Measuring static complexity

Abstract

The concept of “pattern” is introduced, formally defined, and used to analyze various measures of the complexity of finite binary sequences and other objects. The standard Kolmogoroff-Chaitin-Solomonoff complexity measure is considered, along with Bennett's ‘logical depth’, Koppel's ‘sophistication'’, and Chaitin's analysis of the complexity of geometric objects. The pattern-theoretic point of view illuminates the shortcomings of these measures and leads to specific improvements, it gives rise to two novel mathematical concepts--“orders” of complexity and “levels” of pattern, and it yields a new measure of complexity, the “structural complexity”, which measures the total amount of structure an entity possesses.

Keywords

• Kolmogorov complexity
• algorithmic information
• pattern
• sophistication
• structure
• depth.