A constructive version of Hausdorff dimension is developed using constructive supergales, which are betting strategies that generalize the constructive supermartingales used in the theory of individual random sequences. This constructive dimension is used to assign every individual (infinite, binary) sequence S a dimension, which is a real number dim(S) in the interval [0, 1]. Sequences that

Resource-bounded dimension is a complexity-theoretic extension of classical Hausdorff dimension introduced by Lutz (2000) in order to investigate the fractal structure of sets that have resource-bounded measure 0. For example, while it has long been known that the Boolean circuit-size complexity class SIZE � α 2n n has measure 0 in ESPACE for all 0 ≤ α ≤ 1, we now know that SIZE � α 2n n has dimension α in ESPACE for all 0 ≤ α ≤ 1. The present paper furthers this program by developing a natural hierarchy of “rescaled” resource-bounded dimensions. For each integer i and each set X of decision problems, we define the ith-order dimension of X in suitable complexity classes. The 0th-order dimension is precisely the dimension of Hausdorff (1919) and Lutz (2000). Higher and lower orders are useful for various sets X. For example, we prove the following for 0 ≤ α ≤ 1 and any polynomial q(n) ≥ n2. 1. The class SIZE(2 αn) and the time- and space-bounded Kolmogorov complexity classes KT q (2 αn) and KS q (2 αn) have 1 st-order dimension α in ESPACE. 2. The classes SIZE(2nα), KT q (2nα), and KS q (2nα) have 2nd-order dimension α in ESPACE.

We study computably enumerable reals (i.e. their left cut is computably enumerable) in terms of their spectra of representations and presentations.

Supergales, generalizations of supermartingales, have been used by Lutz (2002) to define the constructive dimensions of individual binary sequences. Here it is shown that gales, the corresponding generalizations of martingales, can be equivalently used to define constructive dimension. 1

Given a set X of sequences over a nite alphabet, we investigate the following three quantities. (i) The feasible predictability of X is the highest success ratio that a polynomial-time randomized predictor can achieve on all sequences in X.

A binary sequence A = A(0)A(1) ... is called i.o. Turing-autoreducible if A is reducible to itself via an oracle Turing machine that never queries its oracle at the current input, outputs either A(x) or a don't-know symbol on any given input x, and outputs A(x) for infinitely many x. If in addition the oracle Turing machine terminates on all inputs and oracles, A is called i.o. truth-table-autoreducible.

We give a direct and rather simple construction of Martin-Löf random and rec-random sets with certain additional properties. First, reviewing the result of Gacs and Kucera, given any set X we construct a Martin-Löf random set R from which X can be decoded effectively. Second, by essentially the same construction we obtain a Martin-Löf random set R that is computably enumerable selfreducible. Alternatively, using the observation that a set is computably enumerable selfreducible if and only if its associated real is computably enumerable, the existence of such a set R follows from the known fact that every Chaitin real is Martin-Löf random and computably enumerable. Third, by a variant of the basic construction we obtain a rec-random set that is weak truthtable autoreducible. The mentioned results on self- and autoreducibility complement work of Ebert, Merkle, and Vollmer [7-9], from which it follows that no Martin-Löf random set is Turing-autoreducible and that no rec-random set is truth-table autoreducible.

The word "martingale " has related, but different, meanings in probability theory and theoretical computer science. In computational complexity and algorithmic information theory, a martingale is typically a function d on strings such that E(d(wb)|w) = d(w) for all strings w, where the conditional expectation is computed over all possible values of the next symbol b. In modern probability theory a martingale is typically a sequence,0,,1,,2,... of random variables such that E(,n+1|,0,...,,n) =,n for all n.

This thesis studies average-case complexity theory and polynomial-time reducibilities. The issues in average-case complexity arise primarily from Cai and Selman's extension of Levin's denition of average polynomial time. We study polynomial-time reductions between distributional problems. Under strong but reasonable hypotheses we separate ordinary NP-completeness notions.

Motivated by the observation that executions of a probabilistic system almost surely are fair, we interpret concepts of fairness for nondeterministic processes as partial descriptions of probabilistic behavior. We propose computable fairness as a very strong concept of fairness, attempting to capture all the qualitative properties of probabilistic behavior that we might reasonably expect to see in the behavior of a nondeterministic system. It is shown that computable fairness does describe probabilistic behavior by proving that runs of a probabilistic system almost surely are computable fair. We then turn to the question of how sharp an approximation of randomness is obtained by computable fairness by discussing completeness of computable fairness for certain classes of path properties.