Semi-continuous Sized Types and Termination

TL;DR

This paper develops a semantic and syntactic framework for semi-continuous sized types, ensuring termination in recursive functions with higher-kinded data types and impredicative polymorphism.

Contribution

It introduces a semantics interpreting sized types as functions from ordinals to sets of normalizing terms and establishes semi-continuity as a criterion for admissibility.

Findings

Semantic criterion based on upper semi-continuity for admissibility

Development of a calculus for semi-continuous functions

Enhanced understanding of termination in complex type systems

Abstract

Some type-based approaches to termination use sized types: an ordinal bound for the size of a data structure is stored in its type. A recursive function over a sized type is accepted if it is visible in the type system that recursive calls occur just at a smaller size. This approach is only sound if the type of the recursive function is admissible, i.e., depends on the size index in a certain way. To explore the space of admissible functions in the presence of higher-kinded data types and impredicative polymorphism, a semantics is developed where sized types are interpreted as functions from ordinals into sets of strongly normalizing terms. It is shown that upper semi-continuity of such functions is a sufficient semantic criterion for admissibility. To provide a syntactical criterion, a calculus for semi-continuous functions is developed.