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Logic, semigroups and automata on words

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Abstract

This is a survey article on the connections between the “sequential calculus” of Büchi, a system which allows to formalize properties of words, and the theory of automata. In the sequential calculus, there is a predicate for each letter and the unique extra non logical predicate is the relation symbol <, which is interpreted as the usual order on the integers. Several famous classes have been classified within this logic. We briefly review the main results concerning second order, which covers classes like PH, NP, P, etc., and then study in more detail the results concerning the monadic second-order and first-order logic. In particular, we survey the results and fascinating open problems dealing with the first-order quantifier hierarchy. We also discuss the first-order logic of one successor and the linear temporal logic. There are in fact three levels of results, since these logics can be interpreted on finite words, infinite words or bi-infinite words. The paper is self-contained. In particular, the necessary background on automata and finite semigroups is presented in a long introductory section, which includes some very recent results on the algebraic theory of infinite words.

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References

  1. J. Almeida, Implicit operations on finiteJ-trivial semigroups and a conjecture of I. Simon, J. Pure Appl. Algebra 69(1990)205–218.

    Google Scholar 

  2. M. Arfi, Polynomials operations and relational languages,4th STACS, Lecture Notes in Computer Science 247 (Springer, 1987) pp. 198–206.

    Google Scholar 

  3. M. Arfi, Opérations polynomiales et hiérarchies de concaténation, Theor. Comput. Sci. 91(1991)71–84.

    Google Scholar 

  4. A. Arnold, A syntactic congruence for rational ω-languages, Theor. Comput. Sci. 39(1985)333–335.

    Google Scholar 

  5. D. Beauquier, Bi-limites de langages reconnaissables, Theor. Comput. Sci. 33(1984)335–342.

    Google Scholar 

  6. D. Beauquier, Ensembles reconnaissables de mots bi-infinis, in:Automata on Infinite Words, ed. M. Nivat, Lecture Notes in Computer Science 192 (Springer, 1985) pp. 28–46.

    Google Scholar 

  7. D. Beauquier and M. Nivat, About rational sets of factors of a bi-infinite word, in:Automata, Languages and Programming, ed. W. Brauer, Lecture Notes in Computer Science 194 (Springer, 1985) pp. 33–42.

    Google Scholar 

  8. D. Beauquier and J.E. Pin, Factors of words, in:Automata, Languages and Programming, eds. G. Ausiello, M. Dezani-Ciancaglini and S. Ronchi Della Rocca, Lecture Notes in Computer Science 372 (Springer, 1989) pp. 63–79.

    Google Scholar 

  9. D. Beauquier and J.E. Pin, Languages and scanners, Theor. Comput. Sci. 84(1991)3–21.

    Google Scholar 

  10. J.A. Brzozowski and R. Knast, The dot-depth huerarchy of star-free languages is infinite, J. Comput. Syst. Sci. 16(1978)37–55.

    Google Scholar 

  11. J.A. Brzozowski and I. Simon, Characterizations of locally testable languages, Discr. Math. 4(1973)243–271.

    Google Scholar 

  12. J.R. Büchi, Weak second-order arithmetic and finite automata, Z. Math. Logik und Grundl. Math. 6(1960)66–92.

    Google Scholar 

  13. J.R. Büchi, On a decision method in restricted second-order arithmetic,Proc. 1960 Int. Congr. for Logic, Methodology and Philosophy of Science (Stanford University Press, Stanford, 1962) pp. 1–11.

    Google Scholar 

  14. J.R. Büchi and L.H. Landweber, Definability in the monadic second-order theory of successors, J. Symb. Logic 34(1969)166–170.

    Google Scholar 

  15. S. Cho and D.T. Huynh, Finite automaton aperiodicity is PSPACE-complete, Theor. Comput. Sci. 88(1991)99–116.

    Google Scholar 

  16. J. Cohen, On the expressive power of temporal logic for infinite words, Theor. Comput. Sci. 83(1991)301–312.

    Google Scholar 

  17. J. Cohen, D. Perrin and J.E. Pin, On the expressive power of temporal logic, J. Comput. Syst. Sci. (1993).

  18. K. Compton, On rich words, in:Combinatoric Words, Progress and Perspectives, ed. L. Cummings (Academic Press, 1983) pp. 39–62

  19. D. Cowan, Inverse monoids of dot-depth 2, Int. J. Alg. Comp. 3(1993)411–424.

    Google Scholar 

  20. S. Eilenberg,Automata, Languages and Machines, Vol. A (Academic Press, New York, 1974).

    Google Scholar 

  21. S. Eilenberg,Automata, Languages and Machines, Vol. B (Academic Press, New York, 1976).

    Google Scholar 

  22. C.C. Elgot, Decision problems of finite automata design and related arithmetics, Trans. Amer. Math. Soc. 98(1961)21–52.

    Google Scholar 

  23. E.A. Emerson, Temporal and modal logic, in:Handbook of Theoretical Computer Science, Vol. B:Formal Models and Semantics (Elsevier, 1990) pp. 995–1072.

  24. R. Fagin, Generalized first-order spectra and polynomial-time recognizable sets,SIAM-AMS Proc. 7(1974)43–73.

    Google Scholar 

  25. R. Fagin, Finite-model theory — a personal perspective, Theor. Comput. Sci. 116(1993)3–31.

    Google Scholar 

  26. H. Gaifman, On local and non-local properties,Proc. Herbrandt Symposium, Logic Colloquium '81, ed. J. Stern, Studies in Logic 107 (North-Holland, Amsterdam, 1982) pp. 105–135.

    Google Scholar 

  27. D. Gabbay, A. Pnueli, S. Shelah and J. Stavi, On the temporal analysis of fairness,Proc. 7th ACM Symp. Princ. Progr. Lang. (1980) pp. 163–173.

  28. B.R. Hodgson, Décidabilité par automate fini, Ann. Sci. Math. Québec 7(1983)39–57.

    Google Scholar 

  29. J.A. Kamp, Tense logic and the theory of linear order, Ph.D. Thesis, University of California, Los Angeles (1968).

    Google Scholar 

  30. S.C. Kleene, Representation of events in nerve nets and finite automata, in:Automata Studies, eds. C.E. Shannon and J. McCarthy (Princeton University Press, Princeton, NJ, 1956) pp. 3–42.

    Google Scholar 

  31. N. Immerman, Languages that capture complexity classes, SIAM J. Comput. 16(1987)760–778.

    Google Scholar 

  32. N. Immerman, Nondeterministic space is closed under complement, SIAM J. Comput. 17(1988)935–938.

    Google Scholar 

  33. J. Justin and G. Pirillo, On a natural extension of Jacob's rank, J. Combin. Theory 43(1986)205–218.

    Google Scholar 

  34. R. Ladner, Application of model theoretic games to discrete linear orders and finite automata, Inform. Contr. 33(1977)281–303.

    Google Scholar 

  35. G. Lallement,Semigroups and Combinatorial Applications (Wiley, New York, 1979).

    Google Scholar 

  36. H. Landweber, Finite state games — a solvability algorithm for restricted second-order arithmetic, Notices Amer. Math. Soc. 14(1967)129–130.

    Google Scholar 

  37. H. Landweber, Decision problems for ω-automata, Math. Syst. Theor. 3(1969)376–384.

    Google Scholar 

  38. R. McNaughton, Testing and generating infinite sequences by a finite automaton, Inform. Contr. 9(1966)521–530.

    Google Scholar 

  39. R. McNaughton, Algebraic decision procedures for local testability, Math. Syst. Theor. 8(1974)60–76.

    Google Scholar 

  40. R. McNaughton and S. Pappert,Counter-free Automata (MIT Press, 1971).

  41. A.R. Meyer, Weak monadic second order theory of successor is not elementary recursive,Proc. Boston University Logic Colloquium, Lecture Notes in Mathematics 453 (Springer, Berlin/Heidelberg/New York, 1975) pp. 132–154.

    Google Scholar 

  42. C. Michaux and F. Point, Les ensemblesk-reconnaissables sont définissables dans (ℕ, +,V k ), C.R. Acad. Sci. Paris 303(1986)939–942.

    Google Scholar 

  43. M. Nivat and D. Perrin, Ensembles reconnaissables de mots biinfinis, Can. J. Math 38(1986)513–537.

    Google Scholar 

  44. J.P. Pécuchet, Variétés de semigroupes et mots infinis,STACS 86, eds. B. Monien and G. Vidal-Naquet, Lecture Notes in Computer Science 210 (Springer, 1986) pp. 180–191.

    Google Scholar 

  45. J.P. Pécuchet, Etude syntaxique des parties reconnaissables de mots infinis,Proc. 13th ICALP, ed. L. Kott, Lecture Notes in Computer Science 226 (1986) pp. 294–303.

    Google Scholar 

  46. D. Perrin, Variétés de semigroupes et mots infinis, C.R. Acad. Sci. Paris 295(1982)595–598.

    Google Scholar 

  47. D. Perrin, Recent results on automata and infinite words, in:Mathematical Foundations of Computer Science, Lecture Notes in Computer Science 176 (Springer, New York/Berlin, 1984) pp. 134–148.

    Google Scholar 

  48. D. Perrin, An introduction to automata on infinite words, in:Automata on Infinite Words, ed. M. Nivat, Lecture Notes in Computer Science 192 (Springer, 1984) pp. 2–17.

    Google Scholar 

  49. D. Perrin, Automata, in:Handbook of Theoretical Computer Science, ed. J. Van Leeuwen, Vol. B:Formal Models and Semantics (Elsevier, 1990) chap. 1.

  50. D. Perrin and J.E. Pin, First order logic and star-free sets, J. Comput. Syst. Sci. 32(1986)393–406.

    Google Scholar 

  51. D. Perrin and J.E. Pin, Mots infinis, LITP Report 93-40, to appear.

  52. D. Perrin and P.E. Schupp, Automata on the integers, recurrence distinguishability and the equivalence and decidability of monadic theories,Proc. 1st IEEE Symp. on Logic in Computer Science (1986) pp. 301–304.

  53. J.-E. Pin, Hiérarchies de concaténation, RAIRO Inform. Théor. 18(1984)23–46.

    Google Scholar 

  54. J.-E. Pin,Variétés de Langages Formels (Masson, Paris, 1984); English translation:Varieties of Formal Languages (Plenum, New York, 1986).

    Google Scholar 

  55. J.-E. Pin and H. Straubing, Monoids of upper triangular matrices,Colloquia Mathematica Societatis Janos Bolyai 39 (Semigroups, Szeged, 1981) pp. 259–272.

    Google Scholar 

  56. J.-E. Pin and P. Weil, Polynomial closure and unambiguous products,Proc. 22nd ICALP, Lecture Notes in Computer Science 944 (Springer, Berlin, 1995) pp. 348–359.

    Google Scholar 

  57. W.V. Quine, Concatenation as a basis for arithmetic, J. Symb. Logic 11(1946)105–114.

    Google Scholar 

  58. F.D. Ramsey, On a problem of formal logic, Proc. London Math. Soc. 30(1929)338–384.

    Google Scholar 

  59. M.P. Schützenberger, On finite monoids having only trivial subgroups, Inform. Contr. 8(1965)190–194.

    Google Scholar 

  60. A.L. Semenov, On certain extensions of the arithmetic of addition of natural numbers, Math. USSR Izvestiya 15(1980)401–418.

    Google Scholar 

  61. D. Siefkes, Büchi's monadic second order successor arithmetic, Lecture Notes in Math. 120 (Springer, Berlin, 1970).

    Google Scholar 

  62. I. Simon, Piecewise testable events,Proc. 2nd GI Conf., Lecture Notes in Computer Science 33 (Springer, Berlin, 1975) pp. 214–222.

    Google Scholar 

  63. I. Simon, Factorization forests of finite height, Theor. Comput. Sci. 72(1990)65–94.

    Google Scholar 

  64. J. Stern, Characterization of some classes of regular events, Theor. Comput. Sci. 35(1985)17–42.

    Google Scholar 

  65. J. Stern, Complexity of some problems from the theory of automata, Inform. Contr. 66(1985)163–176.

    Google Scholar 

  66. L. Stockmeyer, The polynomial time hierarchy, Theor. Comput. Sci. 3(1977)1–22.

    Google Scholar 

  67. L.J. Stockmeyer and A.R. Meyer, Word problems requiring exponential time: Preliminary report,Proc. 5th Annual ACM Symposium on the Theory of Computing (1977) pp. 1–9.

  68. H. Straubing, Aperiodic homomorphisms and the concatenation product of recognizable sets, J. Pure Appl. Algebra 15(1979)319–327.

    Google Scholar 

  69. H. Straubing, Semigroups and languages of dot-depth two, Theor. Comput. Sci. 58(1988)361–378.

    Google Scholar 

  70. H. Straubing and D. Thérien, Partially ordered finite monoids and a theorem of I. Simon, J. Algebra 119(1985)393–399.

    Google Scholar 

  71. H. Straubing, D. Thérien and W. Thomas, Regular languages defined with generalized quantifiers,Proc. 15th ICALP, Lecture Notes in Computer Science 317 (Springer, 1988) pp. 561–575.

    Google Scholar 

  72. H. Straubing and P. Weil, On a conjecture concerning dot-depth two languages, Theor. Comput. Sci. 104(1992)161–183.

    Google Scholar 

  73. R. Szelepcsényi, The method of forced enumeration for nondeterministic automata, Acta Informatica 26(1988)279–284.

    Google Scholar 

  74. D. Thérien and A. Weiss, Graph congruences and wreath products, J. Pure Appl. Algebra 35(1985)205–215.

    Google Scholar 

  75. W. Thomas, Star-free regular sets of ω-sequences, Inform. Contr. 42(1979)148–156.

    Google Scholar 

  76. W. Thomas, A combinatorial approach to the theory of ω-automata, Inform. Contr. 48(1981)261–283.

    Google Scholar 

  77. W. Thomas, Classifying regular events in symbolic logic, J. Comput. Syst. Sci. 25(1982)360–375.

    Google Scholar 

  78. W. Thomas, Automata on infinite objects, in:Handbook of Theoretical Computer Science, Vol. B:Formal Models and Semantics (Elsevier, 1990) pp. 135–191.

  79. W. Thomas, On logics, tilings, and automata,Proc. 18th ICALP, Madrid, eds. J. Leach Albert et al., Lecture Notes in Computer Science 510 (Springer, Berlin, 1991) pp. 441–454.

    Google Scholar 

  80. W. Thomas, On the Ehrenfeucht-Fraïssé game in theoretical computer science,TAPSOFT '93, eds. M.C. Gaudel and J.P. Jouannaud, Lecture Notes in Computer Science 668 (Springer, Berlin, 1993) pp. 559–568.

    Google Scholar 

  81. P. Weil, Inverse monoids and the dot-depth hierarchy, Ph.D. Dissertation, University of Nebraska, Lincoln (1988).

    Google Scholar 

  82. P. Weil, Inverse monoids of dot-depth two, Theor. Comput. Sci. 66(1989)233–245.

    Google Scholar 

  83. P. Weil, Some results on the dot-depth hierarchy, Semigroup Forum 46(1993)352–370.

    Google Scholar 

  84. T. Wilke, An Eilenberg theorem for ∞-languages, automata, languages and programming,Proc. 18th ICALP Conf., Lecture Notes in Computer Science 510 (Springer, Berlin, 1991) pp. 588–599.

    Google Scholar 

  85. Th. Wilke, An algebraic theory for regular languages of finite and infinite words, Int. J. Algebra Comput. 3(1993)447–489.

    Google Scholar 

  86. Th. Wilke, Locally threshold testable languages of infinite words,STACS 93, eds. P. Enjalbert, A. Finkel and K.W. Wagner, Lecture Notes in Computer Science 665 (Springer, Berlin, 1993) pp. 607–616.

    Google Scholar 

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Pin, JE. Logic, semigroups and automata on words. Ann Math Artif Intell 16, 343–384 (1996). https://doi.org/10.1007/BF02127803

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