The complexity and distribution of computationally useful problems
David W. Juedes · 1994
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.