computational complexity

, Volume 25, Issue 4, pp 775–814 | Cite as

The complexity of intersecting finite automata having few final states

  • Michael BlondinEmail author
  • Andreas Krebs
  • Pierre McKenzie


The problem of determining whether several finite automata accept a word in common is closely related to the well-studied membership problem in transformation monoids. We raise the issue of limiting the number of final states in the automata intersection problem. For automata with two final states, we show the problem to be \({\oplus}\)L-complete or NP-complete according to whether a nontrivial monoid other than a direct product of cyclic groups of order 2 is allowed in the automata. We further consider idempotent commutative automata and (Abelian, mainly) group automata with one, two, or three final states over a singleton or larger alphabet, elucidating (under the usual hypotheses on complexity classes) the complexity of the intersection nonemptiness and related problems in each case.


Finite automata intersection problem monoids logspace NP-complete point-spread problem 

Subject classification

68Q15 68Q25 68Q17 03D15 68Q70 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. E. Allender & M. Ogihara (1996). Relationships Among PL, #L, and the Determinant. In RAIRO - Theoretical Information and Application, 267–278.Google Scholar
  2. V. Arvind & T. C. Vijayaraghavan (2010). Classifying Problems on Linear Congruences and Abelian Permutation Groups Using Logspace Counting Classes. Computational Complexlity 19, 57–98. ISSN 1016-3328.Google Scholar
  3. L. Babai, E. M. Luks & A. Seress (1987). Permutation groups in NC. In Proc. 19th annual ACM symposium on Theory of computing, 409–420. ISBN 0-89791-221-7.Google Scholar
  4. S. Bala (2002). Intersection of Regular Languages and Star Hierarchy. In Proc. 29th International Colloquium on Automata, Languages and Programming, 159–169. ISBN 3-540-43864-5.Google Scholar
  5. David A. Mix Barrington (1989). Bounded-Width Polynomial-Size Branching Programs Recognize Exactly Those Languages in NC1. J. Comput. Syst. Sci. 38(1), 150–164.Google Scholar
  6. David A. Mix Barrington, Neil Immerman & Howard Straubing (1990). On Uniformity within NC1. J. Comput. Syst. Sci. 41(3), 274–306.Google Scholar
  7. David A. Mix Barrington & Denis Thérien (1988). Finite monoids and the fine structure of NC1. J. ACM 35(4), 941–952Google Scholar
  8. M. Beaudry (1988a). Membership testing in commutative transformation semigroups. Information and Computation 79(1), 84–93. ISSN 0890-5401.Google Scholar
  9. M. Beaudry (1988b). Membership testing in transformation monoids. Ph.D. thesis, McGill University.Google Scholar
  10. M. Beaudry, P. McKenzie & D. Thérien (1992). The Membership Problem in Aperiodic Transformation Monoids. J. ACM 39(3), 599–616.Google Scholar
  11. Christoph Behle & Klaus-Jörn Lange (2006). FO[<]-Uniformity. In IEEE Conference on Computational Complexity, 183–189.Google Scholar
  12. M. Blondin & P. McKenzie (2012). The complexity of intersecting finite automata having few final states. In Proc. 7th International Computer Science Symposium in Russia, 31–42. Springer Berlin Heidelberg.Google Scholar
  13. Allan Borodin (1977). On Relating Time and Space to Size and Depth. SIAM J. Comput. 6(4), 733–744.Google Scholar
  14. G. Buntrock, C. Damm, U. Hertrampf & C. Meinel (1992). Structure and importance of logspace-MOD class. Theory of Computing Systems 25, 223–237. ISSN 1432-4350.Google Scholar
  15. K. Conrad (2013). Characters of Finite Abelian Groups. Lecture Notes. Available at
  16. S. A. Cook & P. McKenzie (1987). Problems Complete for Deterministic Logarithmic Space. J. Algorithms 8(3), 385–394.Google Scholar
  17. M. L. Furst, J. E. Hopcroft & E. M. Luks (1980). Polynomial-Time Algorithms for Permutation Groups. In Proc. 21st Annual Symposium on Foundations of Computer Science, 36–41.Google Scholar
  18. Z. Galil (1976). Hierarchies of complete problems. Acta Informatica 6, 77–88. ISSN 0001-5903.Google Scholar
  19. M.R. Garey & D.S. Johnson (1979). Computers and intractability: a guide to the theory of NP-completeness. W. H. Freeman and Company.Google Scholar
  20. L. M. Goldschlager (1977). The Monotone and Planar Circuit Value Problems Are Log Space Complete for P. SIGACT News 9(2), 25–29.Google Scholar
  21. U. Hertrampf, S. Reith & H. Vollmer (2000). A note on closure properties of logspace MOD classes. Information Processing Letters 75, 91–93. ISSN 0020-0190.Google Scholar
  22. M. Holzer & M. Kutrib (2011). Descriptional and computational complexity of finite automata – A survey. Information and Computation 209(3), 456–470.Google Scholar
  23. N. D. Jones, Y. E. Lien & W. T. Laaser (1976). New problems complete for nondeterministic log space. Theory of Computing Systems 10, 1–17. ISSN 1432-4350.Google Scholar
  24. G. Karakostas, R. J. Lipton & A. Viglas (2003). On the complexity of intersecting finite state automata and NL versus NP. Theoretical Computer Science 302(1-3), 257–274. ISSN 0304-3975.Google Scholar
  25. D.E. Knuth (1981). The Art of Computer Programming: Seminumerical Algorithms, volume 2. Addidon-Wesley, 2nd edition. ISBN 9780201038224.Google Scholar
  26. Michal Koucký, Pavel Pudlák & Denis Thérien (2005). Bounded-depth circuits: separating wires from gates. In STOC, 257–265.Google Scholar
  27. D. Kozen (1977). Lower bounds for natural proof systems. In Proc. 18th Annual Symposium on Foundations of Computer Science, 254–266. ISSN 0272-5428.Google Scholar
  28. K.-J. Lange & P. Rossmanith (1992). The emptiness problem for intersections of regular languages. In Mathematical Foundations of Computer Science, volume 629 of Lecture Notes in Computer Science, 346–354.Google Scholar
  29. E. M. Luks (1986). Parallel Algorithms for Permutation Groups and Graph Isomorphism. In Proc. 27th Annual Symposium on Foundations of Computer Science, 292–302.Google Scholar
  30. E. M. Luks (1990). Lectures on polynomial-time computation in groups. Technical report. College of Computer Science, Northeastern University. Available at
  31. E. M. Luks & P. McKenzie (1988). Parallel Algorithms for Solvable Permutation Groups. Journal of Computer and System Sciences 37(1), 39–62.Google Scholar
  32. B. Luong (2009). Fourier Analysis on Finite Abelian Groups. Birkhäuser. ISBN 9780817649159.Google Scholar
  33. P. McKenzie & S. A. Cook (1987). The parallel complexity of Abelian permutation group problems. SIAM Journal on Computing 16, 880–909. ISSN 0097-5397.Google Scholar
  34. K. Mulmuley (1987). A fast parallel algorithm to compute the rank of a matrix over an arbitrary field. Combinatorica 7, 101–104. ISSN 0209-9683.Google Scholar
  35. Jean-Eric Pin (1986). Varieties of Formal Languages. Plenum Press.Google Scholar
  36. O. Reingold (2005). Undirected ST-connectivity in log-space. In Proc. 37th annual ACM symposium on Theory of computing, 376–385. ISBN 1-58113-960-8.Google Scholar
  37. W. J. Savitch (1970). Relationships between nondeterministic and deterministic tape complexities. Journal of Computer and System Sciences 4(2), 177–192. ISSN 0022-0000.Google Scholar
  38. Marcel Paul Schützenberger (1965). On Finite Monoids Having Only Trivial Subgroups. Information and Control 8(2), 190–194.Google Scholar
  39. Howard Straubing (1994). Finite Automata, Formal Logic and Circuit Complexity. Birkhauser.Google Scholar
  40. Pascal Tesson & Denis Thérien (2005). Complete Classifications for the Communication Complexity of Regular Languages. Theory Comput. Syst. 38(2), 135–159.Google Scholar
  41. Pascal Tesson & Denis Thérien (2007). Logic Meets Algebra: the Case of Regular Languages. Logical Methods in Computer Science 3(1).Google Scholar
  42. Denis Thérien & Thomas Wilke (1998). Over Words, Two Variables Are as Powerful as One Quantifier Alternation. In Symposium on Theory of Computing, 234–240.Google Scholar
  43. H. Vollmer (1999). Introduction to Circuit Complexity – A Uniform Approach. Texts in Theoretical Computer Science. Springer Verlag.Google Scholar
  44. H. T. Wareham (2001). The Parameterized Complexity of Intersection and Composition Operations on Sets of Finite-State Automata. In Implementation and Application of Automata, volume 2088, 302–310.Google Scholar
  45. H.J. Zassenhaus (1999). The Theory of Groups. Dover Books on Mathematics. Dover Publications. ISBN 9780486409221.Google Scholar

Copyright information

© Springer Basel 2014

Authors and Affiliations

  • Michael Blondin
    • 1
    • 2
    Email author
  • Andreas Krebs
    • 3
  • Pierre McKenzie
    • 1
    • 2
  1. 1.Département d’informatique et de recherche opérationnelleUniversité de MontréalMontrealCanada
  2. 2.Laboratoire Spécification et VérificationENS CachanCachanFrance
  3. 3.Wilhelm-Schickard-Institut für InformatikUniversität TübingenTübingenGermany

Personalised recommendations