Computation as a Game

TL;DR

This paper introduces a game-theoretic framework for computation, modeling algorithms and nature as players in a game, which offers new insights into complexity classes and the P vs NP problem.

Contribution

It unifies computation and game theory using domain theory, defining complexity classes through equilibrium concepts and framing P vs NP as equilibrium equivalence.

Findings

Defines a game-theoretic model of computation grounded in domain theory.

Characterizes complexity classes like P and NP via equilibria in the game.

Reframes the P vs NP problem as an equilibrium comparison.

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.