This is my talk for the Santa Fe Institute workshop on Statistical Mechanics, Information Processing and Biology:
It’s about the link between computation and entropy. I take the idea of a Turing machine for granted, but starting with that I explain recursive functions, the Church-Turing thesis, Kolomogorov complexity, the relation between Kolmogorov complexity and Shannon entropy, the uncomputability of Kolmogorov complexity, the ‘complexity barrier’, Levin’s computable version of complexity, and finally my work with Mike Stay on algorithmic thermodynamics.
In my talk slides I mention the ‘complexity barrier’, and state this theorem:
Theorem. Choose your favorite set of axioms for math. If it’s finite and consistent, there exists C ≥ 0, the complexity barrier, such that for no natural number n can you prove the Kolmogorov complexity of n exceeds C.
For a sketch of the proof of this result, go here:
In my talk I showed a movie related to this: an animated video created in 2009 using a program less than 4 kilobytes long that runs on a Windows XP machine:
For more details, read our paper:
• John Baez and Mike Stay, Algorithmic thermodynamics, Math. Struct. Comp. Sci. 22 (2012), 771-787.
or these blog articles:
They all emphasize slightly different aspects!