Y. Toyama (1987), Counterexamples to termination for the direct sum of term rewriting systems, Inf. Process. Lett. 25, pp. 141-143.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....2, when using the extension of the reduction pair we obtain the same size change graphs as with ,# V) Ex. 4 shows that this TRS is size change terminating w.r.t. this reduction pair and hence, by Thm. 8, this proves innermost 4 termination. However, a variant of Toyama s example [15] shows that Thm. 7 and Thm. 8 are not su#cient to prove full (non innermost) termination. f(c(a, b, x) f(c(x, x, x) g(x, y) x, g(x, y) y . We define #=# # restricted to is the terminating TRS c(a, b, x) c(x, x, x) The only maximal multigraph is 1 f # 1 f . ....

Y. Toyama. Counterexamples to the termination for the direct sum of term rewriting systems. Information Processing Letters, 25:141--143, 1987. 15

....only sound if # is a quasi simplification order. Theorem 9 (Improved Termination Proofs with DPs) Let n be a separation of R. The TRS is terminating if for all 1 n and any cycle of the (estimated) dependency graph of DPR (R i ) there is To see this, consider Toyama s TRS [28] where f(0, 1, x) f(x, x, x) and g(x, x, g(x, y) y . # 1 s and # 2 s dependency graphs are empty, whereas the dependency graph of R1 #R2 has a cycle. Hence, if one only considers the graphs of # 1 and # 2 , one could falsely prove termination. However, if ....

Y. Toyama. Counterexamples to the termination for the direct sum of term rewriting systems. Information Processing Letters, 25:141--143, 1987.

....no shared symbols, that is, the theories to be combined are built over disjoint signatures Modularity properties for term rewriting systems over disjoint signatures have been studied in detail. It has turned out that confluence is a modular property [18] but unfortunately termination is not. In [17] it is shown that there exist two confluent and terminating rewrite systems over disjoint signatures such that their union is not terminating. Thus, the union of systems that provide a decision procedure for the word problem in the single theories does not yield a decision procedure for the word ....

Yoshihito Toyama. Counterexamples to termination for the direct sum of term rewriting systems. Information Processing Letters, 25:141-143, 1987.

....at least with the standard path orderings amenable to automation) whereas with our modularity results innermost termination can easily be verified automatically. 4. 1 Toyama example A famous example of a TRS that is innermost terminating, but not terminating, is the following system by Toyama [Toy87] f(O,l,x) f(x,x,x) g(z, y) z 42 This TRS has only one dependency pair, viz. F(0, 1, x) F(x, x, x) This depen dency pair does not occur on a cycle in the innermost dependency graph, since F(Xl, xl, xl) does not unify with F(0, 1, x2) Thus, no inequalities are generated and therefore ....

....because their left hand sides contain redexes. Hence, the only constraint in this example is A(b( b(x) which is satisfied by the recursive path ordering. 4. 19 An innermost terminating system which requires modularity The following system is a variant of the well known example of Toyama [Toy87] which requires modularity results for its innermost termination proof. f(s(x) y,z) f(x,s(c(y) c(z) f(c(x) x,y) c(y) g(x, y) x The system is not terminating as can be seen from the following infinite (cycling) reduction. f(x,c(x) c(g(x,c(x) ....

Y. Toyama. Counterexamples to the termination for the direct sum of term rewriting systems. Information Processing Letters, 25:141-143, 1987.

....n) 1; 4n) 1; n 2) fA (f A (1; n) gA (f A (0; n) 0; 4n) 0; n 2) g A (g A (0; n) gA (f A (1; n) 0; 2n 2) 0; 2n 1) g A (g A (1; n) for all n 2 IN, proving termination. Example 9 Let the TRS consist of the rule: The origin of this TRS is the example of [62] showing that termination is not modular, after that it has served as a counter example for many properties. De ne 0A = 0; 1) 1A = 1; 1) fA ( a; n) b; m) c; k) 0; n m k) if a = b (0; n m 3k) if a 6= b 16 The function fA is strictly monotone in all three arguments. For all ....

....= 1, the second if (x) 2. Since we have a model we may choose discrete orders and obtain that Decr is empty. Termination of R is easily proved by counting the number of f 2 symbols, or by recursive path order. Using theorem 81 we conclude that the original system R is terminating too. from [62]. This system is not simply terminating as we saw in Example 11. For proving termination we want to use the observation that in the left hand side the 54 rst and the second argument of f are distinct while in the right hand side they are equal. This distinction is made by choosing S f = fa; bg ....

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Toyama, Y. Counterexamples to termination for the direct sum of term rewriting systems. Information Processing Letters 25 (1987), 141-143.

....at least with the standard path orderings amenable to automation) whereas with our modularity results innermost termination can easily be veri ed automatically. 4. 1 Toyama example A famous example of a TRS that is innermost terminating, but not terminating, is the following system by Toyama [Toy87]. 42 This TRS has only one dependency pair, viz. F(0; 1; x) F(x; x; x) This dependency pair does not occur on a cycle in the innermost dependency graph, since F(x 1 ; x 1 ; x 1 ) does not unify with F(0; 1; x 2 ) Thus, no inequalities are generated and therefore the TRS is innermost ....

....their left hand sides contain redexes. Hence, the only constraint in this example is A(b(a(b(x) A(b(x) which is satis ed by the recursive path ordering. 4. 19 An innermost terminating system which requires modularity The following system is a variant of the well known example of Toyama [Toy87] which requires modularity results for its innermost termination proof. f(c(x) x; y) c(y) The system is not terminating as can be seen from the following in nite (cycling) reduction. f(x; c(x) c(g(x; c(x) f(g(x; c(x) g(x; c(x) f(g(x; c(x) x; g(x; c(x) f(x; c(x) ....

Y. Toyama. Counterexamples to the termination for the direct sum of term rewriting systems. Information Processing Letters, 25:141-143, 1987.

....automated termination proofs incrementally, that is to use normalization information about basic rules to show that adding new ones will preserve termination. Unfortunately, the termination property does not behave as well as we could expect when dealing with unions of TRS. As shown by Toyama [16] if two TRS are strongly normalizing, their union does not necessarily terminate, even if the two systems do not share any symbol. The significant work of Gramlich showed that projections were to blame and gave sufficient conditions for ensuring termination of unions of TRS possibly sharing ....

.... constructors of F by the module consisting of defined symbols 2 (as signature part) together with all rules of R (as rules part) 4 Termination and unions of TRS Unfortunately, termination is not a modular property of TRS, even for disjoint unions, as shown by Toyama s famous counter example [16]. Example 2 (Toyama) These two TRS (over disjoint signatures) are terminating: R 1 : f(0; 1; x) f(x; x; x) R 2 : g(x; y) x g(x; y) y: Their union nevertheless allows infinite reductions, for instance: f(g(0; 1) g(0; 1) g(0; 1) R2 f(0; g(0; 1) g(0; 1) R2 f(0; 1; g(0; ....

[Article contains additional citation context not shown here]

Y. Toyama. Counterexamples to termination for the direct sum of term rewriting systems. Inf. Process. Lett., 25:141--143, Apr. 1987.

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Y. Toyama (1987), Counterexamples to termination for the direct sum of term rewriting systems, Inf. Process. Lett. 25, pp. 141-143.

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Y. Toyama. Counterexamples to termination for the direct sum of term rewriting systems. Information Processing Letters, 25(3):141143, 1987.

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Y. Toyama. Counterexamples to termination for the direct sum of term rewriting systems. Information Processing Letters, 25(3):141-143, 1987.

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Toyama, Y.: Counterexamples to termination for the direct sum of term rewriting systems, Information Processing Letters, 25(3), 1987, 141--143.

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Y. Toyama. Counterexamples to termination for the direct sum of term rewriting systems. Information Processing Letters, 25(3):141143, 1987.

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Y. Toyama. Counterexamples to termination for the direct sum of term rewriting systems. Information Processing Letters, 25(3):141143, 1987.

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Y. Toyama. Counterexamples to Termination for the Direct Sum of Term Rewriting Systems. Information Processing Letters, 25:141 -- 143, 1987.

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Y. Toyama. Counterexamples to termination for the direct sum of term rewriting systems. Information Processing Letters, 25(3):141--143, 1987.

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Y. Toyama. Counterexamples to the termination for the direct sum of term rewriting systems. Information Processing Letters, 25:141--143, 1987.

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Yoshihito Toyama. Counterexamples to termination for the direct sum of term rewriting systems. Information Processing Letters, 25:141--143, 1987.

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Y. Toyama. Counterexamples to the termination for the direct sum of term rewriting systems. Information Processing Letters, 25:141--143, 1987.

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Y. Toyama. Counterexamples to the termination for the direct sum of term rewriting systems. Information Processing Letters, 25:141--143, 1987.

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Y. Toyama. Counterexamples to the termination for the direct sum of term rewriting systems. Information Processing Letters, 25:141--143, 1987.

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Y. Toyama. Counterexamples to termination for the direct sum of term rewriting systems. Inf. Process. Lett., 25:141--143, 1987.

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Yoshihito Toyama. Counterexamples to termination for the direct sum of term rewriting systems. Information Processing Letters, 25:141-143, 1987.

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Y. Toyama. Counterexamples to termination for the direct sum of term rewriting systems. Information Processing Letters, 25:141143, April 1987.

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Toyama, Y.: 1987, `Counterexamples to Termination for the Direct Sum of Term Rewriting Systems'. Information Processing Letters 25, 141--143.

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Toyama, Y.: 1987, `Counterexamples to Termination for the Direct Sum of Term Rewriting Systems'. Information Processing Letters 25, 141--143.

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