Abstract
The Semantic Web lays its foundations on the study of graph and tree logics. One of the most expressive graph logics is the fully enriched μ-calculus, which is a modal logic equipped with least and greatest fixed-points, nominals, inverse programs and graded modalities. Although it is well-known that the fully enriched μ-calculus is undecidable, it was recently shown that this logic is decidable when its models are finite trees. In the present work, we study the fully-enriched μ-calculus for trees extended with Presburger constraints. These constraints generalize graded modalities by restricting the number of children nodes with respect to Presburger arithmetic expressions. We show that the logic is decidable in EXPTIME. This is achieved by the introduction of a satisfiability algorithm based on a Fischer-Ladner model construction that is able to handle binary encodings of Presburger constraints.
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References
Bárcenas, E., Genevès, P., Layaïda, N., Schmitt, A.: Query reasoning on trees with types, interleaving, and counting. In: Walsh, T. (ed.) IJCAI, pp. 718–723. IJCAI/AAAI (2011)
Bonatti, P.A., Lutz, C., Murano, A., Vardi, M.Y.: The complexity of enriched μ-calculi. In: Bugliesi, M., Preneel, B., Sassone, V., Wegener, I. (eds.) ICALP 2006. LNCS, vol. 4052, pp. 540–551. Springer, Heidelberg (2006)
Bonatti, P.A., Peron, A.: On the undecidability of logics with converse, nominals, recursion and counting. Artif. Intell. 158(1), 75–96 (2004)
Calvanese, D., Giacomo, G.D., Lenzerini, M., Vardi, M.Y.: Node selection query languages for trees. In: Fox, M., Poole, D. (eds.) AAAI. AAAI Press (2010)
Demri, S., Lugiez, D.: Complexity of modal logics with Presburger constraints. J. Applied Logic 8(3), 233–252 (2010)
Habermehl, P., Iosif, R., Vojnar, T.: Automata-based verification of programs with tree updates. Acta Inf. 47(1), 1–31 (2010)
Hosoya, H., Vouillon, J., Pierce, B.C.: Regular expression types for XML. ACM Trans. Program. Lang. Syst. 27(1), 46–90 (2005)
Manna, Z., Sipma, H.B., Zhang, T.: Verifying balanced trees. In: Artemov, S., Nerode, A. (eds.) LFCS 2007. LNCS, vol. 4514, pp. 363–378. Springer, Heidelberg (2007)
Seidl, H., Schwentick, T., Muscholl, A.: Numerical document queries. In: Neven, F., Beeri, C., Milo, T. (eds.) PODS, pp. 155–166. ACM (2003)
Seidl, H., Schwentick, T., Muscholl, A., Habermehl, P.: Counting in trees for free. In: Díaz, J., Karhumäki, J., Lepistö, A., Sannella, D. (eds.) ICALP 2004. LNCS, vol. 3142, pp. 1136–1149. Springer, Heidelberg (2004)
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Bárcenas, E., Lavalle, J. (2013). Expressive Reasoning on Tree Structures: Recursion, Inverse Programs, Presburger Constraints and Nominals. In: Castro, F., Gelbukh, A., González, M. (eds) Advances in Artificial Intelligence and Its Applications. MICAI 2013. Lecture Notes in Computer Science(), vol 8265. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-45114-0_7
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DOI: https://doi.org/10.1007/978-3-642-45114-0_7
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