Abstract
In the paper, we introduce a new tree automata framework, called propositional tree automata, capturing the class of tree languages that are closed under an equational theory and Boolean operations. This framework originates in work on developing a sufficient completeness checker for specifications with rewriting modulo an equational theory. Propositional tree automata recognize regular equational tree languages. However, unlike regular equational tree automata, the class of propositional tree automata is closed under Boolean operations. This extra expressiveness does not affect the decidability of the membership problem. This paper also analyzes in detail the emptiness problem for propositional tree automata with associative theories. Though undecidable in general, we present a semi-algorithm for checking emptiness based on machine learning that we have found useful in practice.
Research supported by ONR Grant N00014-02-1-0715, NSF CAREER CCF-0448178, and NSF CCF-0429639.
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References
Angluin, D.: Learning Regular Sets from Queries and Counterexamples. Information and Computation 75, 87–106 (1987)
Armando, A., Basin, D., Boichut, Y., Chevalier, Y., Compagna, L., Cuellar, J., Hankes Drielsma, P., Heám, P.-C., Kouchnarenko, O., Mantovani, J., Mödersheim, S., von Oheimb, D., Rusinowitch, M., Santiago, J., Turuani, M., Viganò, L., Vigneron, L.: The AVISPA Tool for the Automated Validation of Internet Security Protocols and Applications. In: Etessami, K., Rajamani, S.K. (eds.) CAV 2005. LNCS, vol. 3576, pp. 281–285. Springer, Heidelberg (2005)
Autebert, J., Berstel, J., Boasson, L.: Context-Free Languages and Push-Down Automata, Handbook of Formal Languages, vol. 1, pp. 111–174. Springer, Heidelberg (1997)
Baader, F., Nipkow, T.: Term Rewriting and All That. Cambridge University Press, Cambridge (1998)
Boichut, Y., Heám, P.-C., Kouchnarenko, O.: Automatic Verification of Security Protocols Using Approximations, technical report RR-5727, INRIA (October 2005)
Bouhoula, A., Jouannaud, J.P., Meseguer, J.: Specification and Proof in Membership Equational Logic. TCS, vol. 236, pp. 35–132. Elsevier, Amsterdam (2000)
Comon, H., Dauchet, M., Gilleron, R., Jacquemard, F., Lugiez, D., Tison, S., Tommasi, M.: Tree Automata Techniques and Applications, incomplete draft (2005), available at: http://www.grappa.univ-lille3.fr/tata
Devienne, P., Talbot, J.-M., Tison, S.: Set-Based Analysis for Logic Programming and Tree Automata. In: Van Hentenryck, P. (ed.) SAS 1997. LNCS, vol. 1302, pp. 127–140. Springer, Heidelberg (1997)
Du, D.-Z., Ko, K.: Theory of Computational Complexity. John Wiley and Sons, Chichester (2000)
Gallagher, J.P., Puebla, G.: Abstract Interpretation over Non-deterministic Finite Tree Automata for Set-Based Analysis of Logic Programs. In: Krishnamurthi, S., Ramakrishnan, C.R. (eds.) PADL 2002. LNCS, vol. 2257, pp. 243–261. Springer, Heidelberg (2002)
Genet, T., Klay, F.: Rewriting for Cryptographic Protocol Verification. In: McAllester, D. (ed.) CADE 2000. LNCS, vol. 1831, pp. 271–290. Springer, Heidelberg (2000)
Ginsburg, S.: The Mathematical Theory of Context-Free Languages. McGraw-Hill, New York (1966)
Hendrix, J., Ohsaki, H., Meseguer, J.: Sufficient Completeness Checking with Propositional Tree Automata, technical report UIUCDCS-R-2005-2635, Department of Computer Science, University of Illinois at Urbana-Champaign (2005), available at: http://texas.cs.uiuc.edu/
Hendrix, J., Ohsaki, H., Viswanathan, M.: Propositional Tree Automat, technical report UIUCDCS-R-2006-2695, University of Illinios at Urbana-Champaign (2005), available at: http://texas.cs.uiuc.edu/
Hendrix, J.: CETA: A Library for Equational Tree Automata, Department of Computer Science, University of Illinois at Urbana-Champaign (2006), Software available under GPL license at: http://texas.cs.uiuc.edu/ceta/
Hopcroft, J.E., Ullman, J.D.: Introduction to Automata Theory, Languages, and Computation. Addison-Wesley Publishing Company, Reading (1979)
Hosoya, H., Vouillon, J., Pierce, B.C.: Regular Expression Types for XML. In: Proc. of 5th ICFP, SIGPLAN, Montreal (Canada), vol. 35(9), pp. 11–22. ACM, New York (2000)
Kearns, M., Vazirani, U.: An Introduction to Computational Learning Theory. MIT Press, Cambridge (1994)
Klarlund, N., Møller, A.: MONA Version 1.4 User Manual, BRICS Notes Series NS-01-1, Department of Computer Science, University of Aarhus (2001)
Lugiez, D.: Multitree Automata That Count. TCS, vol. 333. Elsevier, Amsterdam (2005)
Nederhof, M.: Practical Experiments with Regular Approximation of Context-Free Languages. Computational Linguistics 26(1), 17–44 (2000)
Ohsaki, H., Talbot, J.-M., Tison, S., Roos, Y.: Monotone AC-Tree Automata. In: Sutcliffe, G., Voronkov, A. (eds.) LPAR 2005. LNCS (LNAI), vol. 3835, pp. 337–351. Springer, Heidelberg (2005)
Ohsaki, H., Takai, T.: ACTAS: A System Design for Associative and Commutative Tree Automata Theory. In: Proc. of 5th RULE, Aachen (Germany). ENTCS, vol. 124, pp. 97–111. Elsevier, Amsterdam (2005)
Ohsaki, H., Takai, T.: Decidability and Closure Properties of Equational Tree Languages. In: Tison, S. (ed.) RTA 2002. LNCS, vol. 2378, pp. 114–128. Springer, Heidelberg (2002)
Ohsaki, H.: Beyond Regularity: Equational Tree Automata for Associative and Commutative Theories. In: Fribourg, L. (ed.) CSL 2001 and EACSL 2001. LNCS, vol. 2142, pp. 539–553. Springer, Heidelberg (2001)
Parikh, R.: On Context-Free Languages. JACM 13(4), 570–581 (1966)
Seidl, H., Schwentick, T., Muscholl, A.: Numerical Document Queries. In: Proc. of 22nd PODS, SanDiego (USA), pp. 155–166. ACM, New York (2003)
Yagi, I., Takata, Y., Seki, H.: A Static Analysis Using Tree Automata for XML Access Control. In: Peled, D.A., Tsay, Y.-K. (eds.) ATVA 2005. LNCS, vol. 3707, pp. 234–247. Springer, Heidelberg (2005)
Verma, K.N.: Two-Way Equational Tree Automata for AC-Like Theories: Decidability and Closure Properties. In: Nieuwenhuis, R. (ed.) RTA 2003. LNCS, vol. 2706, pp. 180–197. Springer, Heidelberg (2003)
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Hendrix, J., Ohsaki, H., Viswanathan, M. (2006). Propositional Tree Automata. In: Pfenning, F. (eds) Term Rewriting and Applications. RTA 2006. Lecture Notes in Computer Science, vol 4098. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11805618_5
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