Marco Carmosino

Marco Carmosino, Ngu Dang, Tim Jackman

Gate elimination is the primary technique for proving explicit lower bounds against general Boolean circuits, including Li and Yang's state-of-the-art $3.1n - o(n)$ bound for affine dispersers (STOC 2022). Every circuit lower bound is implicitly existential: every circuit that is too small to compute $f$ must err on some input. This raises a natural question: are these lower bounds \emph{constructive}? That is, can we efficiently produce such errors? Chen, Jin, Santhanam, and Williams showed that constructivity plays a central role in many longstanding open problems in complexity theory, and explicitly raised the question of which circuit lower bound techniques can be made constructive (FOCS 2021). We show that a variety of gate elimination arguments yield refuters -- efficient algorithms that, when given a circuit that is too small to compute a function $f$, produce an input on which the circuit errs. Our results range from elementary lower bounds for $XOR$ and the multiplexer to more sophisticated arguments for affine dispersers. Underlying these results is a shift in perspective: gate elimination arguments \emph{are} algorithms. Each step either simplifies the circuit or reveals a violation of some structural or functional property, from which, with a little additional work, explicit counterexamples can be extracted. We further strengthen the $XOR$ result to handle circuits that \emph{match} the lower bound: given any DeMorgan circuit of size $3(n-1)$ that fails to compute $XOR_n$, we can efficiently produce a counterexample. While refuters follow from the gate elimination arguments themselves, this refinement requires a complete characterization of the set of optimal circuits computing $XOR$ -- a requirement rarely met by other explicit functions.

Eric Binnendyk, Marco Carmosino, Antonina Kolokolova et al.

We generalize the "indirect learning" technique of Furst et. al., 1991 to reduce from learning a concept class over a samplable distribution $μ$ to learning the same concept class over the uniform distribution. The reduction succeeds when the sampler for $μ$ is both contained in the target concept class and efficiently invertible in the sense of Impagliazzo & Luby, 1989. We give two applications. - We show that AC0[q] is learnable over any succinctly-described product distribution. AC0[q] is the class of constant-depth Boolean circuits of polynomial size with AND, OR, NOT, and counting modulo $q$ gates of unbounded fanins. Our algorithm runs in randomized quasi-polynomial time and uses membership queries. - If there is a strongly useful natural property in the sense of Razborov & Rudich 1997 -- an efficient algorithm that can distinguish between random strings and strings of non-trivial circuit complexity -- then general polynomial-sized Boolean circuits are learnable over any efficiently samplable distribution in randomized polynomial time, given membership queries to the target function