# Alternation Hierarchies of First Order Logic with Regular Predicates

• Luc Dartois
• Charles Paperman
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9210)

## 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.

## Keywords

Quantiﬁer Alternation Hierarchy Regular Predicate Definability Problem Delay Question Regular Languages
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## References

1. 1.
Almeida, J.: A syntactical proof of locality of DA. Internat. J. Algebra Comput. 6(2), 165–177 (1996)
2. 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)
3. 3.
Büchi, J.R.: Weak second-order arithmetic and finite automata. Z. Math. Logik Grundlagen Math. 6, 66–92 (1960)Google Scholar
4. 4.
Chaubard, L., Pin, J.-É., Straubing, H.: First order formulas with modular predicates. In: LICS, pp. 211–220. IEEE (2006)Google Scholar
5. 5.
Cohen, R.S., Brzozowski, J.A.: Dot-depth of star-free events. J. Comput. Syst. Sci. 5(1), 1–16 (1971)
6. 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. 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)
8. 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)
9. 9.
Knast, R.: A semigroup characterization of dot-depth one languages. RAIRO Inform. Théor. 17(4), 321–330 (1983)
10. 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. 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. 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. 13.
McNaughton, R., Papert, S.: Counter-free Automata. The M.I.T. Press, Cambridge (1971)
14. 14.
Nerode, A.: Linear automaton transformation. Proc. AMS 9, 541–544 (1958)
15. 15.
Péladeau, P.: Logically defined subsets of $${ N}^k$$. Theoret. Comput. Sci. 93(2), 169–183 (1992)
16. 16.
Perrin, D., Pin, J.É.: First-order logic and star-free sets. J. Comput. System Sci. 32(3), 393–406 (1986)
17. 17.
Pin, J.-É.: Syntactic semigroups. In: Rozenberg, G., Salomaa, A. (eds.) Handbook of Formal Languages, vol. 1, pp. 679–746. Springer, Heidelberg (1997)
18. 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. 19.
Schützenberger, M.P.: On finite monoids having only trivial subgroups. Inf. Control 8, 190–194 (1965)
20. 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)
21. 21.
Straubing, H.: A generalization of the Schützenberger product of finite monoids. Theoret. Comput. Sci. 13(2), 137–150 (1981)
22. 22.
Straubing, H.: Finite semigroup varieties of the form $$V\ast D$$. J. Pure Appl. Algebra 36(1), 53–94 (1985)
23. 23.
Straubing, H.: Finite Automata, Formal Logic, and Circuit Complexity. Birkhäuser Boston Inc., Boston (1994)
24. 24.
Thérien, D.: Classification of finite monoids: the language approach. Theoret. Comput. Sci. 14(2), 195–208 (1981)
25. 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. 26.
Thomas, W.: Classifying regular events in symbolic logic. J. Comput. System Sci. 25(3), 360–376 (1982)
27. 27.
Tilson, B.: Categories as algebra: an essential ingredient in the theory of monoids. J. Pure Appl. Algebra 48(1–2), 83–198 (1987)
28. 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