We present the first complete problems for dynamic complexity classes including the classes Dyn-FO and Dyn-ThC/sup 0/, the dynamic classes corresponding to relational calculus and (polynomially bounded) SQL, respectively. The first problem we show complete for Dyn-FO is a single-step version of the circuit value problem (SSCV). Of independent interest, our construction also produces a first-order formula, /spl zeta/, that is in a sense universal for all first-order formulas. Since first-order formulas are stratified by quantifier depth, the first-order formula /spl zeta/ emulates formulas of greater depth by iterated application. As a corollary we obtain a fixed quantifier block, QBC, that is complete for all first-order quantifier blocks.

Integer division has been known to lie in P-uniform TC/sup 0/ since the mid-1980s, and recently this was improved to L-uniform TC/sup 0/. At the time that the results in this paper were proved and submitted for conference presentation, it was unknown whether division lay in DLOGTIME-uniform TC/sup 0/ (also known as FOM). We obtain tight bounds on the uniformity required for division, by showing that division is complete for the complexity class FOM+POW obtained by augmenting FOM with a predicate for powering modulo small primes. We also show that, under a well-known number-theoretic conjecture (that there are many "smooth" primes), POW (and hence division) lies in FOM. Building on this work, Hesse has shown recently that division is in FOM [17]. The essential idea in the fast parallel computation of division and related problems is that of Chinese remainder representation (CRR)-storing a number in the form of its residues modulo many small primes. The fact that CRR operations can be carried out in log space has interesting implications for small space classes. We define two versions of s(n) space for s(n)=o(log n): dspace(s(n)) as the traditional version where the worktape begins blank, and DSPACE(s(n)) where the space bound is established by endmarkers before the computation starts. We present a new translational lemma, and derive as a consequence that (for example), if one can improve the result of Hartmanis and Berman (1976) that {0/sup n/: n is prime} /spl notin/ dspace (log log n) to show that {0/sup n/: n is prime} /spl notin/ DSPACE (log log n), it would follow that L/spl ne/NP.