Finite-State Dimension

Classical Hausdorff dimension was recently effectivized using gales (betting strategies that generalize martingales), thereby endowing various complexity classes with dimension structure and also defining the constructive dimensions of individual binary (infinite) sequences. In this paper we use gales computed by multi-account finite-state gamblers to develop the finite-state dimensions of sets of binary sequences and individual binary sequences. Every rational sequence (binary expansion of a rational number) has finite-state dimension 0, but every rational number in [0; 1] is the finite-state dimension of a sequence in the low-level complexity class AC₀. Our main theorem shows that the finite-state dimension of a sequence is precisely the infimum of all compression ratios achievable on the sequence by information-lossless finite-state compressors.