems with disjoint signatures, only a few also cover shared symbols [17, 20]. For the union of TRS with shared symbols, many approaches are based on commutation criteria, see Figure 1 for the common definitions, where R = R 1 R 2 is assumed. For an overview see [16] For convergent R, some criteria for modularity of normalization follow from known results. Our main ....

Aart Middeldorp and Y. Toyama. Completeness of combinations of constructor systems. In Proc. 4th Int. Conf. Rewriting Techniques and Applications. LNCS 488, 1991.

....are shared. We will recall two modularity results for unions of term rewriting systems with shared defined functions that are easy to apply to interaction nets, and deduce from them two further modularity results for interaction nets. 4.1. 1 Unions of Constructor Systems Middeldorp and Toyama [27] studied the modularity of termination of unions of constructor systems (recall that in a constructor system, left hand sides of rules have the form f(l 1 ; l n ) where f 2 D and l 1 ; l n 2 T (C; X ) In particular, they considered composable unions: two constructor systems (D 1 ....

....) where f 2 D and l 1 ; l n 2 T (C; X ) In particular, they considered composable unions: two constructor systems (D 1 ; C 1 ; R 1 ) D 2 ; C 2 ; R 2 ) are composable if D 1 C 2 = D 2 C 1 = and for any d 2 D 1 D 2 , the same rules defining d appear in R 1 and R 2 . As shown in [27], unions of pairwise composable systems are not modular with respect to termination, but confluent composable constructor systems are. This result applies directly to non dependent semi simple interaction nets: the result of applying Theta to a union of interaction nets is a union of confluent ....

A. Middeldorp and Y. Toyama. Completeness of combinations of constructor systems. In Proc. 4th Rewriting Techniques and Applications, number 488 in LNCS. Springer-Verlag, 1991.

.... y) l) sort(nil) nil sort(cons(x; l) insert(x; sort(l) Here, insert must have a lexicographic status, from right to left, while min; max and sort may have a multiset status (or a lexicographic one) Note that most examples of hierarchical systems found in the literature (e.g. [15]) and the above one, are indeed constructor systems, but our result does not require this restriction. We now turn to the main result of this section: Theorem14. Let R = R 0 R 1 : Rm be a hierarchical union such that R 0 is terminating and alien decreasing (resp. cap decreasing) and R 1 ....

A. Middeldorp and Y. Toyama. Completeness of combinations of constructor systems. In Proc. 4th Rewriting Techniques and Applications, LNCS 488, Como, Italy, 1991.

....These first results have been extended by Middeldorp [676] who also considered the case of conditional rewrite systems. The disjointness assumption was relaxed in the case of constructor systems that are allowed to share constructors, while preserving the confluence and termination properties [678]. A survey of properties of rewrite systems preserved under (disjoint) unions can be found in [677] In [629] term rewriting in structured algebraic specifications is addressed, in particular the question whether properties such as confluence and termination are preserved by structuring ....

A. Middeldorp and Y. Toyama. Completeness of Combinations of Constructor Systems. In R. Book, editor, Rewriting Techniques and Applications. 4th International Conference, RTA-91, pages 188-- 199. Springer LNCS 488, 1991.

....; R) D]C;R) is called a constructor system if every left hand side f(t 1 ; t n ) of a rewrite rule of R satisfies t 1 ; t n 2 T (C; V) Notice that the TRSs in the above example are constructor systems. An interesting result in this regard was obtained by Middeldorp and Toyama [MT91]; they proved that convergence is preserved under the combination of constructor systems which do not share defined symbols (in fact, they proved a more general result) Thus hierarchical systems in this limited sense are more an extension of combinations of constructor systems than a ....

A. Middeldorp and Y. Toyama. Completeness of Combinations of Constructor Systems. In Proceedings of the 4th International Conference on Rewriting Techniques and Applications, pages 188--199. Lecture Notes in Computer Science 488, Springer Verlag, 1991.

....general kind of combination: so called composable systems. As far as conditional term rewriting systems are concerned, all known modularity result but one apply only to disjoint systems. Here we investigate conditional systems which may share constructors. Furthermore, we refute a conjecture of Middeldorp (1990, 1993). 1. Introduction Term rewriting has applications in various fields of computer science such as symbolic computation, functional programming, abstract data type specifications, program verification, program synthesis, and automated theorem proving. In an outstanding paper, Knuth and Bendix (1970) ....

....investigated constructor sharing systems; constructors are function symbols that do not occur at the root position of the left hand side of any rewrite rule, the others are called defined symbols. Among other things, they showed that confluence is not modular for constructor sharing systems. Middeldorp and Toyama (1993) introduced composable systems which have to contain all rewrite rules that define a defined symbol whenever that symbol is shared. The authors, however, restricted their investigations to constructor systems (where no proper subterm of a left hand side of a rewrite rule is allowed to contain ....

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Middeldorp, A., Toyama, Y. (1993). Completeness of combinations of constructor systems. J. Symbolic Computation 15(3), 331--348.

....henceforth be written as l r. 2.3. Definition. A term rewrite system (TRS) is a pair (F ; R) consisting of a signature F and a set R of rewrite rules between terms in T (F ; V) A TRS is called finite if both F and R are finite. Constructor systems are a subclass of TRSs. 2.4. Definition (cf. MT91] Gra93] A constructor system (CS for short) is a TRS (F ; R) with the property that F can be partitioned into disjoint sets D and C such that every left hand side f(t 1 ; t n ) of a rewrite rule of R satisfies f 2 D and t 1 ; t n 2 T (C; V) Function symbols in D are called ....

Aart Middeldorp and Yoshihito Toyama. Completeness of combinations of constructor systems. Proceedings of RTA-91, Lecture Notes in Computer Science(488):188--199, April 1991.

.... completeness (i.e. termination plus confluence) and of other interesting properties of TRSs under disjoint combinations (cf. e.g. Rus87] Mid89] TKB89] Mid90] KO90] Gra92a] Gra92b] Ohl93b] Non disjoint unions of TRSs with common constructors have been considered e.g. in [MT91], KO92] Gra92a] Gra92b] Ohl93b] Ohl93a] More general hierarchical combinations of TRSs have recently been dealt with in [Kri93] Der93] Kri94] Gra93b] FJ93] Some preservation results for (disjoint and non disjoint, but non hierarchical) combinations of conditional TRSs (CTRSs for ....

A. Middeldorp and Y. Toyama. Completeness of combinations of constructor systems. In R.V. Book, editor, Proc. 4th RTA, LNCS 488, pp. 174--187. Springer, 1991.

....EVENODD(x; y) to x, the recursive path ordering satisfies these constraints. 8.29. Modularity, Version 1 The following example demonstrates the usefulness of modularity results. f(c(x; s(y) f(c(s(x) y) g(c(s(x) y) g(c(x; s(y) Modularity results (such as Thm. 8. 1 or a result from [MT91] stating that completeness is modular for constructor systems with disjoint sets of defined symbols) allow us to prove innermost normalisation (and thereby, termination) of both rules separately. So we may use different well founded orderings for the constraint F(c(x; s(y) F(c(s(x) y) and ....

A. Middeldorp and Y. Toyama. Completeness of combinations of constructor systems. In R.V. Book, editor, Proceedings of the 4th International Conference on Rewriting Techniques and Applications, RTA-91, volume 488 of Lecture Notes in Computer Science, pages 188--199, Como, Italy, April 1991. Springer Verlag, Berlin.

....has been done on modular properties of term rewrite systems, some of which is summarized in Theorem 2.3. Unfortunately, applicative systems are never disjoint, as they all contain the symbol Ap. Thus none of these modularity theorems apply to unions of applicative systems. Some authors (e.g. Middeldorp and Toyama, 1991; Krishna Rao, 1993) have extended the notion to allow certain types of shared symbols, but unions of applicative systems still fall outside the scope of known results. Definition 8.1. Two applicative systems are applicatively disjoint if their only common function symbol is Ap. A property P of ....

Middeldorp, A. and Toyama, Y. (1991). Completeness of combinations of constructor systems. J.

....F . If for all rewrite rules l r of R one has l rpo r, then the TRS (F ; R) is terminating. Constructor systems are a subclass of TRSs. It turns out that the transformations of well moded logic programs as presented in Section 2 always yields a constructor system. 1.19. Definition (cf. [MT91], Gra93] A constructor system (CS for short) is a TRS (F ; R) with the property that F can be partitioned into disjoint sets D and C such that every left hand side F (t 1 ; t n ) of a rewrite rule of R satisfies F 2 D and t 1 ; t n 2 T (C; V) Function symbols in D are called ....

Aart Middeldorp and Yoshihito Toyama. Completeness of combinations of constructor systems. Proceedings of RTA-91, Lecture Notes in Computer Science(488):188--199, April 1991.

....Unfortunately, duplicating and collapsing rules occur quite often and naturally in many cases. Hence, practical applicability of conditions (a) c) is rather limited. For further work on modular aspects of termination and of other important properties of TRSs we refer to [15] 11] 5] 13] and [16]. In this paper we shall provide a unifying structural approach to modularity of termination by means of a careful analysis of potential counterexamples. More details and missing proofs can be found in the extended version [6] Extensions and further related results are presented in [8] 7] 3 ....

A. Middeldorp and Y. Toyama. Completeness of combinations of constructor systems. In R.V. Book, editor, Proc. of the 4th Int. Conf. on Rewriting Techniques and Applications, volume 488 of Lecture Notes in Computer Science, pages 174--187. Springer, 1991.

No context found.

Aart Middeldorp and Yoshihito Toyama. Completeness of combinations of constructor systems. In Book [Boo91], pages 174{ 187.

No context found.

A. Middeldorp and Y. Toyama. Completeness of combinations of constructor systems. Journal of Symbolic Computation, 15:331--348, Sept. 1993.

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Middeldorp, A. and Y. Toyama: 1993, `Completeness of Combinations of Constructor Systems'. Journal of Symbolic Computation 15, 331--348.

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Middeldorp, A. and Y. Toyama: 1991, `Completeness of combinations of constructor systems'. In: Proc. 4th Rewriting Techniques and Applications, Como, LNCS 488. Como, Italy.

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Middeldorp, A. and Y. Toyama: 1990, `Completeness of combinations of constructor systems'. Technical Report R9058, CWI.

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Aart Middeldorp and Yoshihito Toyama. Completeness of combinations of constructor systems. In R. Book, editor, Proceedings of the Fourth InternationalConferenceon Rewriting Techniques and Applications #Como, Italy#,volume 488 of Lecture Notes in Computer Science, pages 174#187, Berlin, April 1991. Springer-Verlag.

No context found.

Aart Middeldorp and Yoshihito Toyama. Completeness of combinations of constructor systems. Technical Report R9058, CWI, October 1990.

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