D.S. Lankford. On proving term rewriting systems are Noetherian. Technical report, Louisiana Technical University, Ruston, LA, 1979.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....on concretisations of recursive pairs. This means that the path between a recursive pair minimizes an expression. Here we can plug in modules which try to find one of the numerous different orderings for termination proofs proposed in the literature, e.g. polynomial orderings as proposed in [Ste92, Lan79], one of the orderings presented in [Der87] or some sort of generalized orderings based on multi sets as presented in [Mar87] A simple ordering which in many cases is sufficient enough is the ordering which is based on the number of constructors the normal form of an expression has. be a ....

D. S. Lankford. On proving term-rewriting systems are noetherian. Technical Report MTP-3, Mathematics Department, Louisiana Technical University, 1979.

....then we say that R is simply terminating. Although simple termination is also undecidable (see [33] it covers most usual automatizable orderings for proving termination of rewriting (e.g. recursive path orderings (rpo [9] Knuth Bendix orderings (kbo [23] and polynomial orderings (poly [26]) see [37] for a survey on simplification orderings) Moreover, simple termination has interesting properties regarding modularity: in contrast to the general case, simple termination is modular for disjoint, constructor sharing, and (some classes of) hierarchical unions of TRS s [36] obj ....

D. Lankford. On proving term rewriting systems are noetherian. Technical Report MTP 3, Louisiana Technical University, Ruston, 1979.

....term rewriting to generate suitable reduction pairs automatically. 4 Comparison with Orderings from Term Rewriting Traditional techniques for TRSs prove simple termination where i# it is compatible with a simplification ordering (e.g. LPO or RPOS [5, 9] KBO [10] most polynomial orderings [12]) Equivalently, EmbF terminates for signature . Similar to traditional techniques, the size change principle essentially only verifies simple termination. Theorem 12 (Size Change Principle and Simple Termination) a) A TRS is size change terminating w.r.t. a reduction EmbF ....

D. Lankford. On proving term rewriting systems are Noetherian. Technical Report MTP-3, Louisiana Technical University, Ruston, LA, USA, 1979.

....then we say that R is simply terminating. Although simple termination is also undecidable (see [33] it covers most usual automatizable orderings for proving termination of rewriting (e.g. recursive path orderings (rpo [9] Knuth Bendix orderings (kbo [23] and polynomial orderings (poly [26]) see [37] for a survey on simplification orderings) Moreover, simple termination has interesting properties regarding modularity: in contrast to the general case, simple termination is modular for disjoint, constructor sharing, and (some classes of) hierarchical unions of TRS s [36] 29 obj ....

D. Lankford. On proving term rewriting systems are noetherian. Technical Report MTP 3, Louisiana Technical University, Ruston, 1979.

.... to prove termination of TRSs (automatically) use simplification orders [8, 26] where a term is greater than its proper subterms (subterm property) Examples for simplification orders include lexicographic or recursive path orders [7, 17] the Knuth Bendix order [18] and (most) polynomial orders [20]. However, there are numerous important TRSs which are not simply terminating, i.e. their termination cannot be shown by simplification orders. Therefore, the dependency pair approach [2, 11, 12] was developed which allows the application of simplification orders to non simply terminating TRSs. ....

D. Lankford. On proving term rewriting systems are Noetherian. Technical Report MTP-3, Louisiana Technical University, Ruston, LA, USA, 1979. Available from http://aib.informatik.rwth-aachen.de 30

....automatically. 4 Comparison with Orderings from Term Rewriting Most traditional techniques for TRSs are based on so called simpli cation orderings (like lexicographic or recursive path orderings (possibly with status) RPOS [5, 8] Knuth Bendix orderings KBO [9] and most polynomial orderings [11]) A TRS is simply terminating i it is compatible with a simpli cation ordering. Equivalently, a TRS R over a signature F is simply terminating i R [ F emb terminates. Thm. 13 shows that similar to these traditional techniques, the sizechange principle can essentially only verify simple ....

D. Lankford. On proving term rewriting systems are Noetherian. Technical Report MTP-3, Louisiana Technical University, Ruston, LA, USA, 1979.

....implies termination of the original CSRS (i.e. all these transformations are sound ) Direct approaches to termination analysis of CSRSs and transformational approaches both have their advantages. Techniques for proving termination of ordinary term rewriting have been studied extensively (e.g. [21, 22, 7, 3, 30, 31, 1, 4]) and the main advantage of the transformational approach is that in this way, all termination techniques for ordinary TRSs including future developments can be used to infer termination of CSRSs. For instance, the methods of [5, 20] are unable to handle systems like Example 1. Of the ve ....

D. Lankford. On proving term rewriting systems are Noetherian. Technical Report MTP-3, Louisiana Technical University, Ruston, LA, USA, 1979.

....well known simplification orders like the recursive path order (rpo) 6] to a one rule rewrite system of the form 0 , we find that rpo can handle the case p s for arbitrary q, r, using the precedence 0 1. The same is done by polynomial interpretation [0] x) r 1)x; 1] x) x 1 [14]. So in this case obviously Z is simply terminating. Moreover, since the interpretation is linear, the derivation length D(n) is at most exponential in n [15] Here D(n) is defined to be the maximal number of steps in a reduction starting with a string of length n. Below we show that there are ....

Dallas S. Lankford. On proving term rewriting systems are noetherian. Technical Report MTP-3, Louisiana Technical University, Math. Dept., Ruston, LA, 1979.

....of semantical methods a hierarchy of kinds of termination will be given, based on di erent kinds of well founded orders in which the weights are taken. For most methods the notion of order plays an important role. Before the various methods are discussed some basic facts due to Lankford ([46]) about the connection between termination of TRSs and orders are treated. A strict order is called wellfounded if it does not admit an in nite descending sequence t 0 t 1 t 2 t 3 . Proposition 1 A TRS ( R) is terminating if and only if there exists a wellfounded order on ....

Lankford, D. S. On proving term rewriting systems are noetherian. Tech. Rep. MTP{3, Louisiana Technical University, Ruston, 1979.

....the TRS is terminating. Moreover, such a termination proof yields an upper bound on the length of any derivation starting with a particular term: the length is bounded by the interpretation of the starting term. A standard technique is the interpretation of the function symbols by polynomials, see [5, 1]. For polynomials it is well known that the derivation length of a term is bounded by a function doubly exponential in the size of the term; this bound is shaxp ( 6, 4] In this paper we give a simpler proof of this result. However, we do not restrict to polynomials. The same doubly exponential ....

....Moreover, Hofbauer ( 3] showed that for any TRS for which a termination proof can be given by a recursive path order, a corresponding interpretation in the natural numbers can be given, using primitive recursive interpretations. The standard technique of polynomial interpretations, see [5, 1], is nothing else than this technique in which all interpretation functions are chosen to be polynomials. It appears that such an interpretation in the naturals not only gives information on termination in itself, but also on maximum derivation lengths. 3 Derivation lengths from interpretation ....

LANKFORD, D. S. On proving term rewriting systems are noetherian. Tech. Rep. MTP-3, Louisiana Technical University, Ruston, 1979.

.... properties of a term rewriting system is termination, cf. e.g. DJ90] While in general this problem is undecidable [HL78] several methods for proving termination have been developed (e.g. path orderings [Pla78, Der82, DH95, Ste95b] Knuth Bendix orderings [KB70, Mar87] semantic interpretations [Lan79, BCL87, BL93, Ste94, Zan94, Gie95b], transformation orderings [BD86, BL90, Ste95a] semantic labelling [Zan95] etc. for surveys see e.g. Der87, Ste95b] In this paper we are concerned with the automation of termination proofs for constructor systems (CS for short) Due to the special form of these rewrite systems it is ....

....standard techniques for the automated generation of well founded quasiorderings fail here (and the same problem appears with most other examples) Hence, demanding DP [ EQ is too strong, i.e. in this way most termination proofs will not succeed. DP [ EQ is not satisfied by polynomial orderings [Lan79] either (which do not have to be quasi simplification orderings) 3.3. Constraints Without Defined Symbols In Sect. 3.1 we showed that the existence of a well founded quasi ordering satisfying DP is in general not sufficient for the termination of R, because does not necessarily respect the ....

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D. S. Lankford. On proving term rewriting systems are noetherian. Tech. Report Memo MTP-3, Louisiana Tech. University, Ruston, LA, 1979.

....operators as AC operators and terms using these operators as AC terms in the rest of the paper. Developing well founded orderings on AC terms which are useful for proving termination of AC rewrite systems has been quite a challenge. A number of attempts have been reported in [1, 15, 3] see also [11, 2] for polynomial orderings as well as [5] for other approaches. Two properties can be used to distinguish between most of the related previous work and to motivate the work done in this paper. The first property is RPO compatibility. That is, whether an AC ordering behaves exactly like RPO on ....

Lankford, D.S. (1979): On proving term rewriting systems are Noetherian. Memo MTP-3, Lousiana State University.

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D.S. Lankford. On proving term rewriting systems are Noetherian. Technical report, Louisiana Technical University, Ruston, LA, 1979.

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D. S. Lankford. On proving term rewriting systems are noetherian. Tech. Rep. MTP-3, Louisiana Tech. Univ., Ruston, 1979.

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D. S. Lankford. On proving term rewriting systems are noetherian. Technical Report MTP-3, Math. Dept., Louisiana Tech. Univ., Ruston, May 1979.

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D. S. Lankford. On proving term rewriting systems are noetherian. Technical report, Mathematics Department, Louisiana Tech. University, Ruston, LA, 1979.

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D. S. Lankford. On proving term rewriting systems are noetherian. Technical report, Mathematics Department, Louisiana Tech. University, Ruston, LA, 1979.

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Dallas S. Lankford. On proving term rewriting systems are Noetherian. Memo MTP-3, Mathematics Department, Louisiana Tech. University, Ruston, LA, May 1979. Revised October 1979.

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D. Lankford. On proving term rewriting systems are Noetherian. Technical Report MTP-3, Louisiana Technical University, Ruston, LA, USA, 1979.

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D. Lankford. On proving term rewriting systems are Noetherian. Technical Report MTP-3, Louisiana Technical University, Ruston, LA, USA, 1979. 30

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D.S. Lankford. On proving term rewriting systems are noetherian. Technical Report, Louisiana Technological University, Ruston, LA, 1979.

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D. Lankford. On proving term rewriting systems are Noetherian. Technical Report MTP-3, Louisiana Technical University, Ruston, LA, USA, 1979.

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D. Lankford. On proving term rewriting systems are Noetherian. Technical Report MTP-3, Louisiana Technical University, Ruston, LA, USA, 1979.

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D.S. Lankford. On proving term rewriting systems are Noetherian. Technical Report Memo MTP-3, Louisiana Technology University, 1979.

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D. S. Lankford. On Proving Term Rewriting Systems are Noetherian. Technical Report Memo MTP-3, Louisiana Technical University, Ruston, LA, 1979.

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