Reachability and the Power of Local Ordering

The L NL question remains one of the major unresolved problems in complexity theory Both L and NL have logical characterizations as the sets of totally ordered structures expressible in rst order logic augmented with the appropriate Transitive Closure operator I 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 GM An apparently quite di erent structured model of logspace machines is the Jumping Automaton on Graphs JAG CR We show that the JAG model is intimately related to these logics on one way locally ordered LO 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 de ne Furthermore the language FO DTC LO over two way locally ordered LO 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 one way locally ordered graphs and three lower bounds on DTC