The minimization problem for Boolean formulas

We investigate the computational complexity of the minimization problem for Boolean formulas. Depending on the definition, these problems are trivially in Σ₂^P or II₂ ^P, and these are the best upper bounds known. The only previously known lower bounds are also trivial, and are coNP lower bounds at best, thus leaving quite a large gap between the upper and lower bounds. In this paper, we prove much better lower bounds: hardness for parallel access to NP for those cases in which coNP was the best previously known lower bound, and coNP-hardness for the case in which no lower bound was previously known