Circuit-size Lower Bounds and Non-reducibility to Sparse Sets
Name
MIT-LCS-TM-205.pdf
Size
4.17 MB
Format
Adobe PDF
Checksum (MD5)
5d6be9046286f292e4cfe5473cfbc533
Author(s)
Kannan, Ravindran
Date Issued
October 1981
Series/Report no.
MIT-LCS-TM-205
Abstract
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). In this paper we show that first for each nonnegative integer k, there is a language Lk in Σ2∩π2 (of Meyer and Stockmeyer (1972) hierarchy) which does not have 0(n^k)-size circuits. Using the same techniques, one is able to prove several similar results. For example, we show that for each nonnegative integer k, there is a language Lk in NP that does not have 0(n^k)-size uniform circuits. This follows as a corollary of a stronger result shown in the paper. Finally, we note that existence of "small circuits" is in suitable contexts equivalent to being reducible to sparse sets. Using this, we are able to prove for example that for any time-constructible super-polynomial function f(n), NTIME(f(n)) contains a language which is not many-to-one p-time reducible to any sparse set.
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