A notion of dimension in intersection typed lambda-calculi is presented. The dimension of a typed lambda-term is given by the minimal norm of an elaboration (a proof theoretic decoration) necessary for typing the term at its type, and, intuitively, measures intersection introduction as a resource.
Bounded-dimensional intersection type calculi are shown to enjoy subject reduction, since terms can be elaborated in non-increasing norm under beta-reduction. We prove that a multiset interpretation (corresponding to a non-idempotent and non-linear interpretation of intersection) of dimensionality corresponds to the number of simulateneous constraints required during search for inhabitants. As a consequence, the inhabitation problem is decidable in bounded multiset dimension, and it is proven to be EXPSPACE-complete. This result is a substantial generalization of inhabitation for the rank 2-fragment, yielding a calculus with decidable inhabitation which is independent of rank.
Our results give rise to a new criterion (dimensional bound) for subclasses of intersection type calculi with a decidable inhabitation problem, which is orthogonal to previously known criteria. Additionally, our notion of dimension should be of interest as a proof theoretic measure of complexity, and we give examples of dimensional analysis of some well-known fragments of the intersection type system, including conservativity over simple types, rank 2-types, and normal form typings.
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