Combinatorial Criteria Over Graphs of Specification to Decide Synthesis by Sequential Circuits

  • Y. P. Tison
  • P. Simonnet


Here we present some algorithms which decide, for a given functional specification, whether the function is continuous and whether the function is sequential. When the specification is synchronous (i.e the graph of the function is realized by a synchronous automata) then these two notions coincide with asynchronous sequential functions with bounded delay. We give an example where Büchi’s synthesis by a synchronous sequential function is not possible, but synthesis by an asynchronous sequential function with bounded delay is possible. When the specification is asynchronous, we present an example of a continuous but not sequential function, and we give a sufficient criterion to prove that a function is not sequential.


Finite Automaton Winning Strategy Sequential Circuit Computation Tree Logic Finite Alphabet 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Arnold, A., “Rational languages are unambiguous”, Theor. Comp. Sci. 26, 1983 pp. 221–223.Google Scholar
  2. [2]
    Berstel, J., “Transductions and Context-Free Languages”, Stuttgart, Teubner, 1979.MATHGoogle Scholar
  3. [3]
    Büchi, J.R., “On a decision method in restricted second order arithmetic”, in Proc. Int. Congr. Logic, Method. and Philos. of Science ( E. Nagel et al., eds.), Stanford Univ. Press, Stanford, 1962, pp. 1–11.Google Scholar
  4. [4]
    Büchi, J.R., Landweber, L.H., “Solving sequential conditions by finite-state strategies”, Trans. Amer. Math. Soc. 138, 1969, pp. 295–311.Google Scholar
  5. [5]
    Burch, J.R., Clarke, E.M., Mc Millan, K.L., Dill, D.L., Hwang, L.J., “Symbolic model checking: 1020 states and beyond”, Informations and Computation 98 (2), 1992, pp. 142–170.MATHCrossRefGoogle Scholar
  6. [6]
    Büttner, W., Winkelmann, K., “Equation solving over 2-adic integers and applications to the specification, verification and synthesis of finite state machines”, 1995, to be published.Google Scholar
  7. [7]
    Choffrut, C., “Une caractérisation des fonctions séquentielles et des fonctions sous-séquentielles en tant que relations rationnelles”, Theoretical Computer Science 5, 1977, pp. 325–338.MathSciNetCrossRefGoogle Scholar
  8. [8]
    Church, A., “Logic, arithmetic and automata”, Proc. Intern. Congr. Math. 1962, Almquist and Wiksells, Uppsala, 1963, pp. 21–35.Google Scholar
  9. [9]
    Clarke, E., Grumberg, O., Long, D., “Verification tools for finite-state concurrent systems”, A Decade of Concurrency (J.W. de Bakker et al., eds), Lecture Notes in Computer Science 803, Springer-Verlag, Berlin, 1994, pp. 124–175.Google Scholar
  10. [10]
    Eilenberg, S., “Automata, Languages and Machines”, vol A, Academic Press, New York, 1974.MATHGoogle Scholar
  11. [11]
    Frougny, C., Sakarovitch, J., “Rational relations with bounded delay”, Rapport d’activité Laboratoire Informatique Théorique et Programmation 90.83, Institut Blaise Pascal, 1990.Google Scholar
  12. [12]
    Ginsburg, S., Rose, G.F., “A Characterization of machine mappings”, Canadian Journal of Mathematics 18, 1966, pp. 381–388.MathSciNetMATHCrossRefGoogle Scholar
  13. [13]
    Landweber, L.H., “Decision problems for w-automata”, Math. Syst. Theory 3, 1969, pp. 376–384.Google Scholar
  14. [14]
    Latteux, M., Timmerman, E., “Rational w-transductions”, Laboratoire d’Informatique fondamentale de Lille, Publication n IT 176 90, 1990.Google Scholar
  15. [15]
    Mc Naughton, R., “Testing and generating infinite sequences by a finite automaton”, Inform. and Control 9, 1966, pp. 521–530.CrossRefGoogle Scholar
  16. [16]
    Moschovakis, Y.N., “Descriptive set theory”, North-Holland, 1980.Google Scholar
  17. [17]
    Nivat, M., Perrin, D., “Automata on infinite words”, Ecole de Printemps LNCS 192, 1984.Google Scholar
  18. [18]
    Nökel, K., Winkelmann, K., “Controller synthesis and verification: a case study”, C. Lewerentz, Th. Linder, Formal Development of Reactive Systems, Case Study Production Cell, Springer Lecture Notes in Computer Science, 891, Berlin, Heidelberg, 1995.Google Scholar
  19. [19]
    Perrin, D., Pin, J.E., “Mots infinis”, Laboratoire Informatique Théorique et Programmation, LITP 93. 40, 1993.Google Scholar
  20. [20]
    Staiger, L., “Sequential mappings of w-languages”, Math. Syst. Theory 3, 1987, pp. 376–384.Google Scholar
  21. [21]
    Thomas, W., 1990, “Automata on infinite objects”, Handbook of Theoretical Computer Science, Vol B, North-Holland, Amsterdam.Google Scholar
  22. [22]
    Thomas, W., 1994, “On the synthesis of strategies in infinite games”, Institut für Informatik und Praktische Matematik, Christian-AlbrechtsUniversität Kiel.Google Scholar
  23. [23]
    Trakhtenbrot, B.A., Barzdin, Y.M., “Finite automata”, North-Holland, Amsterdam, 1973.MATHGoogle Scholar
  24. [24]
    Vuillemin, J., “On circuits and numbers”, IEEE Trans. on Computers 43:8: 868–27, 79, 1994.MathSciNetCrossRefGoogle Scholar
  25. [25]
    Wagner, K., “On ω-regular sets, Information and control 43, 1979, pp. 123–177.Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 1997

Authors and Affiliations

  • Y. P. Tison
    • 1
  • P. Simonnet
    • 1
  1. 1.Centre de Mathématiques et de Calcul ScientifiqueUniversité de CorseCorteFrance

Personalised recommendations