On the Complexity of Membership and Counting in Height-Deterministic Pushdown Automata
While visibly pushdown languages properly generalise regular languages and are properly contained in deterministic context-free languages, the complexity of their membership problem is equivalent to that of regular languages. However, the corresponding counting problem could be harder than counting paths in a non-deterministic finite automaton: it is only known to be in LogDCFL.
We investigate the membership and counting problems for generalisations of visibly pushdown automata, defined using the notion of height-determinism. We show that, when the stack-height of a given PDA can be computed using a finite transducer, both problems have the same complexity as for visibly pushdown languages. We also show that when allowing pushdown transducers instead of finite-state ones, both problems become LogDCFL-complete; this uses the fact that pushdown transducers are sufficient to compute the stack heights of all real-time height-deterministic pushdown automata, and yields a candidate arithmetization of LogDCFL that is no harder than LogDCFL(our main result).
KeywordsRegular Language Height Function Language Class Input Word Counting Problem
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- 4.Holzer, M., Lange, K.J.: On the complexities of linear LL(1) and LR(1) grammars. In: Ésik, Z. (ed.) FCT 1993. LNCS, vol. 710, pp. 299–308. Springer, Heidelberg (1993)Google Scholar
- 6.Lange, K.J.: Complexity and structure in formal language theory. In: 8th CoCo, pp. 224–238. IEEE Computer Society, Los Alamitos (1993)Google Scholar
- 7.Mehlhorn, K.: Pebbling mountain ranges and its application to DCFL recognition. In: de Bakker, J.W., van Leeuwen, J. (eds.) ICALP 1980. LNCS, vol. 85, pp. 422–432. Springer, Heidelberg (1980)Google Scholar
- 8.Braunmuhl, B.V., Verbeek, R.: Input-driven languages are recognized in log n space. In: Karpinski, M. (ed.) FCT 1983. LNCS, vol. 158, pp. 40–51. Springer, Heidelberg (1983)Google Scholar
- 10.Alur, R., Madhusudan, P.: Visibly pushdown languages. In: 36th STOC, pp. 202–211. ACM, New York (2004)Google Scholar
- 13.Limaye, N., Mahajan, M., Rao, B.V.R.: Arithmetizing classes arround NC1 and L. Technical Report ECCC TR07- (2007) submitted to TCS (spl.issue for STACS 2007) (2007)Google Scholar
- 17.Blass, A., Gurevich, Y.: A note on nested words. Technical Report MSR-TR-2006-139, Microsoft Research (October 2006)Google Scholar
- 18.Buss, S.: The Boolean formula value problem is in ALOGTIME. In: 19th STOC, pp. 123–131. ACM, New York (1987)Google Scholar
- 19.Vollmer, H.: Introduction to Circuit Complexity: A Uniform Approach. Springer, Heidelberg (1999)Google Scholar