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.

[1]  Stephen R. Mahaney.  Sparse complete sets for NP: Solution of a conjecture of Berman and Hartmanis , 1980, 21st Annual Symposium on Foundations of Computer Science (sfcs 1980).

[2]  John E. Savage,et al.  Computational Work and Time on Finite Machines , 1972, JACM.

[3]  Stephen A. Cook,et al.  The complexity of theorem-proving procedures , 1971, STOC.

[4]  Leonard M. Adleman,et al.  Two theorems on random polynomial time , 1978, 19th Annual Symposium on Foundations of Computer Science (sfcs 1978).

[5]  Neil Immerman,et al.  Upper and lower bounds for first order expressibility , 1980, 21st Annual Symposium on Foundations of Computer Science (sfcs 1980).

[6]  Albert R. Meyer,et al.  The Equivalence Problem for Regular Expressions with Squaring Requires Exponential Space , 1972, SWAT.

[7]  V. M. Khrapchenko.  Complexity of the realization of a linear function in the class of II-circuits , 1971 .

[8]  John E. Savage,et al.  The Complexity of Computing , 1976 .

[9]  Dexter Kozen,et al.  On parallelism in turing machines , 1976, 17th Annual Symposium on Foundations of Computer Science (sfcs 1976).

[10]  Richard J. Lipton,et al.  Some connections between nonuniform and uniform complexity classes , 1980, STOC '80.