Extension Complexity of Formal Languages

Abstract

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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Notes

  1. 1.

    Assume some (arbitrary but fixed) encoding of boolean formulae as binary strings.

  2. 2.

    Perfect Matching remains an easy problem despite exponential lower bound on the extension complexity of the perfect matching polytope. What does an exponential lower bound for the cut polytope tell us about the difficulty of the MAX-CUT problem?

  3. 3.

    This usage, however, is common among number theorists.

  4. 4.

    The description is required only to identify the function uniquely and need not be explicit.

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Acknowledgments

The author would like to acknowledge the support of grant GA15-11559S of GAČR. We also thank Mateus De Oliveira Oliveira for finding a critical flaw in a previous proof of Theorem 7 and the anonymous referees for many valuable suggestions.

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Correspondence to Hans Raj Tiwary.

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Tiwary, H.R. Extension Complexity of Formal Languages. Theory Comput Syst 64, 735–753 (2020). https://doi.org/10.1007/s00224-019-09951-x

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Keywords

  • Extended formulations
  • Formal languages