Formal Methods in System Design

, Volume 36, Issue 3, pp 198–222 | Cite as

Partially-shared zero-suppressed multi-terminal BDDs: concept, algorithms and applications

  • Kai Lampka
  • Markus Siegle
  • Joern Ossowski
  • Christel Baier
Article

Abstract

Multi-Terminal Binary Decision Diagrams (MTBDDs) are a well accepted technique for the state graph (SG) based quantitative analysis of large and complex systems specified by means of high-level model description techniques. However, this type of Decision Diagram (DD) is not always the best choice, since finite functions with small satisfaction sets, and where the fulfilling assignments possess many 0-assigned positions, may yield relatively large MTBDD based representations. Therefore, this article introduces zero-suppressed MTBDDs and proves that they are canonical representations of multi-valued functions on finite input sets. For manipulating DDs of this new type, possibly defined over different sets of function variables, the concept of partially-sharedzero-suppressed MTBDDs and respective algorithms are developed. The efficiency of this new approach is demonstrated by comparing it to the well-known standard type of MTBDDs, where both types of DDs have been implemented by us within the C++-based DD-package JINC. The benchmarking takes place in the context of Markovian analysis and probabilistic model checking of systems. In total, the presented work extends existing approaches, since it not only allows one to directly employ (multi-terminal) zero-suppressed DDs in the field of quantitative verification, but also clearly demonstrates their efficiency.

Keywords

Binary Decision Diagrams and their algorithms Quantitative verification of systems Symbolic data structures for performance analysis 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Formal Methods in System Design (1997) 10(2–3). Special Issue on Multi-Terminal Binary Decision Diagrams Google Scholar
  2. 2.
    Akers SB (1978) Binary decision diagrams. IEEE Trans Comput C-27(6):509–516 CrossRefGoogle Scholar
  3. 3.
    Balbo G, Conte G, Donatelli S, Franceschinis G, Ajmone Marsan M, Ajmone Marsan M (1995) Modelling with generalized stochastic Petri nets. Wiley, New York MATHGoogle Scholar
  4. 4.
    Bryant RE (1986) Graph-based algorithms for Boolean function manipulation. IEEE Trans Comput C-35(8):677–691 CrossRefGoogle Scholar
  5. 5.
    Ciardo G, Lüttgen G, Miner AS (2007) Exploiting interleaving semantics in symbolic state-space generation. Form Methods Syst Des 31(1):63–100 MATHCrossRefGoogle Scholar
  6. 6.
    de Alfaro L, Kwiatkowska M, Norman G, Parker D, Segala R (2000) Symbolic model checking for probabilistic processes using MTBDDs and the Kronecker representation. In: Graf S, Schwartzbach M (eds) Proc. of the 6th int. conference on tools and algorithms for the construction and analysis of systems (TACAS’00). LNCS, vol 1785. Springer, Berlin, pp 395–410 CrossRefGoogle Scholar
  7. 7.
    Hermanns H, Herzog U, Mertsiotakis V (1998) Stochastic process algebras—between LOTOS and Markov chains. Comput Netw ISDN Syst 30(9–10):901–924 CrossRefGoogle Scholar
  8. 8.
    Hermanns H, Kwiatkowska M, Norman G, Parker D, Siegle M (2003) On the use of MTBDDs for performability analysis and verification of stochastic systems. J Log Algebr Program 56(1–2):23–67 MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    JINC BDD package. www.jossowski.de
  10. 10.
    Kam T, Villa T, Brayton R, Sangiovanni-Vincentelli A (1998) Multi-valued decision diagrams: theory and applications. Mult Valued Log 4(1–2):9–62 MATHMathSciNetGoogle Scholar
  11. 11.
    Kuntz M, Siegle M, Werner E (2004) Symbolic performance and dependability evaluation with the tool CASPA. In: Proc. of EPEW. LNCS, vol 3236. Springer, Berlin, pp 293–307 Google Scholar
  12. 12.
    Lampka K, Siegle M (2006) Activity-local state graph generation for high-level stochastic models. In: Meassuring, modeling, and evaluation of systems 2006, April 2006, pp 245–264 Google Scholar
  13. 13.
    Lampka K, Siegle M (2006) Analysis of Markov reward models using zero-supressed multi-terminal decision diagrams. In: Proceedings of VALUETOOLS 2006 (CD-edition), October 2006 Google Scholar
  14. 14.
    Lee CY (1959) Representation of switching circuits by binary-decision programs. Bell Syst Tech J 38:985–999 Google Scholar
  15. 15.
    Minato S (1993) Zero-suppressed BDDs for set manipulation in combinatorial problems. In: Proc. of the 30th ACM/IEEE design automation conference (DAC), Dallas (Texas), USA, June 1993, pp 272–277 Google Scholar
  16. 16.
    Minato S (2001) Zero-suppressed BDDs and their applications. Int J Softw Tools Technol Transf 3(2):156–170 MATHGoogle Scholar
  17. 17.
    Miner A, Parker D (2004) Symbolic representations and analysis of large state spaces. In: Baier Ch, Haverkort B, Hermanns H, Katoen J-P, Siegle M (eds) Validation of stochastic systems, Dagstuhl (Germany), 2004. LNCS, vol 2925. Springer, Berlin, pp 296–338 CrossRefGoogle Scholar
  18. 18.
    Möbius web page. www.moebius.crhc.uiuc.edu
  19. 19.
    Ossowski J, Baier C (2008) A uniform framework for weighted decision diagrams and its implementation. Int J Softw Tools Technol Transf 10(5):425–441 CrossRefGoogle Scholar
  20. 20.
  21. 21.
    PROMOC modeling tool. www.jossowski.de
  22. 22.
    Sasao T, Fujita M (eds) (1996) Representations of discrete functions, vol 1. Kluwer Academic, Dordrecht MATHGoogle Scholar
  23. 23.
    Shannon CS (2000) Eine symbolische Analyse von Relaisschaltkreisen. In: Ein/Aus. Brinkmann und Bose, Berlin. The article originally appeared with the title: A symbolic analysis of switching circuits in Trans. AIEE 57 (1938), 713 Google Scholar
  24. 24.
    Siegle M (2001) Advances in model representation. In: de Alfaro L, Gilmore S (eds) Proc. of the joint int. workshop, PAPM-PROBMIV 2001, Aachen (Germany). LNCS, vol 2165. Springer, Berlin, pp 1–22 Google Scholar
  25. 25.
  26. 26.
    Somenzi F (1998) CUDD: Colorado University decision diagram package release Google Scholar
  27. 27.
    Wegener I (2000) Branching programs and binary decision diagrams. SIAM, Philadelphia MATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  • Kai Lampka
    • 1
  • Markus Siegle
    • 2
  • Joern Ossowski
    • 3
  • Christel Baier
    • 3
  1. 1.Computer Engineering and Communication Networks Lab.ETH ZurichZurichSwitzerland
  2. 2.Inst. for Comp. Eng.Univ. of the German Federal Armed Forces MunichMunichGermany
  3. 3.Inst. for Theoretical Comp. Sc.Technical University DresdenDresdenGermany

Personalised recommendations