Lattices of Logical Fragments over Words

(Extended Abstract)
  • Manfred Kufleitner
  • Alexander Lauser
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7392)


This paper introduces an abstract notion of fragments of monadic second-order logic. This concept is based on purely syntactic closure properties. We show that over finite words, every logical fragment defines a lattice of languages with certain closure properties. Among these closure properties are residuals and inverse \(\mathcal C\)-morphisms. Here, depending on certain closure properties of the fragment, \(\mathcal C\) is the family of arbitrary, non-erasing, length-preserving, length-multiplying, or lengthreducing morphisms. In particular, definability in a certain fragment can often be characterized in terms of the syntactic morphism. This work extends a result of Straubing in which he investigated certain restrictions of first-order formulae.

As motivating examples, we present (1) a fragment which captures the stutter-invariant part of piecewise-testable languages and (2) an acyclic fragment of Σ2. As it turns out, the latter has the same expressive power as two-variable first-order logic FO2.


Expressive Power Atomic Formula Regular Language Comparison Graph Closure Property 
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.


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  1. 1.
    Büchi, J.R.: Weak second-order arithmetic and finite automata. Z. Math. Logik Grundlagen Math. 6, 66–92 (1960)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Chaubard, L., Pin, J.-É., Straubing, H.: Actions, wreath products of \(\mathcal{C}\)-varieties and concatenation product. Theor. Comput. Sci. 356, 73–89 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Chaubard, L., Pin, J.-É., Straubing, H.: First order formulas with modular predicates. In: LICS 2006, pp. 211–220. IEEE Computer Society (2006)Google Scholar
  4. 4.
    Diekert, V., Gastin, P.: First-order definable languages. In: Flum, J., Grädel, E., Wilke, T. (eds.) Logic and Automata: History and Perspectives. Texts in Logic and Games, pp. 261–306. Amsterdam University Press (2008)Google Scholar
  5. 5.
    Diekert, V., Gastin, P., Kufleitner, M.: A survey on small fragments of first-order logic over finite words. Int. J. Found. Comput. Sci. 19(3), 513–548 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Eilenberg, S.: Automata, Languages, and Machines, vol. B. Academic Press (1976)Google Scholar
  7. 7.
    Elgot, C.C.: Decision problems of finite automata design and related arithmetics. Trans. Amer. Math. Soc. 98, 21–51 (1961)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Gehrke, M., Grigorieff, S., Pin, J.-É.: Duality and Equational Theory of Regular Languages. In: Aceto, L., Damgård, I., Goldberg, L.A., Halldórsson, M.M., Ingólfsdóttir, A., Walukiewicz, I. (eds.) ICALP 2008, Part II. LNCS, vol. 5126, pp. 246–257. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  9. 9.
    Kamp, J.A.W.: Tense Logic and the Theory of Linear Order. PhD thesis, University of California (1968)Google Scholar
  10. 10.
    Kufleitner, M., Lauser, A.: Lattices of logical fragments over words. CoRR, abs/1202.3355 (2012),
  11. 11.
    Kunc, M.: Equational description of pseudovarieties of homomorphisms. Theor. Inform. Appl. 37, 243–254 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    McNaughton, R., Papert, S.: Counter-Free Automata. The MIT Press (1971)Google Scholar
  13. 13.
    Pin, J.-É.: A variety theorem without complementation. Russian Mathematics (Iz. VUZ) 39, 80–90 (1995)MathSciNetGoogle Scholar
  14. 14.
    Pin, J.-É.: Expressive power of existential first-order sentences of Büchi’s sequential calculus. Discrete Math. 291(1-3), 155–174 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Pin, J.-É., Straubing, H.: Some results on \(\mathcal{C}\)-varieties. Theor. Inform. Appl. 39, 239–262 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Schützenberger, M.P.: On finite monoids having only trivial subgroups. Inf. Control 8, 190–194 (1965)zbMATHCrossRefGoogle Scholar
  17. 17.
    Simon, I.: Piecewise Testable Events. In: Brakhage, H. (ed.) GI-Fachtagung 1975. LNCS, vol. 33, pp. 214–222. Springer, Heidelberg (1975)Google Scholar
  18. 18.
    Straubing, H.: Finite Automata, Formal Logic, and Circuit Complexity. Birkhäuser (1994)Google Scholar
  19. 19.
    Straubing, H.: On Logical Descriptions of Regular Languages. In: Rajsbaum, S. (ed.) LATIN 2002. LNCS, vol. 2286, pp. 528–538. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  20. 20.
    Straubing, H., Thérien, D., Thomas, W.: Regular languages defined with generalized quantifiers. Inf. Comput. 118(2), 289–301 (1995)zbMATHCrossRefGoogle Scholar
  21. 21.
    Tesson, P., Thérien, D.: Diamonds are forever: The variety DA. In: Gomes, G., et al. (eds.) Semigroups, Algorithms, Automata and Languages 2001, pp. 475–500. World Scientific (2002)Google Scholar
  22. 22.
    Tesson, P., Thérien, D.: Logic meets algebra: The case of regular languages. Log. Methods Comput. Sci. 3(1), 1–37 (2007)Google Scholar
  23. 23.
    Thérien, D., Wilke, T.: Over words, two variables are as powerful as one quantifier alternation. In: STOC 1998, pp. 234–240. ACM Press (1998)Google Scholar
  24. 24.
    Thomas, W.: Classifying regular events in symbolic logic. J. Comput. Syst. Sci. 25, 360–376 (1982)zbMATHCrossRefGoogle Scholar
  25. 25.
    Trakhtenbrot, B.A.: Finite automata and logic of monadic predicates. Dokl. Akad. Nauk SSSR 140, 326–329 (1961) (in Russian)Google Scholar

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© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Manfred Kufleitner
    • 1
  • Alexander Lauser
    • 1
  1. 1.FMIUniversity of StuttgartGermany

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