The Limits of Mathematics

This condensed version of chao-dyn/9509010 will be the main hand-out for a course on algorithmic information theory to be given 22-29 May 1996 at the Rovaniemi Institute of Technology, Rovaniemi, Finland (see announcement at http://www.rotol.fi/ ).Comment: LaTeX, 45 page

An engineering approach to automatic programming

An exploratory study of the automatic generation and optimization of symbolic programs using DECOM - a prototypical requirement specification model implemented in pure LISP was undertaken. It was concluded, on the basis of this study, that symbolic processing languages such as LISP can support a style of programming based upon formal transformation and dependent upon the expression of constraints in an object-oriented environment. Such languages can represent all aspects of the software generation process (including heuristic algorithms for effecting parallel search) as dynamic processes since data and program are represented in a uniform format

Providing Self-Aware Systems with Reflexivity

We propose a new type of self-aware systems inspired by ideas from higher-order theories of consciousness. First, we discussed the crucial distinction between introspection and reflexion. Then, we focus on computational reflexion as a mechanism by which a computer program can inspect its own code at every stage of the computation. Finally, we provide a formal definition and a proof-of-concept implementation of computational reflexion, viewed as an enriched form of program interpretation and a way to dynamically "augment" a computational process.Comment: 12 pages plus bibliography, appendices with code description, code of the proof-of-concept implementation, and examples of executio

Very Simple Chaitin Machines for Concrete AIT

In 1975, Chaitin introduced his celebrated Omega number, the halting probability of a universal Chaitin machine, a universal Turing machine with a prefix-free domain. The Omega number's bits are {\em algorithmically random}--there is no reason the bits should be the way they are, if we define ``reason'' to be a computable explanation smaller than the data itself. Since that time, only {\em two} explicit universal Chaitin machines have been proposed, both by Chaitin himself. Concrete algorithmic information theory involves the study of particular universal Turing machines, about which one can state theorems with specific numerical bounds, rather than include terms like O(1). We present several new tiny Chaitin machines (those with a prefix-free domain) suitable for the study of concrete algorithmic information theory. One of the machines, which we call Keraia, is a binary encoding of lambda calculus based on a curried lambda operator. Source code is included in the appendices. We also give an algorithm for restricting the domain of blank-endmarker machines to a prefix-free domain over an alphabet that does not include the endmarker; this allows one to take many universal Turing machines and construct universal Chaitin machines from them