Leonid A. Levin's Research on Godel's Theorem and Complexity Theory

Title: Leonid A. Levin's Research on Godel's Theorem and Complexity Theory

Abstract: Leonid A. Levin, a renowned computer scientist, has made significant contributions to the field of complexity theory through his research on Godel's Theorem. His work has shed new light on the relationship between the theorem and its usual interpretation, providing a more comprehensive understanding of the concept. Levin's research also extends to other unsolvability results, such as non-recursive tilings, further solidifying his impact on the field.

Main Research Question: Can the gap between the usual interpretation of Godel's Theorem and what is actually proven be closed?

Methodology: Levin's research involves the use of complexity theory and the concept of mutual information, a measure of the amount of information shared between two specific sequences. He applies this methodology to various tasks, such as generating strings of linear Kolmogorov complexity and listing infinite subsets of an immune co-recursive set.

Results: Levin's research has led to the following key findings:

1. The absence of recursive solutions does not make the task impossible when the requirements do not make the solution unique. 2. No probabilistic algorithm can solve a task with a unique non-recursive solution with a positive probability. 3. The concept of mutual information satisfies conservation laws, meaning it cannot be increased by deterministic algorithms or in random processes. 4. The Godel task of extending a formal system to a complete one is possible, as shown by the existence of a specific sequence with infinite mutual information with all total extensions of a universal partial recursive predicate. 5. Some palettes for tiling allow the tiling of an infinite plane, despite being non-recursive. This result challenges the standard interpretation that these borders are easy to tile with dice.

Implications: Levin's research has important implications for the field of complexity theory. His work on Godel's Theorem has provided a more nuanced understanding of the theorem, challenging the usual interpretation and opening up new avenues of research. Additionally, his work on tiling has further solidified the connection between complexity theory and other fields, such as mathematics and physics.

Link to Article: https://arxiv.org/abs/0203029v1 Authors: arXiv ID: 0203029v1