Verifying Balanced Trees

  • Zohar Manna
  • Henny B. Sipma
  • Ting Zhang
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4514)


Balanced search trees provide guaranteed worst-case time performance and hence they form a very important class of data structures. However, the self-balancing ability comes at a price; balanced trees are more complex than their unbalanced counterparts both in terms of data structure themselves and related manipulation operations. In this paper we present a framework to model balanced trees in decidable first-order theories of term algebras with Presburger arithmetic. In this framework, a theory of term algebras (i.e., a theory of finite trees) is extended with Presburger arithmetic and with certain connecting functions that map terms (trees) to integers. Our framework is flexible in the sense that we can obtain a variety of decidable theories by tuning the connecting functions. By adding maximal path and minimal path functions, we obtain a theory of red-black trees in which the transition relation of tree self-balancing (rotation) operations is expressible. We then show how to reduce the verification problem of the red-black tree algorithm to constraint satisfiability problems in the extended theory.


Decision Procedure Transition Relation Balance Tree Tree Automaton Local Update 
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.
    Backofen, R.: A complete axiomatization of a theory with feature and arity constraints. Journal of Logical Programming 24(1-2), 37–71 (1995)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Baldan, P., Corradini, A., Esparza, J., Heindel, T., König, B., Kozioura, V.: Verifying red-black trees. In: Proceedings of the 1st International Workshop on the Verification of Concurrent Systems with Dynamic Allocated Heaps, COSMICAH’05 (2005)Google Scholar
  3. 3.
    Calcagno, C., Gardner, P., Zarfaty, U.: Context logic and tree update. In: Proceedings of the 32nd ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages, pp. 271–282. ACM Press, New York (2005)Google Scholar
  4. 4.
    Comon, H., Dauchet, M., Gilleron, R., Lugiez, D., Tison, S., Tommasi, M.: Tree Automata Techniques and Applications (2002), Electronic edition at
  5. 5.
    Comon, H., Delor, C.: Equational formulae with membership constraints. Information and Computation 112(2), 167–216 (1994)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Cooper, D.C.: Theorem proving in arithmetic without multiplication. In: Machine Intelligence, vol. 7, pp. 91–99. Elsevier, New York (1972)Google Scholar
  7. 7.
    Cormen, T.H., Leiserson, C.E., Rivest, R.L., Stein, C.: Introduction to Algorithms. The MIT Press, Cambridge (2001)zbMATHGoogle Scholar
  8. 8.
    Downey, J., Sethi, R., Tarjan, R.E.: Variations of the common subexpression problem. Journal of the ACM 27, 758–771 (1980)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Enderton, H.B.: A Mathematical Introduction to Logic. Academic Press, London (2001)zbMATHGoogle Scholar
  10. 10.
    Habermehl, P., Iosif, R., Vojnar, T.: Automata-based verification of programs with tree updates. In: Hermanns, H., Palsberg, J. (eds.) TACAS 2006 and ETAPS 2006. LNCS, vol. 3920, pp. 350–364. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  11. 11.
    Korovin, K., Voronkov, A.: A decision procedure for the existential theory of term algebras with the Knuth-Bendix ordering. In: Proceedings of 15th IEEE Symposium on Logic in Computer Science, pp. 291–302. IEEE Computer Society Press, Los Alamitos (2000)Google Scholar
  12. 12.
    Korovin, K., Voronkov, A.: Knuth-Bendix constraint solving is NP-complete. In: Orejas, F., Spirakis, P.G., van Leeuwen, J. (eds.) ICALP 2001. LNCS, vol. 2076, pp. 979–992. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  13. 13.
    Kuncak, V., Rinard, M.: The structural subtyping of non-recursive types is decidable. In: Proceedings of 18th IEEE Symposium on Logic in Computer Science, pp. 96–107. IEEE Computer Society Press, Los Alamitos (2003)CrossRefGoogle Scholar
  14. 14.
    Maher, M.J.: Complete axiomatizations of the algebras of finite, rational and infinite tree. In: Proceedings of the 3rd IEEE Symposium on Logic in Computer Science, pp. 348–357. IEEE Computer Society Press, Los Alamitos (1988)Google Scholar
  15. 15.
    Mal’cev, A.I.: Axiomatizable classes of locally free algebras of various types. In: The Metamathematics of Algebraic Systems, Collected Papers, pp. 262–281. North-Holland, Amsterdam (1971)Google Scholar
  16. 16.
    Nelson, G., Oppen, D.C.: Fast decision procedures based on congruence closure. Journal of the ACM 27(2), 356–364 (1980)zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Oppen, D.C.: Reasoning about recursively defined data structures. Journal of the ACM 27(3), 403–411 (1980)zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Reddy, C.R., Loveland, D.W.: Presburger arithmetic with bounded quantifier alternation. In: Proceedings of the 10th Annual Symposium on Theory of Computing, pp. 320–325. ACM Press, New York (1978)Google Scholar
  19. 19.
    Rugina, R.: Quantitative shape analysis. In: Giacobazzi, R. (ed.) SAS 2004. LNCS, vol. 3148, pp. 228–245. Springer, Heidelberg (2004)Google Scholar
  20. 20.
    Rybina, T., Voronkov, A.: A decision procedure for term algebras with queues. ACM Transactions on Computational Logic 2(2), 155–181 (2001)CrossRefMathSciNetGoogle Scholar
  21. 21.
    Zhang, T., Sipma, H.B., Manna, Z.: Decision procedures for recursive data structures with integer constraints. In: Basin, D., Rusinowitch, M. (eds.) IJCAR 2004. LNCS (LNAI), vol. 3097, pp. 152–167. Springer, Heidelberg (2004)Google Scholar
  22. 22.
    Zhang, T., Sipma, H.B., Manna, Z.: The decidability of the first-order theory of term algebras with Knuth-Bendix order. In: Nieuwenhuis, R. (ed.) CADE 2005. LNCS (LNAI), vol. 3632, pp. 131–148. Springer, Heidelberg (2005)Google Scholar
  23. 23.
    Zhang, T., Sipma, H.B., Manna, Z.: Decision procedures for term algebras with integer constraints. Information and Computation 204, 1526–1574 (2006)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Berlin Heidelberg 2007

Authors and Affiliations

  • Zohar Manna
    • 1
  • Henny B. Sipma
    • 1
  • Ting Zhang
    • 2
  1. 1.Stanford University 
  2. 2.Microsoft Research Asia 

Personalised recommendations