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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, Cambridge, MA
amal@eecs.harvard.edu
Abstract. We present a sound and complete proof technique, based on
syntactic logical relations, for showing contextual equivalence of expres-
sions in a -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 re-
spect 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 es-
sential for handling various circularities, from recursive functions, to re-
cursive types, to impredicative polymorphism. The resulting construction

  

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

 

Collections: Computer Technologies and Information Sciences