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The complexity of intersecting finite automata having few final states

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Abstract

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.

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References

  • E. Allender & M. Ogihara (1996). Relationships Among PL, #L, and the Determinant. In RAIRO - Theoretical Information and Application, 267–278.

  • 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.

  • 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.

  • 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.

  • 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.

  • David A. Mix Barrington, Neil Immerman & Howard Straubing (1990). On Uniformity within NC1. J. Comput. Syst. Sci. 41(3), 274–306.

  • David A. Mix Barrington & Denis Thérien (1988). Finite monoids and the fine structure of NC1. J. ACM 35(4), 941–952

  • M. Beaudry (1988a). Membership testing in commutative transformation semigroups. Information and Computation 79(1), 84–93. ISSN 0890-5401.

  • M. Beaudry (1988b). Membership testing in transformation monoids. Ph.D. thesis, McGill University.

  • M. Beaudry, P. McKenzie & D. Thérien (1992). The Membership Problem in Aperiodic Transformation Monoids. J. ACM 39(3), 599–616.

  • Christoph Behle & Klaus-Jörn Lange (2006). FO[<]-Uniformity. In IEEE Conference on Computational Complexity, 183–189.

  • 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.

  • Allan Borodin (1977). On Relating Time and Space to Size and Depth. SIAM J. Comput. 6(4), 733–744.

  • 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.

  • K. Conrad (2013). Characters of Finite Abelian Groups. Lecture Notes. Available at http://www.math.uconn.edu/~kconrad/blurbs/grouptheory/charthy.pdf.

  • S. A. Cook & P. McKenzie (1987). Problems Complete for Deterministic Logarithmic Space. J. Algorithms 8(3), 385–394.

  • 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.

  • Z. Galil (1976). Hierarchies of complete problems. Acta Informatica 6, 77–88. ISSN 0001-5903.

  • M.R. Garey & D.S. Johnson (1979). Computers and intractability: a guide to the theory of NP-completeness. W. H. Freeman and Company.

  • L. M. Goldschlager (1977). The Monotone and Planar Circuit Value Problems Are Log Space Complete for P. SIGACT News 9(2), 25–29.

  • 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.

  • M. Holzer & M. Kutrib (2011). Descriptional and computational complexity of finite automata – A survey. Information and Computation 209(3), 456–470.

  • 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.

  • 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.

  • D.E. Knuth (1981). The Art of Computer Programming: Seminumerical Algorithms, volume 2. Addidon-Wesley, 2nd edition. ISBN 9780201038224.

  • Michal Koucký, Pavel Pudlák & Denis Thérien (2005). Bounded-depth circuits: separating wires from gates. In STOC, 257–265.

  • D. Kozen (1977). Lower bounds for natural proof systems. In Proc. 18th Annual Symposium on Foundations of Computer Science, 254–266. ISSN 0272-5428.

  • 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.

  • E. M. Luks (1986). Parallel Algorithms for Permutation Groups and Graph Isomorphism. In Proc. 27th Annual Symposium on Foundations of Computer Science, 292–302.

  • E. M. Luks (1990). Lectures on polynomial-time computation in groups. Technical report. College of Computer Science, Northeastern University. Available at http://ix.cs.uoregon.edu/~luks/northeasterncourse.pdf.

  • E. M. Luks & P. McKenzie (1988). Parallel Algorithms for Solvable Permutation Groups. Journal of Computer and System Sciences 37(1), 39–62.

  • B. Luong (2009). Fourier Analysis on Finite Abelian Groups. Birkhäuser. ISBN 9780817649159.

  • P. McKenzie & S. A. Cook (1987). The parallel complexity of Abelian permutation group problems. SIAM Journal on Computing 16, 880–909. ISSN 0097-5397.

  • 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.

  • Jean-Eric Pin (1986). Varieties of Formal Languages. Plenum Press.

  • 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.

  • W. J. Savitch (1970). Relationships between nondeterministic and deterministic tape complexities. Journal of Computer and System Sciences 4(2), 177–192. ISSN 0022-0000.

  • Marcel Paul Schützenberger (1965). On Finite Monoids Having Only Trivial Subgroups. Information and Control 8(2), 190–194.

  • Howard Straubing (1994). Finite Automata, Formal Logic and Circuit Complexity. Birkhauser.

  • Pascal Tesson & Denis Thérien (2005). Complete Classifications for the Communication Complexity of Regular Languages. Theory Comput. Syst. 38(2), 135–159.

  • Pascal Tesson & Denis Thérien (2007). Logic Meets Algebra: the Case of Regular Languages. Logical Methods in Computer Science 3(1).

  • Denis Thérien & Thomas Wilke (1998). Over Words, Two Variables Are as Powerful as One Quantifier Alternation. In Symposium on Theory of Computing, 234–240.

  • H. Vollmer (1999). Introduction to Circuit Complexity – A Uniform Approach. Texts in Theoretical Computer Science. Springer Verlag.

  • 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.

  • H.J. Zassenhaus (1999). The Theory of Groups. Dover Books on Mathematics. Dover Publications. ISBN 9780486409221.

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Authors and Affiliations

  1. Département d’informatique et de recherche opérationnelle, Université de Montréal, Montreal, QC, Canada

    Michael Blondin & Pierre McKenzie

  2. Laboratoire Spécification et Vérification, ENS Cachan, Cachan, France

    Michael Blondin & Pierre McKenzie

  3. Wilhelm-Schickard-Institut für Informatik, Universität Tübingen, Tübingen, Germany

    Andreas Krebs

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  1. Michael Blondin
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  2. Andreas Krebs
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  3. Pierre McKenzie
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Correspondence to Michael Blondin.

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Blondin, M., Krebs, A. & McKenzie, P. The complexity of intersecting finite automata having few final states. comput. complex. 25, 775–814 (2016). https://doi.org/10.1007/s00037-014-0089-9

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