Reachability and the Power of Local Ordering

The L \(L \mathop = \limits^? NL\) NL question remains one of the major unresolved problems in complexity theory. Both L and NL have logical characterizations as the sets of ordered structures expressible in first-order logic augmented with the appropriate Transitive Closure operator [I87]: Over ordered structures, (FO + DTC) captures L and (FO + TC) captures NL. On the other hand, in the absence of ordering, (FO + TC) is strictly more powerful than (FO + DTC) [GM92]. An apparently quite different “structured” model of logspace machines is the Jumping Automaton on Graphs (JAG), [CR80]. We show that the JAG model is intimately related to these logics on “locally ordered” structures. We argue that the usual JAG model is unreasonably weak and should be replaced, wherever possible, by the two-way JAG model, which we define. Furthermore, we have shown that the language (FO + DTC) over two-way locally ordered graphs is more robust than even the two-way JAG model, and yet lower bounds remain accessible. We prove an upper bound on the power of TC over locally ordered graphs, and three lower bounds on DTC.

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

[2]  Ronald Fagin,et al.  Reachability is harder for directed than for undirected finite graphs , 1988, [Proceedings 1988] 29th Annual Symposium on Foundations of Computer Science.

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

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

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

[6]  Neil Immerman,et al.  An optimal lower bound on the number of variables for graph identification , 1992, Comb..

[7]  Neil Immerman,et al.  Descriptive and Computational Complexity , 1989, FCT.

[8]  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).

[9]  Erich Grädel,et al.  Deterministic vs. nondeterministic transitive closure logic , 1992, [1992] Proceedings of the Seventh Annual IEEE Symposium on Logic in Computer Science.

[10]  Neil Immerman,et al.  Tree canonization and transitive closure , 1995, Proceedings of Tenth Annual IEEE Symposium on Logic in Computer Science.

[11]  Allan Borodin,et al.  Time-space tradeoffs for undirected graph traversal , 1990, Proceedings [1990] 31st Annual Symposium on Foundations of Computer Science.

[12]  Armin Hemmerling,et al.  Traps for Jumping Multihead Counter Automata , 1992, J. Inf. Process. Cybern..

[13]  H. Gaifman.  On Local and Non-Local Properties , 1982 .

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

[15]  Jeff Edmonds.  Time-space trade-offs for undirected st-connectivity on a JAG , 1993, STOC '93.

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

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

[18]  Steven Lindell.  A logspace algorithm for tree canonization (extended abstract) , 1992, STOC '92.