Structural metrics: An epistemology

Structural Metrics is an epistemological survey of methods of comparing structures. The text shows how fundamental limits of knowledge affect our ability to objectively compare the intrinsic rules and relationships that govern two given objects. Several ways of analyzing, generating, comparing and mutating structures are examined throughout bringing four fields together: music, communication theory, algorithmic information theory and graph theory. Even though Structural Metrics is rooted in music, it applies to many other fields and is quintessentially pandisciplinary.;The epistemological survey results in two important and novel findings. The first finding came from asking whether it is possible to objectively determine structural similarity of two objects (such as pieces of music). What are the limits of such metrics? A survey of structural metrics reveals a fundamental tradeoff of objectivity for practicality in structural analysis. While we may know that two objects share information, it is basically impossible to know exactly how much information they share. The most objective metric is based on the minimal programs of two objects. A minimal program, defined by Gregory Chaitin as the smallest computer program that calculates a given object, is a maximally compressed encoding of an object. Knowing whether or not a program is minimal is essentially impossible due to the halting problem, which is the inability to compute whether an arbitrary program halts. The halting problem was first pointed out by Alan Turing in 1936. That is, what we need to know to objectively determine differences in structure, we cannot know. All other methods of structural analysis that do not use maximally compressed representations of the objects under examination sacrifice the objectivity of the analysis. What is perhaps often gained are more practically efficient methods. The second finding resulted from the initial goal of implementing structural metrics in evolutionary algorithms. Practical structural mutations cannot be made to the genome of an object (the computer program that generates it) because a minor mutation may result in a non-halting program. This raised the question of how an abstract mathematical theory of evolution can incorporate mutations to the genome and model 'real-world' properties of evolution (regardless of practicality). The result is a new abstract mathematical theory of evolution that takes into account the halting problem as the reason for evolutionary advancement in both minor and major evolutionary leaps.;The text is organized as follows. First, preliminary discussions on music, music related philosophies and information theory are provided. Then, ways to generate and represent musical structures are illustrated followed by a survey of several structural metrics. Finally, it is shown how these metrics can be implemented in 'artificial' evolutionary algorithms in order to reveal how such algorithms do not mimic several important properties of 'real-world' evolution. Thus, the text itself has a trajectory from the generative to the analytical and back. In conclusion, a new abstract mathematical definition of 'real-world' evolution is given along with a return to the more philosophical discussions set, forth in the beginning. At the end, I present several musical works that informed and were informed by the research.