A circuit-size lower bound

As remarked in Cook (1980), we do not know any nonlinear lower bound on the circuit size of a language in P or even in NP. The best known lower bound seems to be due to Paul (1975). Instead of trying to prove lower bounds on the circuit-size of a "natural" language, this note raises the question of whether some language in a class is of provably high circuit complexity. We show that for each nonnegative integer k, there is a language Lk in Σ2P ∩ π2P (of the Meyer and Stockmeyer (1972) hierarchy) Which does not have O(nk)-size circuits. The method is indirect and does not produce the language Lk. Other results of a similar nature are presented and several questions raised.