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 ....

[Article contains additional citation context not shown here]

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.

.... remain terminating when combined with the (nonconfluent, nonbright) system fh(x; y) x; h(x; y) yg (for new function symbol h) Gramlich, 1994 ] c) The systems do not share constructors and each remains terminating when combined with fh(x; y) x; h(x; y) yg (for new function symbol h) Ohlebusch, 1994b ] d) Both systems bright [ Gramlich, 1994; Ohlebusch, 1994b ] e) The systems are both non duplicating (that is, each rule s right side contains no more occurrences of any variable than does the left) Dershowitz, 1995; Ohlebusch, 1994b ] f) One of the systems is both bright and ....

.... system fh(x; y) x; h(x; y) yg (for new function symbol h) Gramlich, 1994 ] c) The systems do not share constructors and each remains terminating when combined with fh(x; y) x; h(x; y) yg (for new function symbol h) Ohlebusch, 1994b ] d) Both systems bright [ Gramlich, 1994; Ohlebusch, 1994b ] e) The systems are both non duplicating (that is, each rule s right side contains no more occurrences of any variable than does the left) Dershowitz, 1995; Ohlebusch, 1994b ] f) One of the systems is both bright and non duplicating [ Dershowitz, 1995; Ohlebusch, 1994b ] The necessity ....

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

....signatures. In Section 8 we investigate the behaviour of simple termination under combinations of term rewriting systems. We show that our notion of simple termination is preserved under constructor sharing combinations. This is not true for the earlier notion of simple termination (Ohlebusch [36]) In two appendices we present some useful facts about partial well orders and, for completeness sake, a proof of Kruskal s Tree Theorem. 2 Preliminaries In order to fix our notations and terminology, we start with a very brief introduction to term rewriting. Term rewriting is surveyed in ....

....So for TRSs over finite signatures the notions of simplifyingness, pseudo simple termination, and simple termination coincide. All areas are inhabited. The TRS R 1 = fa i a i 1 j i 2 Ng we encountered before. For R 2 we can take ff i (a) f i 1 (g(a) j i 2 Ng. This TRS, due to Ohlebusch [35, 36], is simplifying but not pseudo simply terminating because the extension with the embedding rules ff i (x) x j i 2 Ng [ fg(x) xg results in an acyclic TRS that is not terminating. Clearly R 2 is terminating. Note that not every pseudosimply terminating TRS is simply terminating. Later in this ....

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

....for modularity of termination are presented. Some extensions and generalizations of these known results on modular termination to the case of conditional TRSs as well as to some restricted classes of non disjoint unions of TRSs can be found in [Mid90] KO90b] Mid93] Gra92c] Gra93] and [Ohl93]. An interesting result not subsumed by other ones is due to [TKB89] where it is shown that completeness is modular for left linear TRSs. The main link between our abstract results proved in the previous section and the problem of modular termination of rewriting is provided by the following easy ....

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.

....shared constructor, and hierarchical cases. We are primarily interested in the shared constructor case and in results that impose minimal restrictions on R (other than termination, of course) There are few such: for constructor based systems in [19] for right linear and non duplicating S in [21] (for the disjoint case, it conjectured by [25] and proved in [16] Some results for the hierarchical case are contained in [6] and [14] A standard method of proving termination of a union R[S is to find a convergent transformation function T , containing S, such that Q(R; R ; T ) R R ....

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

.... three dioeerent approaches concerning the essential ideas and proof structures (cf. 7] ffl the 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 modi ed generic version of one of the systems ( 5] [16]) ffl the modular approach via modularity of innermost termination where suOEcient criteria for the equivalence of innermost termination (SIN) and general termination (SN) are combined with the (not so diOEcult to establish) modularity of SIN ( 6] 6 ) and ffl the syntactic approach via ....

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

....dependency pairs. Although in general, termination is not modular for the direct sum [Toy87, Dro89, TKB95] i.e. the partition of a TRS into subsystems with disjoint signatures, this modularity property holds for TRSs of a special form [Rus87, Mid89, Gra94, TKB95, SMP95] For a survey see e.g. [Mid90, Ohl94, Gra96a]. However, a TRS often cannot be split into subsystems with disjoint signatures. Therefore, partitions into subsystems which may at least have constructors This work was partially supported by the Deutsche Forschungsgemeinschaft under grants no. Wa 652 7 1,2 as part of the focus program ....

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

....in the following, written A = ff 1 ; ff n ; Recently, term rewriting has arrived at a rather general condition for the modularity of strong normalisation. 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. The term rewriting proof is a rather intricate encoding construction (so much so that it was first only proven for a finitely branching systems[Gra92] In ....

....4) SN is modular for simplifying systems. Proof. The first is obvious. Non duplicating and simplifying systems are strongly normalising under deterministic collapses [Gra92] showing the other three. Hence, all of the conditions listed after Definition 12 follow as corollaries from Theorem 49. [Ohl94] contains further derived criteria. 7 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 seem particularly suited to the study of ....

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

....By the modularity of termination for non overlapping composable TRSs ( 5, 27] Theorem 6.4 Ultra termination is modular for non ultraduplicating composable DCTRSs. Proof By the modularity of termination for non duplicating composable TRSs ( 12, 27] The modularity of C E termination (cf. [26, 12]) is one of the most powerful results ever obtained for TRSs. Recall that a TRS T is said to be C E terminating if T Phi for(X; Y ) X; or(X; Y ) Y g is terminating (cf. 26] The lifted ultraproperty can be expressed in a similar way: it can be proved that T is ultra C E terminating iff T ....

....of termination for non duplicating composable TRSs ( 12, 27] The modularity of C E termination (cf. 26, 12] is one of the most powerful results ever obtained for TRSs. Recall that a TRS T is said to be C E terminating if T Phi for(X; Y ) X; or(X; Y ) Y g is terminating (cf. [26]) The lifted ultraproperty can be expressed in a similar way: it can be proved that T is ultra C E terminating iff T Phifor(X; Y ) X; or(X; Y ) Y g is ultra terminating . Thus we get: Theorem 6.5 Ultra C E termination is modular for DCTRSs. Proof By the modularity of C E termination ....

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

.... the starting point of 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. 30] 19] 33] 20] 14] 7] 8] [25]) Non disjoint unions of TRSs with common constructors have been considered e.g. in [23] 15] 7] 8] 25] 27] More general hierarchical combinations of TRSs have recently been dealt with in [17] 2] 18] 8] 6] Some preservation results for (disjoint and non disjoint, but ....

.... of termination, completeness (i.e. termination plus confluence) and of other interesting properties of TRSs under disjoint combinations (cf. e.g. 30] 19] 33] 20] 14] 7] 8] 25] Non disjoint unions of TRSs with common constructors have been considered e.g. in [23] 15] 7] 8] [25], 27] More general hierarchical combinations of TRSs have recently been dealt with in [17] 2] 18] 8] 6] Some preservation results for (disjoint and non disjoint, but non hierarchical) combinations of conditional TRSs (CTRSs for short) finally have been obtained in [20] 22] 21] ....

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

....is a modular property for TRSs [40] but that in general termination is not [41] Klop and Barendregt gave a counter example which shows that completeness is not modular, but Toyama, Klop and Barendregt [42] proved that completeness in combination with left linearity is modular for TRSs. Ohlebusch [34] showed that if a combination of two TRSs does not terminate, then one of the TRSs is not C E terminating, while the other TRS is collapsing. This generalizes a similar result for finitely branching TRSs of Gramlich [27] Middeldorp [33] presented a panorama of positive and negative results on ....

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

....IBN 97 45, TU Darmstadt. Extended version of a paper presented at RTA 98. This work was partially supported by the Deutsche Forschungsgemeinschaft under grants no. Wa 652 7 1,2 as part of the focus program Deduktion . for direct sums [Rus87, Mid89, Gra94, TKB95, SMP95] For a survey see e.g. [Mid90, Ohl94, Gra96a]. However, a TRS often cannot be split into subsystems with disjoint signatures. Therefore, other partitions have also been considered. In many cases it is desirable to have at least constructors in common in both parts. For the subclass of constructor systems, termination is modular provided that ....

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

....noncollapsing, non duplicating, left linear, non overlapping systems: x s(y) s(x y) p(x) Gamma p(y) x Gamma y ; 2) where is red and Gamma is blue. Generalizations of Theorems 2, 3, 4, and 7 will be proved in the sequel. A more semantic approach was developed in [ Gramlich, 1994; Ohlebusch, 1993 ] based on the syntactic ideas in [ Kurihara and Ohuchi, 1990 ] Theorem8 [ Ohlebusch, 1993 ] The union of red and blue systems that are each terminating when joined with the system fh(x; y) x; h(x; y) yg, for new function symbol h not appearing in either system, is terminating. This ....

.... Gamma p(y) x Gamma y ; 2) where is red and Gamma is blue. Generalizations of Theorems 2, 3, 4, and 7 will be proved in the sequel. A more semantic approach was developed in [ Gramlich, 1994; Ohlebusch, 1993 ] based on the syntactic ideas in [ Kurihara and Ohuchi, 1990 ] Theorem8 [ Ohlebusch, 1993 ] The union of red and blue systems that are each terminating when joined with the system fh(x; y) x; h(x; y) yg, for new function symbol h not appearing in either system, is terminating. This is an undecidable property. Theorem9 [ Ohlebusch, 1993 ] The union of a non duplicating red ....

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

....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.

.... of Toyama s Theorem was given by Klop et al. KMTV91] In contrast to this encouraging result, termination and completeness turned out to lack a modular behavior (see [Toy87a] and also [Ohl93b] Thus several sufficient criteria ensuring their modularity have been given (for an overview see e.g. [Mid90, Gra93, Ohl93a]) In order to prove modularity of completeness, one can of course use the confluence of the combined system to show its termination. For example the deep theorem that completeness is modular for left linear disjoint TRSs [TKB89] crucially depends on Toyama s Theorem. In recent investigations (cf. ....

....In order to prove modularity of completeness, one can of course use the confluence of the combined system to show its termination. For example the deep theorem that completeness is modular for left linear disjoint TRSs [TKB89] crucially depends on Toyama s Theorem. In recent investigations (cf. [KO92, MT93, Ohl93a, Gra93]) one tries to weaken the disjointness requirement. One way to do it is to allow shared con structors (function symbols that do not occur at the root position of the left hand side of any rewrite rule) Unfortunately, confluence is not preserved under the combination of constructor sharing ....

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E. Ohlebusch. On the Modularity of Termination of Term Rewriting Systems. Report Nr. 11, Universitat Bielefeld, 1993.

.... conditions ensuring the preservation of termination under disjoint union were obtained by investigating the distribution of collapsing and duplicating rules among the TRSs see [Rus87, Mid89] The results were extended, mutatis mutandis, to disjoint CTRSs [Mid93] and constructorsharing TRSs [Ohl94c]. In the second part of this paper, we provide a relatively simple proof for analogous results for constructor sharing CTRSs. Furthermore, a simple counterexample disproves a conjecture of Middeldorp [Mid93] which is related to the above mentioned results. More sufficient conditions for the ....

....simple proof for analogous results for constructor sharing CTRSs. Furthermore, a simple counterexample disproves a conjecture of Middeldorp [Mid93] which is related to the above mentioned results. More sufficient conditions for the modularity of termination of TRSs can for instance be found in [TKB89, KO92, Gra94a, Gra94b, Ohl94c, Mar95]. The reader is referred to [Mid90, Mid93, Mid94, Ohl93, Gra93] for modularity results of CTRSs. In recent investigations, one tries to weaken the constructorsharing requirement. On the one hand, MT93, Mid94, Ohl94a, KO94] consider composable systems systems which have to contain all rewrite ....

E. Ohlebusch. On the Modularity of Termination of Term Rewriting Systems. Theoretical Computer Science 136, pages 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.

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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.

No context found.

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

No context found.

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.

No context found.

Ohlebusch, E. (1993b). On the Modularity of Termination of Term Rewriting Systems. Report nr. 11, Universitat Bielefeld.

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