In this article we undertake a study of extension complexity from the perspective of formal languages. We define a natural way to associate a family of polytopes with binary languages. This allows us to define the notion of extension complexity of formal languages. We prove several closure properties of languages admitting compact extended formulations. Furthermore, we give a sufficient machine characterization of compact languages. We demonstrate the utility of this machine characterization by obtaining upper bounds for polytopes for problems in nondeterministic logspace; lower bounds in streaming models; and upper bounds on extension complexities of several polytopes.
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