A complexity theory for feasible closure properties

The authors propose and develop a complexity theory of feasible closure properties. For each of the classes Hash P, SpanP, OptP, and MidP, they establish complete characterizations-in terms of complexity class collapses-of the conditions under which the class has all feasible closure properties. In particular, Hash P is P-closed if and only if PP=UP; SpanP is P-closed if and only if R-MidP is P-closed if and only if P/sup PP/=NP; and OptP is P-closed if and only if NP=co-NP. Furthermore, for each of these classes, the authors show natural operations-such as subtraction and division-to be hard closure properties, in the sense that if a class is closed under one of these, then it has all feasible closure properties. They also study potentially intermediate closure properties for Hash P. These properties-maximum, minimum, median, and decrement-seem neither to be possessed by Hash P nor to be Hash P-hard.<>