### On Uniformity and Circuit Lower Bounds

Rahul Santhanam, Ryan Williams
2014 Computational Complexity
We explore relationships between circuit complexity, the complexity of generating circuits, and algorithms for analyzing circuits. Our results can be divided into two parts: 1. Lower bounds against medium-uniform circuits. Informally, a circuit class is "medium uniform" if it can be generated by an algorithmic process that is somewhat complex (stronger than LOGTIME) but not infeasible. Using a new kind of indirect diagonalization argument, we prove several new unconditional lower bounds against
more » ... medium-uniform circuit classes, including: • For all k, P is not contained in P-uniform SIZE(n k ). That is, for all k, there is a language L k ∈ P that does not have O(n k )-size circuits constructible in polynomial time. This improves Kannan's lower bound from 1982 that NP is not in P-uniform SIZE(n k ) for any fixed k. • For all k, NP is not in P NP || -uniform SIZE(n k ). This also improves Kannan's theorem, but in a different way: the uniformity condition on the circuits is stronger than that on the language itself. • For all k, LOGSPACE does not have LOGSPACE-uniform branching programs of size n k . Eliminating non-uniformity and (non-uniform) circuit lower bounds. We complement these results by showing how to convert any potential simulation of LOGTIME-uniform NC 1 in ACC 0 /poly or TC 0 /poly into a medium-uniform simulation using small advice. This lemma can be used to simplify the proof that faster SAT algorithms imply NEXP circuit lower bounds and leads to the following new connection: • Consider the following task: given a TC 0 circuit C of n O(1) size, output yes when C is unsatisfiable, and output no when C has at least 2 n−2 satisfying assignments. (Behavior on other inputs 178 Santhanam & Williams cc 23 (2014) can be arbitrary.) Clearly, this problem can be solved efficiently using randomness. If this problem can be solved deterministically in 2 n−ω(log n) time, then NEXP ⊂ TC 0 /poly. Another application is to derandomize randomized TC 0 simulations of NC 1 on almost all inputs: • Suppose NC 1 ⊆ BPTC 0 . Then, for every ε > 0 and every language L in NC 1 , there is a LOGTIME-uniform TC 0 circuit family of polynomial size recognizing a language L such that L and L differ on at most 2 n ε inputs of length n, for all n. On uniformity and circuit lower bounds 179 Lower bounds by amplifying uniformity. In the first part of the paper, we prove new lower bounds against "medium-uniform" circuits. Our key insight is that if we assume that a "medium complexity" class has small medium-uniform circuits, this assumption can be applied in multiple ways: not only is it applicable to a language L in the medium complexity class, but it is also applicable to another (medium complexity) language encoding circuits for L. Applying the hypothesis multiple times allows us to simulate medium-uniformity with a very small amount of nonuniform advice and diagonalize against the small-advice simulation to derive a contradiction. This strategy leads to new lower bounds against notions of uniformity for which it is not possible to directly obtain lower bounds by diagonalization. Consider for example, the class of linear-size circuits. If a LOGTIME-uniformity condition is imposed on the class, then it can be simulated in nearly-linear deterministic time, and we can easily diagonalize to find a function in (for example) n 2 time that does not have such circuits. However, suppose we wish to find a function in P that does not have small circuits constructible via a "medium" uniformity notion, such as P-uniformity. Then, the notion of uniformity (which allows for an arbitrary polynomial-time bound) can be more powerful than the function itself (which must lie in some fixed polynomial time bound), so it is no longer possible to directly diagonalize. Nevertheless, we can "indirectly" diagonalize, by reducing the assumption that P has small P-uniform circuits to another time hierarchy result. To describe our results, let us first set up some notation informally (definitions are in Section 1.1). Given a class C of languages, recall a language L is said to have C-uniform circuits of size s(n) if there is a size-s(n) circuit family {D n } such that the description of D n is computable in C. (There are several possible choices about what "description" means; in this paper, our notions of uniformity will be so powerful that these choices are all essentially equivalent.) Our first main result strengthens both the deterministic time hierarchy theorem and the result of Kannan (1982) that for any k, NP is not in P-uniform SIZE(n k ). 2