Abstract
This paper defines an expressive class of constrained equational rewrite systems that supports the use of semantic data structures (e.g., sets or multisets) and contains built-in numbers, thus extending our previous work presented at CADE 2007 [6]. These rewrite systems, which are based on normalized rewriting on constructor terms, allow the specification of algorithms in a natural and elegant way. Built-in numbers are helpful for this since numbers are a primitive data type in every programming language. We develop a dependency pair framework for these rewrite systems, resulting in a flexible and powerful method for showing termination that can be automated effectively. Various powerful techniques are developed within this framework, including a subterm criterion and reduction pairs that need to consider only subsets of the rules and equations. It is well-known from the dependency pair framework for ordinary rewriting that these techniques are often crucial for a successful automatic termination proof. Termination of a large collection of examples can be established using the presented techniques.
Partially supported by NSF grant CCF-0541315.
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References
Arts, T., Giesl, J.: Termination of term rewriting using dependency pairs. TCS 236(1–2), 133–178 (2000)
Baader, F., Nipkow, T.: Term Rewriting and All That. Cambridge University Press, Cambridge (1998)
Blanqui, F., Hardin, T., Weis, P.: On the implementation of construction functions for non-free concrete data types. In: De Nicola, R. (ed.) ESOP 2007. LNCS, vol. 4421, pp. 95–109. Springer, Heidelberg (2007)
Contejean, E., Marché, C., Tomás, A.P., Urbain, X.: Mechanically proving termination using polynomial interpretations. JAR 34(4), 325–363 (2005)
Falke, S., Kapur, D.: Dependency pairs for rewriting with built-in numbers and semantic data structures. Technical Report TR-CS-2007-21, Department of Computer Science, University of New Mexico (2007), http://www.cs.unm.edu/research/tech-reports/
Falke, S., Kapur, D.: Dependency pairs for rewriting with non-free constructors. In: Pfenning, F. (ed.) CADE 2007. LNCS (LNAI), vol. 4603, pp. 426–442. Springer, Heidelberg (2007)
Falke, S., Kapur, D.: Operational termination of conditional rewriting with built-in numbers and semantic data structures. Technical Report TR-CS-2007-22, Department of Computer Science, University of New Mexico (2007), http://www.cs.unm.edu/research/tech-reports/
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)
Giesl, J., Kapur, D.: Dependency pairs for equational rewriting. In: Middeldorp, A. (ed.) RTA 2001. LNCS, vol. 2051, pp. 93–108. Springer, Heidelberg (2001)
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.: The dependency pair framework: Combining techniques for automated termination proofs. In: Baader, F., Voronkov, A. (eds.) LPAR 2004. LNCS (LNAI), vol. 3452, pp. 301–331. Springer, Heidelberg (2005)
Giesl, J., Thiemann, R., Schneider-Kamp, P.: Proving and disproving termination of higher-order functions. In: Gramlich, B. (ed.) FroCos 2005. LNCS (LNAI), vol. 3717, pp. 216–231. Springer, Heidelberg (2005)
Giesl, J., Thiemann, R., Schneider-Kamp, P., Falke, S.: Mechanizing and improving dependency pairs. JAR 37(3), 155–203 (2006)
Giesl, J., Thiemann, R., Swiderski, S., Schneider-Kamp, P.: Proving termination by bounded increase. In: Pfenning, F. (ed.) CADE 2007. LNCS (LNAI), vol. 4603, pp. 443–459. Springer, Heidelberg (2007)
Hirokawa, N., Middeldorp, A.: Tyrolean termination tool: Techniques and features. IC 205(4), 474–511 (2007)
Kusakari, K., Nakamura, M., Toyama, Y.: Argument filtering transformation. In: Nadathur, G. (ed.) PPDP 1999. LNCS, vol. 1702, pp. 47–61. Springer, Heidelberg (1999)
Kusakari, K., Toyama, Y.: On proving AC-termination by AC-dependency pairs. IEICE Transactions on Information and Systems E84-D(5), 604–612 (2001)
Lankford, D.S.: On proving term rewriting systems are Noetherian. Memo MTP-3, Mathematics Department, Louisiana Tech University, Ruston (1979)
Marché, C.: Normalized rewriting: An alternative to rewriting modulo a set of equations. JSC 21(3), 253–288 (1996)
Marché, C., Urbain, X.: Modular and incremental proofs of AC-termination. JSC 38(1), 873–897 (2004)
Peterson, G.E., Stickel, M.E.: Complete sets of reductions for some equational theories. J. ACM 28(2), 233–264 (1981)
Presburger, M.: Über die Vollständigkeit eines gewissen Systems der Arithmetik ganzer Zahlen, in welchem die Addition als einzige Operation hervortritt. In: Comptes Rendus du Premier Congrès de Mathématiciens des Pays Slaves, pp. 92–101 (1929)
Steinbach, J., Kühler, U.: Check your ordering—termination proofs and open problems. Technical Report SR-90-25, Universität Karlsruhe (1990)
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Falke, S., Kapur, D. (2008). Dependency Pairs for Rewriting with Built-In Numbers and Semantic Data Structures. In: Voronkov, A. (eds) Rewriting Techniques and Applications. RTA 2008. Lecture Notes in Computer Science, vol 5117. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-70590-1_7
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DOI: https://doi.org/10.1007/978-3-540-70590-1_7
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