Pure Pointer Programs and Tree Isomorphism

  • Martin Hofmann
  • Ramyaa Ramyaa
  • Ulrich Schöpp
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7794)


In a previous work, Hofmann and Schöpp have introduced the programming language purple to formalise the common intuition of logspace-algorithms as pure pointer programs that take as input some structured data (e.g. a graph) and store in memory only a constant number of pointers to the input (e.g. to the graph nodes). It was shown that purple is strictly contained in logspace, being unable to decide st-connectivity in undirected graphs.

In this paper we study the options of strengthening purple as a manageable idealisation of computation with logarithmic space that may be used to give some evidence that ptime-problems such as Horn satisfiability cannot be solved in logarithmic space.

We show that with counting, purple captures all of logspace on locally ordered graphs. Our main result is that without a local ordering, even with counting and nondeterminism, purple cannot solve tree isomorphism. This generalises the same result for Transitive Closure Logic with counting, to a formalism that can iterate over the input structure, furnishing a new proof as a by-product.


Boolean Variable Predicate Symbol Horn Clause Input Structure Relativised Separation 
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.


  1. 1.
    Bonfante, G.: Some Programming Languages for Logspace and Ptime. In: Johnson, M., Vene, V. (eds.) AMAST 2006. LNCS, vol. 4019, pp. 66–80. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  2. 2.
    Cai, J.-Y., Sivakumar, D.: Sparse hard sets for P: Resolution of a conjecture of Hartmanis. J. Comput. Syst. Sci. 58(2), 280–296 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Cook, S.A., Rackoff, C.: Space lower bounds for maze threadability on restricted machines. SIAM J. Comput. 9(3), 636–652 (1980)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Ebbinghaus, H.-D., Flum, J.: Finite model theory. Springer (1995)Google Scholar
  5. 5.
    Etessami, K., Immerman, N.: Tree canonization and transitive closure. In: IEEE Symp. Logic in Comput. Sci., pp. 331–341 (1995)Google Scholar
  6. 6.
    Grädel, E., McColm, G.L.: On the power of deterministic transitive closures. Inf. Comput. 119(1), 129–135 (1995)zbMATHCrossRefGoogle Scholar
  7. 7.
    Grohe, M., Grußien, B., Hernich, A., Laubner, B.: L-recursion and a new logic for logarithmic space. In: CSL, pp. 277–291 (2011)Google Scholar
  8. 8.
    Hofmann, M., Schöpp, U.: Pointer programs and undirected reachability. In: LICS, pp. 133–142 (2009)Google Scholar
  9. 9.
    Hofmann, M., Schöpp, U.: Pure pointer programs with iteration. ACM Trans. Comput. Log. 11(4) (2010)Google Scholar
  10. 10.
    Immerman, N.: Progress in descriptive complexity. In: Curr. Trends in Th. Comp. Sci., pp. 71–82 (2001)Google Scholar
  11. 11.
    Jones, N.D.: LOGSPACE and PTIME characterized by programming languages. Theor. Comput. Sci. 228(1-2), 151–174 (1999)zbMATHCrossRefGoogle Scholar
  12. 12.
    Richard, E.: Ladner and Nancy A. Lynch. Relativization of questions about log space computability. Mathematical Systems Theory 10, 19–32 (1976)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Lindell, S.: A logspace algorithm for tree canonization (extended abstract). In: STOC 1992, pp. 400–404. ACM, New York (1992)Google Scholar
  14. 14.
    Lu, P., Zhang, J., Poon, C.K., Cai, J.-Y.: Simulating Undirected st-Connectivity Algorithms on Uniform JAGs and NNJAGs. In: Deng, X., Du, D.-Z. (eds.) ISAAC 2005. LNCS, vol. 3827, pp. 767–776. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  15. 15.
    Reingold, O.: Undirected connectivity in log-space. J. ACM 55(4) (2008)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Martin Hofmann
    • 1
  • Ramyaa Ramyaa
    • 1
  • Ulrich Schöpp
    • 1
  1. 1.Ludwig-Maximilians Universität MünchenMunichGermany

Personalised recommendations