Abstract
Model theoretic syntax is concerned with studying the descriptive complexity of grammar formalisms for natural languages by defining their derivation trees in suitable logical formalisms. The central tool for model theoretic syntax has been monadic second-order logic (MSO). Much of the recent research in this area has been concerned with finding more expressive logics to capture the derivation trees of grammar formalisms that generate non-context-free languages. The motivation behind this search for more expressive logics is to describe formally certain mildly context-sensitive phenomena of natural languages. Several extensions to MSO have been proposed, most of which no longer define the derivation trees of grammar formalisms directly, while others introduce logically odd restrictions. We therefore propose to consider first-order transitive closure logic. In this logic, derivation trees can be defined in a direct way. Our main result is that transitive closure logic, even deterministic transitive closure logic, is more expressive in defining classes of tree languages than MSO. (Deterministic) transitive closure logics are capable of defining non-regular tree languages that are of interest to linguistics.
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References
Bargury Y., Makowsky J.A. (1992) The expressive power of transitive closue and 2-way multihead automata. In: Börger E., Jäger G., Büning H.K., Richter M.M. (eds) Computer science logic, CSL ’91, LNCS Vol. 626. Springer, Berlin, pp 1–14
Bresnan J., Kaplan R., Peters S., Zaenen A. (1987) Cross-serial dependencies in Dutch. In: Savitch W.J., Bach E., Marsh W., Safran-Naveh G. (eds) The formal complexity of natural language. Reidel, Dordrecht, pp 286–319
Cornelle T., Rogers J. (2000) Model theoretic syntax. In: Cheng L., Sybesma R. (eds) The first GLOT international state-of-the article book, SGG (Vol. 48). Mouton de Gruyter, Berlin
Courcelle, B. (1990). Graph rewriting: An algebraic and logic approach. In Handbook of theoretical computer science (Vol. B). Amsterdam: Elsevier.
Culy C. (1987) The complexity of the vocabulary of Bambara. In: Savitch W., Bach E., Marsch W., Safran-Naveh G. (eds) The formal complexity of natural language. Reidel, Dordrecht, pp 345–351
Engelfriet J., Hoogeboom H.J. (2006) Nested pebbles and transitive closure. In: Durand B., Thomas W. (eds) STACS 2006, LNCS (Vol. 3884). Springer, Berlin, pp 477–488
Etessami K., Immerman N. (1995) Reachability and the power of local ordering. Theoretical Computer Science 148(2): 261–279
Fagin R. (1975) Monadic generalized spectra. Zeitschrift für Mathematische Logik und Grundlagen der Mathematik 21: 89–96
Huybregts R. (1984) The weak inadequacy of context–free phrase structure grammars. In: Haan G.J., Trommelen M., Zonneveld W. (eds) Van Periferie naar Kern. Foris, Dordrecht, pp 81–99
Immerman N. (1999) Descriptive complexity. Springer, Berlin
Kepser, S. (2006). Properties of binary transitive closure logic over trees. In P. Monachesi, G. Penn, G. Satta & S. Wintner (Eds.), Formal grammar (Vol. 2006, pp. 77–89).
Kleene, S. C. (1956). Representation of events in nerve nets and finite automata. In Automata studies (pp. 3–41). Princeton, N. J.: Princeton University Press.
Kolb H.-P., Michaelis J., Mönnich U., Morawietz F. (2003) An operational and denotational approach to non-context-freeness. Theoretical Computer Science 293(2): 261–289
Langholm T. (2001) A descriptive characterisation of indexed grammars. Grammars 4(3): 205–262
Lautemann C., Schwentick T., Thérien D. (1995) Logics for context-free languages. In: Pacholski L., Tiuryn J. (eds) Computer science logic CSL (Vol 1994). Springer, Berlin
Lohrey M. (2001) On the parallel complexity of tree automata. In: Middeldorp A. (eds) Rewriting techniques and applications, RTA 2001, LNCS (Vol. 2051). Springer, Berlin, pp 201–215
Lynch N. (1977) Log space recognition and translation of parenthesis languages. Journal of the ACM 24: 583–590
Mehlhorn K. (1976) Bracket languages are recognizable in logarithmic space. Information Processing Letter 5: 168–170
Moschovakis Y. (1974) Elementary induction on abstract structures. North-Holland Publishing Company, Amsterdam
Potthoff, A. (1994). Logische Klassifizierung regulärer Baumsprachen. PhD thesis, Christian-Albrechts-Universität zu Kiel.
Pullum G., Gazdar G. (1982) Natural languages and context-free languages. Linguistics and Philosophy 4: 471–504
Rogers J. (1998) A descriptive approach to language theoretic complexity. CSLI Publications, Stanford, CA
Rogers J. (2003) Syntactic structures as multi-dimensional trees. Research on Language and Computation 1(3–4): 265–305
Rosenberg A. (1966) On multi-head finite automata. IBM Journal of Research and Development 10: 388–394
Shieber S. (1985) Evidence against the context-freeness of natural language. Linguistics and Philosophy 8: 333–343
ten Cate B., Segoufin L. (2008) XPath, transitive closure logic, and nested tree walking automata. In: Lenzerini M., Lembo D. (eds) Proceedings of ACM SIGMOD/PODS Conference. ACM, New York
Thatcher J.W. (1967) Characterizing derivation trees of context-free grammars through a generalization of finite automata theory. Journal of Computer and System Sciences 1: 317–322
Tiede H.-J. (2003) Model theoretic syntax and transitive closure logic. In: Libkin L., Penn G. (eds) LICS workshop on logic and computational linguistics. Canada, Ottawa
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Tiede, HJ., Kepser, S. Monadic Second-Order Logic and Transitive Closure Logics Over Trees. Res on Lang and Comput 7, 41–54 (2009). https://doi.org/10.1007/s11168-009-9060-3
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DOI: https://doi.org/10.1007/s11168-009-9060-3