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Summary: SEMI-CONTINUOUS SIZED TYPES AND TERMINATION
ANDREAS ABEL
Institut f¨ur Informatik, Ludwig-Maximilians-Universit¨at M¨unchen
e-mail address: abel@tcs.ifi.lmu.de
Abstract. 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 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 type-level, such as
dependently-typed languages or rich typed intermediate languages in compilers [CW99],
termination of expressions that compute a type is required for type checking and type sound-
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