An upper bound for the circuit complexity of existentially quantified Boolean formulas

The expressive power of existentially quantified Boolean formulas ∃CNF with free variables is investigated. We introduce a hierarchy of subclasses ∃MU∗(k) of ∃CNF formulas based on the maximum deficiency k of minimal unsatisfiable subformulas of the bound part of the formulas. We will establish an upper bound of the size of minimally equivalent circuits. It will be shown, that there are constants a and b, such that for every formula in ∃MU∗(k) of length m of the bound part and length l of the free part of the formula there is an equivalent circuit of size less than l+a⋅mb(log2(m)+k)2.