Lower Bounds against Weakly Uniform Circuits

A family of Boolean circuits {Cn}n>0 is called γ(n)-weakly uniform if there is a polynomial-time algorithm for deciding the directconnection language of every Cn, given advice of size γ(n). This is a relaxation of the usual notion of uniformity, which allows one to interpolate between complete uniformity (when γ(n) = 0) and complete nonuniformity (when γ(n) > |Cn|). Weak uniformity is essentially equivalent to succinctness introduced by Jansen and Santhanam [12]. Our main result is that Permanent is not computable by polynomialsize n-weakly uniform TC circuits. This strengthens the results by Allender [2] (for uniform TC) and by Jansen and Santhanam [12] (for weakly uniform arithmetic circuits of constant depth). Our approach is quite general, and can be used to extend to the “weakly uniform” setting all currently known circuit lower bounds proved for the “uniform” setting. For example, we show that Permanent is not computable by polynomial-size (logn)-weakly uniform threshold circuits of depth o(log log n), generalizing the result by Koiran and Perifel [16].