Dimension in Complexity Classes

A theory of resource-bounded dimension is developed using gales, which are natural generalizations of martingales. When the resource bound ∆ (a parameter of the theory) is unrestricted, the resulting dimension is precisely the classical Hausdorff dimension (sometimes called “fractal dimension”). Other choices of the parameter ∆ yield internal dimension theories in E, E2, ESPACE, and other complexity classes, and in the class of all decidable problems. In general, if C is such a class, then every set X of languages has a dimension in C, which is a real number dim(X | C) ∈ [0, 1]. Along with the elements of this theory, two preliminary applications are presented: 1. For every real number 0 ≤ α ≤ 1 2 , the set FREQ(≤ α), consisting of all languages that asymptotically contain at most α of all strings, has dimension H(α) — the binary entropy of α — in E and in E2. 2. For every real number 0 ≤ α ≤ 1, the set SIZE(α n n ), consisting of all languages decidable by Boolean circuits of at most α n n gates, has dimension α in ESPACE.