Dimension in complexity classes

A theory of resource-bounded dimension is developed using gales, which are natural generalizations of martin-gales. When the resource bound /spl Delta/(a parameter of the theory) is unrestricted, the resulting dimension is precisely the classical Haludolff dimension (sometimes called "fractal dimension"). Other choices of the parameter /spl Delta/ yield internal dimension theories in E, E/sub 2/, 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)/spl isin/[0, 1]. Along with the elements of this theory two preliminary applications are presented: 1. For every real number 0/spl les//spl alpha//spl les/ 1/2 , the set FREQ(/spl ap//spl alpha/), consisting of all languages that asymptotically contain at most /spl alpha/ of all strings, has dimension /spl Hscr/(/spl alpha/)-the binary entropy of /spl alpha/-in E and in E/sub 2/. 2. For every real number 0/spl les//spl alpha//spl les/1, the set SIZE(/spl alpha/2/sup n//n), consisting of all languages decidable by Boolean circuits of at most /spl alpha/2/sup n//n gates, has dimension /spl alpha/ in ESPACE.