CIDEC     ÜIK Estonian Winter Schools in Computer Science     Eesti arvutiteaduse talvekoolid EWSCS 2003EATTK 2003

### Gregory Chaitin

IBM TJ Watson RC
Yorktown Heights, NY, USA

## Algorithmic information theory

### Abstract

Most work on computational complexity is concerned with time. However this course will try to show that program-size complexity, which measures algorithmic information, is of much greater philosophical significance. In particular, I'll discuss how one can use this complexity measure to study what can and cannot be achieved by formal axiomatic mathematical theories. In particular, I'll show (a) that there are natural information-theoretic constraints on formal axiomatic theories, and that program-size complexity provides an alternative path to incompleteness from the one originally used by Kurt Gödel. Furthermore, I'll show (b) that in pure mathematics there are mathematical facts that are true for no reason, that are true by accident. These have to do with determining the successive binary digits of the precise numerical value of the halting probability for a "self-delimiting" universal Turing machine. I believe that these meta-theorems (a,b) showing (a) that the complexity of axiomatic theories can be characterized information-theoretically and (b) that God plays dice in pure mathematics strongly suggest a quasi-empirical view of mathematics, i.e., that math is different from physics, but perhaps not as different as people usually think. I'll also discuss the convergence of theoretical computer science with theoretical physics, Leibniz's ideas on complexity, Stephen Wolfram's book A New Kind of Science, and how to attempt to use information theory to define what a living being is.

### Course materials

• G. Chaitin. Paradoxes of randomness. Complexity, v. 7, n. 5, pp. 14-21, 2002. [pdf, html]

• G. Chaitin. Elegant Lisp programs. In C. Calude, eds., People and ideas in theoretical computer science, pp. 32-52. Springer-Verlag, Singapore, 1999. [html]

• G. Chaitin. Invitation to algorithmic information theory. In D. S. Bridges, C. Calude, J. Gibbons, S. Reeves, I. Witten, eds., Combinatorics, Complexity & Logic: Proc. of DMTCS'96, pp. 1-23. Springer-Verlag, Singapore, 1997. [html]

• G. Chaitin. Meta-mathematics and the foundations of mathematics. Bulletin EATCS, v. 77, pp. 167-179, 2002. [pdf, html]

• G. Chaitin. On the intelligibility of the universe and the notions of simplicity, complexity and irreducibility. 2002. [pdf, html]

• G. Chaitin. Foils, Computer Science Winter School Estonia, March 2003. [html]