Tree canonization and transitive closure

We prove that tree isomorphism is not expressible in the language (FO+TC+COUNT). This is surprising since in the presence of ordering the language captures NL, whereas tree isomorphism and canonization are in L (Lindell, 1992). Our proof uses an Ehrenfeucht-Fraisse game for transitive closure logic with counting. As a corresponding upper bound, we show that tree canonization is expressible in (FO+COUNT)[log n]. The best previous upper bound had been (FO+COUNT)[n/sup 0(1)/] (Dublish and Maheshwari, 1990). The lower bound remains true for bounded-degree trees, and we show that for bounded-degree trees counting is not needed in the upper bound. These results are the first separations of the unordered versions of the logical languages for NL, AC/sup 1/, and ThC/sup 1/. Our results were motivated by a conjecture in (Etessami and Immerman, 1995) that (FO+TC+COUNT+1LO)=NL, i.e., that a one-way local ordering sufficed to capture NL. We disprove this conjecture, but we prove that a two-way local ordering does suffice, i.e., (FO+TC+COUNT+2LO)=NL.

[1]  Neil Immerman,et al.  On Uniformity within NC¹ , 1990, J. Comput. Syst. Sci..

[2]  Johann A. Makowsky,et al.  The Ehrenfeucht-Fraisse Games for Transitive Closure , 1992, LFCS.

[3]  N. Immerman,et al.  On uniformity within NC 1 . , 1988 .

[4]  Neil Immerman,et al.  Reachability and the Power of Local Ordering , 1995, Theor. Comput. Sci..

[5]  E. Lander,et al.  Describing Graphs: A First-Order Approach to Graph Canonization , 1990 .

[6]  Neil Immerman,et al.  Languages that Capture Complexity Classes , 1987, SIAM J. Comput..

[7]  Martin Otto,et al.  The expressive power of fixed-point logic with counting , 1996, Journal of Symbolic Logic.

[8]  Kousha Etessami.  Counting Quantifiers, Successor Relations, and Logarithmic Space , 1997, J. Comput. Syst. Sci..

[9]  Pratul Dublish,et al.  Query Languages which Express all PTIME Queries for Trees and Unicyclic Graphs , 1990, MFCS.

[10]  Neil Immerman.  Nondeterministic Space is Closed Under Complementation , 1988, SIAM J. Comput..

[11]  Stephen A. Cook,et al.  Space Lower Bounds for Maze Threadability on Restricted Machines , 1980, SIAM J. Comput..

[12]  Steven Lindell.  A Logspace Algorithm for Tree Canonization , 1992 .

[13]  Neil Immerman,et al.  An optimal lower bound on the number of variables for graph identification , 1989, 30th Annual Symposium on Foundations of Computer Science.

[14]  Walter L. Ruzzo,et al.  Tree-size bounded alternation(Extended Abstract) , 1979, J. Comput. Syst. Sci..

[15]  Neil Immerman,et al.  Expressibility and Parallel Complexity , 1989, SIAM J. Comput..

[16]  Richard J. Lipton,et al.  Random walks, universal traversal sequences, and the complexity of maze problems , 1979, 20th Annual Symposium on Foundations of Computer Science (sfcs 1979).

[17]  Neil Immerman.  Upper and lower bounds for first order expressibility , 1980, 21st Annual Symposium on Foundations of Computer Science (sfcs 1980).

[18]  Erich Grädel,et al.  On Transitive Closure Logic , 1991, CSL.