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I am following the book of Arora and Barak book.

We consider Boolean circuits as we do, Specifically, inner nodes are either AND, OR (both – fan-in 2), or NOT (fan-in 1) gates. The fan-out of each gate is 2. The size of a circuit is the number of inner nodes (gates) in it.

Let $ ∶ \{0,1\}^ → \{0,1\}$ be a function, where $ > 1$ is a Natural number. Assume that there exist circuits $_0, _1$ both of size at most $,$ such that for each $ ∈ \{0,1\},$ the circuit $_$ computes the function $_ ∶ \{0,1\}^{−1} → \{0,1\}$, defined by $$_(_1, … , _{−1}) = (_1, … , _{−1}, ).$$

But the book mention that there is a circuit $$ of size at most $2 + 4$ that computes $,$ that I don't understand. How they get size at most $2d+4$?

2 Answers2

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Probably they mean the size for (g0 AND NOT b) OR (g1 AND b).

Hendrik Jan
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I think another answer beside Hendrik Jan's one is $(b\lor g_0)\land (\neg b \lor g_1)$, deduced from "if-else form" $(b\to g_1)\land (\neg b\to g_0)$, i.e. if $b$ then $g_1$ else $g_0$.

minh quý lê
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