Abstract
The existence of a size gap between deterministic and nondeterministic two-way automata is one of the most famous open problems in automata theory. This problem is also related to the famous DLOG vs. NLOG question. An exponential gap between the number of states of two-way nondeterministic automata (2nfas) and their deterministic counterparts (2dfas) has been proved only for some restrictions of 2dfas up to now. It seems that the hardness of this problem lies in the fact that, when trying to prove lower bounds, we must consider every possible automaton, without imposing any particular structure or meaning to the states, while when designing a specific automaton we always assign an unambiguous interpretation to the states. In an attempt to capture the concept of meaning of states, a new model of two-way automata, namely reasonable automaton (ra), was introduced in [6]. In a ra, each state is associated with a logical formula expressing some properties of the input word, and transitions are designed to maintain consistency within this setting. In this paper we extend the study, started in [6], of the descriptional complexity of ras solving the liveness problem, showing several lower and upper bounds for different choices of allowed atomic predicates and connectors.
This work was partially supported by grants SNF 200021_146372/1 and VEGA 1/0979/12.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Berman, P.: A note on sweeping automata. In: de Bakker, J., van Leeuwen, J. (eds.) ICALP. LNCS, vol. 85, pp. 91–97. Springer, Heidelberg (1980)
Berman, P., Lingas, A.: On complexity of regular languages in terms of finite automata. Technical report, Institute of Computer Science, Polish Academy of Sciences, Warsaw (1977)
Chrobak, M.: Finite automata and unary languages. Theor. Comp. Sci. 47(2), 149–158 (1986)
Geffert, V.: Magic numbers in the state hierarchy of finite automata. Information and Computation 205(11), 1652–1670 (2007)
Geffert, V., Mereghetti, C., Pighizzini, G.: Converting two-way nondeterministic unary automata into simpler automata. Theor. Comp. Sci. 295, 189–203 (2003)
Hromkovič, J., Královič, R., Královič, R., Štefanec, R.: Determinism vs. Nondeterminism for Two-Way Automata: Representing the Meaning of States by Logical Formulæ. Int. J. Found. Comput. Sci. 24(7), 955–978 (2013)
Hromkovič, J., Schnitger, G.: Nondeterminism versus determinism for two-way finite automata: generalizations of Sipser’s separation. In: Baeten, J.C.M., Lenstra, J.K., Parrow, J., Woeginger, G.J. (eds.) ICALP 2003. LNCS, vol. 2719, pp. 439–451. Springer, Heidelberg (2003)
Kapoutsis, C.A.: Removing bidirectionality from nondeterministic finite automata. In: Jedrzejowicz, J., Szepietowski, A. (eds.) MFCS 2005. LNCS, vol. 3618, pp. 544–555. Springer, Heidelberg (2005)
Kapoutsis, C.A.: Size complexity of two-way finite automata. In: Diekert, V., Nowotka, D. (eds.) DLT 2009. LNCS, vol. 5583, pp. 47–66. Springer, Heidelberg (2009)
Kapoutsis, C.A.: Nondeterminism is essential in small two-way finite automata with few reversals. Inf. Comput. 222, 208–227 (2013)
Kapoutsis, C.A., Královič, R., Mömke, T.: Size complexity of rotating and sweeping automata. Journal of Computer and System Sciences 78(2), 537–558 (2012)
Kolodin, A.N.: Two-way nondeterministic automata. Cybernetics and Systems Analysis 10(5), 778–785 (1972)
Lupanov, O.: A comparison of two types of finite automata. Problemy Kibernet 9, 321–326 (1993). (in Russian); German translation: Über den Vergleich zweier Typen endlicher Quellen. Probleme der Kybernetik 6, 329–335 (1966)
Micali, S.: Two-way deterministic finite automata are exponentially more succinct than sweeping automata. Inf. Process. Lett. 12, 103–105 (1981)
Moore, F.R.: On the bounds for state-set size in the proofs of equivalence between deterministic, nondeterministic, and two-way finite automata. IEEE Transactions on Computers C–20(10), 1211–1214 (1971)
Rabin, M.O., Scott, D.: Finite automata and their decision problems. IBM J. Res. Dev. 3(2), 114–125 (1959)
Sakoda, W.J., Sipser, M.: Nondeterminism and the size of two way finite automata. In: Proc. of the 10th Annual ACM Symposium on Theory of Computing (STOC 1978), pp. 275–286. ACM Press, New York (1978)
Sipser, M.: Lower bounds on the size of sweeping automata. Journal of Computer and System Sciences 21, 195–202 (1980)
Štefanec, R.: Semantically restricted two-way automata. PhD Thesis, Comenius University in Bratislava (2014)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer International Publishing Switzerland
About this paper
Cite this paper
Bianchi, M.P., Hromkovič, J., Kováč, I. (2015). On the Size of Two-Way Reasonable Automata for the Liveness Problem. In: Potapov, I. (eds) Developments in Language Theory. DLT 2015. Lecture Notes in Computer Science(), vol 9168. Springer, Cham. https://doi.org/10.1007/978-3-319-21500-6_9
Download citation
DOI: https://doi.org/10.1007/978-3-319-21500-6_9
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-21499-3
Online ISBN: 978-3-319-21500-6
eBook Packages: Computer ScienceComputer Science (R0)