Step-Indexed Syntactic Logical Relations for Recursive and Quantified Types

  • Amal Ahmed
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3924)


We present a sound and complete proof technique, based on syntactic logical relations, for showing contextual equivalence of expressions in a λ-calculus with recursive types and impredicative universal and existential types. Our development builds on the step-indexed PER model of recursive types presented by Appel and McAllester. We have discovered that a direct proof of transitivity of that model does not go through, leaving the “PER” status of the model in question. We show how to extend the Appel-McAllester model to obtain a logical relation that we can prove is transitive, as well as sound and complete with respect to contextual equivalence. We then augment this model to support relational reasoning in the presence of quantified types.

Step-indexed relations are indexed not just by types, but also by the number of steps available for future evaluation. This stratification is essential for handling various circularities, from recursive functions, to recursive types, to impredicative polymorphism. The resulting construction is more elementary than existing logical relations which require complex machinery such as domain theory, admissibility, syntactic minimal invariance, and ⊤ ⊤-closure.


Operational Semantic Recursive Function Logical Relation Typing Rule Shade Part 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Amal Ahmed
    • 1
  1. 1.Harvard UniversityCambridge

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