Advertisement

Mathematical systems theory

, Volume 26, Issue 3, pp 237–269 | Cite as

State-complexity of finite-state devices, state compressibility and incompressibility

  • Jean-Camille Birget
Article

Abstract

We study how the number of states may change when we convert between different finite-state devices. The devices that we consider are finite automata that are one-way or two-way, deterministic or nondeterministic or alternating. We obtain several new simulation results (e.g., ann-state 2NFA can be simulated by a 1NFA with ≤ 8 n + 2 states, and by a 1AFA with ≤n 2 states), and state-incompressibility results (e.g., in order to simulate ann-state 2DFA, a 1NFA needs ≥√/2 n−2 states, and a 2AFA needs ≥c√n states for some constant c, in general).

Keywords

Boolean Function Input Port Start State Regular Language Finite Automaton 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [AU]
    A. Aho, J. Ullman,The Theory of Parsing, Translation and Compiling, Vol. 1, Prentice-Hall, Englewood Cliffs, NJ, 1972.Google Scholar
  2. [B1]
    J. C. Birget, Concatenation of inputs in a two-way automaton,Theoret. Comput. Sci.,63 (1989), 141–156.MathSciNetzbMATHCrossRefGoogle Scholar
  3. [B2]
    J. C. Birget, Positional simulation of two-way automata: proof of a conjecture of R. Kannan, and generalizations,J. Comput. System Sci. (special issue onSTOC 89),45 (1992), 154–179.MathSciNetzbMATHGoogle Scholar
  4. [B3]
    J. C. Birget, Two-way automata and length-preserving homomorphisms, Report # 109, Dept. of Computer Science, University of Nebraska (1990) (submitted).Google Scholar
  5. [B4]
    J. C. Birget, The minimum automaton of certain languages (in preparation).Google Scholar
  6. [B5]
    J. C. Birget, Strict local testability of the finite control of two-way automata and of regular picture description languages,Internat. J. Algebra Comput.,1 (1991), 161–175.MathSciNetzbMATHCrossRefGoogle Scholar
  7. [B6]
    J. C. Birget, Partial order on words, minimal elements of a regular language, and state-complexity,Theoret. Comput. Sci. (to appear).Google Scholar
  8. [B7]
    J. C. Birget, Intersection and union of regular languages and state-complexity,Inform. Process. Lett.,43 (1992), 185–190.MathSciNetzbMATHCrossRefGoogle Scholar
  9. [BH]
    M. Blum, C. Hewitt, Automata on a 2-dimensional tape,Proc. 8th IEEE Symp. on Switching and Automata Theory, 1965, pp. 155–160.Google Scholar
  10. [BG]
    R. Book, S. Greibach, Quasi-realtime languages,Math. System Theory,4 (1970), 97–111.MathSciNetzbMATHCrossRefGoogle Scholar
  11. [BL]
    J. Brzozowski, E. Leiss, On equations for regular languages, finite automata, and sequential networks,Theoret. Computer Sci.,10 (1980), 19–35.MathSciNetzbMATHCrossRefGoogle Scholar
  12. [BS]
    J. Brzozowski, C. Seger, Advances in asynchronous circuit theory, Part 1,Bull. EATCS,42 (1990), 198–249.zbMATHGoogle Scholar
  13. [CKS]
    A. Chandra, D. Kozen, L. Stockmeyer, Alternation,J. Assoc. Comput. Mach.,28 (1981), 114–133.MathSciNetzbMATHCrossRefGoogle Scholar
  14. [CS]
    A. Chandra, L. Stockmeyer, Alternation,Proc. 17th IEEE Symp. on Foundations of Computer Sci., 1976, pp. 98–108.Google Scholar
  15. [C]
    M. Chrobak, Finite automata and unary languages,Theoret. Comput. Sci.,47 (1986), 149–158.MathSciNetzbMATHCrossRefGoogle Scholar
  16. [E]
    S. Eilenberg,Automata, Languages, and Machines, Vol. A, Academic Press, New York, 1974.zbMATHGoogle Scholar
  17. [H]
    F. C. Hennie, One-tape off-line Turning machine computations,Inform. and Control,8 (1965), 553–578.MathSciNetCrossRefGoogle Scholar
  18. [HU]
    J. Hopcroft, J. Ullman,Introduction to Automata Theory, Languages and Computation, Addison-Wesely, Reading, MA, 1979.zbMATHGoogle Scholar
  19. [Ka]
    R. Kannan, Alternation and the power of non-determinism,Proc. 15th ACM Symp. on Theory of Computing, 1983, 344–346.Google Scholar
  20. [Ko]
    D. Kozen, On parallelism in Turing machines,Proc. 17th IEEE Symp. on Foundations of Computer Sci 1976, pp. 89–97.Google Scholar
  21. [LLS]
    R. Ladner, R. Lipton, L. Stockmeyer, Alternating pushdown automata,Proc. 19th IEEE Symp. on Foundations of Computer Sciences 1978, pp. 92–106, andSIAM J. Comput.,13(1) (1984), 135–155.Google Scholar
  22. [L1]
    E. Leiss, Succinct representation of regular languages by boolean automata,Theoret. Comput. Sci.,13(1981), 323–330.MathSciNetzbMATHCrossRefGoogle Scholar
  23. [L2]
    E. Leiss, Succinct representation of regular languages by boolean automata,II,Theoret. Comput. Sci.,38 (1985), 133–136.MathSciNetzbMATHCrossRefGoogle Scholar
  24. [L3]
    E. Leiss, A class of tractable unrestricted regular expressions,Theoret. Comput. Sci.,35 (1985), 313–327.MathSciNetzbMATHCrossRefGoogle Scholar
  25. [MP]
    R. McNaughton, S. Papert,Counter-free Automata, MIT Press, Cambridge, MA, 1971.zbMATHGoogle Scholar
  26. [MF]
    A. R. Meyer, M. J. Fischer, Economy of description by automata, grammars, and formal systems,Proc. 21st IEEE Symp. on Switching and Automata Theory, 1971, pp. 188–191.Google Scholar
  27. [RS]
    M. Rabin, D. Scott, Finite automata and their decision problems,IBM J. Res. Develop.,3 (1959), 114–125; also in E. F. Moore (ed.),Sequential Machines: Selected Papers, Addison-Wesley, Reading, MA, 1964.MathSciNetCrossRefGoogle Scholar
  28. [SS]
    W. Sakoda, M. Sipser, Non-determinism and the size of two-way automata,Proc. 10th ACM Symp. on Theory of Computing, 1978, pp. 275–286.Google Scholar
  29. [Sh]
    J. C. Shepherdson, The reduction of two-way automata to one-way automata,IBM J. Res. Develop.,3 (1959), 198–200; also in E. F. Moore (ed.),Sequential Machines: Selected Papers, Addison-Wesley, Reading, MA, 1964.MathSciNetCrossRefGoogle Scholar
  30. [Si1]
    M. Sipser, Lower bounds on the size of sweeping machines,Proc. 11th ACM Symp. on Theory of Computing, 1979, pp. 360–364; andJ. Comput. System Sci.,21 (1980), 195–202.Google Scholar
  31. [Si2]
    M. Sipser, Halting space-bounded computations,Theoret. Comput. Sci.,10 (1980), 335–338 alsoProc. 19th IEEE Symp. on Foundations of Computer Science, 1978.MathSciNetzbMATHCrossRefGoogle Scholar
  32. [V]
    M. Vardi, A note on the reduction of two-way automata to one-way automata,Inform. Process. Lett.,30 (1989), 261–264.MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag New York Inc 1993

Authors and Affiliations

  • Jean-Camille Birget
    • 1
  1. 1.Department of Computer Science and EngineeringUniversity of NebraskaLincolnUSA

Personalised recommendations