Remarks on "Random Sequences"

We show that standard statistical tests for randomness of finite sequences are language-dependent in an inductively pernicious way

Notes on Hierarchies and Inductive Inference

The following notes rework a discussion due to Kevin Kelly on the application of topological notions in the context of learning (see Kelly (1990)). All the results except for (2), (4) and (9) are due to Kelly, but are proved differently

Relevant Consequence and Empirical Inquiry

A criterion of adequacy is proposed for theories of relevant consequence. According to the criterion, scientists whose deductive reasoning is limited to some proposed subset of the standard consequence relation must not thereby suffer a reduction in scientific competence. A simple theory of relevant consequence is introduced and shown to satisfy the criterion with respect to a formally defined paradigm of empirical inquiry

Synthesizing inductive expertise

AbstractWe consider programs that accept descriptions of inductive inference problems and return machines that solve them. Several design specifications for synthesizers of this kind are considered from a recursion-theoretic perspective

Uniform Inductive Improvement

We examine uniform procedures for improving the scientific competence of inductive inference machines. Formally, such procedures are construed as recursive operators. Several senses of improvement are considered, including (a) enlarging the class of functions on which success is certain, and (b) transforming probable success into certain success

A Universal Inductive Inference Machine

A paradigm of scientific discovery is defined within a first-order logical framework. It is shown that within this paradigm there exists a formal scientist that is Turing computable and universal in the sense that it solves every problem that any scientist can solve. It is also shown that universal scientists exist for no regular logics that extend first order logic and satisfy the Lowenheim-Skolem condition

Logic and Learning

The theory of first-order logic - or Model Theory - appears in few studies of learning and scientific discovery. We speculate about the reasons for this omission, and then argue for the utility of Model Theory in the analysis and design of automated systems of scientific discovery. One scientific task is treated from this perspective in detail, namely, concept discovery. Two formal paradigms bearing on this probleni are presented and investigated using the tools of logical theory. One paradigm bears on PAC learning, the other on identification in the limit