Semi-continuous Sized Types and Termination

A type-based approach to termination uses 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 semicontinuous functions is developed.

• Type: Journal
• Year: 2008
• Where: CORR
• Authors: Andreas Abel
• Keywords: CORR 2008, Education, Ordinal Bound, Recursive Function, Type-based Approach