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In this paper we propose the design of a more robust General Game Player, able to successfully play the class of synchronous independent games using Propositional Automata, a framework for reasoning about discrete, dynamic, multiagent systems, developed by the Stanford Logic Group. We prove conditions under which a Propositional Automata represents multiple, synchronous independent subgames, and in doing so, prove when a Propositional Automata can be separated, or factored into multiple automata. A General Game Player able to recognize such independences can computationally save, by searching the smaller game trees represented by the independent factored automata for solutions. Additionally we explore the concept of independent substructures appearing within Propositional Automata given certain states, and prove conditions under which a Propositional Automata is contingently factorable into multiple Propositional Automata.