I know that boolean satisfiability is NP-Complete, but is the minimization/simplification of a boolean expression, by which I mean taking a given expression in symbolic form and producing an equivalent but simplified expression in symbolic form, NP-Complete? I'm not sure that there's a reduction from satisfiability to minimization, but I feel like there probably is. Does anyone know for sure?
Well, look at it this way: using a minimisation algorithm, you can compact any non-satisfiable expression to the literal false, right? This effectively solves SAT. So at least a complete minimisation algorithm must be NP-hard.
As per the comment by Para Black, the problem (which is also known as the minimum equivalent DNF problem) was actually shown to be non-NP and Σ2P-complete by Chris Umans in 1998.