Amplifying lower bounds by means of self-reducibility

Abstract

We observe that many important computational problems in NC 1 share a simple self-reducibility property. We then show that, for any problem A having this self-reducibility property, A has polynomial size TC 0 circuits if and only if it has TC 0 circuits of size n 1+ for every > 0 (counting the number of wires in a circuit as the size of the circuit). As an example of what this observation yields, consider the Boolean Formula Evaluation problem (BFE), which is complete for NC 1 and has the self-reducibility property. It follows from a lower bound of Impagliazzo, Paturi, and Saks, that BFE requires depth d TC 0 circuits of size n 1+ d . If one were able to improve this lower bound to show that there is some constant > 0 such that every TC 0 circuit family recognizing BFE has size n 1+ , then it would follow that TC 0 = NC 1 . We show that proving lower bounds of the form n 1+ is not ruled out by the Natural Proof framework of Razborov and Rudich and hence there is currently no known barrier for separating classes such as ACC 0 , TC 0 and NC 1 via existing "natural" approaches to proving circuit lower bounds.We also show that problems with small uniform constant-depth circuits have algorithms that simultaneously have small space and time bounds. We then make use of known time-space tradeoff lower bounds to show that SAT requires uniform depth d TC 0 and AC 0 [6] circuits of size n 1+c for some constant c depending on d.

Uniform derandomization from pathetic lower bounds

“…Hence, as observed in [32, corollary 4.3], for any given g > 0, an io-TC 0 circuit of polynomial size for FWP can be combined with the self-reduction for FWP (for a suitably chosen e) to obtain an io-TC 0 circuit of size n 1+g . Theorem 3.2 is not stated in terms of io-TC 0 in [32], but the proof there shows that if there are infinitely many input lengths n where FWP has circuits of size n k , then there are infinitely many input lengths m where FWP has circuits of size m 1+g . The strong downward self-reducibility property allows small circuits for inputs of size m to be constructed by efficiently using circuits for size n < m as subcomponents.…”

“…An oracle arithmetic circuit is called pure (following [32]) if all non-oracle gates are of bounded fan-in. (Note that this use of the term 'pure' is not related to the 'pure' arithmetic circuits defined by Nisan and Wigderson [46].…”

“…In this paper, we combine these two threads of derandomization with the recent insight that, in some cases, extremely modest-sounding (or even 'pathetic') lower bounds can be amplified to obtain superpolynomial bounds [32]. In order to carry out this combination, we need to identify and exploit some special properties of certain functions in and near NC 1 .…”

Binary Pattern Tile Set Synthesis Is NP-Hard

“…This kind of method is likely to become common when it comes to proving correctness of biological or chemical systems, due to the complexity of these objects. Moreover, the "natural proofs" line of research [1,4,30,34] also suggests that our ability to produce and verify large proofs is likely to become fundamental in complexity theory.…”

Cracks in the Defenses: Scouting Out Approaches on Circuit Lower Bounds

“…It turns out that this would have significant consequences. It is shown in [11] that, if NC 1 = TC 0 , then for every > 0, the Boolean Formula Evaluation problem has TC 0 circuits of size n 1+ . That is, proving even a size lower bound of n 1.01 would separate TC 0 from NC 1 .…”