The Degree of Irreversibility in Deterministic Finite Automata

  • Holger Bock Axelsen
  • Markus Holzer
  • Martin Kutrib
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9705)


Recently, Holzer et al. gave a method to decide whether the language accepted by a given deterministic finite automaton (DFA) can also be accepted by some reversible deterministic finite automaton (REV-DFA), and eventually proved NL-completeness. Here, we show that the corresponding problem for nondeterministic finite state automata (NFA) is PSPACE-complete. The recent DFA method essentially works by minimizing the DFA and inspecting it for a forbidden pattern. We here study the degree of irreversibility for a regular language, the minimal number of such forbidden patterns necessary in any DFA accepting the language, and show that the degree induces a strict infinite hierarchy of languages. We examine how the degree of irreversibility behaves under the usual language operations union, intersection, complement, concatenation, and Kleene star, showing tight bounds (some asymptotically) on the degree.


  1. 1.
    Axelsen, H.B., Kutrib, M., Malcher, A., Wendlandt, M.: Boosting reversible pushdown machines by preprocessing. In: RC 2016, LNCS. Springer (2016)Google Scholar
  2. 2.
    Glushkov, V.M.: The abstract theory of automata. Russ. Math. Surv. 16, 1–53 (1961)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Harrison, M.A.: Introduction to Formal Language Theory. Addison-Wesley, Boston (1978)zbMATHGoogle Scholar
  4. 4.
    Holzer, M., Jakobi, S., Kutrib, M.: Minimal reversible deterministic finite automata. In: Potapov, I. (ed.) DLT 2015. LNCS, vol. 9168, pp. 276–287. Springer, Heidelberg (2015)CrossRefGoogle Scholar
  5. 5.
    Kondacs, A., Watrous, J.: On the power of quantum finite state automata. In: FOCS 1997, pp. 66–75. IEEE (1997)Google Scholar
  6. 6.
    Kutrib, M.: Aspects of reversibility for classical automata. In: Calude, C.S., Freivalds, R., Kazuo, I. (eds.) Gruska Festschrift. LNCS, vol. 8808, pp. 83–98. Springer, Heidelberg (2014)Google Scholar
  7. 7.
    Landauer, R.: Irreversibility and heat generation in the computing process. IBM J. Res. Dev. 3, 183–191 (1961)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Meyer, A.R., Stockmeyer, L.J.: The equivalence problem for regular expressions with squaring requires exponential time. In: SWAT 1972, pp. 125–129. IEEE (1972)Google Scholar
  9. 9.
    Morita, K.: Reversible computing and cellular automata–a survey. Theoret. Comput. Sci. 395(1), 101–131 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Schultz, U.P., Laursen, J.S., Ellekilde, L., Axelsen, H.B.: Towards a domain-specific language for reversible assembly sequences. In: Krivine, J., Stefani, J.-B. (eds.) RC 2015. LNCS, vol. 9138, pp. 111–126. Springer, Switzerland (2015)CrossRefGoogle Scholar
  11. 11.
    Yu, S., Zhuang, Q., Salomaa, K.: The state complexity of some basic operations on regular languages. Theoret. Comput. Sci. 125, 315–328 (1994)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Holger Bock Axelsen
    • 1
  • Markus Holzer
    • 2
  • Martin Kutrib
    • 2
  1. 1.Department of Computer ScienceUniversity of CopenhagenCopenhagenDenmark
  2. 2.Institut für InformatikUniversität GiessenGiessenGermany

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