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Deciding probabilistic automata weak bisimulation: theory and practice

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Formal Aspects of Computing

Abstract

Weak probabilistic bisimulation on probabilistic automata can be decided by an algorithm that needs to check a polynomial number of linear programming problems encoding weak transitions. It is hence of polynomial complexity. This paper discusses the specific complexity class of the weak probabilistic bisimulation problem, and it considers several practical algorithms and linear programming problem transformations that enable an efficient solution. We then discuss two different implementations of a probabilistic automata weak probabilistic bisimulation minimizer, one of them employing SAT modulo linear arithmetic as the solver technology. Empirical results demonstrate the effectiveness of the minimization approach on standard benchmarks, also highlighting the benefits of compositional minimization.

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References

  1. Arora S, Hazan E, Kale S (2012) The multiplicative weights update method: a meta-algorithm and applications. Theory Comput 8: 121–164

    Article  MathSciNet  MATH  Google Scholar 

  2. Ahuja RK, Magnanti TJ, Orlin JB (1993) Network flows: theory, algorithms, and applications. Prentice Hall, New York

    MATH  Google Scholar 

  3. Anstreicher KM (1999) Linear programming in \({\mathcal{O}(\frac{n^3}{{\rm \ln} n} L)}\) operations. SIAM J. Optim. 9(4): 803–812

    Article  MathSciNet  MATH  Google Scholar 

  4. Beling PA (2001) Exact algorithms for linear programming over algebraic extensions. Algorithmica 31(4): 459–478

    Article  MathSciNet  MATH  Google Scholar 

  5. Bahçeci U, Feyziog̃lu O (2012) A network simplex based algorithm for the minimum cost proportional flow problem with disconnected subnetworks. Optim Lett 6: 1173–1184

    Article  MathSciNet  MATH  Google Scholar 

  6. Böde E, Herbstritt M, Hermanns H, Johr S, Peikenkamp T, Pulungan R, Rakow J, Wimmer R, Becker B (2009) Compositional dependability evaluation for STATEMATE. ITSE 35(2): 274–292

    Article  Google Scholar 

  7. Blum L, Shub M, Smale M (1989) On a theory of computation and complexity over the real numbers; \({NP}\)-completeness, recursive functions and universal machines. Bull Am Math Soc 21(1): 1–46

    Article  MathSciNet  MATH  Google Scholar 

  8. Barrett C, Stump A, Tinelli C (2010) The SMT-LIB standard: version 2.0. In SMT

  9. Bertsimas D, Tsitsiklis JN (1997) Introduction to linear optimization. Athena Scientific, Belmont

    Google Scholar 

  10. Calvete HI (2002) Network simplex algorithm for the general equal flow problem. Eur J Oper Res 150(3): 585–600

    Article  MathSciNet  MATH  Google Scholar 

  11. Chehaibar G, Garavel H, Mounier L, Tawbi N, Zulian F (1996) Specification and verification of the \({{\rm PowerScale^{registered}}}\) bus arbitration protocol: an industrial experiment with LOTOS. In: FORTE, pp 435–450

  12. Crouzen P, Hermanns H (2010) Aggregation ordering for massively compositional models. In: ACSD, pp 171–180,

  13. Coste N, Hermanns H, Lantreibecq E, Serwe W (2009) Towards performance prediction of compositional models in industrial GALS designs. In: CAV. LNCS, vol 5643, pp 204–218

  14. Christiano P, Kelner JA, M¸dry A, Spielman D (2011) Electrical flows, laplacian systems, and faster approximation of maximum flow in undirected graphs. In: STOC, pp 273–282

  15. Crouzen P, Lang F (2011) Smart reduction. In FASE. LNCS, vol 6603, pp 111–126

  16. Cattani S, Segala R (2002) Decision algorithms for probabilistic bisimulation. In: CONCUR of LNCS, vol 2421, pp 371–385

  17. Deng Y (2005) Axiomatisations and types for probabilistic and mobile processes. PhD thesis, École des Mines de Paris

  18. Derman C (1970) Finite state markovian decision processes. Academic Press, Inc, New York

  19. Mendonça de Moura L, Bjørner N (2008) Z3: an efficient SMT solver. In: TACAS. LNCS, vol 4963, pp 337–340

  20. Eisentraut C, Hermanns H, Schuster J, Turrini A, Zhang L (2013) The quest for minimal quotients for probabilistic automata. In: TACAS. LNCS, vol 7795, pp 16–31

  21. Eisentraut C, Hermanns H, Zhang L (2010) Concurrency and composition in a stochastic world. In: CONCUR. LNCS, vol 6269, pp 21–39

  22. Eisentraut C, Hermanns H, Zhang L (2010) On probabilistic automata in continuous time. In: LICS, pp 342–351

  23. Gebler D, Hashemi V, Turrini A (2014) Computing behavioral relations for probabilistic concurrent systems. In: Stochastic model checking. Rigorous dependability analysis using model checking techniques for stochastic systems. LNCS, vol 8453. Springer Berlin Heidelberg, pp 117–155

  24. GNU linear programming kit. http://www.gnu.org/software/glpk/.

  25. Graf S, Steffen B, Lüttgen G (1996) Compositional minimisation of finite state systems using interface specifications. Formal Asp Comput 8(5): 607–616

    Article  MATH  Google Scholar 

  26. Hansson HA (1991) Time and probability in formal design of distributed systems. PhD thesis, Uppsala University

  27. Hashemi V, Hermanns H, Turrini A (2012) On the efficiency of deciding probabilistic automata weak bisimulation. ECEASST, vol 66: 66

    Google Scholar 

  28. Helgason RV, Kennington JL (1995) Primal simplex algorithms for minimum cost network flows. In: Network models. Handbooks in operations research and management science, vol 7, chapter 2. Elsevier, Amsterdam, pp 85–113

  29. Hermanns H, Katoen J-P (2000) Automated compositional Markov chain generation for a plain-old telephone system. Sci Comput Program 36(1): 97–127

    Article  MATH  Google Scholar 

  30. Hochbaum DS, Naor JS (1994) Simple and fast algorithms for linear and integer programs with two variables per inequality. SIAM J Comput 23(6): 1179–1192

    Article  MathSciNet  MATH  Google Scholar 

  31. Jonsson B, Larsen KG (1991) Specification and refinement of probabilistic processes. In: LICS, pp 266–277

  32. Jones WB, Thron WB (1980) Continued fractions: analytic theory and applications. In: Encyclopedia of mathematics and its applications. Addison-Wesley, New York

    Google Scholar 

  33. Karmarkar N (1984) A new polynomial-time algorithm for linear programming. Combinatorica 4(4): 373–395

    Article  MathSciNet  MATH  Google Scholar 

  34. Khachyan LG (1979) A polynomial algorithm in linear programming. Sov Math Doklady 20(1): 191–194

    Google Scholar 

  35. Katoen J-P, Kemna T, Zapreev IS, Jansen DN (2007) Bisimulation minimisation mostly speeds up probabilistic model checking. In: TACAS. LNCS, vol 4424, pp 76–92

  36. Klee V, Minty GJ (1972) How good is the simplex algorithm? In: Inequalities, vol III, pp 159–175. Defense Technical Information Center, USA

  37. Krimm J-P, Mounier L (2000) Compositional state space generation with partial order reductions for asynchronous communicating systems. In: TACAS. LNCS, vol 1785, pp 266–282

  38. Kwiatkowska M, Norman G, Parker D (2011) PRISM 4.0: verification of probabilistic real-time systems. In: CAV. LNCS, vol 6806, pp 585–591

  39. Kanellakis PC, Smolka SA (1990) CCS expressions, finite state processes, and three problems of equivalence. I&C. 86(1):43–68

  40. LpSolve mixed integer linear programming solver. http://lpsolve.sourceforge.net.

  41. Milner R (1989) Communication and concurrency. Prentice-Hall International, Englewood Cliffs

    MATH  Google Scholar 

  42. Morrison DR, Sauppe JJ, Jacobson SH (2011) A network simplex algorithm for the equal flow problem on a generalized network. INFORMS J Comput 25(1): 2–12

    Article  MathSciNet  Google Scholar 

  43. Morrison DR, Sauppe JJ, Jacobson SH (2013) An algorithm to solve the proportional network flow problem. Optim Lett 8(3): 801–809

    Article  MathSciNet  MATH  Google Scholar 

  44. Philippou A, Lee I, Sokolsky O (2000) Weak bisimulation for probabilistic systems. In: CONCUR. LNCS, vol 1877, pp 334–349

  45. PRISM model checker. http://www.prismmodelchecker.org/

  46. Parma A, Segala R (2004) Axiomatization of trace semantics for stochastic nondeterministic processes. In: QEST, pp 294–303

  47. Paige R, Tarjan RE (1987) Three partition refinement algorithms. SIAM J Comput 16(6): 973–989

    Article  MathSciNet  MATH  Google Scholar 

  48. Pulat PS (1989) A decomposition algorithm to determine the maximum flow in a generalized network. Comput Oper Res 16: 161–172

    Article  MathSciNet  MATH  Google Scholar 

  49. Schrijver A (2003) Combinatorial optimization: polyhedra and efficiency. In: Algorithms and combinatorics, vol 24. Springer, Berlin

    Google Scholar 

  50. Segala R (1995) Modeling and verification of randomized distributed real-time systems. PhD thesis, MIT

  51. Segala R (2006) Probability and nondeterminism in operational models of concurrency. In: CONCUR. LNCS, vol 4137, pp 64–78

  52. Shamir R (1987) The efficiency of the simplex method: a survey. Manag Sci 33(3): 301–334

    Article  MathSciNet  MATH  Google Scholar 

  53. Segala R, Lynch NA (1995) Probabilistic simulations for probabilistic processes. Nordic J Comput 2(2): 250–273

    MathSciNet  MATH  Google Scholar 

  54. Turrini A, Hermanns H (2015) Polynomial time decision algorithms for probabilistic automata. I&C 244:134–171

  55. Vardi MY (1985) Automatic verification of probabilistic concurrent finite-state programs. In: FOCS, pp 327–338

  56. Vazirani VV (2004) Approximation algorithms. Springer, Berlin

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Authors and Affiliations

  1. Department of Computer Science, Saarland University, Saarbrücken, Germany

    Luis María Ferrer Fioriti, Vahid Hashemi & Holger Hermanns

  2. Max Planck Institute for Informatics, Saarbrücken, Germany

    Vahid Hashemi

  3. State Key Laboratory of Computer Science, Institute of Software, Chinese Academy of Sciences, Beijing, China

    Andrea Turrini

Authors
  1. Luis María Ferrer Fioriti
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  2. Vahid Hashemi
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  3. Holger Hermanns
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  4. Andrea Turrini
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Correspondence to Andrea Turrini.

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Communicated by Joachim Parrow

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Ferrer Fioriti, L.M., Hashemi, V., Hermanns, H. et al. Deciding probabilistic automata weak bisimulation: theory and practice. Form Asp Comp 28, 109–143 (2016). https://doi.org/10.1007/s00165-016-0356-4

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