Parameterized circuit complexity and the W hierarchy

A parameterized problem 〈L, k〉 belongs to W[t] if there exists k′ computed from k such that 〈L, k〉 reduces to the weight-k′ satisfiability problem for weft-t circuits. We relate the fundamental question of whether the W[t] hierarchy is proper to parameterized problems for constant-depth circuits. We define classes G[t] as the analogues of AC⁰ depth-t for parameterized problems, and N[t] by weight-k′ existential quantification on G[t], by analogy with NP = ∃ · P. We prove that for each t, W[t] equals the closure under fixed-parameter reductions of N[t]. Then we prove, using Sipser's results on the AC⁰ depth-t hierarchy, that both the G[t] and the N[t] hierarchies are proper. If this separation holds up under parameterized reductions, then the W[t] hierarchy is proper. We also investigate the hierarchy H[t] defined by alternating quantification over G[t]. By trading weft for quantifiers we show that H[t] coincides with H[1]. We also consider the complexity of unique solutions, and show a randomized reduction from W[t] to Unique W[t].