Kolmagorav Complexity, Complexity Cores, and the Distribution of Hardness

Problems that are complete for exponential space are provably intractable and known to be exceedingly complex in several technical respects. However, every problem decidable in exponential space is efficiently reducible to every complete problem, so each complete problem must have a highly organized structure. The authors have recently exploited this fact to prove that complete problems are, in two respects, unusually simple for problems in expontential space. Specifically, every complete problem must have unusually small complexity cores and unusually low space-bounded Kolmogorov complexity. It follows that the complete problems form a negligibly small subclass of the problems decidable in exponential space. This paper explains the main ideas of this work

The Structure of logarithmic advice complexity classes

A nonuniform class called here Full-P/log, due to Ko, is studied. It corresponds to polynomial time with logarithmically long advice. Its importance lies in the structural properties it enjoys, more interesting than those of the alternative class P/log; specifically, its introduction was motivated by the need of a logarithmic advice class closed under polynomial-time deterministic reductions. Several characterizations of Full-P/log are shown, formulated in terms of various sorts of tally sets with very small information content. A study of its inner structure is presented, by considering the most usual reducibilities and looking for the relationships among the corresponding reduction and equivalence classes defined from these special tally sets.Postprint (published version

The complexity and distribution of computationally useful problems

The solutions of certain natural decision problems such as the halting problem and the boolean satisfiability problem contain large amounts of useful information about computation that is highly organized and readily available to efficient computational processes. Such problems are computationally useful. This dissertation investigates the complexity and distribution of these computationally useful problems. The main results of this dissertation are of the following three general types. (1) Useful problems contain highly organized information. (2) Very useful problems are so highly organized that they are unusually simple and hence rare. (3) Useful problems are, as a whole, not rare and thus are not necessarily simple;A result of type (1) is proven in Chapter 3. Bennett recently extended algorithmic information theory to include a notion of computational depth that appears to quantify the level of organization in binary strings and sequences. The main result of Chapter 3 states that every weakly useful sequence is strongly deep. (A sequence x is weakly useful if a non-negligible set of recursive problems are decidable within a fixed recursive time bound when given access to x.);Results of type (2) are presented in Chapters 4 and 5. These results say that the ≤[subscript]sp m P-complete problems for E = DTIME(2[superscript] linear) and the ≤[subscript]sp m p/poly-complete problems for ESPACE = DSPACE(2[superscript] linear) are unusually simple and hence rare. Complete problems are very useful because every problem in E or ESPACE is efficiently decidable when given access to one of these problems;Chapter 6 develops a result of type (3). This result says that the weakly ≤[subscript]sp m P-complete problems for E and ESPACE are not rare and hence are not necessarily simple. Weakly complete problems are useful because every problem in a non-negligible subset of E or ESPACE is efficiently decidable when given access to one of these problems;The above results (and others along the way) are obtained through a systematic investigation of the measure-theoretic structure of complexity classes