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Alternation Hierarchies of First Order Logic with Regular Predicates

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Fundamentals of Computation Theory (FCT 2015)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9210))

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Abstract

We investigate the decidability of the definability problem for fragments of first order logic over finite words enriched with regular numerical predicates. In this paper, we focus on the quantifier alternation hierarchies of first order logic. We obtain that deciding this problem for each level of the alternation hierarchy of both first order logic and its two-variable fragment when equipped with all regular numerical predicates is not harder than deciding it for the corresponding level equipped with only the linear order.

Relying on some recent results, this proves the decidability for each level of the alternation hierarchy of the two-variable first order fragment while in the case of the first order logic the question remains open for levels greater than two.

The main ingredients of the proofs are syntactic transformations of first-order formulas as well as the infinitely testable property, a new algebraic notion on varieties that we define.

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References

  1. Almeida, J.: A syntactical proof of locality of DA. Internat. J. Algebra Comput. 6(2), 165–177 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  2. Barrington, D.A.M., Compton, K., Straubing, H., Thérien, D.: Regular languages in \({\rm{NC}}^1\). J. Comput. System Sci. 44(3), 478–499 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  3. Büchi, J.R.: Weak second-order arithmetic and finite automata. Z. Math. Logik Grundlagen Math. 6, 66–92 (1960)

    Google Scholar 

  4. Chaubard, L., Pin, J.-É., Straubing, H.: First order formulas with modular predicates. In: LICS, pp. 211–220. IEEE (2006)

    Google Scholar 

  5. Cohen, R.S., Brzozowski, J.A.: Dot-depth of star-free events. J. Comput. Syst. Sci. 5(1), 1–16 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  6. Dartois, L., Paperman, C.: Two-variable first order logic with modular predicates over words. In: Portier, N., Wilke, T. (eds.) STACS, pp. 329–340. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, Dagstuhl (2013)

    Google Scholar 

  7. Diekert, V., Gastin, P., Kufleitner, M.: A survey on small fragments of first-order logic over finite words. Internat. J. Found. Comput. Sci. 19(3), 513–548 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  8. Ésik, Z., Ito, M.: Temporal logic with cyclic counting and the degree of aperiodicity of finite automata. Acta Cybernet. 16(1), 1–28 (2003)

    MathSciNet  MATH  Google Scholar 

  9. Knast, R.: A semigroup characterization of dot-depth one languages. RAIRO Inform. Théor. 17(4), 321–330 (1983)

    MathSciNet  MATH  Google Scholar 

  10. Krebs, A., Straubing, H.: An effective characterization of the alternation hierarchy in two-variable logic. In: FSTTCS, pp. 86–98 (2012)

    Google Scholar 

  11. Kufleitner, M., Lauser, A.: Quantifier alternation in two-variable first-order logic with successor is decidable. In: STACS, pp. 305–316 (2013)

    Google Scholar 

  12. Kufleitner, M., Weil, P.: The \({\rm {FO}}^2\) alternation hierarchy is decidable. In: Computer Science Logic 2012, Volume 16 of LIPIcs. Leibniz Int. Proc. Inform., pp. 426–439. Schloss Dagstuhl. Leibniz-Zent. Inform., Wadern (2012)

    Google Scholar 

  13. McNaughton, R., Papert, S.: Counter-free Automata. The M.I.T. Press, Cambridge (1971)

    MATH  Google Scholar 

  14. Nerode, A.: Linear automaton transformation. Proc. AMS 9, 541–544 (1958)

    Article  MathSciNet  MATH  Google Scholar 

  15. Péladeau, P.: Logically defined subsets of \({ N}^k\). Theoret. Comput. Sci. 93(2), 169–183 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  16. Perrin, D., Pin, J.É.: First-order logic and star-free sets. J. Comput. System Sci. 32(3), 393–406 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  17. Pin, J.-É.: Syntactic semigroups. In: Rozenberg, G., Salomaa, A. (eds.) Handbook of Formal Languages, vol. 1, pp. 679–746. Springer, Heidelberg (1997)

    Chapter  Google Scholar 

  18. Place, T., Zeitoun, M.: Going higher in the first-order quantifier alternation hierarchy on words. In: Esparza, J., Fraigniaud, P., Husfeldt, T., Koutsoupias, E. (eds.) ICALP 2014, Part II. LNCS, vol. 8573, pp. 342–353. Springer, Heidelberg (2014)

    Google Scholar 

  19. Schützenberger, M.P.: On finite monoids having only trivial subgroups. Inf. Control 8, 190–194 (1965)

    Article  MATH  Google Scholar 

  20. Simon, I.: Piecewise testable events. In: Brakhage, H. (ed.) Automata Theory and Formal Languages (Second GI Conference Kaiserslautern, 1975). LNCS, vol. 33, pp. 214–222. Springer, Heidelberg (1975)

    Chapter  Google Scholar 

  21. Straubing, H.: A generalization of the Schützenberger product of finite monoids. Theoret. Comput. Sci. 13(2), 137–150 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  22. Straubing, H.: Finite semigroup varieties of the form \(V\ast D\). J. Pure Appl. Algebra 36(1), 53–94 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  23. Straubing, H.: Finite Automata, Formal Logic, and Circuit Complexity. Birkhäuser Boston Inc., Boston (1994)

    Book  MATH  Google Scholar 

  24. Thérien, D.: Classification of finite monoids: the language approach. Theoret. Comput. Sci. 14(2), 195–208 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  25. Thérien, D., Wilke, T.: Over words, two variables are as powerful as one quantifier alternation. In: STOC 1998 (Dallas. TX), pp. 234–240. ACM, New York (1999)

    Google Scholar 

  26. Thomas, W.: Classifying regular events in symbolic logic. J. Comput. System Sci. 25(3), 360–376 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  27. Tilson, B.: Categories as algebra: an essential ingredient in the theory of monoids. J. Pure Appl. Algebra 48(1–2), 83–198 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  28. Weis, P., Immerman, N.: Structure theorem and strict alternation hierarchy for \({\rm {FO}}^2\) on words. Log. Methods Comput. Sci. 5(3:3:4), 23 (2009)

    Google Scholar 

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Author information

Authors and Affiliations

  1. University of Warsaw, Warsaw, Poland

    Charles Paperman

  2. École Centrale Marseille, Marseille, France

    Luc Dartois

  3. LIF, UMR7279, Aix-Marseille Université and CNRS, Marseille, France

    Luc Dartois

Authors
  1. Luc Dartois
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  2. Charles Paperman
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Correspondence to Charles Paperman .

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Editors and Affiliations

  1. Université Paris Diderot, Paris, France

    Adrian Kosowski

  2. Université Bordeaux 1, Talence, France

    Igor Walukiewicz

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Dartois, L., Paperman, C. (2015). Alternation Hierarchies of First Order Logic with Regular Predicates. In: Kosowski, A., Walukiewicz, I. (eds) Fundamentals of Computation Theory. FCT 2015. Lecture Notes in Computer Science(), vol 9210. Springer, Cham. https://doi.org/10.1007/978-3-319-22177-9_13

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  • DOI: https://doi.org/10.1007/978-3-319-22177-9_13

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-22176-2

  • Online ISBN: 978-3-319-22177-9

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