We investigate finite-rank intersection type systems, analyzing the complexity of their type inference problems and their relation to the problem of recognizing semantically equivalent terms. Intersection types allow something of type T1 /\ T2 to be used in some places at type T1 and in other places at type T2 . A finite-rank intersection type system bounds how deeply the /\ can appear in type expressions. Such type systems enjoy strong normalization, subject reduction, and computable type inference, and they support a pragmatics for implementing parametric polymorphism. As a consequence, they provide a conceptually simple and tractable alternative to the impredicative polymorphism of System F and its extensions, while typing many more programs than the Hindley-Milner type system found in ML and Haskell. While type inference is computable at every rank, we show that its complexity grows exponentially as rank increases. Let K(0, n) = n and K(t + 1, n) = 2^K(t,n); we prove that recognizing the pure lambda-terms of size n that are typable at rank k is complete for dtime[K(k-1, n)]. We then consider the problem of deciding whether two lambda-terms typable at rank k have the same normal form, Generalizing a well-known result of Statman from simple types to finite-rank intersection types. ...

Postponement of K -contractions and the conservation theorem do not hold for ordinary but have been established by de Groote for a mixture of with another reduction relation. In this paper, de Groote's results are generalised for a single reduction relation e which generalises . We show morever, that e has the Preservation of Strong Normalisation property. Keywords: Generalised -reduction, Postponement of K-contractions, Generalised Conservation, Preservation of Strong Normalisation. 1 The -calculus with generalized reduction In the term (( x : y :N)P )Q, the abstraction starting with x and the argument P form the redex ( x : y :N)P . When this redex is contracted, the abstraction starting with y and Q will in turn form a redex. It is important to note that Q (or some residual of Q) is the only argument that the abstraction (or some residual of the abstraction) starting with y can ever have. This fact has been exploited by many researchers. Reduction has been ex...

A higher order unification (HOU) method based on the ...-style of explicit substitution is proposed. The method is based on the treatment introduced by Dowek, Hardin and Kirchner in [DHK95] using the ...-style of explicit substitution. Correctness and completeness properties of the proposed approach are shown and advantages of the method, inherited from the qualities of the ... calculus, are pointed out.