Complete problems for dynamic complexity classes

We present the first complete problems for dynamic complexity classes including the classes Dyn-FO and Dyn-ThC⁰, 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, ζ, that is in a sense universal for all first-order formulas. Since first-order formulas are stratified by quantifier depth, the first-order formula ζ 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.