Abstract
It cannot be decided whether a pushdown automaton accepts using constant pushdown height, with respect to the input length, or not. Furthermore, in the case of acceptance in constant height, the height cannot be bounded by any recursive function in the size of the description of the machine. In contrast, in the restricted case of pushdown automata over a one-letter input alphabet, i.e., unary pushdown automata, the above property becomes decidable. Moreover, if the height is bounded by a constant in the input length, then it is at most exponential with respect to the size of the description of the pushdown automaton. This bound cannot be reduced. Finally, if a unary pushdown automaton uses nonconstant height to accept, then the height should grow at least as the logarithm of the input length. This bound is optimal.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
In some papers pdas are presented in different forms. As pointed out in [2], it is possible to turn the definition of pdas into these equivalent forms, with a polynomial increase in size and by preserving the property of being constant height.
- 2.
We point out that for unambiguous pdas, the property is decidable [10].
- 3.
Notice that here H(n) is a function of the size of the pda and not of the input.
References
Alberts, M.: Space complexity of alternating Turing machines. In: Budach, L. (ed.) FCT 1985. LNCS, vol. 199, pp. 1–7. Springer, Heidelberg (1985). https://doi.org/10.1007/BFb0028785
Bednárová, Z., Geffert, V., Mereghetti, C., Palano, B.: Removing nondeterminism in constant height pushdown automata. Inform. Comput. 237, 257–267 (2014)
Bednárová, Z., Geffert, V., Reinhardt, K., Yakaryilmaz, A.: New results on the minimum amount of useful space. Internat. J. Found. Comput. Sci. 27(2), 259–282 (2016). https://doi.org/10.1142/S0129054116400098
Chomsky, N.: A note on phrase structure grammars. Inform. Control 2(4), 393–395 (1959). https://doi.org/10.1016/S0019-9958(59)80017-6
Geffert, V., Mereghetti, C., Palano, B.: More concise representation of regular languages by automata and regular expressions. Inform. Comput. 208(4), 385–394 (2010). https://doi.org/10.1016/j.ic.2010.01.002
Ginsburg, S., Rice, H.G.: Two families of languages related to ALGOL. J. ACM 9(3), 350–371 (1962). https://doi.org/10.1145/321127.321132
Guillon, B., Pighizzini, G., Prigioniero, L.: Non-self-embedding grammars, constant-height pushdown automata, and limited automata. In: Câmpeanu, C. (ed.) CIAA 2018. LNCS, vol. 10977, pp. 186–197. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-94812-6_16
Hartmanis, J.: Context-free languages and Turing machine computations. In: Mathematical Aspects of Computer Science. Proceedings of Symposia in Applied Mathematics, vol. 19, pp. 42–51. American Mathematical Society (1967)
Hopcroft, J.E., Ullman, J.D.: Introduction to Automata Theory, Languages and Computation. Addison-Wesley, Boston (1979)
Malcher, A., Meckel, K., Mereghetti, C., Palano, B.: Descriptional complexity of pushdown store languages. J. Autom. Lang. Comb. 17(2–4), 225–244 (2012)
Mereghetti, C., Pighizzini, G.: Optimal simulations between unary automata. SIAM J. Comput. 30(6), 1976–1992 (2001)
Meyer, A.R., Fischer, M.J.: Economy of description by automata, grammars, and formal systems. In: Proceedings of 12th Annual Symposium on Switching and Automata Theory, pp. 188–191. IEEE Computer Society (1971)
Pighizzini, G., Shallit, J., Wang, M.: Unary context-free grammars and pushdown automata, descriptional complexity and auxiliary space lower bounds. J. Comput. Syst. Sci. 65(2), 393–414 (2002). https://doi.org/10.1006/jcss.2002.1855
Rado, T.: On non-computable functions. Bell Syst. Tech. J. 41(3), 877–884 (1962). https://doi.org/10.1002/j.1538-7305.1962.tb00480.x
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2019 IFIP International Federation for Information Processing
About this paper
Cite this paper
Pighizzini, G., Prigioniero, L. (2019). Pushdown Automata and Constant Height: Decidability and Bounds. In: Hospodár, M., Jirásková, G., Konstantinidis, S. (eds) Descriptional Complexity of Formal Systems. DCFS 2019. Lecture Notes in Computer Science(), vol 11612. Springer, Cham. https://doi.org/10.1007/978-3-030-23247-4_20
Download citation
DOI: https://doi.org/10.1007/978-3-030-23247-4_20
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-23246-7
Online ISBN: 978-3-030-23247-4
eBook Packages: Computer ScienceComputer Science (R0)