Easiness Amplification and Uniform Circuit Lower Bounds

We present new consequences of the assumption that time-bounded algorithms can be “compressed” with non-uniform circuits. Our main contribution is an “easiness amplification” lemma for circuits. One instantiation of the lemma says: if n1+ε-time, Õ(n)-space computations have n1+o(1) size (non-uniform) circuits for some ε > 0, then every problem solvable in polynomial time and Õ(n) space has n1+o(1) size (non-uniform) circuits as well. This amplification has several consequences: An easy problem without small LOGSPACE-uniform circuits. For all ε > 0, we give a natural decision problem General Circuit n-Composition that is solvable in n1+ε time, but we prove that polynomial-time and logarithmic-space preprocessing cannot produce n1+o(1)-size circuits for the problem. This shows that there are problems solvable in n1+ε time which are not in LOGSPACE-uniform n1+o(1) size, the first result of its kind. We show that our lower bound is non-relativizing, by exhibiting an oracle relative to which the result is false. Problems without low-depth LOGSPACE-uniform circuits. For all ε > 0, 1 < d < 2, and e < d we give another natural circuit composition problem computable in Õ(n1+ε) time, or in O((logn)) space (though not necessarily simultaneously) that we prove does not have SPACE[(logn)]-uniform circuits of Õ(n) size and O((logn)) depth. We also show SAT does not have circuits of Õ(n) size and log2−o(1) n depth that can be constructed in log2−o(1) n space. A strong circuit complexity amplification. For every ε > 0, we give a natural problem Circuit n-Composition and show that if it has Õ(n)-size circuits (uniform or not), then every problem solvable in 2O(n) time and 2O( √ n logn) space (simultaneously) has 2O( √ n logn)size circuits (uniform or not). We also show the same consequence holds assuming SAT has Õ(n)-size circuits. As a corollary, if n1.1 time computations (or O(n) nondeterministic time computations) have Õ(n)-size circuits, then all problems in exponential time and subexponential space (such as quantified Boolean formulas) have significantly subexponential-size circuits. This is a new connection between the relative circuit complexities of easy and hard problems. 1998 ACM Subject Classification F.1.3 Relations Among Complexity Classes