Exploring Randomness (Discrete Mathematics and Theoretical Computer Science)

This book presents the technical core of Chaitin's theory of program-size complexity, also known as algorithmic information theory. LISP is used to present the key algorithms and to enable computer users to interact with the author's proofs and discover for themselves how they work. In The Unknowable I use LISP to compare my work on incompleteness with that of G6del and Turing, and in The Limits of Mathematics I use LISP to discuss my work on incompleteness in more detail. In this book we'll use LISP to explore my theory of randomness, called algorithmic information theory (AIT). And when I say'explore'I mean it! This book is full of exercises for the reader, ranging from the mathematical equivalent oftrivial'fin­ ger warm-ups'for pianists, to substantial programming projects, to questions I can formulate precisely but don't know how to answer, to questions that I don't even know how to formulate precisely! I really want you to follow my example and hike offinto the wilder­ ness and explore AIT on your own! You can stay on the trails that I've blazed and explore the well-known part of AIT, or you can go off on your own and become a fellow researcher, a colleague of mine! One way or another, the goal of this book is to make you into a participant, not a passive observer of AlT. In other words, it's too easy to just listen to a recording of AIT, that's not the way to learn music.

This essential companion to Chaitin's successful books The Unknowable and The Limits of Mathematics, presents the technical core of his theory of program-size complexity. The two previous volumes are more concerned with applications to meta-mathematics. LISP is used to present the key algorithms and to enable computer users to interact with the authors proofs and discover for themselves how they work. The LISP code for this book is available at the author's Web site together with a Java applet LISP interpreter. "No one has looked deeper and farther into the abyss of randomness and its role in mathematics than Greg Chaitin. This book tells you everything hes seen. Don miss it." John Casti, Santa Fe Institute, Author of Goedel: A Life of Logic.'