Theory of Computing Systems

, Volume 63, Issue 3, pp 367–385 | Cite as

Better Complexity Bounds for Cost Register Automata

  • Eric AllenderEmail author
  • Andreas Krebs
  • Pierre McKenzie


Cost register automata (CRAs) are one-way finite automata whose transitions have the side effect that a register is set to the result of applying a state-dependent semiring operation to a pair of registers. Here it is shown that CRAs over the tropical semiring \((\mathbb {N}\cup \{\infty \},\min ,+)\) can simulate polynomial time computation, proving along the way that a naturally defined width-k circuit value problem over the tropical semiring is \(\textsf {P}\)-complete. Then the copyless variant of the CRA, requiring that semiring operations be applied to distinct registers, is shown no more powerful than \(\textsf {NC}^{1}\) when the semiring is \((\mathbb {Z},+,\times )\) or \(({\Gamma }^{*}\cup \{\bot \},\max ,\text {concat})\). This relates questions left open in recent work on the complexity of CRA-computable functions to long-standing class separation conjectures in complexity theory, such as \(\textsf {NC}\) versus \(\textsf {P}\) and NC1 versus GapNC1.


Computational complexity Circuit complexity Cost register automata Arithmetic circuits Tropical semiring 



Some of this research was performed at the 29th McGill Invitational Workshop on Computational Complexity, held at the Bellairs Research Institute of McGill University, in February, 2017.


  1. 1.
    Allender, E., Mertz, I.: Complexity of regular functions. Journal of Computer and System Sciences, 2017. To appear; LATA 2015 Special Issue. Earlier version appeared in Proceedings of 9th International Conference on Language and Automata Theory and Applications (LATA’15), Springer Lecture Notes in Computer Science, vol. 8977, pp. 449–460 (2015)Google Scholar
  2. 2.
    Allender, E.: Arithmetic circuits and counting complexity classes. In: Krajíček, J. (ed.) Complexity of Computations and Proofs, volume 13 of Quaderni di Matematica, pp 33–72. Seconda Università di Napoli (2004)Google Scholar
  3. 3.
    Allender, E., Krebs, A., McKenzie, P.: Better complexity bounds for cost register automata. In: 42nd International Symposium on Mathematical Foundations of Computer Science (MFCS), volume 83 of LIPIcs, pp 24:1–24:14. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik (2017)Google Scholar
  4. 4.
    Alur, R., Cerný, P.: Streaming transducers for algorithmic verification of single-pass list-processing programs. In: 38th ACM, SIGPLAN-SIGACT Symposium on Principles of Programming Languages, pp. 599–610. POPL (2011)Google Scholar
  5. 5.
    Alur, R., D’Antoni, L., Deshmukh, J.V., Raghothaman, M., Yuan, Y.: Regular functions, cost register automata, and generalized min-cost problems. CoRR, arXiv:1111.0670 (2011)
  6. 6.
    Alur, R., D’Antoni, L., Deshmukh, J.V., Raghothaman, M., Yuan, Y.: Regular functions and cost register automata. In: 28th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), pp.13–22. See also the expanded version, [5] (2013)Google Scholar
  7. 7.
    Alur, R., Freilich, A., Raghothaman, M.: Regular combinators for string transformations. In: Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science, (CSL-LICS), p. 9. ACM (2014)Google Scholar
  8. 8.
    Alur, R., Raghothaman, M.: Decision problems for additive regular functions. In Proc. 40th International Colloquium on Automata, Languages, and Programming (ICALP), Lecture Notes in Computer Science, pp. 37–48. Springer (2013)Google Scholar
  9. 9.
    Arora, S., Barak, B.: Computational complexity: a modern approach, vol. 1. University Press, Cambridge (2009)Google Scholar
  10. 10.
    Barrington, D. A.: Bounded-width polynomial-size branching programs recognize exactly those languages in NC1. J. Comput. Syst. Sci. 38, 150–164 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Barrington, D. A. M., Lu, C.-J., Miltersen, P. B., Skyum, S.: Searching constant width mazes captures the AC0 hierarchy. In: Proceedings of 15th International Symposium on Theoretical Aspects of Computer Science (STACS), number 1373 in Lecture Notes in Computer Science, pp. 73–83. Springer (1998)Google Scholar
  12. 12.
    Beaudry, M., McKenzie, P., Péladeau, P., Thérien, D.: Finite monoids: from word to circuit evaluation. SIAM J. Comput. 26, 138–152 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Ben-Or, M., Cleve, R.: Computing algebraic formulas using a constant number of registers. SIAM J. Comput. 21(1), 54–58 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Cadilhac, M., Krebs, A., Limaye, N.: Value automata with filters. CoRR, arXiv:1510.02393 (2015)
  15. 15.
    Colcombet, T.: The theory of stabilisation monoids and regular cost functions. In: Proceedings of the 36th International Colloquium, Automata, Languages and Programming, ICALP 2009, Part II, pp. 139–150 (2009)Google Scholar
  16. 16.
    Colcombet, T.: Regular cost functions, part I: logic and algebra over words, Log. Methods Comput. Sci.,9, 3 (2013)Google Scholar
  17. 17.
    Colcombet, T., Kuperberg, D., Manuel, A., Toruṅczyk, S.: Cost functions definable by min/max automata. In: 33rd Symposium on Theoretical Aspects of Computer Science, STACS 2016, pp. 29:1– 29:13. Orléans, France (2016)Google Scholar
  18. 18.
    Daviaud, Laure, Reynier, P., Talbot, J.: A generalised twinning property for minimisation of cost register automata. In: Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’16, pp. 857–866. New York (2016)Google Scholar
  19. 19.
    Droste, M., Kuich, W., Vogler, H.: Handbook of Weighted Automata. Springer-Verlag, New York (2009)CrossRefzbMATHGoogle Scholar
  20. 20.
    Greenlaw, R., Hoover, H.J., Ruzzo, W.L.: Limits to Parallel Computation: P-Completeness Theory. Oxford University Press, Oxford (1995)zbMATHGoogle Scholar
  21. 21.
    Hesse, W., Allender, E., Barrington, D.A.: Uniform constant-depth threshold circuits for division and iterated multiplication. J. Comput. Syst. Sci. 65, 695–716 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Karloff, H.J., Ruzzo, W.L.: The iterated mod problem. Inf. Comput. 80(3), 193–204 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Koucký, M.: Unpublished Manuscript (2003)Google Scholar
  24. 24.
    Krebs, A., Limaye, N., Ludwig, M.: Cost register automata for nested words. In: Proceedings of 22nd International Computing and Combinatorics Conference - (COCOON), number 9797 in Lecture Notes in Computer Science, pp. 587–598. Springer (2016)Google Scholar
  25. 25.
    Lynch, N.A.: Straight-line program length as a parameter for complexity analysis. J. Comput. Syst. Sci. 21(3), 251–280 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Mahajan, M., Nimbhorkar, P., Tawari, A. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik (2017)Google Scholar
  27. 27.
    Mazowiecki, F., Riveros, C.: Maximal partition logic: Towards a logical characterization of copyless cost register automata. In: 24th EACSL Annual Conference on Computer Science Logic, CSL, pp. 144–159. Berlin (2015)Google Scholar
  28. 28.
    Mazowiecki, F., Riveros, C.: Copyless cost-register automata: Structure, expressiveness, and closure properties. In: 33rd Symposium on Theoretical Aspects of Computer Science, STACS, pp. 53:1–53:13. Orléans (2016)Google Scholar
  29. 29.
    Pin, J.-E.: Tropical Semirings. Cambridge University Press, Cambridge (1998)CrossRefzbMATHGoogle Scholar
  30. 30.
    Pippenger, N.: On simultaneous resource bounds. In: IEEE Symposium on Foundations of Computer Science (FOCS), pp. 307–311 (1979)Google Scholar
  31. 31.
    Straubing, H.: Finite Automata, Formal Logic, and Circuit Complexity. Birkhäuser, Boston (1994)CrossRefzbMATHGoogle Scholar
  32. 32.
    Vollmer, H.: Introduction to Circuit Complexity: A Uniform Approach. Springer-Verlag, New York (1999)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Computer ScienceRutgers UniversityNew BrunswickUSA
  2. 2.WSIUniversität TübingenTübingenGermany
  3. 3.Université de MontréalQuébecCanada

Personalised recommendations