Relative Termination via Dependency Pairs

[EN] A term rewrite system is terminating when no infinite reduction sequences are possible. Relative termination generalizes termination by permitting infinite reductions as long as some distinguished rules are not applied infinitely many times. Relative termination is thus a fundamental notion that has been used in a number of different contexts, like analyzing the confluence of rewrite systems or the termination of narrowing. In this work, we introduce a novel technique to prove relative termination by reducing it to dependency pair problems. To the best of our knowledge, this is the first significant contribution to Problem #106 of the RTA List of Open Problems. We first present a general approach that is then instantiated to provide a concrete technique for proving relative termination. The practical significance of our method is illustrated by means of an experimental evaluation.Open access funding provided by Austrian Science Fund (FWF). We would like to thank Nao Hirokawa, Keiichirou Kusakari, and the anonymous reviewers for their helpful comments and suggestions in early stages of this work.Iborra, J.; Nishida, N.; Vidal Oriola, G.; Yamada, A. (2017). Relative Termination via Dependency Pairs. Journal of Automated Reasoning. 58(3):391-411. https://doi.org/10.1007/s10817-016-9373-5391411583Alarcón, B., Lucas, S., Meseguer, J.: A dependency pair framework for A $$\vee$$ ∨ C-termination. In: WRLA 2010, LNCS, vol. 6381, pp. 36–52. Springer (2010)Arts, T., Giesl, J.: Termination of term rewriting using dependency pairs. Theor. Comput. Sci. 236(1–2), 133–178 (2000)Arts, T., Giesl, J.: A collection of examples for termination of term rewriting using dependency pairs. Technical report AIB-2001-09, RWTH Aachen (2001)Baader, F., Nipkow, T.: Term Rewriting and All That. Cambridge University Press, Cambridge (1998)Bachmair, L., Dershowitz, N.: Critical pair criteria for completion. J. Symb. 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Termination of rewriting strategies: a generic approach

We propose a generic termination proof method for rewriting under strategies, based on an explicit induction on the termination property. Rewriting trees on ground terms are modeled by proof trees, generated by alternatively applying narrowing and abstracting steps. The induction principle is applied through the abstraction mechanism, where terms are replaced by variables representing any of their normal forms. The induction ordering is not given a priori, but defined with ordering constraints, incrementally set during the proof. Abstraction constraints can be used to control the narrowing mechanism, well known to easily diverge. The generic method is then instantiated for the innermost, outermost and local strategies.Comment: 49 page

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Polaron Effective Mass, Band Distortion, and Self-Trapping in the Holstein Molecular Crystal Model

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Crystal chemical principles and transport theory have been used to predict structures and specific compounds which might find application as solid electrolytes in rechargeable high energy and high power density batteries operating at temperatures less than 200 C. Structures with 1-, 2-, and 3-dimensional channels were synthesized and screened by nuclear magnetic resonance, dielectric loss, and conductivity. There is significant conductivity at room temperature in some of the materials but none attain a level that is comparable to beta-alumina. Microwave and fast pulse methods were developed to measure conductivity in powders and in small crystals

Termination of Narrowing with Dependency Pairs

In this work, we generalize the Dependency Pairs approach for automated proofs of termination to prove the termination of narrowing.We identify the phenomenon of echoing in infinite narrowing derivations and demonstrate that the new narrowing dependency pairs faithfully capture the shape of such derivations and provide a termination criterion.Iborra López, J. (2008). Termination of Narrowing with Dependency Pairs. http://hdl.handle.net/10251/13622Archivo delegad