It is known that for all monotone functions f : {0, 1}n → {0, 1}, if x ∈ {0, 1}n is chosen uniformly at random and y is obtained from x by flipping each of the bits of x independently with probability = n, then P[f(x) 6= f(y)] < cn, for some c > 0. Previously, the best construction of monotone functions satisfying P[fn(x) 6= fn(y)] ≥ δ, where 0 < δ < 1/2, required ≥ c(δ)n, where α = 1 − ln 2/ l...

Recall the online learning model we discussed in the previous lecture: N = # experts For t = 1, 2, . . . , T rounds: 1) each expert i, 1 ≤ i ≤ N , makes a prediction ξi ∈ {0, 1} 2) learner makes a prediction ŷ ∈ {0, 1} 3) observe outcome y ∈ {0, 1} (a mistake happens if ŷ 6= y) With this framework in hand, we investigated a particular algorithm, Weighted Majority Algorithm (WMA), as follows: N ...

The initial majority identification task is a fundamental test problem in cellular automaton research. To pass the test, two-state has to attain uniform configuration consisting of only state that was initially majority. It does so solely through its local, internal dynamics—i.e., success an example emergent computation. Finding new, better-performing automata continues be interest for what add...