Inductive Reasoning and Komogorov Complexity

This is a sloppy first draft of [J. Comp. System Sciences, 44:2(1992), 343-384]. Also, there are some problems with the pictures and references due to the obsolete troff processing. Reasoning to obtain the ‘truth’ about reality, from external data, is an important, controversial, and complicated issue in man’s effort to understand nature. (Yet, today, we try to make machines do this.) There have been old useful principles, new exciting models, and intricate theories scattered in vastly different areas including philosophy of science, statistics, computer science, and psychology. We focus on inductive reasoning in correspondence with ideas of R.J. Solomonoff. While his proposals result in perfect procedures, they involve the noncomputable notion of Kolmogorov complexity. In this paper we develop the thesis that Solomonoff’s method is fundamental in the sense that many other induction principles can be viewed as particular ways to obtain computable approximations to it. We demonstrate this explicitly in the cases of Gold’s paradigm for inductive inference, Rissanen’s minimum description length (MDL) principle, Fisher’s maximum likelihood principle, and Jaynes’ maximum entropy principle. We