# Lecture 6

Today we talked about Hardness-Randomness tradeoffs. We stated, without a proof, that if E requires circuits of size S(n) then one can construct a pseudorandom generator with stretch S(n)^c for some constant c < 1. We talked informally about the proof strategy and mentioned that usually the theorem is proven in two steps. In the first step one constructs the PRG under a stronger assumption, namely, that E cannot be approximated by small circuits (see Theorem 20.6 in Arora-Barak book). Then one shows that the assumption can be weakened to the desired assumption, that is, a computational assumption rather than an approximation one.

A key ingerdient in the second step is local decodable codes, or local list decodable codes (see Chapter 19 in Arora-Barak book). A somewhat more recent construction that doesn’t go through these two steps was given by Umans in a beautiful paper. This paper is one of the papers for the end of the semester talks.

As mentioned, in this course we focus on the “other direction”, namely, showing that derandomization implies circuit lower bounds. We stated a beautiful result by Impagliazzo and Wigderson (see also Theorem 20.16 in Arora-Barak book) that shows that you can get some sort of derandomization under uniform assumptions only (BPP not equal to EXP). Nevertheless, the theorem we started to prove, by Impagliazzo and Kabanets (see Theorem 20.17 in Arora-Barak and references therein) shows that derandomizing ZEROP will yield some circuit lower bound – either NEXP is not in P/poly or the permanent requires super-polynomial arithmetic circuits.

To prove this theorem we took a short detour into interactive proofs, introduced the classes MA, IP and MIP (the latter just for fun, as it equals NEXP – a star complexity class in this course). Using the famous result IP=PSPACE (Theorem 8.19 in Arora-Barak book and references therein). We deduced that PSPACE is contained in P/poly implies PSPACE = MA. We used this to improves upon Meyer to show that if EXP is in P/poly then EXP collapses to MA (and not just to the second level of PH). Finally, we used this last statement to show that MA_EXP (~NEXP + randomness) is not contained in P/poly – this is the best known uniform lower bound for P/poly to my knowledge.

We then switched to talk about succinct problems (such as succinct-SAT) and more generally, about strings that their truth table can be computed by a small circuit. For a much better non-formal description than what I gave, see Williams’ casual tour around a circuit complexity bound (which will be useful also for the rest of the course). We started to prove a theorem by Imapagliazzo, Kabanets and Wigderson (Lemma 2.20 in Arora-Barak) that shows that if NEXP is contained in P/poly then NEXP=EXP (and thus =MA). The proof uses the easy witness method introduced by Kabanets (see his paper, which also appears at the end of the semester talks). Next time we will continue the proof.