Jack Jie Dai

Finite-state dimension

Classical Hausdorff dimension (sometimes called fractal 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. The theorem of Eggleston (Quart. J. Math. Oxford Ser. 20 (1949) 31-36) relating Hausdorff dimension to entropy is shown to hold for finite-state dimension, both in the space of all sequences and in the space of all rational sequences (binary expansions of rational numbers). Every rational sequence 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.

A result regarding convergence of random logistic maps

Let {Ci}[infinity]0 be i.i.d. random variables with values in [1,4]. Define random variables {Xn}[infinity]0 with values in [0,1] by Xn+1=CnXn(1-Xn). Then under some mild conditions, the probability distribution of Xn converges in variation norm.

Random random walks on the integers mod n

This paper considers typical random walks on the integers mod n such that the random walk is supported on constant k values. This paper extends a result of Hildebrand to show that for any integer n, roughly n2/(k-1) steps usually suffice to get the random walk close to uniformly distributed if the k values satisfy some conditions needed for the random walk to get close to uniformly distributed.

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.

A stronger Kolmogorov zero-one law for resource-bounded measure

Resource-bounded measure has been defined on the classes E, E2, ESPACE, E2SPACE, REC, and the class of all languages. It is shown here that if C is any of these classes and X is a set of languages that is closed under finite variations and has outer measure > 1 in C, then X has measure 0 in C. This result strengthens Lutzs resource-bounded generalization of the classical Kolmogorov zero-one law. It also gives a useful sufficient condition for proving that a set has measure 0 in a complexity class.