On Probabilistic Term Rewriting

  • Martin AvanziniEmail author
  • Ugo Dal Lago
  • Akihisa Yamada
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10818)


We study the termination problem for probabilistic term rewrite systems. We prove that the interpretation method is sound and complete for a strengthening of positive almost sure termination, when abstract reduction systems and term rewrite systems are considered. Two instances of the interpretation method—polynomial and matrix interpretations—are analyzed and shown to capture interesting and nontrivial examples when automated. We capture probabilistic computation in a novel way by means of multidistribution reduction sequences, thus accounting for both the nondeterminism in the choice of the redex and the probabilism intrinsic in firing each rule.


Term Rewriting Abstract Reduction Systems (ARS) Surement Results Polynomial Interpretations Bournez 
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.



We thank the anonymous reviewers for their constructive remarks that improved the paper. Example 12 is due to one of them. We thank Luis María Ferrer Fioriti for the analysis of a counterexample in [14]. This work is partially supported by the ANR projects 14CE250005 ELICA and 16CE250011 REPAS, the FWF project Y757, and JST ERATO HASUO Metamathematics for Systems Design Project (No. JPMJER1603).


  1. 1.
    Agha, G., Meseguer, J., Sen, K.: PMaude: rewrite-based specification language for probabilistic object systems. Electr. Notes Theor. Comput. Sci. 153(2), 213–239 (2006)CrossRefGoogle Scholar
  2. 2.
    Avanzini, M.: Verifying polytime computability automatically. Ph.D. thesis, University of Innsbruck (2013)Google Scholar
  3. 3.
    Avanzini, M., Dal Lago, U., Yamada, A.: On probabilistic term rewriting (Technical report). CoRR cs/CC/1802.09774 (2018).
  4. 4.
    Baader, F., Nipkow, T.: Term Rewriting and All That. Cambridge University Press, Cambridge (1998)CrossRefGoogle Scholar
  5. 5.
    Bournez, O., Garnier, F.: Proving positive almost-sure termination. In: Giesl, J. (ed.) RTA 2005. LNCS, vol. 3467, pp. 323–337. Springer, Heidelberg (2005). Scholar
  6. 6.
    Bournez, O., Garnier, F.: Proving positive almost sure termination under strategies. In: Pfenning, F. (ed.) RTA 2006. LNCS, vol. 4098, pp. 357–371. Springer, Heidelberg (2006). Scholar
  7. 7.
    Bournez, O., Kirchner, C.: Probabilistic rewrite strategies. Applications to ELAN. In: Proceedings of 13th RTA, pp. 252–266 (2002)CrossRefGoogle Scholar
  8. 8.
    Brémaud, P.: Marcov Chains. Springer, New York (1999). Scholar
  9. 9.
    Chatterjee, K., Fu, H., Goharshady, A.K.: Termination analysis of probabilistic programs through Positivstellensatz’s. In: Chaudhuri, S., Farzan, A. (eds.) CAV 2016. LNCS, vol. 9779, pp. 3–22. Springer, Cham (2016). Scholar
  10. 10.
    Dal Lago, U., Zorzi, M.: Probabilistic operational semantics for the lambda calculus. RAIRO - TIA 46(3), 413–450 (2012)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Dal Lago, U., Grellois, C.: Probabilistic termination by monadic affine sized typing. In: Proceedings of 26th ESOP, pp. 393–419 (2017)CrossRefGoogle Scholar
  12. 12.
    Dal Lago, U., Martini, S.: On constructor rewrite systems and the lambda calculus. LMCS 8(3), 1–27 (2012)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Endrullis, J., Waldmann, J., Zantema, H.: Matrix interpretations for proving termination of term rewriting. JAR 40(3), 195–220 (2008)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Ferrer Fioriti, L.M., Hermanns, H.: Probabilistic termination: soundness, completeness, and compositionality. In: Proceedings of 42nd POPL, pp. 489–501. ACM (2015)Google Scholar
  15. 15.
    Fuhs, C., Giesl, J., Middeldorp, A., Schneider-Kamp, P., Thiemann, R., Zankl, H.: SAT solving for termination analysis with polynomial interpretations. In: Marques-Silva, J., Sakallah, K.A. (eds.) SAT 2007. LNCS, vol. 4501, pp. 340–354. Springer, Heidelberg (2007). Scholar
  16. 16.
    Gnaedig, I.: Induction for positive almost sure termination. In: PPDP 2017, pp. 167–178. ACM (2007)Google Scholar
  17. 17.
    Goldwasser, S., Micali, S.: Probabilistic encryption. JCSS 28(2), 270–299 (1984)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Goodman, N.D., Mansinghka, V.K., Roy, D.M., Bonawitz, K., Tenenbaum, J.B.: Church: a language for generative models. In: Proceedings of 24th UAI, pp. 220–229. AUAI Press (2008)Google Scholar
  19. 19.
    Hirokawa, N., Moser, G.: Automated complexity analysis based on context-sensitive rewriting. In: Dowek, G. (ed.) RTA 2014. LNCS, vol. 8560, pp. 257–271. Springer, Cham (2014). Scholar
  20. 20.
    Kaminski, B.L., Katoen, J.: On the hardness of almost-sure termination. In: MFCS 2015, Proceedings, Part I, Milan, Italy, 24–28 August 2015, pp. 307–318 (2015)CrossRefGoogle Scholar
  21. 21.
    Lankford, D.: Canonical algebraic simplification in computational logic. Technical report ATP-25, University of Texas (1975)Google Scholar
  22. 22.
    Lucas, S.: Polynomials over the reals in proofs of termination: from theory to practice. ITA 39(3), 547–586 (2005)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Puterman, M.L.: Markov Decision Processes: Discrete Stochastic Dynamic Programming, 1st edn. Wiley, New York (1994)CrossRefGoogle Scholar
  24. 24.
    Rabin, M.O.: Probabilistic automata. Inf. Control 6(3), 230–245 (1963)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Saheb-Djahromi, N.: Probabilistic LCF. In: MFCS, pp. 442–451 (1978)CrossRefGoogle Scholar
  26. 26.
    Santos, E.S.: Probabilistic turing machines and computability. Proc. Am. Math. Soc. 22(3), 704–710 (1969)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Terese (ed.): Term Rewriting Systems. Cambridge Tracts in Theoretical Computer Science, vol. 55. Cambridge University Press, Cambridge (2003)Google Scholar
  28. 28.
    Yamada, A., Kusakari, K., Sakabe, T.: Nagoya termination tool. In: Dowek, G. (ed.) RTA 2014. LNCS, vol. 8560, pp. 466–475. Springer, Cham (2014). Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Martin Avanzini
    • 1
    Email author
  • Ugo Dal Lago
    • 1
    • 2
  • Akihisa Yamada
    • 3
  1. 1.Inria Sophia AntipolisValbonneFrance
  2. 2.Department of Computer ScienceUniversity of BolognaBolognaItaly
  3. 3.National Institute of InformaticsTokyoJapan

Personalised recommendations