By Bruck R.H.

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This is in sharp contrast with the fact, proved more than 60 years ago by Riordan and Shannon (1942) that most boolean functions require formulas of leafsize about 2n = log n. Then Shannon (1949) showed a lower bound 2n =n for circuits. Their arguments were the first applications of counting arguments in boolean function complexity: count how many different boolean functions of n variables can be computed using a given number of elementary operations, and n compare this number with the total number 22 of all boolean functions.

Comparing these bounds, we can conclude that the sequence F cannot be contained in any invariant class Q with < 1. t u This result serves as an indication that there (apparently) is no other way to construct a most-complex sequence of boolean function other than to do a “brute n force search” (or “perebor” in Russian): just try all 22 boolean functions. 5 So Where are the Complex Functions? Unfortunately, the results above are not quite satisfactory: we know that almost all boolean functions are complex, but no specific (or explicit) complex function is known.

13). 1 On Explicitness We are not going to introduce the classes of the complexity hierarchy. Instead, we will use the following simple definitions of “explicitness”. x; y/ in time polynomial in n C m. x; y/ D 1 for at least one y 2 f0; 1gm. In this case, simple functions correspond to the class P, and explicit functions form ˚ xn is “very explicit”: to the class NP. 5 So Where are the Complex Functions? 37 determine its value, it is enough just to sum up all bits and divide the result by 2. A classical example of an explicit function (a function in NP) which is not known to be in P is the clique function.

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