Density Condensation of Boolean Formulas

Backgrounds and Motivations. Conventional complexity theory gives us only asymptotic information and does not give us any information about the complexity of each individual instance. It is also true, however, that many of us are feeling that the complexity of each instance is quite different from one to another. Instance complexity, denoted by ic(x: A), has been thus introduced [7, 9] as a measure of the complexity of an individual instance x for a decision problem A. ic(x: A) is defined as the length of the shortest program that gives the correct answer to the query “x ∈ A? ” and that does not make any mistake for other inputs (although “don’t know ” answers are permitted). It is closely related with (at least, upper bounded by) Kolmogorov complexity, which is the length of the shortest program that computes x from the empty input. Under this new measure, each element in A can have a different instance complexity; some are easy and some are hard. Now it is obviously desirable if we can convert a hard instance into an easy one by reducing its instance complexity. More concretely, this can be done by designing a mapping (algorithm) δ such that for each instance x, (i) δ(x) ∈ A iff x ∈ A and (ii) ic(δ(x) : A) < ic(x: A). Note that determining the answer (yes/no) of an instance x is a special case of a complexity reduction, i.e., the complete reduction which converts x into a trivial instance whose answer is instantly known. Thus