Abstract
Building upon the known generalized-quantifier-based first-order characterization of LOGCFL, we lay the groundwork for a deeper investigation. Specifically, we examine subclasses of LOGCFL arising from varying the arity and nesting of groupoidal quantifiers. Our work extends the elaborate theory relating monoidal quantifiers to NC1 and its subclasses. In the absence of the BIT predicate, we resolve the main issues: we show in particular that no single outermost unary groupoidal quantifier with FO can capture all the context-free languages, and we obtain the surprising result that a variant of Greibach’s “hardest context-free language” is LOGCFL-complete under quantifier-free BIT-free interpretations. We then prove that FO with unary groupoidal quantifiers is strictly more expressive with the BIT predicate than without. Considering a particular groupoidal quantifier, we prove that first-order logic with majority of pairs is strictly more expressive than first-order with majority of individuals. As a technical tool of independent interest, we define the notion of an aperiodic nondeterministic finite automaton and prove that FO translations are precisely the mappings computed by single-valued aperiodic nondeterministic finite transducers.
Research performed while on leave at the Universität Tübingen. Supported by the (German) DFG, the (Canadian) NSERC and the (Québec) FCAR.
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References
J.-M. Autebert, J. Berstel, and L. Boasson. Context-free languages and pushdown automata. In R. Rozenberg and A. Salomaa, editors, Handbook of Formal Languages, volume I, chapter 3. Springer Verlag, Berlin Heidelberg, 1997.
D. A. Mix Barrington, K. Compton, H. Straubing, and D. Thérien. Regular languages in NC1. Journal of Computer and System Sciences, 44:478–499, 1992.
D. A. Mix Barrington, N. Immerman, and H. Straubing. On uniformity within NC1. Journal of Computer and System Sciences, 41:274–306, 1990.
F. Bédard, F. Lemieux, and P. McKenzie. Extensions to Barrington’s M-program model. Theoretical Computer Science, 107:31–61, 1993.
H.-J. Burtschick and H. Vollmer. Lindström quantifiers and leaf language definability. International Journal of Foundations of Computer Science, 9:277–294, 1998.
S. A. Cook. Characterizations of pushdown machines in terms of time-bounded computers. Journal of the ACM, 18:4–18, 1971.
S. Greibach. The hardest context-free language. SIAM Journal on Computing, 2:304–310, 1973.
N. Immerman. Descriptive Complexity. Springer Verlag, New York, 1998.
S. Lindell. manuscript, 1994. e-mail communication by Kenneth W. Regan.
P. Lindström. First order predicate logic with generalized quantifiers. Theoria, 32:186–195, 1966.
A. Mateescu and A. Salomaa. Aspects of classical language theory. In R. Rozenberg and A. Salomaa, editors, Handbook of Formal Languages, volume I, chapter 4. Springer Verlag, Berlin Heidelberg, 1997.
R. McNaughton and S. Papert. Counter-Free Automata. MIT Press, 1971.
P. Péladeau P. McKenzie and D. Thérien. NC1: The automata-theoretic viewpoint. Computational Complexity, 1:330–359, 1991.
M. P. Schützenberger. On finite monoids having only trivial subgroups. Information & Control, 8:190–194, 1965.
R. Smolensky. Algebraic methods in the theory of lower bounds for Boolean circuit complexity. In Proceedings 19th Symposium on Theory of Computing, pages 77–82. ACM Press, 1987.
J. Stern. Complexity of some problems from the theory of automata. Information & Computation, 66:163–176, 1985.
H. Straubing. Finite Automata, Formal Logic, and Circuit Complexity. Birkhäuser, Boston, 1994.
I. H. Sudborough. On the tape complexity of deterministic context-free languages. Journal of the ACM, 25:405–414, 1978.
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Lautemann, C., McKenzie, P., Schwentick, T., Vollmer, H. (1999). The Descriptive Complexity Approach to LOGCFL. In: Meinel, C., Tison, S. (eds) STACS 99. STACS 1999. Lecture Notes in Computer Science, vol 1563. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-49116-3_42
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DOI: https://doi.org/10.1007/3-540-49116-3_42
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