Abstract
The paper at hand introduces a refinement of interpretation based termination criteria for term rewrite systems in the dependency pair setting. Traditional methods share the property that—in order to be successful—all rewrite rules must (weakly) decrease with respect to some measure. The novelty of our approach is that we allow some rules to increase the interpreted value. These rules are found by simultaneously searching for adequate polynomial interpretations while considering the information of the dependency graph. We prove that our method extends the termination proving power of linear natural interpretations. Furthermore, this generalization perfectly fits the recursive SCC decomposition algorithm which is implemented in virtually every termination prover dealing with term rewrite systems.
This research is supported by FWF (Austrian Science Fund) project P18763.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Arts, T., Giesl, J.: Termination of term rewriting using dependency pairs. Theoretical Computer Science 236, 133–178 (2000)
Baader, F., Nipkow, T.: Term Rewriting and All That. Cambridge University Press, Cambridge (1998)
Eén, N., Sörensson, N.: An extensible SAT-solver. In: Giunchiglia, E., Tacchella, A. (eds.) SAT 2003. LNCS, vol. 2919, pp. 502–518. Springer, Heidelberg (2004)
Endrullis, J.: Jambox (2007), http://joerg.endrullis.de
Endrullis, J., Waldmann, J., Zantema, H.: Matrix interpretations for proving termination of term rewriting. Journal of Automated Reasoning 40(2-3), 195–220 (2008)
Fuhs, C., Giesl, J., Middeldorp, A., Schneider-Kamp, P., Thiemann, R., Zankl, H.: SAT solving for termination analysis with polynomial interpretations. In: Marques-Silva, J., Sakallah, K.A. (eds.) SAT 2007. LNCS, vol. 4501, pp. 340–354. Springer, Heidelberg (2007)
Geser, A., Hofbauer, D., Waldmann, J., Zantema, H.: On tree automata that certify termination of left-linear term rewriting systems. Information and Computation 205(4), 512–534 (2007)
Giesl, J., Schneider-Kamp, P., Thiemann, R.: AProVE 1.2: Automatic termination proofs in the dependency pair framework. In: Furbach, U., Shankar, N. (eds.) IJCAR 2006. LNCS (LNAI), vol. 4130, pp. 281–286. Springer, Heidelberg (2006)
Giesl, J., Thiemann, R., Schneider-Kamp, P., Falke, S.: Mechanizing and improving dependency pairs. Journal of Automated Reasoning 37(3), 155–203 (2006)
Hirokawa, N., Middeldorp, A.: Dependency pairs revisited. In: van Oostrom, V. (ed.) RTA 2004. LNCS, vol. 3091, pp. 249–268. Springer, Heidelberg (2004)
Hirokawa, N., Middeldorp, A.: Polynomial interpretations with negative coefficients. In: Buchberger, B., Campbell, J.A. (eds.) AISC 2004. LNCS (LNAI), vol. 3249, pp. 185–198. Springer, Heidelberg (2004)
Hirokawa, N., Middeldorp, A.: Automating the dependency pair method. Information and Computation 199(1,2), 172–199 (2005)
Hirokawa, N., Middeldorp, A.: Tyrolean termination tool: Techniques and features. Information and Computation 205(4), 474–511 (2007)
Hong, H., Jakuš, D.: Testing positiveness of polynomials. Journal of Automated Reasoning 21(1), 23–38 (1998)
Koprowski, A., Waldmann, J.: Arctic termination ⋯ below zero. In: Voronkov, A. (ed.) RTA 2008. LNCS, vol. 5117. Springer, Heidelberg (to appear, 2008)
Korp, M., Middeldorp, A.: Proving termination of rewrite systems using bounds. In: Baader, F. (ed.) RTA 2007. LNCS, vol. 4533, pp. 273–287. Springer, Heidelberg (2007)
Kurihara, M., Kondo, H.: Efficient BDD encodings for partial order constraints with application to expert systems in software verification. In: Orchard, B., Yang, C., Ali, M. (eds.) IEA/AIE 2004. LNCS (LNAI), vol. 3029, pp. 827–837. Springer, Heidelberg (2004)
Lankford, D.: On proving term rewrite systems are noetherian. Technical Report MTP-3, Louisiana Technical University, Ruston, LA, USA (1979)
Lucas, S.: On the relative power of polynomials with real, rational, and integer coefficients in proofs of termination of rewriting. Applicable Algebra in Engineering, Communication and Computing 17(1), 49–73 (2006)
Waldmann, J.: Matchbox: A tool for match-bounded string rewriting. In: van Oostrom, V. (ed.) RTA 2004. LNCS, vol. 3091, pp. 85–94. Springer, Heidelberg (2004)
Zantema, H.: Reducing right-hand sides for termination. In: Middeldorp, A., van Oostrom, V., van Raamsdonk, F., de Vrijer, R. (eds.) Processes, Terms and Cycles: Steps on the Road to Infinity. LNCS, vol. 3838, pp. 173–197. Springer, Heidelberg (2005)
Zantema, H., Waldmann, J.: Termination by quasi-periodic interpretations. In: Baader, F. (ed.) RTA 2007. LNCS, vol. 4533, pp. 404–418. Springer, Heidelberg (2007)
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 2008 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Zankl, H., Middeldorp, A. (2008). Increasing Interpretations. In: Autexier, S., Campbell, J., Rubio, J., Sorge, V., Suzuki, M., Wiedijk, F. (eds) Intelligent Computer Mathematics. CICM 2008. Lecture Notes in Computer Science(), vol 5144. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-85110-3_16
Download citation
DOI: https://doi.org/10.1007/978-3-540-85110-3_16
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-85109-7
Online ISBN: 978-3-540-85110-3
eBook Packages: Computer ScienceComputer Science (R0)