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 a (partial) complexity reduction should be useful if it can be done in a significantly smaller amount of time compared to its complete version. A little surprisingly, however, we have virtually no literature on this problem. The reason is probably this: Considering the (very abstract) nature of the above definition of instance complexity, it appears to be quite difficult to design nontrivial algorithms for its reduction. Our Contribution. In this paper, we focus on the Boolean satisfiability problem and our basic goal is exactly as mentioned above, i.e., we wish to convert a hard instance (formula) into an easy one. What we do, however, is not a complexity reduction but a density extension (DE): Let f be a Boolean formula of n variables. Then its density, denoted by d(f), is defined as d(f) = (the number of f ’s satisfying assignments) / 2. One can see that d(f) is the probability that random sampling of an assignment hits a satisfying assignment. This intuitively means that a formula of high density is easier than a formula of low density; it is certainly desirable if we can extend the density of a formula f . (We sometimes say that “we condense f .”) Now we should be more careful: When we use the random testing as mentioned above, its time complexity depends not only on d(f) but also on |f | (= the length of f) since we need to evaluate the value of f for each assignment. Therefore, |f |/d(f)