Church Synthesis Problem with Parameters

  • Alexander Rabinovich
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4207)


The following problem is known as the Church Synthesis problem:

  • Input: an \({\mathit{MLO}}\) formula ψ(X,Y).

  • Task: Check whether there is an operator Y = F(X) such that
    $$Nat \models \forall X \psi(X,F(X))$$
    and if so, construct this operator.

Büchi and Landweber proved that the Church synthesis problem is decidable; moreover, they proved that if there is an operator F which satisfies ([1]), then ([1]) can be satisfied by the operator defined by a finite state automaton. We investigate a parameterized version of the Church synthesis problem. In this version ψ might contain as a parameter a unary predicate P. We show that the Church synthesis problem for P is computable if and only if the monadic theory of \(\langle{\mathit{Nat},<,P}\rangle\) is decidable. We also show that the Büchi-Landweber theorem can be extended only to ultimately periodic parameters.


Synthesis Problem Winning Strategy Unary Predicate Game Graph Deterministic Automaton 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [Bu60]
    Büchi, J.R.: On a decision method in restricted second order arithmetic. In: Nagel, E., et al. (eds.) Proc. International Congress on Logic, Methodology and Philosophy of Science, pp. 1–11. Stanford University Press (1960)Google Scholar
  2. [BL69]
    Büchi, J.R., Landweber, L.H.: Solving sequential conditions by finitestate strategies. Transactions of the AMS 138(27), 295–311 (1969)CrossRefGoogle Scholar
  3. [CDT02]
    Cachat, T., Duparc, J., Thomas, W.: Solving pushdown games with a Σ3 winning condition. In: Bradfield, J.C. (ed.) CSL 2002. LNCS, vol. 2471, pp. 322–336. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  4. [CT02]
    Carton, O., Thomas, W.: The Monadic Theory of Morphic Infinite Words and Generalizations. Inf. Comput. 176(1), 51–65 (2002)MATHCrossRefMathSciNetGoogle Scholar
  5. [Ch69]
    Choueka, Y.: Finite Automata on Infinite Structure. Ph.D Thesis, Hebrew University (1970)Google Scholar
  6. [ER66]
    Elgot, C., Rabin, M.O.: Decidability and Undecidability of Extensions of Second (First) Order Theory of (Generalized) Successor. J. Symb. Log. 31(2), 169–181 (1966)MATHCrossRefGoogle Scholar
  7. [EJ91]
    Emerson, E.A., Jutla, C.S.: Tree Automata, Mu-Calculus and Determinacy (Extended Abstract). In: FOCS 1991, pp. 368–377 (1991)Google Scholar
  8. [GTW02]
    Grädel, E., Thomas, W., Wilke, T.: Automata, Logics, and Infinite Games. In: Grädel, E., Thomas, W., Wilke, T. (eds.) Automata, Logics, and Infinite Games. LNCS, vol. 2500. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  9. [KV97]
    Kupferman, O., Vardi, M.Y.: Synthesis with incomplete information. In: 2nd International Conference on Temporal Logic, pp. 91–106 (1997)Google Scholar
  10. [PP04]
    Perrin, D., Pin, J.E.: Infinite Words Automata, Semigroups, Logic and Games. Pure and Applied Mathematics, vol. 141. Elsevier, Amsterdam (2004)MATHGoogle Scholar
  11. [Rab72]
    Rabin, M.O.: Automata on Infinite Objects and Church’s Problem. Amer. Math. Soc., Providence (1972)MATHGoogle Scholar
  12. [RT98]
    Rabinovich, A., Trakhtenbrot, B.A.: From Finite Automata toward Hybrid Systems. In: Brim, L., Gruska, J., Zlatuška, J. (eds.) MFCS 1998. LNCS, vol. 1450, pp. 411–422. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  13. [Rab05]
    Rabinovich, A.: On decidability of Monadic logic of order over the naturals extended by monadic predicates (submitted, 2005)Google Scholar
  14. [Rab06]
    Rabinovich, A.: The Church problem over ω expanded by factorial numbers (in preparation, 2006)Google Scholar
  15. [Sem84]
    Semenov, A.: Logical theories of one-place functions on the set of natural numbers. Mathematics of the USSR - Izvestia 22, 587–618 (1984)MATHCrossRefGoogle Scholar
  16. [Rob58]
    Robinson, R.M.: Restricted Set-Theoretical Definitions in Arithmetic. Proceedings of the AMS 9(2), 238–242 (1958)MATHCrossRefGoogle Scholar
  17. [Se04]
    Serre, O.: Games With Winning Conditions of High Borel Complexity. In: Díaz, J., Karhumäki, J., Lepistö, A., Sannella, D. (eds.) ICALP 2004. LNCS, vol. 3142, pp. 1150–1162. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  18. [Shel75]
    Shelah, S.: The monadic theory of order. Ann. of Math. 102, 379–419 (1975)CrossRefMathSciNetGoogle Scholar
  19. [Sie75]
    Siefkes, D.: The recursive sets in certain monadic second order fragments of arithmetic. Arch. Math. Logik 17, 71–80 (1975)MATHCrossRefMathSciNetGoogle Scholar
  20. [Th75]
    Thomas, W.: Das Entscheidungsproblem für einige Erweiterungen der Nachfalger-Arithmetic. Ph. D. Thesis Albert-Ludwigs Universität (1975)Google Scholar
  21. [Th95]
    Thomas, W.: On the synthesis of strategies in infinite games. In: Mayr, E.W., Puech, C. (eds.) STACS 1995. LNCS, vol. 900, pp. 1–13. Springer, Heidelberg (1995)Google Scholar
  22. [Th03]
    Thomas, W.: Constructing infinite graphs with a decidable MSO-theory. In: Rovan, B., Vojtáš, P. (eds.) MFCS 2003. LNCS, vol. 2747, pp. 113–124. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  23. [Trak61]
    Trakhtenbrot, B.A.: Finite automata and the logic of one-place predicates (Russian version 1961). AMS Transl. 59, 23–55 (1966)Google Scholar
  24. [Wal01]
    Walukiewicz, I.: Pushdown processes: games and model checking. Information and Computation 164, 234–263 (2001)MATHCrossRefMathSciNetGoogle Scholar
  25. [Wal02]
    Walukiewicz, I.: Monadic second order logic on tree-like structures. TCS 1, 311–346 (2002)CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Alexander Rabinovich
    • 1
  1. 1.Dept. of CSTel Aviv Univ. 

Personalised recommendations