Behavioral Rewrite Systems and Behavioral Productivity

  • Grigore Roşu
  • Dorel Lucanu
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8373)


This paper introduces behavioral rewrite systems, where rewriting is used to evaluate experiments, and behavioral productivity, which says that each experiment can be fully evaluated, and investigates some of their properties. First, it is shown that, in the case of (infinite) streams, behavioral productivity generalizes and may bring to a more basic rewriting setting the existing notion of stream productivity defined in the context of infinite rewriting and lazy strategies; some arguments are given that in some cases one may prefer the behavioral approach. Second, a behavioral productivity criterion is given, which reduces the problem to conventional term rewrite system termination, so that one can use off-the-shelf termination tools and techniques for checking behavioral productivity in general, not only for streams. Finally, behavioral productivity is shown to be equivalent to a proof-theoretic (rather than model-theoretic) notion of behavioral well-specifiedness, and its difficulty in the arithmetic hierarchy is shown to be \(\Pi_2^0\)-complete. All new concepts are exemplified over streams, infinite binary trees, and processes.


Binary Tree Behavioral Signature Stream Productivity Termination Tool Arithmetic Hierarchy 
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.
    Bidoit, M., Hennicker, R.: Observer complete definitions are behaviourally coherent. In: OBJ/CAFEOBJ/MAUDE AT FORMAL METHODS 1999, pp. 83–94. THETA (1999)Google Scholar
  2. 2.
    Bidoit, M., Hennicker, R., Kurz, A.: Observational logic, constructor-based logic, and their duality. Theoretical Computer Science 3(298), 471–510 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Buss, S., Roşu, G.: Incompleteness of behavioral logics. In: Proceeding of CMCS 2000. ENTCS, vol. 33, pp. 61–79. Elsevier (2000)Google Scholar
  4. 4.
    Diaconescu, R., Futatsugi, K.: CafeOBJ Report. AMAST Series in Computing, vol. 6. World Scientific (1998)Google Scholar
  5. 5.
    Dijkstra, E.W.: On the productivity of recursive definitions. EWD749 (September 1980)Google Scholar
  6. 6.
    Durán, F., Lucas, S., Meseguer, J.: Mtt: The maude termination tool (system description). In: Armando, A., Baumgartner, P., Dowek, G. (eds.) IJCAR 2008. LNCS (LNAI), vol. 5195, pp. 313–319. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  7. 7.
    Endrullis, J., Geuvers, J., Zantema, H.: Degrees of undecidability in term rewriting. In: Grädel, E., Kahle, R. (eds.) CSL 2009. LNCS, vol. 5771, pp. 255–270. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  8. 8.
    Endrullis, J., Grabmayer, C., Hendriks, D.: Data-oblivious stream productivity. In: Cervesato, I., Veith, H., Voronkov, A. (eds.) LPAR 2008. LNCS (LNAI), vol. 5330, pp. 79–96. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  9. 9.
    Endrullis, J., Grabmayer, C., Hendriks, D.: Complexity of fractran and productivity. In: Schmidt, R.A. (ed.) CADE 2009. LNCS (LNAI), vol. 5663, pp. 371–387. Springer, Heidelberg (2009)Google Scholar
  10. 10.
    Endrullis, J., Grabmayer, C., Hendriks, D., Isihara, A., Klop, J.W.: Productivity of stream definitions. Theor. Comput. Sci. 411(4-5), 765–782 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Endrullis, J., Hendriks, D., Bakhshi, R.: On the Complexity of Equivalence of Specifications of Infinite Objects. In: Proc. ACM SIGPLAN Int. Conf. on Functional Programming (ICFP 2013), pp. 153–164. ACM (2012)Google Scholar
  12. 12.
    Giesl, J., Thiemann, R., Schneider-Kamp, P., Falke, S.: Automated Termination Proofs with AProVE. In: van Oostrom, V. (ed.) RTA 2004. LNCS, vol. 3091, pp. 210–220. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  13. 13.
    Goguen, J.A., Diaconescu, R.: Towards an algebraic semantics for the object paradigm. In: Ehrig, H., Orejas, F. (eds.) Abstract Data Types 1992 and COMPASS 1992. LNCS, vol. 785, pp. 1–29. Springer, Heidelberg (1994)CrossRefGoogle Scholar
  14. 14.
    Hausmann, D., Mossakowski, T., Schröder, L.: Iterative Circular Coinduction for CoCASL in Isabelle/HOL. In: Cerioli, M. (ed.) FASE 2005. LNCS, vol. 3442, pp. 341–356. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  15. 15.
    Kapur, D., Narendran, P., Rosenkrantz, D.J., Zhang, H.: Sufficient-completeness, ground-reducibility and their complexity. Acta Inf. 28(4), 311–350 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Lucanu, D., Goriac, E.-I., Caltais, G., Roşu, G.: CIRC: A behavioral verification tool based on circular coinduction. In: Kurz, A., Lenisa, M., Tarlecki, A. (eds.) CALCO 2009. LNCS, vol. 5728, pp. 433–442. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  17. 17.
    Mossakowski, T., Schröder, L., Roggenbach, M., Reichel, H.: Algebraic-coalgebraic specification in CoCASL. J. Log. Alg. Program. 67(1-2), 146–197 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Reichel, H.: Behavioural equivalence – a unifying concept for initial and final specifications. In: The 3rd Hungarian Comp. Sci. Conference, Akademiai Kiado (1981)Google Scholar
  19. 19.
    Roşu, G.: Hidden Logic. PhD thesis, University of California at San Diego (2000)Google Scholar
  20. 20.
    Roşu, G.: Equality of streams is a \(\Pi_2^0\)-complete problem. In: Proceedgins of ICFP 2006, pp. 184–191. ACM (2006)Google Scholar
  21. 21.
    Roşu, G., Goguen, J.: Hidden congruent deduction. In: Caferra, R., Salzer, G. (eds.) FTP 1998. LNCS (LNAI), vol. 1761, pp. 251–266. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  22. 22.
    Roşu, G., Lucanu, D.: Circular Coinduction – A Proof Theoretical Foundation. In: Kurz, A., Lenisa, M., Tarlecki, A. (eds.) CALCO 2009. LNCS, vol. 5728, pp. 127–144. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  23. 23.
    Rogers, H.: Theory of Recursive Functions and Effective Computability. Paperback edn. The MIT Press (1987)Google Scholar
  24. 24.
    Sijtsma, B.A.: On the productivity of recursive list definitions. ACM Trans. Program. Lang. Syst. 11(4), 633–649 (1989)CrossRefGoogle Scholar
  25. 25.
    Silva, A., Rutten, J.: Behavioural differential equations and coinduction for binary trees. In: Leivant, D., de Queiroz, R. (eds.) WoLLIC 2007. LNCS, vol. 4576, pp. 322–336. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  26. 26.
    Simonsen, J.G.: The \(\Pi_2^0\)-completeness of most of the properties of rewriting systems you care about (and productivity). In: Treinen, R. (ed.) RTA 2009. LNCS, vol. 5595, pp. 335–349. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  27. 27.
    Terese: Term Rewriting Systems. Cambridge Tracts in Theoretical Computer Science, vol. 55. Cambridge University Press (2003)Google Scholar
  28. 28.
    Zantema, H.: A tool proving well-definedness of streams using termination tools. In: Kurz, A., Lenisa, M., Tarlecki, A. (eds.) CALCO 2009. LNCS, vol. 5728, pp. 449–456. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  29. 29.
    Zantema, H.: Well-definedness of streams by termination. In: Treinen, R. (ed.) RTA 2009. LNCS, vol. 5595, pp. 164–178. Springer, Heidelberg (2009)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Grigore Roşu
    • 1
    • 2
  • Dorel Lucanu
    • 2
  1. 1.University of Illinois at Urbana-ChampaignUSA
  2. 2.Alexandru Ioan Cuza UniversityIaşiRomania

Personalised recommendations