Enno Ohlebusch. On the modularity of termination of term rewriting systems. Theoretical Computer Science, 136:333--360, 1994.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....all shared constructors fully replacing, and let (R ; be their union (with R = F ; R ) As for TRS s, this result holds not only for finite CSRS s, but also for finitely branching ones. But, in contrast to the disjoint union case, it doesn t hold any more for infinitely branching systems, cf. [35] for a counterexample. F = F 1 ]F 2 , 1 t 2 ) Moreover suppose that there exists an f i 2 F i with j( f i )j 2, for i = 1;2. Then, if (R 1 ; 1 ) and (R 2 ; 2 ) are simply terminating, then (R ; is simply terminating, too. Note that also other symmetric and asymmetric results can ....

E. Ohlebusch. On the modularity of termination of term rewriting systems. Theoretical Computer Science, 136:333--360, 1994.

....all shared constructors fully replacing, and let (R , be their union (with R = F , R ) As for TRS s, this result holds not only for finite CSRS s, but also for finitely branching ones. But, in contrast to the disjoint union case, it doesn t hold any more for infinitely branching systems, cf. [35] for a counterexample. F = F 1 2 ) Moreover suppose that there exists an f i 2, for i = 1,2. Then, if (R 1 , 1 ) and (R 2 , 2 ) are simply terminating, then (R , is simply terminating, too. Note that also other symmetric and asymmetric results can be easily obtained from ....

E. Ohlebusch. On the modularity of termination of term rewriting systems. Theoretical Computer Science, 136:333--360, 1994.

....a new (regarding all involved signatures) symbol; we denote the projective TRS fG(x; y) x; G(x; y) yg. A system R is said C E terminating if R [ is terminating. 2 Constructors and defined symbols are here Arts Giesl s notions. 3 We use here the terminology introduced by Olhebusch [14]. That notion gives precious results. In particular: C E termination is a modular property for disjoint union of TRS [8, 15] as well as for union of composable TRS as shown by Kurihara Ohushi [9] Note that before the introduction of dependency pairs termination proofs were usually performed ....

E. Ohlebusch. On the modularity of termination of term rewriting systems. Theoretical Comput. Sci., 136:333--360, 1994.

....conceptual levels. At one end, combination of logical systems is studied with an emphasis on formal properties, using tools from mathematics and logics. Examples of results obtained on this level are transfer results for modal logics [43, 28] and modularity results for term rewriting systems [74, 52]. On the other end of the spectrum, the combination of software tools necessitates considering physical connections and appropriate communication languages [19, 44] Between these two extremes lies the combination of constraint systems and the respective solvers, which is the topic of this paper. ....

E. Ohlebusch. On the modularity of termination of term rewriting systems. Theoretical Computer Science, 136:333-360, 1994.

.... following infinite rewrite sequence: f(g(0; 1) g(0; 1) g(0; 1) 2 f(0; 1; g(0; 1) f(g(0; 1) g(0; 1) g(0; 1) Toyama s observation stimulated several researchers to establish sufficient conditions under which the disjoint union of term rewriting systems preserves termination (see [41, 81] and the references given there) The interesting fact, now, is that such conditions are not needed in the case of term graph rewriting. For, termination of ) coll does behave modular with respect to disjoint unions. To demonstrate this by Toyama s example, let us try to simulate by ) the infinite ....

Enno Ohlebusch. On the modularity of termination of term rewriting systems. Theoretical Computer Science, 136:333--360, 1994.

....termination is not modular. One of the (finite) TRSs in his famous counterexample is not simplifying. Kurihara and Ohuchi [7] showed that simplifyingness is a modular property. Gramlich [4] showed that pseudo simple termination is a modular property of so called finitely branching TRSs. Ohlebusch [11] showed that the latter result extends to arbitrary systems. The following result is easily derived from this. Theorem 5. Simple termination is a modular property of TRSs. 4 Because of the disjointness requirement, modularity is a rather restricted property. If we allow the sharing of certain ....

....[8] were the first to study cs modularity. They showed that simplifyingness is cs modular. Gramlich [4] showed that pseudo simple termination is cs modular for finitely branching TRSs. Surprisingly, the latter result does not extend to arbitrary TRSs, as shown by the following example of Ohlebusch [11]: R 1 = ff i (c i ; x) f i 1 (x; x) j i 2 Ng; R 2 = fa c i j i 2 Ng: Both TRSs are pseudo simply terminating. They share constructors fc i j i 2 Ng, but their union is not (pseudo simply) terminating. With help of Theorem 4 it is not difficult to show that TRS R 1 is not simply terminating. ....

E. Ohlebusch, On the Modularity of Termination of Term Rewriting Systems, Theoretical Computer Science, to appear.

....of a strongly innermost normalizing system need not be strongly innermost normalizing. The independent subsystems play a very crucial role in studying modularity of composable and hierarchical combinations, where two systems sharing some de#ned symbols and rewrite rules are combined (see e.g. [3, 9 11, 13, 16]) Two su#cient conditions for strong innermost normalization of independent subsystems are proposed: i) the subsystem is non duplicating or (ii) the omitted rules do not overlap the rules in the chosen subsystem. These two results are tight in the sense that violation of both these conditions ....

E. Ohlebusch, On the modularity of termination of term rewriting systems, Theoret. Comput. Sci. 136 (1994) 333 --360.

....y i ) of minimal length gives rise to a sequence [x 1 ] x 2 ] y n ] in the underlying order R Gamma (TX ) of the coproduct monad at X . See [Lu97] for the details. A term rewriting system Theta is called strongly normalising under deterministic collapses (SNDC or C E terminating) [Ohl94,Gra92] if it is SN and the disjoint union Theta C E is SN, where C E is the term rewriting system C E def = fG( x; y) x; G( x; y) yg. A recent term rewriting result is that the disjoint union is not SN if one system is SNDC and the other collapsing. The term rewriting proof is a rather ....

....4) SN is modular for simplifying systems. Proof. The first is obvious. For the rest, non duplicating and simplifying systems are strongly normalising under deterministic collapses [Gra92] Hence, all of the conditions listed in the introduction follow as corollaries from Theorem 45. [Ohl94] contains further derived criteria. 8 Conclusions and Further Work We have shown how monads can be used to give a semantics to term rewriting systems by generalising the well known equivalence between universal algebra and finitary monads on Set. Monads are well suited to the study of modular ....

E. Ohlebusch. On the modularity of termination of term rewriting systems. TCS 136:333-- 360, 1994.

.... infinite rewrite sequence: f(g(0; 1) g(0; 1) g(0; 1) 2 f(0; 1; g(0; 1) f(g(0; 1) g(0; 1) g(0; 1) ut Toyama s observation stimulated several researchers to establish sufficient conditions under which the disjoint union of term rewriting systems preserves termination (see [41,82] and the references given there) The interesting fact, now, is that such conditions are not needed in the case of term graph rewriting. For, termination of ) coll does behave modular with respect to disjoint unions. To demonstrate this by Toyama s example, let us try to simulate by ) the infinite ....

Enno Ohlebusch. On the modularity of termination of term rewriting systems. Theoretical Computer Science, 136:333--360, 1994.

....automated. A first system based on MSPO has been developed by which many examples, including the ones in the paper, have been checked. The software and examples are available at www.lsi.upc.es albert. Additionally, applying known abstract sufficient conditions ensuring modularity of termination [Ohl94, Gra94], modularity results for termination based on MSPO are obtained. In particular, for TRS proved terminating using MSPO with the aforementioned classes of underlying quasi orderings, termination is proved to be modular for disjoint systems and for finite constructor sharing systems. Note that these ....

....constructor sharing unions of TRS s, i.e. systems which share only constructors. A TRS R is called terminating under non deterministic collapses, denoted C terminating, if R [ fG(x; y) x; G(x; y) yg terminates for some new symbol G. For C termination we have the following results: ffl [Ohl94] C termination is a modular property for disjoint unions of TRS. ffl [Gra94] C termination is a modular property for constructor sharing unions of finite TRS. Lemma 6.1. If Xi B and R is included in mspo then R is C terminating. Note that since B includes the subterm relation and G ....

E. Ohlebusch. On modularity of termination of term rewriting systems. Theoretical Computer Science, 136(2):333--360, 1994.

.... The above results are successfully applied to the study of the modularity of important properties of term rewriting systems: termination, completeness and uniqueness of normal forms (the only main properties of TRSs that are not modular) In particular, we show that C E termination (cf. [10, 21]) is a maximal criterion, and provide a formal justification in terms of complexity of the difficulty of the study of the modularity of termination in TRS. Moreover, we completely solve the problem of the modularity of termination for left linear TRSs, providing the only two optimal criteria. We ....

....property P =Termination (Termination will be also indicated with the acronym SN, after Strong Normalization) Lemma 9.1 (TRSs; Phi) is SN acid. Among the many results on the modularity of termination (see e.g. 20, 13, 22, 24] for a panoramic) the best results so far obtained are the ones in [21] and [14] We will come back to the result of [14] in the next subsection. In [21] Ohlebusch, generalizing a previous result of Gramlich for finitely branching TRSs ( 10] proved that C E termination is modular. It is straightforward to see that the class of C E terminating TRSs coincides ....

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E. Ohlebusch. On the modularity of termination of term rewriting systems. Theoretical Computer Science, 136(2):333--360, 1994.

.... a couple of investigations about how to obtain sufficient criteria for the preservation of termination, 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 ....

.... 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 short) finally have been obtained in ....

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E. Ohlebusch. On the modularity of termination of term rewriting systems. Technical Report 11, Universitat Bielefeld, March 1993.

.... on three different approaches concerning the essential ideas and proof structures (cf. 9] 1) a general approach via an abstract structure theorem where the basic idea is to reduce non termination in the union to non termination of a slightly extended generic version of one of the systems ( 7] [19]) 2) a modular approach via modularity of innermost termination where sufficient criteria for the equivalence of innermost termination and general termination are combined with the modularity of innermost termination ( 8] and (3) a syntactic approach via left linearity which in essence is ....

E. Ohlebusch. On the modularity of termination of term rewriting systems. Theoretical Computer Science, 136:333--360, 1994.

....a transformation function 4 Phi into a reduction sequence Phi(uoe) R 1 Phi(voe) R1[C E Phi(uoe) of terms over F 1 [ fConsg. The transformation function Phi satisfies Phi(t) C [ Phi(t 1 ) Phi(t n ) for every term t with root(t) 2 F 1 and t = C[ t 1 ; t n ] cf. Ohlebusch, 1994b ] In this case, we first define oe 0 = fx i 7 Phi(x i oe) j 1 i ng and obtain uoe 0 = Phi(uoe) R 1 Phi(voe) voe 0 R1[C E Phi(uoe) uoe 0 : Let uoe 0 = u 0 ; u 1 ; u k = uoe 0 be the sequence of terms occurring in the above reduction sequence. Now in each ....

E. Ohlebusch. On the Modularity of Termination of Term Rewriting Systems. Theoretical Computer Science, 136:333--360, 1994.

....seems to be essential. A TRS R is called variable preserving or non erasing if all variables that occur at the left hand side l of some rule l r 2 R also occur in r. Of course sufficient conditions can be given which ensure the modularity of termination. The interested reader is referred to [6, 3] where it is shown that most of the known results are covered by one powerful theorem. Another deep theorem of Toyama, Klop and Barendregt [9] says that termination is a modular property of confluent left linear TRSs. Their result, in conjunction with the above observation, suggests to examine ....

E. Ohlebusch. On the Modularity of Termination of Term Rewriting Systems. Report Nr. 11, Universitat Bielefeld, 1993. Revised version to appear in Theoretical Computer Science.

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Enno Ohlebusch. On the modularity of termination of term rewriting systems. Theoretical Computer Science, 136:333--360, 1994.

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E. Ohlebusch. On the modularity of termination of term rewriting systems. Theoretical Computer Science, 136:333-- 360, 1994.

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Enno Ohlebusch. On the modularity of termination of term rewriting systems. Theoretical Computer Science, 136:333--360, 1994. In essence, a proof in the spirit of [18] (still) seems to fail, since the deletion part of the pile-and-delete technique of [18], [22] appears not to work anymore (as argued already in [18]). 24

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E. Ohlebusch. On the modularity of termination of term rewriting systems. Theoretical Comput. Sci., 136:333--360, 1994.

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Enno Ohlebusch. On the modularity of termination of term rewriting systems. Theoretical Computer Science, 136:333360, 1994.

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Ohlebusch, E.: 1994, `On the Modularity of Termination of Term Rewriting Systems'. Theoretical Computer Science 136, 333--360.

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Ohlebusch, E.: 1994, `On the Modularity of Termination of Term Rewriting Systems'. Theoretical Computer Science 136, 333--360.

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E. Ohlebusch. On modularity of termination of term rewriting systems. Theoretical Computer Science, 136(2):333--360, 1994.

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Enno Ohlebusch. On the modularity of termination of term rewriting systems. Theoretical Computer Science, 136:333--360, 1994.

No context found.

Enno Ohlebusch. On the modularity of termination of term rewriting systems. Report 11, Abteilung Informationstechnik, Universit#at Bielefeld, Bielefeld, Germany, 1993.

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