Second-Order Equational Logic (Extended Abstract)

  • Marcelo Fiore
  • Chung-Kil Hur
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6247)

Abstract

We extend universal algebra and its equational logic from first to second order as follows.

  1. 1

    We consider second-order equational presentations as specified by identities between second-order terms, with both variables and parameterised metavariables over signatures of variable-binding operators.

     
  2. 1

    We develop an algebraic model theory for second-order equational presentations, generalising the semantics of (first-order) algebraic theories and of (untyped and simply-typed) lambda calculi.

     
  3. 1

    We introduce a deductive system, Second-Order Equational Logic, for reasoning about the equality of second-order terms. Our development is novel in that this equational logic is synthesised from the model theory. Hence it is necessarily sound.

     
  4. 1

    Second-Order Equational Logic is shown to be a conservative extension of Birkhoff’s (First-Order) Equational Logic.

     
  5. 1

    Two completeness results are established: the semantic completeness of equational derivability, and the derivability completeness of (bidirectional) Second-Order Term Rewriting.

     

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Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Marcelo Fiore
    • 1
  • Chung-Kil Hur
    • 2
  1. 1.Computer LaboratoryUniversity of Cambridge 
  2. 2.Laboratoire PPSUniversité Paris 7 

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