Abstract
We introduce a new kind of tree automaton, a dependency tree automaton, that is suitable for deciding properties of classes of terms with binding. Two kinds of such automaton are defined, nondeterministic and alternating. We show that the nondeterministic automata have a decidable nonemptiness problem and leave as an open question whether this is true for the alternating version. The families of trees that both kinds recognise are closed under intersection and union. To illustrate the utility of the automata, we apply them to terms of simply typed lambda calculus and provide an automata-theoretic characterisation of solutions to the higher-order matching problem.
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Alur, R., Madhusudan, P.: Adding nested structure to words. In: H. Ibarra, O., Dang, Z. (eds.) DLT 2006. LNCS, vol. 4036, pp. 1–13. Springer, Heidelberg (2006)
Alur, R., Chaudhuri, S., Madhusudan, P.: Languages of nested trees. In: Ball, T., Jones, R.B. (eds.) CAV 2006. LNCS, vol. 4144, pp. 329–342. Springer, Heidelberg (2006)
Barendregt, H.: Lambda calculi with types. In: Abramsky, S., Gabbay, D., Maibaum, T. (eds.) Handbook of Logic in Computer Science, vol. 2, pp. 118–309. Oxford University Press, Oxford (1992)
Comon, H., Dauchet, M., Gilleron, R., Jacquemard, F., Lugiez, D., Tison, S., Tommasi, M.: Tree Automata Techniques and Applications. Draft Book (2002), http://l3ux02.univ-lille3.fr/tata/
Comon, H., Jurski, Y.: Higher-order matching and tree automata. In: Nielsen, M. (ed.) CSL 1997. LNCS, vol. 1414, pp. 157–176. Springer, Heidelberg (1998)
Joly, T.: The finitely generated types of the lambda calculus. In: Abramsky, S. (ed.) TLCA 2001. LNCS, vol. 2044, pp. 240–252. Springer, Heidelberg (2001)
Ong, C.-H.L.: On model-checking trees generated by higher-order recursion schemes. In: Procs. LICS 2006, pp. 81–90 (2006); (Longer version available from Ong’s web page, 55 pages (preprint, 2006)
Padovani, V.: Decidability of fourth-order matching. Mathematical Structures in Computer Science, vol. 10(3), pp. 361–372 (2001)
Segoufin, L.: Automata and logics for words and trees over an infinite alphabet. In: Ésik, Z. (ed.) CSL 2006. LNCS, vol. 4207, pp. 41–57. Springer, Heidelberg (2006)
Schubert, A.: Linear interpolation for the higher-order matching problem. In: Bidoit, M., Dauchet, M. (eds.) CAAP 1997, FASE 1997, and TAPSOFT 1997. LNCS, vol. 1214, pp. 441–452. Springer, Heidelberg (1997)
Statman, R.: Completeness, invariance and λ-definability. The Journal of Symbolic Logic 47, 17–26 (1982)
Stirling, C.: Higher-order matching and games. In: Ong, L. (ed.) CSL 2005. LNCS, vol. 3634, pp. 119–134. Springer, Heidelberg (2005)
Stirling, C.: A game-theoretic approach to deciding higher-order matching. In: Bugliesi, M., Preneel, B., Sassone, V., Wegener, I. (eds.) ICALP 2006. LNCS, vol. 4052, pp. 348–359. Springer, Heidelberg (2006)
Stirling, C.: Higher-order matching, games and automata. In: Procs. LICS 2007, pp. 326–335 (2007)
Støvring, K.: Higher-order beta matching with solutions in long beta-eta normal form. Nordic Journal of Computing 13, 117–126 (2006)
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Stirling, C. (2009). Dependency Tree Automata. In: de Alfaro, L. (eds) Foundations of Software Science and Computational Structures. FoSSaCS 2009. Lecture Notes in Computer Science, vol 5504. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-00596-1_8
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DOI: https://doi.org/10.1007/978-3-642-00596-1_8
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