Dimension Extractors

A dimension extractor is an algorithm designed to increase the effective dimension – i.e., the computational information density – of an infinite sequence. A constructive dimension extractor is exhibited by showing that every sequence of positive constructive dimension is Turing equivalent to a sequence of constructive strong dimension arbitrarily close to 1. Similar results are shown for computable dimension and truth-table equivalence, and for pispace dimension and pispace Turing equivalence, where pispace represents Lutz’s hierarchy of super-polynomial space bounds. Thus, with respect to constructive, computable, and pispace information density, any sequence in which almost every prefix has information density bounded away from zero can be used to compute a sequence in which infinitely many prefixes have information density that is nearly maximal. In the constructive dimension case, the reduction is uniform with respect to the input sequence: a single oracle Turing machine, taking as input a rational upper bound on the dimension of the input sequence, works for every input sequence of positive constructive dimension. As an application, the resource-bounded extractors are used to characterize the computable dimension of individual sequences in terms of compression via truthtable reductions and to characterize the pispace dimension of individual sequences in terms of compression via pispace-bounded Turing reductions, in analogy to previous known results connecting effective dimensions to compression with effective reductions. ∗This research was funded in part by grant number 9972653 from the National Science Foundation as part of their Integrative Graduate Education and Research Traineeship (IGERT) program.