Indexed Realizability for Bounded-Time Programming with References and Type Fixpoints

Abstract

The field of implicit complexity has recently produced several bounded-complexity programming languages. This kind of language allows to implement exactly the functions belonging to a certain complexity class. We present a realizability semantics for a higher-order functional language based on a fragment of linear logic called LAL which characterizes the complexity class PTIME. This language features recursive types and higher-order store. Our realizability is based on biorthogonality, indexing and is quantitative. This last feature enables us not only to derive a semantical proof of termination, but also to give bounds on the number of computational steps of typed programs.