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Existential and Universal Width of Alternating Finite Automata

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Descriptional Complexity of Formal Systems (DCFS 2023)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 13918))

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Abstract

The existential width of an alternating finite automaton (AFA) A on a string w is, roughly speaking, the number of nondeterministic choices that A uses in an accepting computation on w that uses least nondeterminism. The universal width of A on string w is the least number of parallel branches an accepting computation of A on w uses. The existential or universal width of A is said to be finite if it is bounded for all accepted strings. We show that finiteness of existential and universal width of an AFA is decidable. Also we give hardness results and give an algorithm to decide whether the existential or universal width of an AFA is bounded by a given integer.

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Notes

  1. 1.

    With binary representation of integers some results, for example, the upper bound of Proposition 15, would change.

  2. 2.

    The order of children of a node is not important and we can assume that elements of \(\delta (q, b)\) are ordered by an arbitrary linear order.

  3. 3.

    Alternatively, it would be possible to indicate a failed computation by a leaf labeled by (q, b). The failure symbol is used for clarity.

  4. 4.

    Fail-leaves are included for the worst-case variant of the measure considered in [16]. For the best-case variant considered here it does not make a difference whether we count also fail-leaves.

References

  1. Chandra, A.K., Kozen, D.C., Stockmeyer, L.J.: Alternation. J. ACM 28(1), 114–133 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  2. Fellah, A., Jürgensen, H., Yu, S.: Constructions for alternating finite automata. Int. J. Comput. Math. 35, 117–132 (1990)

    Article  MATH  Google Scholar 

  3. Geffert, V.: An alternating hierarchy for finite automata. Theor. Comput. Sci. 445, 1–24 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  4. Goldstine, J., Kappes, M., Kintala, C.M.R., Leung, H., Malcher, A., Wotschke, D.: Descriptional complexity of machines with limited resources. J. Univ. Comput. Sci. 8(2), 193–234 (2002)

    MathSciNet  MATH  Google Scholar 

  5. Goldstine, J., Kintala, C.M.R., Wotschke, D.: On measuring nondeterminism in regular languages. Inf. Comput. 86(2), 179–194 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  6. Han, Y.S., Ko, S.K., Salomaa, K.: Deciding path size of nondeterministic (and input-driven) pushdown automata. Theor. Comput. Sci. 939, 170–181 (2023)

    Article  MathSciNet  MATH  Google Scholar 

  7. Han, Y.S., Salomaa, A., Salomaa, K.: Ambiguity, nondeterminism and state complexity of finite automata. Acta Cybernetica 23, 141–157 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  8. Hashiguchi, K.: Limitedness theorem on finite automata with distance functions. J. Comput. Syst. Sci. 24, 223–244 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  9. Holzer, M., Kutrib, M.: Descriptional and computational complexity of finite automata: a survey. Inf. Comput. 209(3), 456–470 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  10. Hromkovič, J.: On the power of alternation in automata theory. J. Comput. Syst. Sci. 31, 28–39 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  11. Hromkovič, J., Schnitger, G.: Ambiguity and communication. Theory Comput. Syst. 48, 517–534 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  12. Hromkovič, J., Seibert, S., Karhumäki, J., Klauck, H., Schnitger, G.: Communication complexity method for measuring nondeterminism in finite automata. Inf. Comput. 172, 202–217 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  13. Kapoutsis, C., Zakzok, M.: Alternation in two-way finite automata. Theor. Comput. Sci. 870, 75–102 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  14. Keeler, C., Salomaa, K.: Alternating finite automata with limited universal branching. In: Leporati, A., Martín-Vide, C., Shapira, D., Zandron, C. (eds.) LATA 2020. LNCS, vol. 12038, pp. 196–207. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-40608-0_13

    Chapter  MATH  Google Scholar 

  15. Keeler, C., Salomaa, K.: Combining limited parallelism and nondeterminism in alternating finite automata. In: Jirásková, G., Pighizzini, G. (eds.) DCFS 2020. LNCS, vol. 12442, pp. 91–103. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-62536-8_8

    Chapter  Google Scholar 

  16. Keeler, C., Salomaa, K.: Width measures of alternating finite automata. In: International Conference Descriptional Complexity of Formal Systems, DCFS. Lecture Notes in Computer Science, vol. 13037, pp. 88–99. Springer, Heidelberg (2021). https://doi.org/10.1007/978-3-030-93489-7_8

  17. Keeler, C., Salomaa, K.: Structural properties of NFAs and growth rates of nondeterminism measures. Inf. Comput. 284, 104690 (2022)

    Google Scholar 

  18. Kozen, D.: Lower bounds for natural proof systems. In: Proceedings of 18th Annual Symposium on Foundations of Computer Science, pp. 254–266 (1977)

    Google Scholar 

  19. Leiss, E.: Succinct representation of regular languages by Boolean automata. Theor. Comput. Sci. 13, 323–330 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  20. Leung, H.: Limitedness theorem on finite automata with distance functions: an algebraic proof. Theor. Comput. Sci. 81, 137–145 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  21. Leung, H.: On finite automata with limited nondeterminism. Acta Informatica 35, 595–624 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  22. Leung, H.: Separating exponentially ambiguous finite automata from polynomially ambiguous finite automata. SIAM J. Comput. 27, 1073–1082 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  23. Leung, H., Podolskiy, V.: The limitedness problem on distance automata: Hashiguchi’s method revisited. Theor. Comput. Sci. 310, 147–158 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  24. Palioudakis, A., Salomaa, K., Akl, S.G.: State complexity of finite tree width NFAs. J. Automata Lang. Comb. 17(2–4), 245–264 (2012)

    MathSciNet  MATH  Google Scholar 

  25. Palioudakis, A., Salomaa, K., Akl, S.G.: Worst case branching and other measures of nondeterminism. Int. J. Found. Comput. Sci. 28(3), 195–210 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  26. Ravikumar, B., Ibarra, O.H.: Relating the type of ambiguity of finite automata to the succinctness of their representation. SIAM J. Comput. 18(6), 1263–1282 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  27. Shallit, J.: A Second Course in Formal Languages and Automata Theory. Cambridge University Press, Cambridge (2008)

    Book  MATH  Google Scholar 

  28. Sipser, M.: Introduction to the Theory of Computation, 3rd edn. Cengage (2013)

    Google Scholar 

  29. Weber, A., Seidl, H.: On the degree of ambiguity of finite automata. Theor. Comput. Sci. 88(2), 325–349 (1991)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

We thank the anonymous referees for a careful reading of the paper and useful suggestions. Han, Kim and Ko were supported by the NRF grant (RS-2023-00208094). Salomaa was supported by Natural Sciences and Engineering Research Council of Canada (NSERC).

Author information

Authors and Affiliations

  1. Department of Computer Science, Yonsei University, 50, Yonsei-Ro, Seodaemun-Gu, Seoul, 120-749, Republic of Korea

    Yo-Sub Han & Sungmin Kim

  2. Department of Computer Science and Engineering, Kangwon National University, 1 Kangwondaehak-gil, Chuncheon-si, Gangwon-do, 24341, Republic of Korea

    Sang-Ki Ko

  3. School of Computing, Queen’s University, Kingston, ON, K7L 3N6, Canada

    Kai Salomaa

Authors
  1. Yo-Sub Han
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  2. Sungmin Kim
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  4. Kai Salomaa
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Corresponding author

Correspondence to Kai Salomaa .

Editor information

Editors and Affiliations

  1. University of Potsdam, Potsdam, Germany

    Henning Bordihn

  2. Santa Clara University, Santa Clara, CA, USA

    Nicholas Tran

  3. University of Debrecen, Debrecen, Hungary

    György Vaszil

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Han, YS., Kim, S., Ko, SK., Salomaa, K. (2023). Existential and Universal Width of Alternating Finite Automata. In: Bordihn, H., Tran, N., Vaszil, G. (eds) Descriptional Complexity of Formal Systems. DCFS 2023. Lecture Notes in Computer Science, vol 13918. Springer, Cham. https://doi.org/10.1007/978-3-031-34326-1_4

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  • DOI: https://doi.org/10.1007/978-3-031-34326-1_4

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