Showing NP-hardness of HALF-SAT
Hint: Take a new variable $x$ and add it to all clauses. This shows that HALF-SAT' is NP-hard, where HALF-SAT' differs from HALF-SAT by strengthening "at least half" to "more than half". HALF-SAT is similar.

The $P$-closure of HALF-SAT' (and HALF-SAT) forms the complexity class $PP$.