Critical Algorithmic Information Theory and the Asymptotic Emergence of Tsallis q-Exponentials in Formal Languages

Abstract

Abstract

We establish an asymptotically exact theoretical correspondence between the Solomonoff–Chaitin algorithmic semi-measure over formal derivation trees and Tsallis non-extensive statistical mechanics, uniquely characterized by a q-exponential distribution with q=5/3. Rather than assuming universal equivalence across arbitrary formal languages, we specify the explicit structural conditions required for this asymptotic emergence. We define the class of Binary-Choice Deterministic CNF Grammars (BCDGs) operating on a deterministic three-tape Turing machine MSISS. Under minimal optimal encoding where binary branching non-terminal nodes consume 1 bit and deterministic terminal leaf productions consume 0 bits, we prove that valid derivation parse trees of total size ℓ=2k+1 consume precisely |p$|=ℓ+12=k+1 bits (including a non-binary halting delimiter $∉{0,1}). Within the asymptotic grammar counting family N(ℓ)∼Cℓ−3/2μℓ, Kraft’s inequality imposes the strict structural bound μ≤2. At the critical phase transition point μ=2—where the topological entropy of the grammar Htop=log22=0.5bits/node exactly matches the asymptotic internal node density capacity of binary derivation trees—the exponential growth factor cancels identically against the tape density factor. This leaves the pure asymptotic power-law decay mAIT(ℓ)∼(2π)−1ℓ−3/2, which corresponds asymptotically to a Tsallis q-exponential distribution P(ℓ)∝eq(−λ(ℓ−ℓmin)) with q=5/3 and λ=32ℓmin. Furthermore, we prove that the truncated Shannon entropy of the derivation space is subextensive with respect to the scale cutoff L. The framework is fully validated through exact Catalan combinatorics and a verified numerical convergence script.