Computational Complexity of Physical Counting

TL;DR

This paper explores the mathematical foundations of physical counting and decision complexity, linking concepts from information theory, thermodynamics, and computational complexity to characterize optimal actions in factored state spaces.

Contribution

It introduces a decision complexity measure called srank, proves its properties through fourteen theorems, and analyzes the computational complexity of sufficiency-related problems.

Findings

Decision complexity measure srank derived from information-theoretic principles.

Complexity results: sufficiency-check is coNP-complete, anchor-sufficiency is $ ext{Sigma}_2^P$-complete.

Empirical links between thermodynamics principles and decision bounds.

Abstract

We characterize which coordinates of a factored state space determine optimal actions. For $\mathcal{D}=(A,S,U)$ with $S=X_1\times\cdots\times X_n$, coordinate set $I$ is sufficient if $s_I=s'_I\Rightarrow\operatorname{Opt}(s)=\operatorname{Opt}(s')$. The decision quotient $Q=S/{\sim}$ ($s\sim s'\Leftrightarrow\operatorname{Opt}(s)=\operatorname{Opt}(s')$) is the minimal abstraction: any abstraction preserving optimal actions factors uniquely through $Q$. We prove fourteen first-principles theorems (thirteen from pure mathematics, one empirical). The chain from counting measure to probability to Bayes' theorem to $Q$ follows from finite set cardinality. Fisher information, entropy, optimal transport, rate-distortion, and thermodynamics each independently recover $\mathrm{srank}$ as decision complexity measure. From $\log x\leq x-1$ alone, Bayesian updating uniquely minimizes expected log loss. Complexity: SUFFICIENCY-CHECK and MINIMUM-SUFFICIENT-SET are coNP-complete; ANCHOR-SUFFICIENCY is $\Sigma_2^P$-complete; stochastic and sequential variants PP- and PSPACE-complete with strict separation. Six subcases admit polynomial algorithms. Under ETH, succinct encodings carry $2^{\Omega(n)}$ lower bounds. Verification requires $\geq 2^{n-1}$ witness pairs. Two results carry empirical conditions. Conditional on Landauer's principle ($k_BT\ln 2$ per bit erasure; experimentally confirmed 2012), $dU\geq\lambda\,dC$ follows by composition with bit-operation bounds; rejecting it requires rejecting Landauer. Conditional on stochastic thermodynamics (Barato--Seifert 2015), $\mathrm{Var}(J)/\langle J\rangle^2\geq 2/\sigma$ bounds decision precision by entropy production, minimal $\sigma$ scaling with $\mathrm{srank}$.