article

Polytool: Polynomial interpretations as a basis for termination analysis of logic programs

Authors:

Manh thang Nguyen,

Danny De schreye,

Peter Schneider-kampAuthors Info & Claims

Theory and Practice of Logic Programming, Volume 11, Issue 1

Pages 33 - 63

https://doi.org/10.1017/S1471068410000025

Published: 01 January 2011 Publication History

Abstract

Our goal is to study the feasibility of porting termination analysis techniques developed for one programming paradigm to another paradigm. In this paper, we show how to adapt termination analysis techniques based on polynomial interpretations—very well known in the context of term rewrite systems—to obtain new (nontransformational) termination analysis techniques for definite logic programs (LPs). This leads to an approach that can be seen as a direct generalization of the traditional techniques in termination analysis of LPs, where linear norms and level mappings are used. Our extension generalizes these to arbitrary polynomials. We extend a number of standard concepts and results on termination analysis to the context of polynomial interpretations. We also propose a constraint-based approach for automatically generating polynomial interpretations that satisfy the termination conditions. Based on this approach, we implemented a new tool, called <monospace>Polytool</monospace>, for automatic termination analysis of LPs.

References

[1]

APT, K. R. 1990. Logic programming. In Handbook of Theoretical Computer Science, Vol. B, Ed. Jan van Leeuwen, MIT Press, 493-574.

Digital Library

[2]

ARTS, T. AND GIESL, J. 2000. Termination of term rewriting using dependency pairs. Theoretical Computer Science 236, 1-2, 133-178.

Digital Library

[3]

BORRALLERAS, C., LUCAS, S., NAVARRO-MARSET, R., RODRÍGUEZ-CARBONELL, E. AND RUBIO, A. 2009. Solving non-linear polynomial arithmetic via SAT modulo linear arithmetic. In Proc. CADE '09. Ed. Renate Schmidt, LNAI, Springer-Verlag, vol. 5663, 294-305.

Digital Library

[4]

BOSSI, A., COCCO, N. AND FABRIS, M. 1991 Proving termination of logic programs by exploiting term properties. In Proc. TAPSOFT '91. Eds. S. Abramsky and T. Maibaum, LNCS, Springer-Verlag, vol. 494, 153-180.

Digital Library

[5]

BRAUBURGER, J. AND GIESL, J. 1998 Termination analysis by inductive evaluation. In Proc. CADE '98. Eds. C. Kirchner and H. Kirchner, LNAI, Springer-Verlag, vol. 1421, 254-269.

Digital Library

[6]

BRUYNOOGHE, M., CODISH, M., GALLAGHER, J. P., GENAIM, S. AND VANHOOF, W. 2007. Termination analysis of logic programs through combination of type-based norms. ACM Transactions on Programming Languages and Systems 29, 2, 10-es.

Digital Library

[7]

BRUYNOOGHE, M., GALLAGHER, J. P. AND VAN HUMBEECK, W. 2005. Inference of well-typings for logic programs with application to termination analysis. In Proc. SAS '05. Eds. C. Hankin and I. Siveroni, LNCS, Springer-Verlag, vol. 3672, 35-51.

Digital Library

[8]

CODISH, M. AND TABOCH, C. 1999. A semantic basis for the termination analysis of logic programs. Journal of Logic Programming 41, 1, 103-123.

[9]

CONTEJEAN, E., MARCHÉ, C., TOMÁS, A. P. AND URBAIN, X. 2005. Mechanically proving termination using polynomial interpretations. Journal of Automated Reasoning 34, 4, 325- 363.

Digital Library

[10]

DECORTE, S., DE SCHREYE, D. AND FABRIS, M. 1993. Automatic inference of norms: A missing link in automatic termination analysis. In Proc. ILPS '93. Ed. D. Miller, MIT Press, 420-436.

Digital Library

[11]

DECORTE, S., DE SCHREYE, D. AND VANDECASTEELE, H. 1999. Constraint-based automatic termination analysis of logic programs. ACM Transactions on Programming Languages and Systems 21, 6, 1137-1195.

Digital Library

[12]

DERSHOWITZ, N. 1995. 33 examples of termination. In Proc. French Spring School in Theoretical Computer Science. Eds. H. Comon and J. Jouannaud, LNCS, Springer-Verlag, vol. 909, 16- 26.

Digital Library

[13]

DERSHOWITZ, N., LINDENSTRAUSS, N. AND SAGIV, Y. 1997. What norms are useful for logic programs? In Proc. WST '97. Transparencies available from URL: http://www.cs.huji. ac.il/~naomil/.

[14]

DE SCHREYE, D. AND SEREBRENIK, A. 2002. Acceptability with general orderings. In Computational Logic: Logic Programming and Beyond. Eds. A. Kakas and F. Sadri, LNCS, Springer-Verlag, vol. 2407, 187-210.

Digital Library

[15]

FUHS, C., GIESL, J., MIDDELDORP, A., SCHNEIDER-KAMP, P., THIEMANN, R. AND ZANKL, H. 2007. SAT solving for termination analysis with polynomial interpretations. In Proc. SAT '07. Eds. J. Marques-Silva, K. Sakallah, LNCS, Springer-Verlag, vol. 4501, 340-354.

Digital Library

[16]

FUHS, C., GIESL, J., MIDDELDORP, A., SCHNEIDER-KAMP, P., THIEMANN, R. AND ZANKL, H. 2008. Maximal termination. In Proc. RTA '08, 110-125.

Digital Library

[17]

GALLAGHER, J. P., HENRIKSEN, K. S. AND BANDA, G. 2005. Techniques for scaling up analyses based on pre-interpretations. In Proc. ICLP '05. Eds. M. Gabbrielli and G. Gupta, LNCS, Springer-Verlag, vol. 3668, 280-296.

Digital Library

[18]

GIESL, J., SCHNEIDER-KAMP, P. AND THIEMANN, R. 2006a. AProVE 1.2: Automatic termination proofs in the dependency pair framework. In Proc. IJCAR '06. Eds. U. Furbach and N. Shankar, LNAI, Springer-Verlag, vol. 4130, 281-286.

Digital Library

[19]

GIESL, J., THIEMANN, R., SCHNEIDER-KAMP, P. AND FALKE, S. 2006b. Mechanizing and improving dependency pairs. Journal of Automated Reasoning 37, 3, 155-203.

Digital Library

[20]

GIESL, J., THIEMANN, R., SWIDERSKI, S. AND SCHNEIDER-KAMP, P. 2007. Proving termination by bounded increase. In Proc. CADE '07. Ed. F. Pfenning, LNAI, Springer-Verlag, vol. 4603, 443-459.

Digital Library

[21]

HEATON, A., ABO-ZAED, M., CODISH, M. AND KING, A. 2000. A simple polynomial groundness analysis for logic programs. Journal of Logic Programming 45, 1-3, 143-156.

[22]

HIROKAWA, N. AND MIDDELDORP, A. 2005. Automating the dependency pair method. Information and Computation 199, 1-2, 172-199.

Digital Library

[23]

HONG, H. AND JAKU¿, D. 1998. Testing positiveness of polynomials. Journal of Automated Reasoning 21, 1, 23-38.

Digital Library

[24]

JANSSENS, G. AND BRUYNOOGHE, M. 1992. Deriving descriptions of possible values of program variables by means of abstract interpretation. Journal of Logic Programming 13, 2-3, 205- 258.

Digital Library

[25]

LANKFORD, D. S. 1979. On Proving Term Rewriting Systems are Noetherian. Tech. Rep., Mathematics Department, Louisiana Tech University, Ruston, LA.

[26]

LINDENSTRAUSS, N. 2000. TermiLog: Termination analyzer for logic programs. URL: http://www.cs.huji.ac.il/~naomil/termilog.php.

[27]

LINDENSTRAUSS, N. AND SAGIV, Y. 1997. Automatic termination analysis of logic programs. In Proc. ICLP '97. Ed. L. Naish, MIT Press, 63-77.

[28]

LINDENSTRAUSS, N., SAGIV, Y. AND SEREBRENIK, A. 2004. Proving termination for logic programs by the query-mapping pairs approach. In Program Development in Computational Logic. Eds. M. Bruynooghe and K. Lau, LNCS, Springer-Verlag, vol. 3049, 453-498.

[29]

LLOYD, J. W. 1987. Foundations of Logic Programming. Springer Verlag, Berlin.

[30]

MESNARD, F. AND BAGNARA, R. 2005. cTI: A constraint-based termination inference tool for ISO-Prolog. Theory and Practice of Logic Programming 5, 1-2, 243-257.

Digital Library

[31]

NGUYEN, M. T. 2009 Termination Analysis: Crossing Paradigm Borders. Ph.D. thesis, Department of Computer Science, K. U. Leuven, Belgium.

[32]

NGUYEN, M. T. AND DE SCHREYE, D. 2005. Polynomial interpretations as a basis for termination analysis of logic programs. In Proc. ICLP '05. Eds. M. Gabbrielli and G. Gupta, LNCS, Springer-Verlag, vol. 3668, 311-325. Extended version appeared as Technical report, Department of Computer Science, K. U. Leuven, Belgium, URL: http://www.cs.kuleuven.be/publicaties/rapporten/cw/CW412.pdf.

Digital Library

[33]

NGUYEN, M. T. AND DE SCHREYE, D. 2007. Polytool: Proving termination automatically based on polynomial interpretations. In Proc. LOPSTR '06. Ed. G. Puebla, LNCS, Springer-Verlag, vol. 4407, 210-218. Extended version appeared as Technical report, Department of Computer Science, K. U. Leuven, Belgium, URL: http://www.cs.kuleuven.be/~manh/ polytool/publications/tech-reports/reportCW442.pdf.

Digital Library

[34]

NGUYEN, M. T., GIESL, J., SCHNEIDER-KAMP, P. AND DE SCHREYE, D. 2008. Termination analysis of logic programs based on dependency graphs. In Proc. LOPSTR '07. Ed. A. King, LNCS, Springer-Verlag, vol. 4915, 8-22.

Digital Library

[35]

OHLEBUSCH, E., CLAVES, C. AND MARCHÉ, C. 2000. TALP: A tool for the termination analysis of logic programs. In Proc. RTA '00. Ed. L. Bachmair, LNCS, Springer-Verlag, vol. 1833, 270-273.

Digital Library

[36]

SCHNEIDER-KAMP, P., GIESL, J. AND NGUYEN, M. T. The dependency triple framework for termination of logic programs. In Proc. LOPSTR '09. LNCS. Forthcoming.

Digital Library

[37]

SCHNEIDER-KAMP, P., GIESL, J., SEREBRENIK, A. AND THIEMANN, R. 2009. Automated termination proofs for logic programs by term rewriting. ACM Transactions on Computational Logic 11, 1. Preliminary version appeared in Proc. LOPSTR '06, Ed. G. Puebla, LNCS, Springer-Verlag, vol. 4407, pp. 117-193, 2007.

Digital Library

[38]

SEREBRENIK, A. 2003. Termination Analysis of Logic Programs. Ph.D. thesis, Department of Computer Science, K. U. Leuven, Belgium. URL: http://www.cs.kuleuven.ac.be/ publicaties/doctoraten/cw/CW2003_02.abs.html.

[39]

SEREBRENIK, A. AND DE SCHREYE, D. 2003. Hasta-La-Vista: Termination analyser for logic programs. In Proc. WLPE '03. Eds. F. Mesnard and A. Serebrenik, 60-74.

[40]

STEINBACH, J. 1992. Proving polynomials positive. In Proc. FSTTCS '92. Ed. R. Shyamasundar, LNCS, Springer-Verlag, vol. 652, 18-20.

Digital Library

[41]

TABOCH, C., GENAIM, S. AND CODISH, M. 2002. TerminWeb: Semantics-based termination analyser for logic programs. URL: http://www.cs.bgu.ac.il/~mcodish/TerminWeb.

[42]

VERSCHAETSE, K. AND DE SCHREYE, D. 1991. Deriving termination proofs for logic programs, using abstract procedures. In Proc. ICLP '91. Ed. K. Furukawa, MIT Press, 301-315.

Cited By

Calautti MGreco STrubitsyna I(2017)Detecting Decidable Classes of Finitely Ground Logic Programs with Function SymbolsACM Transactions on Computational Logic10.1145/314380418:4(1-42)Online publication date: 16-Nov-2017
https://dl.acm.org/doi/10.1145/3143804
Giesl JAschermann CBrockschmidt MEmmes FFrohn FFuhs CHensel JOtto CPlücker MSchneider-Kamp PStröder TSwiderski SThiemann R(2017)Analyzing Program Termination and Complexity Automatically with AProVEJournal of Automated Reasoning10.1007/s10817-016-9388-y58:1(3-31)Online publication date: 1-Jan-2017
https://dl.acm.org/doi/10.1007/s10817-016-9388-y
Drabent W(2016)Correctness and Completeness of Logic ProgramsACM Transactions on Computational Logic10.1145/289843417:3(1-32)Online publication date: 18-May-2016
https://dl.acm.org/doi/10.1145/2898434
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cover image Theory and Practice of Logic Programming

Theory and Practice of Logic Programming Volume 11, Issue 1

January 2011

138 pages

ISSN:1471-0684

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Cambridge University Press

United States

Publication History

Published: 01 January 2011

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Cited By

Calautti MGreco STrubitsyna I(2017)Detecting Decidable Classes of Finitely Ground Logic Programs with Function SymbolsACM Transactions on Computational Logic10.1145/314380418:4(1-42)Online publication date: 16-Nov-2017
https://dl.acm.org/doi/10.1145/3143804
Giesl JAschermann CBrockschmidt MEmmes FFrohn FFuhs CHensel JOtto CPlücker MSchneider-Kamp PStröder TSwiderski SThiemann R(2017)Analyzing Program Termination and Complexity Automatically with AProVEJournal of Automated Reasoning10.1007/s10817-016-9388-y58:1(3-31)Online publication date: 1-Jan-2017
https://dl.acm.org/doi/10.1007/s10817-016-9388-y
Drabent W(2016)Correctness and Completeness of Logic ProgramsACM Transactions on Computational Logic10.1145/289843417:3(1-32)Online publication date: 18-May-2016
https://dl.acm.org/doi/10.1145/2898434
Pettorossi AProietti MSenni V(2012)Constraint-based correctness proofs for logic program transformationsFormal Aspects of Computing10.1007/s00165-012-0233-824:4-6(569-594)Online publication date: 1-Jul-2012
https://dl.acm.org/doi/10.1007/s00165-012-0233-8
Sneyers JDe Schreye D(2011)Probabilistic termination of CHRiSM programsProceedings of the 21st international conference on Logic-Based Program Synthesis and Transformation10.1007/978-3-642-32211-2_15(221-236)Online publication date: 18-Jul-2011
https://dl.acm.org/doi/10.1007/978-3-642-32211-2_15
Ströder TSchneider-Kamp PGiesl J(2010)Dependency triples for improving termination analysis of logic programs with cutProceedings of the 20th international conference on Logic-based program synthesis and transformation10.5555/2008282.2008294(184-199)Online publication date: 23-Jul-2010
https://dl.acm.org/doi/10.5555/2008282.2008294
Codish MFuhs CGiesl JSchneider-Kamp P(2010)Lazy abstraction for size-change terminationProceedings of the 17th international conference on Logic for programming, artificial intelligence, and reasoning10.5555/1928380.1928396(217-232)Online publication date: 10-Oct-2010
https://dl.acm.org/doi/10.5555/1928380.1928396

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