Recall that one motivation for studying non-uniform computation is the hope that it might be easier to prove lower bounds in that setting. (This is somewhat paradoxical, as non-uniform algorithms are more powerful than uniform algorithms; nevertheless, since circuits are more “combinatorial” in nature than uniform algorithms, there may still be justification for such hope.) The ultimate goal here would be to prove that N P ̸ ⊂ P /poly, which would imply P ̸ = N P. Unfortunately, after over two decades of attempts we are unable to prove anything close to this. Here, we show one example of a lower bound that we have been able to prove; we then discuss one “barrier ” that partly explains why we have been unable to prove stronger bounds.

The main purpose of this work is to formally define monotone expanders and motivate their study with (known and new) connections to other graphs and to several computational and pseudorandomness problems. In particular we explain how monotone expanders of constant degree lead to: 1. Constant degree dimension expanders in finite fields, resolving a question of [BISW04]. 2. O(1)-page and O(1)-pushdown expanders, resolving a question of [GKS86], and leading to tight lower bounds on simulation time for certain Turing Machines. Bourgain [Bou09] gave recently an ingenious construction of such constant degree monotone expanders. The first application (1) above follows from a reduction in [DS08]. We give a short exposition of both construction and reduction. The new contributions of this paper are simple. First, we explain the observation leading to the second application (2) above, and some of its consequences. Second, we observe that a variant of the zig-zag graph product preserves monotonicity, and use it to give a simple alternative construction of monotone expanders, with near-constant degree. 1

Abstract. Spira [28] showed that any Boolean formula of size s can be simulated in depth O(log s). We generalize Spira’s theorem and show that any Boolean circuit of size s with segregators of size f(s) can be simulated in depth O(f(s) log s). If the segregator size is at least s ε for some constant ε> 0, then we can obtain a simulation of depth O(f(s)). This improves and generalizes a simulation of polynomial-size Boolean circuits of constant treewidth k in depth O(k 2 log n) by Jansen and Sarma [17]. Since the existence of small balanced separators in a directed acyclic graph implies that the graph also has small segregators, our results also apply to circuits with small separators. Our results imply that the class of languages computed by non-uniform families of polynomial-size circuits that have constant size segregators equals non-uniform NC 1. Considering space bounded Turing machines to generate the circuits, for f(s) log 2 s-space uniform families of Boolean circuits our small-depth simulations are also f(s) log 2 s-space uniform. As a corollary, we show that the Boolean Circuit Value problem for circuits with constant size segregators (or separators) is in deterministic SP ACE(log 2 n). Our results also imply that the Planar Circuit Value problem, which is known to be P-Complete [16], can be solved in deterministic SP ACE ( √ n log n). Key words: Boolean circuits, circuit size, circuit depth, Spira’s theorem, Turing machines, space complexity 1