half-gates in garble circuit
As classical Yao circuit, two-party have their own input independently, why he assumes that they both know neither input?


Consider the circuit $(a_1 \oplus b_1) \land (a_2 \oplus b_2)$ where Alice's input is $a_1,a_2$ and Bob's input is $b_1,b_2$. The bits going into the AND gate are not known to either party, so you can't use one of the half-AND constructions.

Parties know their inputs to the circuit but not the input to every gate in the circuit. The bits going into an internal gate in the circuit can be influenced by both parties' inputs, so that neither knows the value in the clear. (You can indeed replace AND gates with half-AND gates at the input layer of a circuit.)