Computing and evolving variants of computational depth

Abstract

The structure and organization of information in binary strings and (infinite) binary sequences are investigated using two computable measures of complexity related to computational depth. First, fundamental properties of recursive computational depth, a refinement of Bennett's original notion of computational depth, are developed, and it is shown that the recursively weakly (respectively, strongly) deep sequences form a proper subclass of the class of weakly (respectively, strongly) deep sequences. It is then shown that every weakly useful sequence is recursively strongly deep, strengthening a theorem by Juedes, Lathrop, and Lutz. It follows from these results that not every strongly deep sequence is weakly useful, thereby answering an open question posed by Juedes. Second, compression depth, a feasibly computable depth measurement, is developed based on the Lempel-Ziv compression algorithm. LZ compression depth is further formalized by introducing strongly (compression) deep sequences and showing that analogues of the main properties of computational depth hold for compression depth. Critical to these results, it is shown that a sequence that is not normal must be compressible by the Lempei-Ziv algorithm. This yields a new, simpler proof that the Champernowne sequence is normal. Compression depth is also used to measure the organization of genes in genetic algorithms. Using finite-state machines to control the actions of an automaton playing prisoner's dilemma, a genetic algorithm is used to evolve a population of finite-state machines (players) to play prisoner's dilemma against each other. Since the fitness function is based solely on how well a player performs against all other players in the population, any accumulation of compression depth (organization) in the genetic structure of the player can only by attributed to the fact that more fit players have a more highly organized genetic structure. It is shown experimentally that this is the case.