Safraless Compositional Synthesis

  • Orna Kupferman
  • Nir Piterman
  • Moshe Y. Vardi
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4144)


In automated synthesis, we transform a specification into a system that is guaranteed to satisfy the specification. In spite of the rich theory developed for system synthesis, little of this theory has been reduced to practice. This is in contrast with model-checking theory, which has led to industrial development and use of formal verification tools. We see two main reasons for the lack of practical impact of synthesis. The first is algorithmic: synthesis involves determinization of automata on infinite words, and a solution of parity games with highly complex state spaces; both problems have been notoriously resistant to efficient implementation. The second is methodological: current theory of synthesis assumes a single comprehensive specification. In practice, however, the specification is composed of a set of properties, which is typically evolving – properties may be added, deleted, or modified.

In this work we address both issues. We extend the Safraless synthesis algorithm of Kupferman and Vardi so that it handles LTL formulas by translating them to nondeterministic generalized Büchi automata. This leads to an exponential improvement in the complexity of the algorithm. Technically, our algorithm reduces the synthesis problem to the emptiness problem of a nondeterministic Büchi tree automaton \({\cal A}\). The generation of \({\cal A}\) avoids determinization, avoids the parity acceptance condition, and is based on an analysis of runs of universal generalized co-Büchi tree automata. The clean and simple structure of \({\cal A}\) enables optimizations and a symbolic implementation. In addition, it makes it possible to use information gathered during the synthesis process of properties in the process of synthesizing their conjunction.


Winning Strategy Regular Tree Tree Automaton Program Synthesis Parity Game 
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. 1.
    Abadi, M., Lamport, L., Wolper, P.: Realizable and unrealizable concurrent program specifications. In: Ronchi Della Rocca, S., Ausiello, G., Dezani-Ciancaglini, M. (eds.) ICALP 1989. LNCS, vol. 372, pp. 1–17. Springer, Heidelberg (1989)CrossRefGoogle Scholar
  2. 2.
    Althoff, C.S., Thomas, W., Wallmeier, N.: Observations on determinization of büchi automata. In: Farré, J., Litovsky, I., Schmitz, S. (eds.) CIAA 2005. LNCS, vol. 3845, pp. 262–272. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  3. 3.
    Choueka, Y.: Theories of automata on ω-tapes: A simplified approach. JCSS 8, 117–141 (1974)MATHMathSciNetGoogle Scholar
  4. 4.
    Church, A.: Logic, arithmetics, and automata. In: ICM 1962, pp. 23–35 (1963)Google Scholar
  5. 5.
    Clarke, E.M., Grumberg, O., Peled, D.: Model Checking. MIT Press, Cambridge (1999)Google Scholar
  6. 6.
    Dill, D.L.: Trace theory for automatic hierarchical verification of speed independent circuits. MIT Press, Cambridge (1989)Google Scholar
  7. 7.
    Elgaard, J., Klarlund, N., Möller, A.: Mona 1.x: new techniques for WS1S and WS2S. In: Y. Vardi, M. (ed.) CAV 1998. LNCS, vol. 1427, pp. 516–520. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  8. 8.
    Emerson, E.A.: Automata, tableaux, and temporal logics. In: Parikh, R. (ed.) Logic of Programs 1985. LNCS, vol. 193, pp. 79–87. Springer, Heidelberg (1985)Google Scholar
  9. 9.
    Gerth, R., Peled, D., Vardi, M.Y., Wolper, P.: Simple on-the-fly automatic verification of linear temporal logic. In: Protocol Specification, Testing, and Verification, pp. 3–18 (1995)Google Scholar
  10. 10.
    Grädel, E., Thomas, W., Wilke, T.: Automata, Logics, and Infinite Games. LNCS, vol. 2500. Springer, Heidelberg (2002)MATHCrossRefGoogle Scholar
  11. 11.
    Gurumurthy, S., Kupferman, O., Somenzi, F., Vardi, M.Y.: On complementing nondeterministic Büchi automata. In: Geist, D., Tronci, E. (eds.) CHARME 2003. LNCS, vol. 2860, pp. 96–110. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  12. 12.
    Harding, A., Ryan, M., Schobbens, P.Y.: A new algorithm for strategy synthesis in ltl games. In: Halbwachs, N., Zuck, L.D. (eds.) TACAS 2005. LNCS, vol. 3440, pp. 477–492. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  13. 13.
    JurzińSki, M.: Small progress measures for solving parity games. In: Reichel, H., Tison, S. (eds.) STACS 2000. LNCS, vol. 1770, pp. 290–301. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  14. 14.
    Kupferman, O., Vardi, M.Y.: From complementation to certification. In: Jensen, K., Podelski, A. (eds.) TACAS 2004. LNCS, vol. 2988, pp. 591–606. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  15. 15.
    Kupferman, O., Vardi, M.Y.: Safraless decision procedures. In: 46th FOCS (2005)Google Scholar
  16. 16.
    Lichtenstein, O., Pnueli, A.: Checking that finite state concurrent programs satisfy their linear specification. In: 12th POPL, pp. 97–107 (1985)Google Scholar
  17. 17.
    Manna, Z., Pnueli, A.: The Temporal Logic of Reactive and Concurrent Systems: Specification. Springer, Heidelberg (1992)Google Scholar
  18. 18.
    Miyano, S., Hayashi, T.: Alternating finite automata on ω-words. TCS 32, 321–330 (1984)MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Mostowski, A.W.: Regular expressions for infinite trees and a standard form of automata. In: Skowron, A. (ed.) SCT 1984. LNCS, vol. 208, pp. 157–168. Springer, Heidelberg (1985)Google Scholar
  20. 20.
    Muller, D.E., Saoudi, A., Schupp, P.E.: Alternating automata, the weak monadic theory of the tree and its complexity. In: Kott, L. (ed.) ICALP 1986. LNCS, vol. 226. Springer, Heidelberg (1986)Google Scholar
  21. 21.
    Piterman, N.: From nondeterministic Büchi and Streett automata to deterministic parity automata. In: 25th LICS (to appear, 2006)Google Scholar
  22. 22.
    Piterman, N., Pnueli, A., Saar, Y.: Design synthesis in action: Solving a 2exptime-complete problem in n 3. In: Emerson, E.A., Namjoshi, K.S. (eds.) VMCAI 2006. LNCS, vol. 3855, pp. 364–380. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  23. 23.
    Pnueli, A., Rosner, R.: On the synthesis of a reactive module. In: 16th POPL, pp. 179–190 (1989)Google Scholar
  24. 24.
    Rabin, M.O.: Weakly definable relations and special automata. In: Symp. Math. Logic and Foundations of Set Theory, pp. 1–23 (1970)Google Scholar
  25. 25.
    Rabin, M.O.: Automata on infinite objects and Church’s problem. In: AMS (1972)Google Scholar
  26. 26.
    Rosner, R.: Modular Synthesis of Reactive Systems. PhD thesis, Weizmann Institute of Science (1992)Google Scholar
  27. 27.
    Safra, S.: On the complexity of ω-automata. In: 29th FOCS, pp. 319–327 (1988)Google Scholar
  28. 28.
    Safra, S.: Exponential determinization for ω-automata with strong-fairness acceptance condition. In: 24th STOC (1992)Google Scholar
  29. 29.
    Tasiran, S., Hojati, R., Brayton, R.K.: Language containment using non-deterministic omega-automata. In: Camurati, P.E., Eveking, H. (eds.) CHARME 1995. LNCS, vol. 987, pp. 261–277. Springer, Heidelberg (1995)Google Scholar
  30. 30.
    Vardi, M.Y., Wolper, P.: Automata-theoretic techniques for modal logics of programs. JCSS 32(2), 182–221 (1986)MathSciNetGoogle Scholar
  31. 31.
    Vardi, M.Y., Wolper, P.: Reasoning about infinite computations. IC 115(1), 1–37 (1994)MATHMathSciNetGoogle Scholar
  32. 32.
    Yang, J., Seger, C.J.H.: Introduction to generalized symbolic trajectory evaluation. In: 19th DAC, pp. 360–367. IEEE, Los Alamitos (2001)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Orna Kupferman
    • 1
  • Nir Piterman
    • 2
  • Moshe Y. Vardi
    • 3
  1. 1.Hebrew University 
  2. 2.Ecole Polytechnique Fédéral de Lausanne (EPFL) 
  3. 3.Rice University and Microsoft Research 

Personalised recommendations