By Jukna S.

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Gt is the function computed at its last gate gt . So we now assume that every circuit computes only one boolean function. This implies that every class F ⊆ Bn of |F | > φ(n, t) functions must contain a function requiring circuits of size > t. This was the main idea of Riordan–Shannon’s argument. 12 Lemma φ(n, t) ≤ tt e2t+4n . In particular, φ(n, t) ≤ 2t for t ≥ n ≥ 16. Proof. Clearly, we may suppose n, t ≥ 2. Let g1 , . . , gt be names of the gates in a circuit. To describe a concrete circuit, it is suﬃcient to attach to each gate one of the connectives ∧, ∨, ¬ and an unordered pair of names of two other gates or literals.

Thus Lim(Q) exists and is a number in the interval [1, 2]. 24, every invariant class Q of boolean functions deﬁnes the unique real number 0 ≤ σ ≤ 1 such that Lim(Q) = 2σ . This number is an important parameter of the invariant class characterizing its cardinality. It also characterizes the maximum circuit complexity of functions in Q. We will therefore denote this parameter by writing Qσ if σ is the parameter of Q. For example, if P is the class of all linear boolean functions (parity functions), then |P (n)| ≤ 2n+1 , implying that Lim(P ) = 1, and hence, σ = 0.

In particular, if t(n) ≫ n log n, then for any ﬁxed d, φ(n, t) ≥ nd ·φ(n, t/2) for all suﬃciently large n. 8), Chow sets N = 2n and lets A ⊆ {0, 1}N to be the set of all truth tables of boolean functions f ∈ Bn computable circuits of size at most t. A truth table is a 0-1 vector a = (a1 , . . , aN ), and it describes the unique n i−1 function fa ∈ Bn deﬁned by fa (x) = abin(x) where bin(x) = i=1 xi 2 n is the number whose binary code is vector x ∈ {0, 1} . The boundary δ(A) of A ⊂ {0, 1}N is the set of all vectors b ∈ A that diﬀer from at least one a ∈ A in exactly one position.

### Boolean function complexity by Jukna S.

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