On the Power of Built-In Relations in Certain Classes of Program Schemes

One of the fundamental results of finite model theory is Immerman’s characterization [10,11] of the complexity class NL (non-deterministic logspace) as the class of problems definable in transitive closure logic augmented with a built-in successor relation, (±TC)∗[FOs]. Essentially, by “built-in successor relation” we mean that any sentence of the logic has access to a successor relation on the universe of the (finite) structure in which it is interpreted (and where this successor relation does not necessarily come as part of the structure). So that our sentences should define problems, i.e., sets of structures closed under isomorphism, we only ever consider the invariant sentences; that is, those sentences with the property that every (appropriate) structure either satisfies the sentence no matter which successor relation is chosen or it doesn’t satisfy the sentence no matter which successor relation is chosen. Unfortunately, the property of being an invariant sentence is not decidable, even for first-order logic with a built-in successor relation [7], and so one should not really say that one has a truly “logical” characterization of NL. The general question of whether complexity classes such as NL and P (polynomial-time)