A Proof of Weak Termination Providing the Right Way to Terminate

We give an inductive method for proving weak innermost termination of rule-based programs, from which we automatically infer, for each successful proof, a finite strategy for data evaluation. We first present the proof principle, using an explicit induction on the termination property, to prove that any input data has at least one finite evaluation. For that, we observe proof trees built from the rewrite system, schematizing the innermost rewriting tree of any ground term, and generated with two mechanisms: abstraction, schematizing normalization of subterms, and narrowing, schematizing rewriting steps. Then, we show how, for any ground term, a normalizing rewriting strategy can be extracted from the proof trees, even if the ground term admits infinite rewriting derivations.

[1]  Jürgen Giesl,et al.  Proving Innermost Normalisation Automatically , 1997, RTA.

[2]  Hélène Kirchner,et al.  Termination of Rewriting with Local Strategies , 2001, Electron. Notes Theor. Comput. Sci..

[3]  Hélène Kirchner,et al.  Induction for innermost and outermost ground termination , 2001 .

[4]  Hélène Kirchner,et al.  System Presentation -- CARIBOO: An induction based proof tool for termination with strategies , 2002, PPDP '02.

[5]  Salvador Lucas Termination of Rewriting With Strategy Annotations , 2001, LPAR.

[6]  Hélène Kirchner,et al.  Induction for termination with local strategies , 2001 .

[7]  Ataru T. Nakagawa,et al.  An overview of CAFE specification environment-an algebraic approach for creating, verifying, and maintaining formal specifications over networks , 1997, First IEEE International Conference on Formal Engineering Methods.

[8]  Bernhard Gramlich,et al.  On Termination and Confluence Properties of Disjoint and Constructor-Sharing Conditional Rewrite Systems , 1996, Theor. Comput. Sci..

[9]  Jean-Jacques Lévy,et al.  Computations in Orthogonal Rewriting Systems, II , 1991, Computational Logic - Essays in Honor of Alan Robinson.

[10]  Claude Kirchner,et al.  An overview of ELAN , 1998, WRLA.

[11]  Nachum Dershowitz Orderings for Term-Rewriting Systems , 1979, FOCS.

[12]  Nachum Dershowitz,et al.  Orderings for term-rewriting systems , 1979, 20th Annual Symposium on Foundations of Computer Science (sfcs 1979).

[13]  Paul Klint,et al.  A meta-environment for generating programming environments , 1989, TSEM.

[14]  Jean Goubault-Larrecq A Proof of Weak Termination of Typed lambda-sigma-Calculi , 1996, TYPES.

[15]  Hélène Kirchner,et al.  Outermost ground termination , 2004, WRLA.

[16]  Bernhard Gramlich,et al.  Relating Innermost, Weak, Uniform and Modular Termination of Term Rewriting Systems , 1992, LPAR.

[17]  Narciso Martí-Oliet,et al.  Maude: specification and programming in rewriting logic , 2002, Theor. Comput. Sci..

[18]  Hubert Comon-Lundh,et al.  Disunification: A Survey , 1991, Computational Logic - Essays in Honor of Alan Robinson.

[19]  Jean Goubault-Larrecq,et al.  Well-Founded Recursive Relations , 2001, CSL.

[20]  J. Van Leeuwen,et al.  Handbook of theoretical computer science - Part A: Algorithms and complexity; Part B: Formal models and semantics , 1990 .