Advertisement

Incompleteness Theorems, Large Cardinals, and Automata over Finite Words

  • Olivier FinkelEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10185)

Abstract

We prove that one can construct various kinds of automata over finite words for which some elementary properties are actually independent from strong set theories like \(T_n =:\mathbf{ZFC} +\) “There exist (at least) n inaccessible cardinals”, for integers \(n\ge 0\). In particular, we prove independence results for languages of finite words generated by context-free grammars, or accepted by 2-tape or 1-counter automata. Moreover we get some independence results for weighted automata and for some related finitely generated subsemigroups of the set \(\mathbb {Z}^{3\times 3}\) of 3-3 matrices with integer entries. Some of these latter results are independence results from the Peano axiomatic system PA.

Keywords

Automata and formal languages Logic in computer science Finite words Context-free grammars 2-tape automaton Post correspondence problem Weighted automaton Finitely generated matrix subsemigroups of \(\mathbb {Z}^{3\times 3}\) Models of set theory Incompleteness theorems Large cardinals Inaccessible cardinals Independence from the axiomatic system “ZFC + there exist n inaccessible cardinals” Independence from Peano Arithmetic 

References

  1. [Ber79]
    Berstel, J.: Transductions and context free languages. Teubner Studienbücher Informatik (1979). http://www-igm.univ-mlv.fr/~berstel/
  2. [Dra74]
    Drake, F.R.: Set Theory, An Introduction to Large cardinals. Studies in Logic and the Foundations of Mathematics, vol. 76. North-Holland, Amsterdam (1974)zbMATHGoogle Scholar
  3. [EFT94]
    Ebbinghaus, H.-D., Flum, J., Thomas, W.: Mathematical Logic. Undergraduate Texts in Mathematics, 2nd edn. Springer, New York (1994). Translated from the German by Margit MeßmerCrossRefzbMATHGoogle Scholar
  4. [Fin03]
    Finkel, O.: Undecidability of topological and arithmetical properties of infinitary rational relations. RAIRO-Theoret. Inf. Appl. 37(2), 115–126 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  5. [Fin09]
    Finkel, O.: The complexity of infinite computations in models of set theory. Logical Methods Comput. Sci. 5(4:4), 1–19 (2009)MathSciNetzbMATHGoogle Scholar
  6. [Fin11]
    Finkel, O.: Some problems in automata theory which depend on the models of set theory. RAIRO-Theoret. Inf. Appl. 45(4), 383–397 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  7. [Fin15]
    Finkel, O.: Incompleteness theorems, large cardinals, and automata over infinite words. In: Halldórsson, M.M., Iwama, K., Kobayashi, N., Speckmann, B. (eds.) ICALP 2015. LNCS, vol. 9135, pp. 222–233. Springer, Heidelberg (2015). doi: 10.1007/978-3-662-47666-6_18 CrossRefGoogle Scholar
  8. [Fri11]
    Friedman, H.M.: My forty years on his shoulders. In: Gödel, K. (ed.) The Foundations of Mathematics, pp. 399–432. Cambridge Univ. Press, Cambridge (2011)Google Scholar
  9. [Gen36]
    Gentzen, G.: Die Widerspruchsfreiheit der reinen Zahlentheorie. Math. Ann. 112(1), 493–565 (1936)MathSciNetCrossRefzbMATHGoogle Scholar
  10. [Göd63]
    Gödel, K.: On formally undecidable propositions of Principia Mathematica and related systems (1963). Translated by B. Meltzer, with an introduction by R. B. Braithwaite. Basic Books Inc., Publishers, New YorkGoogle Scholar
  11. [Har85]
    Hartmanis, J.: Independence results about context-free languages and lower bounds. Inf. Process. Lett. 20(5), 241–248 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  12. [Har02]
    Harju, T.: Decision questions on integer matrices. In: Kuich, W., Rozenberg, G., Salomaa, A. (eds.) DLT 2001. LNCS, vol. 2295, pp. 57–68. Springer, Heidelberg (2002). doi: 10.1007/3-540-46011-X_5 CrossRefGoogle Scholar
  13. [HH01]
    Halava, V., Harju, T.: Mortality in matrix semigroups. Am. Math. Mon. 108(7), 649–653 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  14. [HMU01]
    Hopcroft, J.E., Motwani, R., Ullman, J.D.: Introduction to Automata Theory, Languages, and Computation. Addison-Wesley Series in Computer Science. Addison-Wesley Publishing Co., Reading (2001)zbMATHGoogle Scholar
  15. [Hor14]
    Horská, A.: Where is the Gödel-point Hiding: Gentzen’s Consistency Proof of 1936 and His Representation of Constructive Ordinals. Springer Briefs in Philosophy. Springer, Cham (2014)CrossRefzbMATHGoogle Scholar
  16. [Iba79]
    Ibarra, O.H.: Restricted one-counter machines with undecidable universe problems. Math. Syst. Theory 13, 181–186 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  17. [Jec02]
    Jech, T.: Set Theory, 3rd edn. Springer, Heidelberg (2002)zbMATHGoogle Scholar
  18. [JY81]
    Joseph, D., Young, P.: Independence results in computer science? J. Comput. Syst. Sci. 23(2), 205–222 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  19. [Kan97]
    Kanamori, A.: The Higher Infinite. Springer, Heidelberg (1997)CrossRefzbMATHGoogle Scholar
  20. [KM87]
    Kanamori, A., McAloon, K.: On Gödel incompleteness and finite combinatorics. Ann. Pure Appl. Logic 33(1), 23–41 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  21. [Kun80]
    Kunen, K.: Set Theory. An Introduction to Independence Proofs. Studies in Logic and the Foundations of Mathematics, vol. 102. North-Holland Publishing Co., Amsterdam, New York (1980)zbMATHGoogle Scholar
  22. [Poi00]
    Poizat, B.: A Course in Model Theory: An Introduction to Contemporary Mathematical Logic. Universitext. Springer, New York (2000). Translated from the French by Moses Klein and revised by the AuthorCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Institut de Mathématiques de Jussieu - Paris Rive GaucheCNRS et Université Paris 7ParisFrance

Personalised recommendations