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Turing Machines and Finite Models

  • Leonid Libkin
Part of the Texts in Theoretical Computer Science book series (TTCS)

Abstract

In this chapter we introduce the technique of coding Turing machines in various logics. It is precisely this technique that gave rise to numerous applications of finite model theory in computational complexity. We start by proving the earliest such result, Trakhtenbrot’s theorem, stating that finite satisfiability is not decidable. For the proof of Trakhtenbrot’s theorem, we code Turing machines with no inputs. By a refinement of this technique, we code nondeterministic polynomial time Turing machines in existential second-order logic (∃SO), proving Fagin’s theorem stating that∃SO-definable properties of finite structures are precisely those whose complexity is NP.

Keywords

Linear Order Turing Machine Complexity Class Finite Model Polynomial Hierarchy 
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Bibliographic Notes

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Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Leonid Libkin
    • 1
  1. 1.Department of Computer ScienceUniversity of TorontoTorontoCanada

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