Abstract
Considering an initial set of terms E, a rewriting relation \(\mathcal{R}\) and a goal set of terms Bad, reachability analysis in term rewriting tries to answer to the following question: does there exists at least one term of Bad that can be reached from E using the rewriting relation \(\mathcal{R}\)?
Some of the approaches try to show that there exists at least one term of Bad reachable from E using the rewriting relation \(\mathcal{R}\) by computing the set of reachable terms. Some others tackle the unreachability problem i.e. no term of Bad is reachable by rewriting from E. For the latter, over-approximations are computed. A main obstacle is to be able to compute an over-approximation precise enough that does not intersect Bad i.e. a conclusive approximation. This notion of precision is often defined by a very technical parameter of techniques implementing this over-approximation approach. In this paper, we propose a new characterization of conclusive approximations by logical formulae generated from a new kind of automata called symbolic tree automata. Solving a such formula leads automatically to a conclusive approximation without extra technical parameters.
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References
Baader, F., Nipkow, T.: Term Rewriting and All That. Cambridge University Press, Cambridge (1998)
Boichut, Y., Genet, T., Jensen, T., Leroux, L.: Rewriting Approximations for Fast Prototyping of Static Analyzers. In: Baader, F. (ed.) RTA 2007. LNCS, vol. 4533, pp. 48–62. Springer, Heidelberg (2007)
Boichut, Y., Héam, P.-C.: A theoretical limit for safety verification techniques with regular fix-point computations. Inf. Process. Lett. 108(1), 1–2 (2008)
Boichut, Y., Héam, P.-C., Kouchnarenko, O.: Approximation-based tree regular model-checking. Nord. J. Comput. 14(3), 216–241 (2008)
Bouajjani, A., Habermehl, P., Rogalewicz, A., Vojnar, T.: Abstract regular tree model checking. ENTCS 149(1), 37–48 (2006)
Boyer, B., Genet, T., Jensen, T.: Certifying a Tree Automata Completion Checker. In: Armando, A., Baumgartner, P., Dowek, G. (eds.) IJCAR 2008. LNCS (LNAI), vol. 5195, pp. 523–538. Springer, Heidelberg (2008)
Comon, H., Dauchet, M., Gilleron, R., Jacquemard, F., Lugiez, D., Löding, C., Tison, S., Tommasi, M.: Tree automata techniques and applications (2008)
Feuillade, G., Genet, T., Viet TriemTong, V.: Reachability Analysis over Term Rewriting Systems. Journal of Automated Reasonning 33(3-4), 341–383 (2004)
Gallagher, J., Rosendahl, M.: Approximating term rewriting systems: a horn clause specification and its implementation. In: Cervesato, I., Veith, H., Voronkov, A. (eds.) LPAR 2008. LNCS (LNAI), vol. 5330, pp. 682–696. Springer, Heidelberg (2008)
Genet, T.: Decidable approximations of sets of descendants and sets of normal forms. In: Nipkow, T. (ed.) RTA 1998. LNCS, vol. 1379, pp. 151–165. Springer, Heidelberg (1998)
Genet, T., Klay, F.: Rewriting for Cryptographic Protocol Verification. In: McAllester, D. (ed.) CADE 2000. LNCS (LNAI), vol. 1831, pp. 271–290. Springer, Heidelberg (2000)
Genet, T., Rusu, R.: Equational tree automata completion. JSC 45, 574–597 (2010)
Gilleron, R., Tison, S.: Regular tree languages and rewrite systems. Fundamenta Informaticae 24, 157–175 (1995)
Gyenizse, P., Vágvölgyi, S.: Linear Generalized Semi-Monadic Rewrite Systems Effectively Preserve Recognizability. TCS 194(1-2), 87–122 (1998)
Henriksen, J., Jensen, J., Jørgensen, M., Klarlund, N., Paige, B., Rauhe, T., Sandholm, A.: Mona: Monadic second-order logic in practice. In: Brinksma, E., Steffen, B., Cleaveland, W.R., Larsen, K.G., Margaria, T. (eds.) TACAS 1995. LNCS, vol. 1019, pp. 89–110. Springer, Heidelberg (1995)
Jacquemard, F.: Decidable approximations of term rewriting systems. In: Ganzinger, H. (ed.) RTA 1996. LNCS, vol. 1103, pp. 362–376. Springer, Heidelberg (1996)
Monniaux, D.: Abstracting Cryptographic Protocols with Tree Automata. In: Cortesi, A., Filé, G. (eds.) SAS 1999. LNCS, vol. 1694, pp. 149–163. Springer, Heidelberg (1999)
Takai, T.: A Verification Technique Using Term Rewriting Systems and Abstract Interpretation. In: van Oostrom, V. (ed.) RTA 2004. LNCS, vol. 3091, pp. 119–133. Springer, Heidelberg (2004)
Takai, T., Kaji, Y., Seki, H.: Right-linear finite-path overlapping term rewriting systems effectively preserve recognizability. In: Bachmair, L. (ed.) RTA 2000. LNCS, vol. 1833, pp. 246–260. Springer, Heidelberg (2000)
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Boichut, Y., Dao, TBH., Murat, V. (2011). Characterizing Conclusive Approximations by Logical Formulae. In: Delzanno, G., Potapov, I. (eds) Reachability Problems. RP 2011. Lecture Notes in Computer Science, vol 6945. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-24288-5_8
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DOI: https://doi.org/10.1007/978-3-642-24288-5_8
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