Advertisement

On the Size of Logical Automata

  • Martin Raszyk
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11376)

Abstract

The state complexity of simulating 1NFA by 2DFA is a long-standing open question, which is of particular interest also due to its connection to the DLOG vs. NLOG problem for Turing machines.

What makes proving lower bounds on the size of deterministic two-way automata particularly hard is the fact that one has to consider any automaton, and unlike the designer, one does not have any meaning of the states at hand. This motivates the notion of logical automata whose states are annotated by formulas representing the meaning of a state.

In the paper at hand, we first introduce the notion of logical automata and present a general approach to proving lower bounds on the number of states of logical automata. We then apply this approach to derive an exponential lower bound on the size of logical automata over formulas with a restricted set of atomic predicates. Finally, we complement the lower bound with an (also exponential) upper bound.

Notes

Acknowledgements

The author would like to thank Hans-Joachim Böckenhauer, Juraj Hromkovič, and the referees for their valuable comments and suggestions.

References

  1. 1.
    Berman, P., Lingas, A.: On complexity of regular languages in terms of finite automata. Technical report 304, Institute of Computer Science, Polish Academy of Sciences (1977)Google Scholar
  2. 2.
    Bianchi, M.P., Hromkovič, J., Kováč, I.: On the size of two-way reasonable automata for the liveness problem. In: Potapov, I. (ed.) DLT 2015. LNCS, vol. 9168, pp. 120–131. Springer, Cham (2015).  https://doi.org/10.1007/978-3-319-21500-6_9CrossRefGoogle Scholar
  3. 3.
    Hromkovič, J., Královič, R., Královič, R., Štefanec, R.: Determinism vs. nondeterminism for two-way automata. In: Yen, H.-C., Ibarra, O.H. (eds.) DLT 2012. LNCS, vol. 7410, pp. 24–39. Springer, Heidelberg (2012).  https://doi.org/10.1007/978-3-642-31653-1_4CrossRefGoogle Scholar
  4. 4.
    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).  https://doi.org/10.1007/3-540-45061-0_36CrossRefzbMATHGoogle Scholar
  5. 5.
    Sakoda, W.J., Sipser, M.: Nondeterminism and the size of two way finite automata. In: Proceedings of the Tenth Annual ACM Symposium on Theory of Computing. ACM (1978)Google Scholar
  6. 6.
    Sipser, M.: Lower bounds on the size of sweeping automata. In: Proceedings of the Eleventh Annual ACM Symposium on Theory of Computing. ACM (1979)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of Computer Science, ETH ZürichZürichSwitzerland

Personalised recommendations