Boolean function
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When the domain is restricted to the n-dimensional hypercube [ 0 , 1 ] n {\displaystyle [0,1]^{n}} , the polynomial f ∗ ∗ ( x ) : [ 0 , 1 ] n → → [ 0 , 1 ] {\displaystyle f^{*}(x):[0,1]^{n}\rightarrow [0,1]} gives the probability of a positive outcome when the Boolean function f is applied to n independent random (Bernoulli) variables, with individual probabilities x. A special case of this fact is the piling-up lemma for parity functions. The polynomial form of a Boolean function can also be used as its natural extension to fuzzy logic.
On the symmetric hypercube