We establish that the decidability of the first order theory of a class of finite structures C is a simple and useful condition for guaranteeing that the expressive power of FO+LFP properly extends that of FO on C, unifying separation results for various classes of structures that have been studied. We then apply this result to show that it encompasses certain constructive pebble game techniques which are widely used to establish separations between FO and FO+LFP, and demonstrate that these same techniques cannot succeed in performing separations from any complexity class that contains DLOGTIME.

The extensions of first-order logic with a least fixed point operator (FO + LFP) and with a partial fixed point operator (FO + PFP) are known to capture the complexity classes P and PSPACE respectively in the presence of an ordering relation over finite structures. Recently, Abiteboul and Vianu (in "Proceedings of the 23rd ACM Symposium on the Theory of Computing," 1991) investigated the relationship of these two logics in the absence of an ordering, using a machine model of generic computation. In particular, they showed that the two languages have equivalent expressive power if and only if P = PSPACE. These languages can also be seen as fragments of an infinitary logic where each formula has a bounded number of variables, Lω∞ω, (see, for instance, Kolaitis and Vardi, in "Proceedings of the 5th IEEE Symposium on Logic in Computer Science," pp. 156-167, 1990). We investigate this logic on finite structures and provide a normal form for it. We also present a treatment of Abiteboul and Vianu′s results from this point of view. In particular, we show that we can write a formula of FO + LFP that defines an ordering of the Lk∞ω, types uniformly over all finite structures. One consequence of this is a generalization of the equivalence of FO + LFP and P from ordered structures to classes of structures where every element is definable. We also settle a conjecture mentioned by Abiteboul and Vianu by showing that FO + LFP is properly contained in the polynomial time computable fragment of Lω∞ω, raising the question of whether the latter fragment is a recursively enumerable class.

We present a solution to the problem of assigning to each directed tree T of size n a unique isomorphism invariant name for T, using only work space O(log n). Hence, tree isomorphism is computable in logspace. As another consequence, we obtain the corollary that the set of logspace computable queries (Lspace) on trees is recursively enumerable. Our results extend easily to undirected trees and even forests.