On Automating Inductive and Non-inductive Termination Methods

The Coq and ProPre systems show the automated termination of a recursive function by first constructing a tree associated with the specification of the function which satisfies a notion of terminal property and then verifying that this construction process is formally correct. However, those two steps strongly depend on inductive principles and hence Coq and ProPre can only deal with the termination proofs that are inductive. There are however many functions for which the termination proofs are non-inductive. In this article, we attempt to extend the class of functions whose proofs can be done automatically a la Coq and ProPre to a larger class including functions whose termination proofs are not inductive. We do this by extending the terminal property notion and replacing the verification step above by one that searches for a decreasing measure which can be used to establish the termination of the function.

[1]  D. Knuth,et al.  Simple Word Problems in Universal Algebras , 1983 .

[2]  Fairouz Kamareddine,et al.  On Formalised Proofs of Termination of Recursive Functions , 1999, PPDP.

[3]  Marianne Simonot,et al.  Automatizing Termination Proofs of Recursively Defined Functions , 1994, Theor. Comput. Sci..

[4]  Hugo Herbelin,et al.  The Coq proof assistant : reference manual, version 6.1 , 1997 .

[5]  Michel Parigot.  Recursive Programming with Proofs , 1992, Theor. Comput. Sci..

[6]  Jürgen Giesl,et al.  Automatically Proving Termination Where Simplification Orderings Fail , 1997, TAPSOFT.

[7]  Hans Zantema,et al.  Simple Termination of Rewrite Systems , 1997, Theor. Comput. Sci..

[8]  Pierre Lescanne,et al.  Termination of Rewriting Systems by Polynomial Interpretations and Its Implementation , 1987, Sci. Comput. Program..

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

[10]  N. A C H U M D E R S H O W I T Z.  Termination of Rewriting' , 2022 .

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

[12]  Pascal Manoury,et al.  Des preuves de totalite de fonctions comme synthese de programmes , 1992 .

[13]  Isabelle Gnaedig,et al.  Termination Proofs Using gpo Ordering Constraints , 1997, TAPSOFT.

[14]  Joachim Steinbach.  Generating Polynomial Orderings , 1994, Inf. Process. Lett..

[15]  Nachum Dershowitz,et al.  Natural Termination , 1995, Theor. Comput. Sci..

[16]  J. Urgen Giesl.  Generating Polynomial Orderings for Termination Proofs ? , 1995 .

[17]  Pascal Manoury.  A User's Friendly Syntax to Define Recursive Functions as Typed lambda-Terms , 1994, TYPES.

[18]  Marianne Simonot,et al.  An ordinal measure based procedure for termination of functions , 2001, Theor. Comput. Sci..

[19]  Donald E. Knuth,et al.  Simple Word Problems in Universal Algebras††The work reported in this paper was supported in part by the U.S. Office of Naval Research. , 1970 .