Abstract
Matrix interpretations can be used to bound the derivational complexity of term rewrite systems. In particular, triangular matrix interpretations over the natural numbers are known to induce polynomial upper bounds on the derivational complexity of (compatible) rewrite systems. Using techniques from linear algebra, we show how one can generalize the method to matrices that are not necessarily triangular but nevertheless polynomially bounded. Moreover, we show that our approach also applies to matrix interpretations over the real (algebraic) numbers. In particular, it allows triangular matrix interpretations to infer tighter bounds than the original approach.
This research is supported by FWF (Austrian Science Fund) project P22467.
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References
Alarcón, B., Lucas, S., Navarro-Marset, R.: Proving termination with matrix interpretations over the reals. In: WST 2009, pp. 12–15 (2009)
Baader, F., Nipkow, T.: Term Rewriting and All That. Cambridge University Press, Cambridge (1998)
Charalambides, C.A.: Enumerative Combinatorics. Chapman & Hall/CRC, Boca Raton (2002)
Chuan-Chong, C., Khee-Meng, K.: Principles and Techniques in Combinatorics. World Scientific Publishing Company, Singapore (1992)
Endrullis, J., Waldmann, J., Zantema, H.: Matrix interpretations for proving termination of term rewriting. JAR 40(2–3), 195–220 (2008)
Gebhardt, A., Hofbauer, D., Waldmann, J.: Matrix evolutions. In: WST 2007, pp. 4–8 (2007)
Geser, A., Hofbauer, D., Waldmann, J., Zantema, H.: On tree automata that certify termination of left-linear term rewriting systems. I&C 205(4), 512–534 (2007)
Hofbauer, D.: Termination proofs by context-dependent interpretations. In: Middeldorp, A. (ed.) RTA 2001. LNCS, vol. 2051, pp. 108–121. Springer, Heidelberg (2001)
Hofbauer, D., Lautemann, C.: Termination proofs and the length of derivations (preliminary version). In: Dershowitz, N. (ed.) RTA 1989. LNCS, vol. 355, pp. 167–177. Springer, Heidelberg (1989)
Hofbauer, D., Waldmann, J.: Termination of string rewriting with matrix interpretations. In: Pfenning, F. (ed.) RTA 2006. LNCS, vol. 4098, pp. 328–342. Springer, Heidelberg (2006)
Jungers, R.M., Protasov, V., Blondel, V.D.: Efficient algorithms for deciding the type of growth of products of integer matrices. LAA 428(10), 2296–2311 (2008)
Koprowski, A., Waldmann, J.: Arctic termination ... below zero. In: Voronkov, A. (ed.) RTA 2008. LNCS, vol. 5117, pp. 202–216. Springer, Heidelberg (2008)
Moser, G., Schnabl, A., Waldmann, J.: Complexity analysis of term rewriting based on matrix and context dependent interpretations. In: FSTTCS 2008. LIPIcs, vol. 2, pp. 304–315 (2008)
Neurauter, F., Middeldorp, A.: Polynomial interpretations over the reals do not subsume polynomial interpretations over the integers. In: RTA 2010. LIPIcs, vol. 6, pp. 243–258 (2010)
Rose, H.E.: Linear Algebra: A Pure Mathematical Approach. Birkhäuser, Basel (2002)
Serre, D.: Matrices: Theory and Applications. Springer, Heidelberg (2002)
Terese: Term Rewriting Systems. Cambridge Tracts in Theoretical Computer Science, vol. 55. Cambridge University Press, Cambridge (2003)
Termination problem data base, version 7.0.2. (2010), http://termcomp.uibk.ac.at/status/downloads/tpdb-7.0.2.tar.gz
Waldmann, J.: Polynomially bounded matrix interpretations. In: RTA 2010. LIPIcs, vol. 6, pp. 357–372 (2010)
Zankl, H., Korp, M.: Modular complexity analysis via relative complexity. In: RTA 2010. LIPIcs, vol. 6, pp. 385–400 (2010)
Zankl, H., Middeldorp, A.: Satisfiability of non-linear (ir)rational arithmetic. In: LPAR-16. LNCS, vol. 6355. Springer, Heidelberg (to appear, 2010)
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Neurauter, F., Zankl, H., Middeldorp, A. (2010). Revisiting Matrix Interpretations for Polynomial Derivational Complexity of Term Rewriting . In: Fermüller, C.G., Voronkov, A. (eds) Logic for Programming, Artificial Intelligence, and Reasoning. LPAR 2010. Lecture Notes in Computer Science, vol 6397. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-16242-8_39
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DOI: https://doi.org/10.1007/978-3-642-16242-8_39
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