Hostname: page-component-cd9895bd7-fscjk Total loading time: 0 Render date: 2025-01-02T19:36:25.123Z Has data issue: false hasContentIssue false
  • English
  • Français

Polynomials over the reals in proofs of termination : from theory to practice

Published online by Cambridge University Press:  15 July 2005

Salvador Lucas
Salvador Lucas*
Affiliation:
DSIC, Universidad Politécnica de Valencia. Camino de Vera s/n, E-46022 Valencia, Spain; [email protected]
Get access

Abstract

This paper provides a framework to address termination problems in term rewriting by using orderings induced by algebras over the reals. The generation of such orderings is parameterized by concrete monotonicity requirements which are connected with different classes of termination problems: termination of rewriting, termination of rewriting by using dependency pairs, termination of innermost rewriting, top-termination of infinitary rewriting, termination of context-sensitive rewriting, etc. We show how to define term orderings based on algebraic interpretations over the real numbers which can be used for these purposes. From a practical point of view, we show how to automatically generate polynomial algebras over the reals by using constraint-solving systems to obtain the coefficients of a polynomial in the domain of the real or rational numbers. Moreover, as a consequence of our work, we argue that software systems which are able to generate constraints for obtaining polynomial interpretations over the naturals which prove termination of rewriting (e.g., AProVE, CiME, and TTT), are potentially able to obtain suitable interpretations over the reals by just solving the constraints in the domain of the real or rational numbers.

Keywords

Algebraic interpretationspolynomial orderingsrewritingtermination.
Type
Research Article
Information
RAIRO - Theoretical Informatics and Applications , Volume 39 , Issue 3: Foundations of Software Science and Computer Structures (FOSSECAS'04) , July 2005 , pp. 547 - 586
Copyright
© EDP Sciences, 2005

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Arts, T. and Giesl, J., Termination of Term Rewriting Using Dependency Pairs. Theor. Comput. Sci. 236 (2000) 133–178. CrossRef
T. Arts and J. Giesl, A collection of examples for termination of term rewriting using dependency pairs. Technical report, AIB-2001-09, RWTH Aachen, Germany (2001).
J. Bochnak, M. Coste and M-F. Roy, Géométrie algébraique réelle. Springer-Verlag, Berlin (1987).
Borralleras, C., Lucas, S. and Rubio, A., Recursive Path Orderings can be Context-Sensitive, in Proc. of 18th International Conference on Automated Deduction, CADE'02, edited by A. Voronkov, Springer-Verlag, Berlin. LNAI 2392 (2002) 314–331.
F. Baader and T. Nipkow, Term Rewriting and All That. Cambridge University Press (1998).
ben Cherifa, A. and Lescanne, P., Termination of rewriting systems by polynomial interpretations and its implementation. Sci. Comput. Program. 9 (1987) 137–160. CrossRef
A. Cichon and P. Lescanne, Polynomial interpretations and the complexity of algorithms, in Proc. of 11th International Conference on Automated Deduction, CADE'92, edited by D. Kapur, Springer-Verlag, Berlin. LNAI 607 (1992) 139–147.
E. Contejean and C. Marché, B. Monate and X. Urbain, Proving termination of rewriting with CiME, in Proc. of 6th International Workshop on Termination, WST'03, edited by A. Rubio, Technical Report DSIC II/15/03, Valencia, Spain (2003) 71–73. Available at http://cime.lri.fr
E. Contejean, C. Marché, A.-P. Tomás and X. Urbain, Mechanically proving termination using polynomial interpretations. Research Report 1382, LRI, Université de Paris-Sud (2004).
Dauchet, M., Simulation of turing machines by a regular rewrite rule. Theor. Comput. Sci. 103 (1992) 409–420. CrossRef
Dershowitz, N., A note on simplification orderings. Inform. Proc. Lett. 9 (1979) 212–215. CrossRef
Dershowitz, N., Orderings for term rewriting systems. Theor. Comput. Sci. 17 (1982) 279–301. CrossRef
Dershowitz, N., Termination of rewriting. J. Symbol. Comput. 3 (1987) 69–115. CrossRef
Dershowitz, N., Kaplan, S. and Plaisted, D., Rewrite, rewrite, rewrite, rewrite, rewrite. Theor. Comput. Sci. 83 (1991) 71–96. CrossRef
Fernández, M.-L., Relaxing monotonicity for innermost termination. Inform. Proc. Lett. 93 (2005) 117–123. CrossRef
Giesl, J., Arts, T. and Ohlebusch, E., Modular Termination Proofs for Rewriting Using Dependency Pairs. J. Symbol. Comput. 38 (2002) 21–58. CrossRef
A. Geser, Relative Termination. Ph.D. Thesis. Fakultät für Mathematik und Informatik. Universität Passau (1990).
Giesl, J., Generating Polynomial Orderings for Termination Proofs, in Proc. of 6th International Conference on Rewriting Techniques and Applications, RTA'95, edited by J. Hsiang, Springer-Verlag, Berlin. Lect. Notes Comput. Sci. 914 (1995) 426–431. CrossRef
B. Gramlich and S. Lucas, Simple termination of context-sensitive rewriting, in Proc. of 3rd ACM SIGPLAN Workshop on Rule-based Programming, RULE'02 ACM Press, New York (2002) 29–41.
Giesl, J. and Middeldorp, A., Transformation Techniques for Context-Sensitive Rewrite Systems. J. Funct. Program. 14 (2004) 379–427. CrossRef
Giesl, J., Thiemann, R., Schneider-Kamp, P. and Falke, S., Automated Termination Proofs with AProVE, in Proc. of 15h International Conference on Rewriting Techniques and Applications, RTA'04, edited by V. van Oostrom, Springer-Verlag, Berlin. Lect. Notes. Comput. Sci. 3091 (2004) 210–220. Available at http://www-i2.informatik.rwth-aachen.de/AProVE CrossRef
Hong, H. and Jakuš, D., Testing Positiveness of Polynomials. J. Automated Reasoning 21 (1998) 23–38. CrossRef
Hofbauer, D. and Lautemann, C., Termination proofs and the length of derivations, in Proc. of the 3rd International Conference on Rewriting Techniques and Applications, RTA'89, edited by N. Dershowitz, Springer-Verlag, Berlin. Lect. Notes Comput. Sci. 355 (1989) 167–177. CrossRef
Hirokawa, N. and Middeldorp, A., Dependency Pairs Revisited, in Proc. of 15h International Conference on Rewriting Techniques and Applications, RTA'04, edited by V. van Oostrom, Springer-Verlag, Berlin. Lect. Notes. Comput. Sci. 3091 (2004) 249–268. CrossRef
Hirokawa, N. and Middeldorp, A., Polynomial Interpretations with Negative Coefficients, in Proc. of the 7th International Conference on Artificial Intelligence and Symbolic Computation, AISC'04, edited by B. Buchberger and J.A. Campbell, Springer-Verlag, Berlin. LNAI 3249 (2004) 185–198.
N. Hirokawa and A. Middeldorp, Tyrolean Termination Tool, in Proc. of 16th International Conference on Rewriting Techniques and Applications, RTA'05, edited by J. Giesl. Lect. Notes. Comput. Sci., to appear (2005). Available at http://cl2-informatik.uibk.ac.at
Hofbauer, D., Termination Proofs by Context-Dependent Interpretations, in Proc. of 12th International Conference on Rewriting Techniques and Applications, RTA'01, edited by A. Middeldorp, Springer-Verlag, Berlin. Lect. Notes Comput. Sci. 2051 (2001) 108–121. CrossRef
D.E. Knuth and P.E. Bendix, Simple Word Problems in Universal Algebra, in Computational Problems in Abstract Algebra, edited by J. Leech, Pergamon Press (1970) 263–297.
Kusakari, K., Nakamura, M. and Toyama, Y., Argument Filtering Transformation, in International Conference on Principles and Practice of Declarative Programming, PPDP'99, edited by G. Nadathur, Springer-Verlag, Berlin. Lect. Notes Comput. Sci. 1702 (1999) 47–61. CrossRef
D.S. Lankford, On proving term rewriting systems are noetherian. Technical Report, Louisiana Technological University, Ruston, LA (1979).
S. Lang, Algebra. Springer-Verlag, Berlin (2004).
Lucas, S., Context-sensitive computations in functional and functional logic programs. J. Funct. Logic Program. 1998 (1998) 1–61.
Lucas, S., Context-Sensitive Rewriting Strategies. Inform. Comput. 178 (2002) 293–343. CrossRef
Lucas, S., Termination of (Canonical) Context-Sensitive Rewriting, in Proc. 13th International Conference on Rewriting Techniques and Applications, RTA'02, edited by S. Tison, Springer-Verlag, Berlin. Lect. Notes Comput. Sci. 2378 (2002) 296–310. CrossRef
Lucas, S., MU-TERM: A Tool for Proving Termination of Context-Sensitive Rewriting, in Proc. of 15h International Conference on Rewriting Techniques and Applications, RTA'04, edited by V. van Oostrom, Springer-Verlag, Berlin. Lect. Notes Comput. Sci. 3091 (2004) 200–209. Available at http://www.dsic.upv.es/~slucas/csr/termination/muterm CrossRef
S. Lucas, Proving Termination of Context-Sensitive Rewriting by Transformation. Technical Report DSIC-II/18/04, DSIC, Universidad Politécnica de Valencia (2004).
E. Ohlebusch, Advanced Topics in Term Rewriting. Springer-Verlag, Berlin (2002).
J.P. Rellier, CON'FLEX. INRA, France, 1996. Main URL: http://www.inra.fr/bia/T/rellier/Logiciels/conflex/welcome.html
Steinbach, J., Generating Polynomial Orderings. Inform. Proc. Lett. 49 (1994) 85–93. CrossRef
Steinbach, J., Simplification orderings: History of results. Fundamenta Informaticae 24 (1995) 47–88.
A. Tarski, A Decision Method for Elementary Algebra and Geometry. Second Edition. University of California Press, Berkeley (1951).
Thiemann, R., Giesl, J. and Schneider-Kamp, P., Improved Modular Termination Proofs Using Dependency Pairs, in Proc. of 2nd International Joint Conference on Automated Reasoning, IJCAR'04, edited by D.A. Basin and M. Rusinowitch, Springer-Verlag, Berlin. Lect. Notes Comput. Sci. 3097 (2004) 75–90. CrossRef
Zantema, H., Termination of Context-Sensitive Rewriting, in Proc. of 8th International Conference on Rewriting Techniques and Applications, RTA'97, edited by H. Comon, Springer-Verlag, Berlin. Lect. Notes Comput. Sci. 1232 (1997) 172–186. CrossRef
H. Zantema, Termination, in Term Rewriting Systems, Chap. 6. edited by TeReSe, Cambridge University Press (2003).