B. Gramlich. On proving termination by innermost termination. In Proc. 7th Int. Conf. on Rewriting Techniques and Applications (RTA '96), volume 1103 of LNCS, pp. 93--107. Springer, 1996.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....However, termination of CSR only approximates termination. For instance, in Maude, the default local strategy associated to a k ary symbol f , is (1 2 Delta Delta Delta k 0) see [Eke98] As in [GWMFJ00] by OBJ we mean OBJ2, OBJ3, CafeOBJ, or Maude. Example 4. Consider the TRS [Gra96]: and let (f) 1 0) and (a) 0) This TRS is terminating, but it is not terminating, since we have: f(a) f(a) Delta Delta Delta. The point here is that computations under the E strategy are basically innermost. Innermost rewriting computations can be terminating even for ....

....f(a) Delta Delta Delta. The point here is that computations under the E strategy are basically innermost. Innermost rewriting computations can be terminating even for nonterminating TRSs. This gives rise to the topic of innermost termination of rewriting which has been studied in e.g. [AG97,Gra96]. For instance, the TRS of Example 4 is nonterminating, but innermost terminating [Gra96] Given a TRS R = Sigma; R) we consider Sigma as the disjoint union Sigma = C ]D of symbols c 2 C, called constructors and symbols f 2 D, called defined functions, where D = froot(l) j l r 2 Rg and C = ....

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B. Gramlich. On Proving Termination by Innermost Termination. In H. Ganzinger, editor, Proc. of 7th International Conference on Rewriting Techniques and Applications, RTA'96, LNCS 1103:97-107, SpringerVerlag, Berlin, 1996.

....showsUpto; g. The termination of (a similar version of) R is proved in [GM99] Theorem 2 connects termination of CSR and termination of OBJ programs with positive evaluation strategies. However, termination of CSR only approximates termination of such OBJ programs. Example 3. Consider the TRS [Gra96]: f(a) f(a) a b and a local strategy such that (f) 1 0) and (a) 0) This TRS is terminating, but it is not terminating, since we have the following: f(a) f(a) Delta Delta Delta The point here is that OBJ computations are basically innermost. Innermost rewriting ....

....f(a) f(a) Delta Delta Delta The point here is that OBJ computations are basically innermost. Innermost rewriting computations can be terminating even though the TRS is not terminating. This gives rise the topic of innermost termination of rewriting which has been studied in e.g. [AG97,AG00,Gra95,Gra96]. Given a TRS R = Sigma; R) we consider Sigma as the disjoint union Sigma = C ] F of symbols c 2 C, called constructors and symbols f 2 F , called defined functions, where F = froot(l) j l r 2 Rg and C = Sigma Gamma F . We say that an E strategy map is elementary if for all f 2 F , ....

B. Gramlich. On Proving Termination by Innermost Termination. In H. Ganzinger, editor, Proc. of 7th International Conference on Rewriting Techniques and Applications, RTA'96, LNCS 1103:97-107, SpringerVerlag, Berlin, 1996.

....paper, we study certain characteristics of strong innermost normalization, i.e. termination under the innermost reduction strategy which closely corresponds to callby value evaluations. To the best of our knowledge, there are just two others works on this topic, namely, the work of Gramlich [5, 6] relating termination and innermost termination and the work of Arts and Giesl [1] on automatic proofs for innermost termination. We concur with the observation in [1] that the results in our paper can be used in signi#cantly improving the power of their method. Finally, we summarize the results ....

B. Gramlich, On proving termination by innermost termination, Proc. RTA'96, Lecture Notes in Computer Science, vol. 1103, Springer, Berlin, 1996, pp. 93 --107.

....are contracted, can be used to model call by value computation semantics. For that reason, there has been an increasing interest in research on properties of rewriting under strategies. In particular, the study of termination is important when regarding such restricted versions of rewriting [35, 36, 45]. To prove innermost termination (also called (strong) innermost normalisation) one has to show that the length of every innermost reduction is finite. Techniques for proving innermost termination can for example be utilized for termination proofs of functional programs (modelled by TRSs) with ....

....improving the technique can be developed (Sect. 3.3 and Sect. 3.4) The automated checking of this criterion enables us to prove innermost termination automatically, even if the TRS is not terminating. Additionally, for several classes of TRSs innermost termination already suffices for termination [35, 36]. Moreover, numerous modularity results exist for innermost termination [4, 5, 6, 35, 44, 45] which do not hold for termination. Therefore, for those classes of TRSs termination can be proved by splitting the TRS and proving innermost termination of the subsystems separately. The advantage of ....

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B. Gramlich, On proving termination by innermost termination, in: Proc. RTA-96, Lecture Notes in Computer Science, Vol. 1103 (Springer, Berlin, 1996) 93--107.

....normalisation can be proved automatically. In Sect. 4 and 5 our technique is refined further and in Sect. 6 we give a summary and comment on connections and possible combinations with related approaches. For several classes of TRSs, innermost normalisation already suffices for termination [Gra95, Gra96]. Moreover, several modularity results exist for innermost normalisation [Kri95, Art96] which do not hold for termination. Therefore, for those classes of TRSs termination can be proved by splitting the TRS and proving innermost normalisation of the subsystems separately. The advantage of this ....

B. Gramlich. On proving termination by innermost termination. In Proceedings of RTA-96, LNCS 1103, pages 93--107, 1996.

....are contracted, can be used to model call by value computation semantics. For that reason, there has been an increasing interest in research on properties of rewriting under strategies. In particular, the study of termination is important when regarding such restricted versions of rewriting [27,28,36]. To prove innermost termination (also called (strong) innermost normalization) one has to show that the length of every innermost reduction is finite. Techniques for proving innermost termination can for example be utilized for termination proofs of functional programs (modelled by TRSs) with ....

....improving the technique can be developed (Sect. 3.3 and Sect. 3.4) The automated checking of this criterion enables us to prove innermost termination automatically, even if the TRS is not terminating. Additionally, for several classes of TRSs innermost termination already suffices for termination [27,28]. Moreover, numerous modularity results exist for innermost termination [5,7,27,35,36] which do not hold for termination. Therefore, for those classes of TRSs termination can be proved by splitting the TRS and proving innermost termination of the sub29 systems separately. The advantage of this ....

[Article contains additional citation context not shown here]

B. Gramlich. On proving termination by innermost termination. In H. Ganzinger, editor, Proceedings of the 7th International Conference on Rewriting Techniques and Applications, RTA-96, volume 1103 of Lecture Notes in Computer Science, pages 93--107, New Brunswick, NJ, USA, July 1996. Springer Verlag, Berlin.

.... One step reduction Hauptsatz WN for reduction Corollary 3 ISN for reduction Figure 2. 6: The Curry Howard correspondence of Sequent Calculus Logic and Explicit Substitution Calculus Theorem 11 ( 32, Theorem 5 (a) For any non overlapping TRS we have: ISN ) SN We can remark that in [34], Gramlich presents several sufficient conditions under which SN follows from ISN. The above is the most immediate. Definition 14 (Term Rewriting Systems) A TRS consists of a countable signature, F , of capital letter function symbols, each with an associated arity and of a disjoint, countable ....

Bernhard Gramlich. On proving termination by innermost termination. In Proceedings of the 7th International Conference on Rewriting Techniques and Applications (RTA-96), volume 1103 of LNCS, pages 93--107. Springer-Verlag, July 27--30 1996.

....variable names but he suggests in [26] how to adapt the correspondence to a calculus a la de Bruijn. This does not alter the specialised nature of the calculus 5 We have omitted all definitions of standard rewriting notions such as SR, WIN, etc. due to space constraints. We refer instead to [3, 23, 37]. 3 A completely different approach has been taken by Di Cosmo Kesner [13] who encode calculi with explicit substitutions into proof nets. They use the encoding to show termination for several calculi with explicit substitutions. They utilise the Curry Howard Correspondence in the opposite ....

Bernhard Gramlich. On proving termination by innermost termination. In Proceedings of the 7th International Conference on Rewriting Techniques and Applications (RTA-96), volume 1103 of LNCS, pages 93--107. Springer-Verlag, July 27--30 1996.

....program Deduktion . x Utrecht University, E mail: thomas cs.ruu.nl FB Informatik, TH Darmstadt, Alexanderstr. 10, 64283 Darmstadt, Germany, E mail: giesl inferenzsysteme.informatik.th darmstadt.de For several classes of TRSs, innermost normalisation already suffices for termination [Gra95, Gra96]. Moreover, several modularity results exist for innermost normalisation [Kri95, Art96] which do not hold for termination. Therefore, for those classes of TRSs termination can be proved by splitting the TRS and proving innermost normalisation of the subsystems separately. The advantage of this ....

....of these term rewriting systems. It is shown how our method can automatically derive innermost normalisation of these term rewriting systems. The examples in the next section are term rewriting systems for which innermost normalisation suffices to guarantee termination by the results of Gramlich [Gra95, Gra96]. Many of these examples are term rewriting systems that are not simply terminating. Therefore, their termination cannot be shown by most other automatic methods. However, by our approach they can be proved terminating. For proving termination of the examples, our technique first transforms the ....

B. Gramlich. On proving termination by innermost termination. In H. Ganzinger, editor, Proceedings of the 7th International Conference on Rewriting Techniques and Applications, RTA-96, volume 1103 of Lecture Notes in Computer Science, pages 93--107, New Brunswick, NJ, USA, July 1996. Springer Verlag, Berlin.

....DKM90] semantic interpretations [Lan79, BL87, BL93, Ste94, Zan94, Gie95] transformation orderings [BD86, BL90, Ste95a] semantic labelling [Zan95] etc. for surveys see e.g. Der87, Ste95b] Moreover, termination properties of rewriting under strategies are also of great importance, cf. e.g. [Gra95, Kri96, Gra96]. For example, innermost rewriting can be used to model call by value semantics of computation formalisms. In this paper we are concerned with the automation of both strong and innermost normalisation proofs. While most methods for automated strong normalisation proofs are restricted to ....

....for proving innermost normalisation, but also for proving termination for TRSs that are nonoverlapping (as are all orthogonal TRSs) or for overlay systems with joinable critical pairs. Further results on classes of TRSs where innermost normalisation implies strong normalisation can be found in [Gra96]. Theorem 3.4 ( Gra95] ffl Any innermost normalising and non overlapping TRS is terminating. ffl Any innermost normalising overlay system with joinable critical pairs is terminating. Example 3.7 Consider the following overlay system for computing the average of two integers: a(0; 0) 0 ....

B. Gramlich. On proving termination by innermost termination. In Proceedings of the 7th International Conference on Rewriting Techniques and Applications, LNCS 1103, New Brunswick, NJ, USA, 1996.

....is modular for direct sums and for TRSs with shared constructors [Gra95] for composable constructor systems [MT93] for composable TRSs [Ohl95] and for proper extensions [KR95] which are special hierarchical combinations. As innermost termination implies termination for several classes of TRSs [Gra95, Gra96b], these results can also be used for termination proofs of such systems. For example, this holds for locally confluent overlay systems (and in particular for non overlapping TRSs) In this paper we show that the modular approach using dependency pairs extends previous modularity results and we ....

B. Gramlich, On proving termination by innermost termination. In Proc. RTA-96, LNCS 1103, pp. 93--107, New Brunswick, NJ, 1996.

....normalisation can be proved automatically. In Sect. 4 and 5 our technique is refined further and in Sect. 6 we give a summary and comment on connections and possible combinations with related approaches. For several classes of TRSs, innermost normalisation already suffices for termination [Gra95, Gra96]. Moreover, several modularity results exist for innermost normalisation [Kri95, Art96] which do not hold for termination. Therefore, for those classes of TRSs termination can be proved by splitting the TRS and proving innermost normalisation of the subsystems separately. The advantage of this ....

B. Gramlich. On proving termination by innermost termination. In Proceedings of RTA-96, LNCS 1103, pages 93--107, 1996.

....for direct sums and for TRSs with shared constructors [Gra95] for composable constructor systems [MT93] for composable TRSs [Ohl95] and for proper extensions [KR95] which are a special class of hierarchical combinations. As innermost termination implies termination for several classes of TRSs [Gra95, Gra96b], these results can also be used for termination proofs of such systems. In particular, this holds for locally confluent overlay systems (and in particular for non overlapping TRSs) In this paper we show that the modular approach using dependency pairs extends previous modularity results and we ....

B. Gramlich, On proving termination by innermost termination. In Proc. RTA-96, LNCS 1103, New Brunswick, NJ, 1996.

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B. Gramlich. On proving termination by innermost termination. In Proc. 7th Int. Conf. on Rewriting Techniques and Applications (RTA '96), volume 1103 of LNCS, pp. 93--107. Springer, 1996.

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B. Gramlich. On proving termination by innermost termination. In Proc. 7th RTA, LNCS 1103, pages 97--107, 1996.

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B. Gramlich. On proving termination by innermost termination. In Proc. 7th RTA, pages 97--107, 1996. LNCS 1103.

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B. Gramlich. On proving termination by innermost termination. In Proc. 7th RTA, LNCS 1103, pages 97--107, 1996.

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B. Gramlich. On proving termination by innermost termination. In Proc. 7th Int. Conf. on Rewriting Techniques and Applications (RTA '96), volume 1103 of LNCS, pp. 93--107. Springer, 1996.

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B. Gramlich. On Proving Termination by Innermost Termination. In Proc. of RTA'96, LNCS 1103:97-107, Springer-Verlag, Berlin, 1996.

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