How does a half adder made of crossbar latches operate?
When the inputs \$\small{\text{A}}\$, \$\mathsf{\small \overline{\text{A}}}\$, \$\small{\text{B}}\$, and \$\mathsf{\small \overline{\text{B}}}\$ are latched in, they create four minterms \$\mathsf{\small \overline{\text{A}}}\$ ⦁ \$\mathsf{\small \overline{\text{B}}}\$, \$\small{\text{A ⦁ B}}\$, \$\mathsf{\small \overline{\text{A}}}\$, and \$\mathsf{\small \overline{\text{B}}}\$ which are generated vertically using a wired-AND configuration as described in the patent.

Two outputs are then generated:  SUM = \$ \overline{ (\small \overline{\text{A}} ⦁ \small \overline{\text{B}}) + (\small {\text{A}} ⦁ \small {\text{B}}) },  \$

and

CARRY = \$ \overline{ \small \overline{\text{A}} + \small \overline{\text{B}} }  \$

because the outputs are generated horizontally using a NOR function as described in the patent.

Then using De Morgan's laws, The first equation simplifies to A xor B, and the second to A ⦁ B which are the definition of a half adder.

You can verify this by typing "simplify not((not a and not b) or (a and b))" into WolframAlpha for the first equation, and "simplify not(not a or not b)" for the second.