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Lecture Notes on Algorithmic Information Theory

Charles Alexandre Bédard

TL;DR

These notes present Algorithmic Information Theory as a unification of computation and information, grounding information content in the shortest descriptions of individual strings rather than probabilities. The core framework builds plain and prefix Kolmogorov complexity, proves the invariance principle, and derives the Coding Theorem linking description length to universal probability, while addressing uncomputability and incompleteness through the Halting Probability $\Omega$. The exposition questions Hilbert-style formalism, demonstrates the limits of formal systems via Chaitin's incompleteness, and highlights the deep connections between information, randomness, and mathematical truth. Together, these ideas illuminate how algorithmic randomness, induction, and the limits of axioms shape our understanding of computation and knowledge.

Abstract

Algorithmic information theory roots the concept of information in computation rather than probability. These lecture notes were constructed in conjunction with the graduate course I taught at Università della Svizzera italiana in the spring of 2023. The course is intended for graduate students and researchers seeking a self-contained journey from the foundations of computability theory to prefix complexity and the information-theoretic limits of formal systems. My exposition ignores boundaries between computer science, mathematics, physics, and philosophy -- an approach I consider essential when explaining inherently multidisciplinary fields. Lecture recordings are available online. Among other topics, the notes cover bit strings, codes, Shannon information theory, computability theory, the universal Turing machine, the Halting Problem, Rice's Theorem, plain algorithmic complexity, the Invariance Theorem, incompressibility, Solomonoff's induction, self-delimiting Turing machines, prefix algorithmic complexity, the halting probability Omega, Chaitin's Incompleteness Theorem, The Coding Theorem, lower semi-computable semi-measures, and the chain rule for algorithmic complexity.

Lecture Notes on Algorithmic Information Theory

TL;DR

These notes present Algorithmic Information Theory as a unification of computation and information, grounding information content in the shortest descriptions of individual strings rather than probabilities. The core framework builds plain and prefix Kolmogorov complexity, proves the invariance principle, and derives the Coding Theorem linking description length to universal probability, while addressing uncomputability and incompleteness through the Halting Probability . The exposition questions Hilbert-style formalism, demonstrates the limits of formal systems via Chaitin's incompleteness, and highlights the deep connections between information, randomness, and mathematical truth. Together, these ideas illuminate how algorithmic randomness, induction, and the limits of axioms shape our understanding of computation and knowledge.

Abstract

Algorithmic information theory roots the concept of information in computation rather than probability. These lecture notes were constructed in conjunction with the graduate course I taught at Università della Svizzera italiana in the spring of 2023. The course is intended for graduate students and researchers seeking a self-contained journey from the foundations of computability theory to prefix complexity and the information-theoretic limits of formal systems. My exposition ignores boundaries between computer science, mathematics, physics, and philosophy -- an approach I consider essential when explaining inherently multidisciplinary fields. Lecture recordings are available online. Among other topics, the notes cover bit strings, codes, Shannon information theory, computability theory, the universal Turing machine, the Halting Problem, Rice's Theorem, plain algorithmic complexity, the Invariance Theorem, incompressibility, Solomonoff's induction, self-delimiting Turing machines, prefix algorithmic complexity, the halting probability Omega, Chaitin's Incompleteness Theorem, The Coding Theorem, lower semi-computable semi-measures, and the chain rule for algorithmic complexity.

Paper Structure

This paper contains 47 sections, 37 theorems, 136 equations, 8 figures, 1 table.

Table of Contents

  1. Introduction
  2. Hilbert's Theory of Everything
  3. Berry's Paradox
  4. Strings and Codes
  5. Strings
  6. Codes
  7. Kraft's Theorem
  8. Prefix codes of $\{0,1\}^*$
  9. Shannon Information Theory
  10. Optimal Codes
  11. Properties of Entropy
  12. Mutual Information
  13. Computability Theory
  14. Turing's Model of Computation
  15. Mathematizing the mathematician
  16. ...and 32 more sections

Key Result

Proposition 1

$\forall x \in \{0,1\}^*$, $|x| = \lfloor \log x \rfloor$.

Figures (8)

  • Figure 1: Contrasting Shannon's information theory with AIT.
  • Figure 2: Classes of codes with the codes from Example \ref{['ex:codes']}.
  • Figure 3: The binary tree representing the codewords of $E_4$.
  • Figure 4: A Venn diagram representing the algebra of entropies, conditional entropies, and mutual information. The entropy $H(X)$ is represented by the whole circular area enclosed by the blue line, while $H(X \rvert Y)$ is represented by the sub-region outside the red circle. The mutual information $I(X;Y)$ is represented by the overlapping region.
  • Figure 5: A generic configuration of a Turing Machine.
  • ...and 3 more figures

Theorems & Definitions (120)

  • Proposition 1
  • proof
  • Example 1
  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Example 2
  • Theorem 1: Kraft
  • ...and 110 more