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Step-Indexed Syntactic Logical Relations for Recursive and Quantified Types

Summary: Step-Indexed Syntactic Logical Relations
for Recursive and Quantified Types
Amal Ahmed
Harvard University
March 2006
We present a proof technique, based on syntactic logical relations, for showing contextual equivalence
of expressions in a -calculus with recursive types and impredicative universal and existential types. We
show that for recursive and polymorphic types, the method is both sound and complete with respect to
contextual equivalence, while for existential types, it is sound but incomplete. 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 func-
tions, 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,


Source: Ahmed, Amal - School of Informatics, Indiana University


Collections: Computer Technologies and Information Sciences