Abstract
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.
This work was supported by Deutsche Forschungsgemeinschaft (dfg) under grant purple.
Chapter PDF
Similar content being viewed by others
Keywords
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.
References
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)
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)
Cook, S.A., Rackoff, C.: Space lower bounds for maze threadability on restricted machines. SIAM J. Comput. 9(3), 636–652 (1980)
Ebbinghaus, H.-D., Flum, J.: Finite model theory. Springer (1995)
Etessami, K., Immerman, N.: Tree canonization and transitive closure. In: IEEE Symp. Logic in Comput. Sci., pp. 331–341 (1995)
Grädel, E., McColm, G.L.: On the power of deterministic transitive closures. Inf. Comput. 119(1), 129–135 (1995)
Grohe, M., Grußien, B., Hernich, A., Laubner, B.: L-recursion and a new logic for logarithmic space. In: CSL, pp. 277–291 (2011)
Hofmann, M., Schöpp, U.: Pointer programs and undirected reachability. In: LICS, pp. 133–142 (2009)
Hofmann, M., Schöpp, U.: Pure pointer programs with iteration. ACM Trans. Comput. Log. 11(4) (2010)
Immerman, N.: Progress in descriptive complexity. In: Curr. Trends in Th. Comp. Sci., pp. 71–82 (2001)
Jones, N.D.: LOGSPACE and PTIME characterized by programming languages. Theor. Comput. Sci. 228(1-2), 151–174 (1999)
Richard, E.: Ladner and Nancy A. Lynch. Relativization of questions about log space computability. Mathematical Systems Theory 10, 19–32 (1976)
Lindell, S.: A logspace algorithm for tree canonization (extended abstract). In: STOC 1992, pp. 400–404. ACM, New York (1992)
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)
Reingold, O.: Undirected connectivity in log-space. J. ACM 55(4) (2008)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Hofmann, M., Ramyaa, R., Schöpp, U. (2013). Pure Pointer Programs and Tree Isomorphism. In: Pfenning, F. (eds) Foundations of Software Science and Computation Structures. FoSSaCS 2013. Lecture Notes in Computer Science, vol 7794. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-37075-5_21
Download citation
DOI: https://doi.org/10.1007/978-3-642-37075-5_21
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-37074-8
Online ISBN: 978-3-642-37075-5
eBook Packages: Computer ScienceComputer Science (R0)