E. Ohlebusch. On the modularity of confluence of constructor-sharing term rewriting systems. In Proceedings of the 19th colloquium on Trees in Algebra and Programming, LNCS 787, pages 261--275, 1994.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....r 2 R 2 g are confluent and disjoint. Then by modularity of confluence for disjoint union of TRSs (e.g. see [3] fl 1 r 1 j l r 2 R 1 g [ fl 2 r 2 j l r 2 R 2 g = R 1 [ R 2 ) is confluent. Therefore R 1 [ R 2 is confluent by Corollary 3.12. 3 The following corollary is given in [7]. Corollary 3.19. Confluence is modular for combinations of constructor systems with shared constructors if the combined system contains neither collapsing nor constructor lifting rules. Proof. This is nothing but a combination of TRSs with a common transparent part R 1 j C = R 2 j C = 3 ....

....we have shown modularity results for two kinds of combinations of TRSs: a combination with a common transparent part and a combination with a common lower sorted part. Our results give extensions of modularity of termination in [2] modularity application of [11] and modularity of confluence in [7]. Also it gives an alternative proof of modularity application of [1] 19 Although we did not present it, it is easily seen that one can similarly lift other modularity results for disjoint union of TRSs (e.g. 4] 9] for these combinations of TRSs. ....

E. Ohlebusch. On the modularity of confluence of constructor-sharing term rewriting systems. In Proceedings of the 19th colloquium on Trees in Algebra and Programming, LNCS 787, pages 261--275, 1994.

....components [8] Following standard terminology, we say a property P is modular if P is inferred from those of its components. However, unlike other important properties in the theory of term rewriting, only few results which relax the disjoint limitation for modularity of confluence is known [6], 3] A property P of TRSs is said to be persistent if for any many sorted TRS hF ; Ri, hF ; Ri has the property P iff its underlying unsorted TRS h Theta(F ) Theta(R)i has the property P. Here Theta denotes the sort elimination function. The notion of persistency is formulated by H. Zantema ....

Ohlebusch, E., On the modularity of confluence of constructorsharing term rewriting systems. In Proceedings of the 19th colloquium on Trees in Algebra and Programming, LNCS 787, pp. 261--275, 1994.

....[9] Following standard terminology, we say a property P is modular if P is inferred from those of its subsystems. However, unlike other important properties in the theory of term rewriting, only few results which relax the disjoint limitation of modularity of confluence are known [2] 5] [8]. A property P of TRSs is said to be persistent if for any many sorted TRS hF ; Ri, hF ; Ri has the property P iff its underlying unsorted TRS h Theta(F ) Theta(R)i has the property P. Here Theta denotes the sort elimination 1 function. The notion of persistency is formulated by H. Zantema ....

.... C (r3) F (x) F (G(x) r4) To show the confluence of this system directly seems difficult from known results, since this system is neither terminating nor left linear. It is easily seen that the modularity for disjoint union can not apply. Also we can neither apply results of [2] 5] nor [8]. Let S be a set f0; 1; 2g of ordered sorts with 1 0 and 2 0 and no other relations. Think of sort attachment : 8 : A : 1 f : 0 Theta 1 1 G : 0 0 C : 0 F : 0 2: It is easily checked this is consistent with R. Now, R is confluent iff any well sorted ....

[Article contains additional citation context not shown here]

E. Ohlebusch. On the modularity of confluence of constructor-sharing term rewriting systems. In Proceedings of the 19th colloquium on Trees in Algebra and Programming, LNCS 787, pages 261--275, 1994.

.... 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 short) finally have been obtained in [Mid90] ....

E. Ohlebusch. On the modularity of confluence of constructor-sharing term rewriting systems. Technical Report 13, Universitat Bielefeld, 1993.

....proof, as an extension of the result on strongly nonoverlapping systems [ 62; 21 ] Related results appear in [ 88; 99; 69 ] but the original conjecture is still open. This is related to Problem 2. This problem is also related with modularity of confluence of systems sharing constructors, see [ 82 ] . Problem 80 (H. Comon) Strong sequentiality is a property of rewrite systems introduced in [ 49 ] see [ 51 ] which ensures the existence of optimal reduction strategies. Is strong sequentiality decidable for arbitrary rewrite systems What is the complexity of strong sequentiality in the ....

E. Ohlebusch. On the modularity of confluence of constructor-sharing term rewriting systems. In Proceedings of the Colloquium on Trees in Algebra and Programming, 1994.

....the simplified proof of Toyama s theorem ( 13] stating that confluence is modular for disjoint unions of (unconditional) TRSs. By guaranteeing the existence of preserved reducts, the proof of [13] essentially carries over to constructor sharing combinations of TRSs as recognized by Ohlebusch ([26]) One straightforward criterion to ensure the crucial preservation property property is to require weak normalization of the involved TRSs which implies the modularity of semi completeness for constructor sharing TRSs ( 26] The preservation property was already implicitly used by Middeldorp ....

.... to constructor sharing combinations of TRSs as recognized by Ohlebusch ( 26] One straightforward criterion to ensure the crucial preservation property property is to require weak normalization of the involved TRSs which implies the modularity of semi completeness for constructor sharing TRSs ([26]) The preservation property was already implicitly used by Middeldorp ( 20] 22] for proving modularity of confluence for disjoint unions of CTRSs. By combining this proof structure of [20] with [26] Ohlebusch recently succeeded in proving the modularity of semi completeness for constructor ....

[Article contains additional citation context not shown here]

E. Ohlebusch, On the modularity of confluence of constructor-sharing term rewriting systems, in: S. Tison, ed., Proc. 19th Coll. on Trees in Algebra and Programming (CAAP'94), Lecture Notes in Computer Science, Vol. 787, (Springer, Berlin, 1994) 261--275.

....the remaining symbols are constructors. The first part of ( states that all the defined symbols are contained in D. The second part states that no rule of S is collapsing, since r 6= and that no rule of S is constructor lifting, since r 2 D Delta Sigma . For example, E. Ohlebusch proves in [Ohl94] that confluence is inherited by the combination of two constructor sharing term rewriting systems provided that none of the participating systems contains any collapsing or constructor lifting rules. In the following let S i be a finite string rewriting system on Sigma i , i = 1; 2, where Sigma ....

....the following. Corollary 7.12. The string rewriting system S 0 is convergent if and only if S 1 and S 2 both are convergent. Theorem 7. 11 is the adaptation of the corresponding result for constructor sharing termrewriting systems that contain neither collapsing nor constructor lifting rules [Ohl94], while Theorem 7.9 corresponds to a result by B. Gramlich [Gra94] Using the same ideas as above it can further be shown easily that S 0 has one of the properties WN, YUN, NF, UN, or UN if and only if S 1 and S 2 both have this property. We conclude the discussion of the presentation ....

E. Ohlebusch. On the modularity of confluence of constructor-sharing termrewriting systems. In S. Tison, editor, Trees in Algebra and Programming, Proc. of CAAP'94, LNCS787, pages 261--275. Springer-Verlag, Berlin, 1994.

....j 1 ) C b f Phi D b (s 1 ) Phi D b (s l Gamma1 ) Phi D b (u 1 ) Phi D b (u p ) Phi D b (s l 1 ) Phi D b (s m )g, where C b f; g = C b [ C b f; g; Now it is a consequence of s 0 l 2 Delta D b (s l ) 22 E. Ohlebusch that Phi D b (s 0 l ) occurs in the term Phi D b (s l ) and hence Phi D b (s l ) Sort(f Phi D b (u) j u 2 Delta D b (s l )g) h: Phi D b (s 0 l ) i CE Phi D b (s 0 l ) C b f Phi D b (u 1 ) Phi D b (u p )g: All in all, Phi D b (t j ) CE Phi D b (t ....

Ohlebusch, E. (1994a). On the modularity of confluence of constructor-sharing term rewriting systems.

....Properties of Constructor Sharing Conditional Term Rewriting Systems Enno Ohlebusch Universitat Bielefeld, 33501 Bielefeld, Germany, e mail: enno techfak.uni bielefeld.de Abstract. First, using a recent modularity result [Ohl94b] for unconditional term rewriting systems (TRSs) it is shown that semi completeness is a modular property of constructor sharing join conditional term rewriting systems (CTRSs) Second, we do not only extend results of Middeldorp [Mid93] on the modularity of termination for disjoint CTRSs to ....

.... Ohuchi [KO92] refuted the modularity of confluence for constructor sharing TRSs systems which may share constructors. Constructors are function symbols that do not occur at the root position in left hand sides of rewrite rules, the others are called defined symbols. Recently, we have shown [Ohl94b] that semi completeness (confluence plus normalization) is modular for constructor sharing TRSs. In the first part of this paper, we extend this result to CTRSs. Unlike confluence, termination lacks a modular behavior for disjoint TRSs see [Toy87a] The first sufficient conditions ensuring the ....

[Article contains additional citation context not shown here]

E. Ohlebusch. On the Modularity of Confluence of Constructor-Sharing Term Rewriting Systems. In Proceedings of the 19th Colloquium on Trees in Algebra and Programming, pages 261--275. Lecture Notes in Computer Science 787, Springer Verlag, 1994.

....y) y) and the other is collapsing. This result has already been achieved by Gramlich [7] for finitely branching term rewriting systems. A more sophisticated approach is necessary, however, to prove it in full generality. Most of the known sufficient criteria for the preservation of termination [24, 15, 13, 7] follow as corollaries from our result, and new criteria are derived. This paper particularly settles the open question whether simple termination is modular in general. We will moreover shed some light on modular properties of combined systems with shared constructors. For instance, it will be ....

....provided that one of the following conditions is satisfied: 1. Neither R 1 nor R 2 contains collapsing rules. 2. Neither R 1 nor R 2 contains duplicating rules. 3. One of the systems contains neither collapsing nor duplicating rules. Statements 1 and 2 were first proved by Rusinowitch in [24] (Drosten obtained parts of these results independently in 2 he required right linearity, cf. 5] The proof of the last statement is due to Middeldorp [15] A very simple intuitive proof of Theorem 1.2 can be found in [20] see also the proof of Theorem 4.13) An equivalent formulation of ....

[Article contains additional citation context not shown here]

E. Ohlebusch. On the Modularity of Confluence of Constructor-Sharing Term Rewriting Systems. In Proceedings of the 19th Colloquium on Trees in Algebra and Programming, pages 261--275. Lecture Notes in Computer Science 787, Berlin: Springer Verlag, 1994.

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC