We investigate the question whether there is a (p-)optimal proof system
for SAT or for TAUT and its relation to completeness and collapse results
for nondeterministic function classes. A p-optimal proof system for SAT is
shown to imply (1) that there exists a complete function for the class of all
total nondeterministic multi-valued functions and (2) that any set
with an optimal proof system has a p-optimal proof system.
By replacing the assumption of the mere existence of a (p-)optimal proof
system by the assumption that certain proof systems are (p-)optimal we
obtain stronger consequences, namely collapse results for various function
classes. Especially we investigate the question whether the standard
proof system for SAT is p-optimal. We show that this assumption is
equivalent to a variety of complexity theoretical assertions studied
before, and to the assumption that every optimal proof system is
p-optimal. Finally, we investigate the question whether there is an optimal
proof system for TAUT that has the effective interpolation property, and
show some relations between various completeness assumptions.