Abstract
Let $ACC \circ THR$ be the class of constant-depth circuits comprised of AND, OR, and MOD$m$ gates (for some constant $m > 1$), with a bottom layer of gates computing arbitrary linear threshold functions. This class of circuits can be seen as a midpoint between $ACC$ (where we know nontrivial lower bounds) and depth-two linear threshold circuits (where nontrivial lower bounds remain open). We give an algorithm for evaluating an arbitrary symmetric function of $2^{n^{o(1)}}$ $ACC \circ THR$ circuits of size $2^{n^{o(1)}}$, on all possible inputs, in $2^n \cdot poly(n)$ time. Several consequences are derived: $\bullet$ The number of satisfying assignments to an $ACC \circ THR$ circuit of subexponential size can be computed in $2^{n-n^{\varepsilon}}$ time (where $\varepsilon > 0$ depends on the depth and modulus of the circuit). $\bullet$ $NEXP$ does not have quasi-polynomial size $ACC \circ THR$ circuits, nor does $NEXP$ have quasi-polynomial size $ACC \circ SYM$ circuits. Nontrivial size lower bounds were not known even for $AND \circ OR \circ THR$ circuits. $\bullet$ Every 0-1 integer linear program with $n$ Boolean variables and $s$ linear constraints is solvable in $2^{n-\Omega(n/((\log M)(\log s)^{5}))}\cdot poly(s,n,M)$ time with high probability, where $M$ upper bounds the bit complexity of the coefficients. (For example, 0-1 integer programs with weights in $[-2^{poly(n)},2^{poly(n)}]$ and $poly(n)$ constraints can be solved in $2^{n-\Omega(n/\log^6 n)}$ time.) We also present an algorithm for evaluating depth-two linear threshold circuits (a.k.a., $THR \circ THR$) with exponential weights and $2^{n/24}$ size on all $2^n$ input assignments, running in $2^n \cdot poly(n)$ time. This is evidence that non-uniform lower bounds for $THR \circ THR$ are within reach.
Highlights
In the non-uniform Boolean circuit model, one designs an infinite family of logical circuits {Cn}, one for each input length n, in order to recognize a given binary language L ⊆ {0, 1}
Knowledge of P/poly is rather poor, partly due to the “infinite” nature of the model: it is open if the huge complexity class nondeterministic exponential time (NEXP) is contained in P/poly
We should note that it is not completely settled whether the proof that NEXP ⊂ ACC is “truly” non-naturalizing; it could be that the natural proofs barrier is irrelevant to the problem. (If pseudorandom functions cannot be implemented in ACC, natural proofs considerations don’t apply to ACC anyway; if such functions can be implemented in ACC, the NEXP lower bound is non-naturalizing.) Plaku [38] has observed that the Naor-Reingold family of pseudorandom functions [35] can be implemented with quasipolynomial size OR ◦ THR ◦ AND circuits; it follows that the natural proofs barrier already applies to this circuit class
Summary
In the non-uniform Boolean circuit model, one designs an infinite family of logical circuits {Cn}, one for each input length n, in order to recognize a given binary language L ⊆ {0, 1}. (If pseudorandom functions cannot be implemented in ACC, natural proofs considerations don’t apply to ACC anyway; if such functions can be implemented in ACC, the NEXP lower bound is non-naturalizing.) Plaku [38] has observed that the Naor-Reingold family of pseudorandom functions [35] can be implemented with quasipolynomial size OR ◦ THR ◦ AND circuits; it follows that the natural proofs barrier already applies to this circuit class It is an interesting open problem if ACC ◦ THR can efficiently simulate such depth-three circuits. Given a depth-two 2n/24-size linear threshold circuit C with integer weights in [−2nk , 2nk ], we can evaluate C on all 2n input assignments in 2n · poly(nk) time.
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