🤖 AI Summary
This paper addresses the fundamental open question of P vs NP. Method: It introduces the novel “computation-as-game” paradigm, modeling any computational problem as a two-player zero-sum game between an algorithm and nature: the algorithm asymptotically refines solutions within a Scott domain, while nature imposes penalties based on bias; correctness is defined as Nash equilibrium in the limit. Complexity classes—including P and NP—are newly characterized via the existence of Nash equilibria under specific informational and temporal constraints; P = NP is thus recast as the equivalence of equilibrium existence under these two constraint regimes. Contribution/Results: The framework unifies domain theory, game theory, and computational complexity theory, yielding a game-theoretic, semantic, and structurally principled characterization of complexity classes—providing a novel analytical pathway for understanding the nature of computation and the P vs NP problem.
📝 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.