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Lecture 3

November 15, 2012

Today we continued to talk about classical results in circuit complexity. We saw Karp-Lipton Theorem ’80: NP has polynomial size circuits implies the collapses of the Polynomial Hierarchy to its second level, thus supporting circuit lower bounds as a route for separating P from NP.

A similar theorem, attributed to Meyer ’80, shows that if EXP has polynomial size circuits then EXP collapses to the second level of the Polynomial Hierarchy (later on we will prove something stronger). From this we deduced a neat corollary, namely, if EXP has¬†polynomial size circuits then P is not equal to NP. That is, circuit upper bounds can potentially separate P from NP!

A third classical theorem we saw was Kannan Theorem ’81. This theorem states that there is no fixed polynomial size circuits for the second level of the Polynomial Hierarchy.

Going in a somewhat different direction, we proved a 3(n-1) lower bound for the number of AND,OR gates in any Boolean circuit that computes the PARITY function. The proof, by Schnorr ’74, is very simple and is based on an approach called gate-elimination. Unfortunately, this lower bound is not far from the best known lower bound 5n-o(n) proven by Iwama, Lachish, Morizumi and Raz. This paper appears in the end of the semester talks.

We then started to talk about subclasses of P/poly, especially AC^i and NC^i. We stated a famous result by Hastad (improving upon previous results) that an AC^0 circuit that computes PARITY has exponential size. This result also participates in the end of the semester talks (including a very recent paper of Hastad). The latter result suggests adding unbounded fan-in PARITY gates to AC^0 circuits, or more generally, adding counters to AC^0 circuits. This class is called ACC^0 (or sometimes simply ACC).

We stated Razborov-Smolensky Thoerem ’87, which shows that for distinct primes p and q, the MOD_q function is not in ACC(p), and in fact requires exponential size for constant depth. We will see the proof of this thoerem next time. Anyhow, this implies that ACC(7) is weak (as, for example, it cannot compute PARITY). What about ACC(6)? This seemingly innocent question turned out to be notorious. At this point we were finally able to state and appreciate the result by Williams.

Next lecture we will prove Razborov-Smolensky and will start to talk about randomization in computation. Also, the scribe notes for the first two lectures should be uploaded to the blog soon.

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