Computation as a Game
Paul Alexander Bilokon
TL;DR
This work reframes computation as a game of approximation between an Algorithm and Nature, rooted in domain theory and game semantics, and then develops three parallel reformulations—domain-theoretic, mean-field probabilistic, and categorical/topos-theoretic—to recast classical complexity classes as equilibrium concepts under resource constraints. It defines a two-player computational game with a cost-penalty tradeoff, extends to mean-field dynamics yielding HJB–Fokker–Planck equilibria, and provides a topos-based semantic account of feasibility and verification. The key contributions include precise game-theoretic and categorical formulations of $\mathbf{P}$ and $\mathbf{NP}$, a mean-field interpretation of complexity via energy landscapes, and a coherent framework linking complexity, learning, and physics-inspired dynamics. This cross-disciplinary perspective offers a novel lens on the $\mathbf{P}$ vs $\mathbf{NP}$ problem and suggests future avenues in quantitative domains and distributed computation.
Abstract
We present a unifying representation of computation as a two-player game between an \emph{Algorithm} and \emph{Nature}, grounded in domain theory and game theory. The Algorithm produces progressively refined approximations within a Scott domain, while Nature assigns penalties proportional to their distance from the true value. Correctness corresponds to equilibrium in the limit of refinement. This framework allows us to define complexity classes game-theoretically, characterizing $\mathbf{P}$, $\mathbf{NP}$, and related classes as sets of problems admitting particular equilibria. The open question $\mathbf{P} \stackrel{?}{=} \mathbf{NP}$ becomes a problem about the equivalence of Nash equilibria under differing informational and temporal constraints.