Summary: SEMI­CONTINUOUS SIZED TYPES AND TERMINATION
ANDREAS ABEL
Institut f˜ur Informatik, Ludwig­Maximilians­Universit˜at M˜unchen
e­mail address: abel@tcs.ifi.lmu.de
Abstract. A type­based 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 higher­kinded 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 semi­continuity of such functions is a su#cient
semantic criterion for admissibility. To provide a syntactical criterion, a calculus for semi­
continuous functions is developed.
1. Introduction
Termination of computer programs has received continuous interest in the history of
computer science, and classical applications are total correctness and termination of par­
tial evaluation. In languages with a notion of computation on the type­level, such as
dependently­typed languages or rich typed intermediate languages in compilers [CW99],
termination of expressions that compute a type is required for type checking and type sound­

Source: Abel, Andreas - Theoretische Informatik, Ludwig-Maximilians-Universität München

Collections: Computer Technologies and Information Sciences