Factoring Derivation Spaces via Intersection Types

Abstract

In typical non-idempotent intersection type systems, proof normalization is not confluent. In this paper we introduce a confluent non-idempotent intersection type system for the λ-calculus. Typing derivations are presented using proof term syntax. The system enjoys good properties: subject reduction, strong normalization, and a very regular theory of residuals. A correspondence with the λ-calculus is established by simulation theorems. The machinery of non-idempotent intersection types allows us to track the usage of resources required to obtain an answer. In particular, it induces a notion of garbage: a computation is garbage if it does not contribute to obtaining an answer. Using these notions, we show that the derivation space of a λ-term may be factorized using a variant of the Grothendieck construction for semilattices. This means, in particular, that any derivation in the λ-calculus can be uniquely written as a garbage-free prefix followed by garbage.In this case, the space of computations is quite easy to understand, because the subexpressions (1 + 1) and (2 * 3 + 1) cannot interact with each other. Indeed, the ⋆ Work partially supported by CONICET. R 2 are residuals of R, and, conversely, R is an ancestor of R 1 and R 2 . The second phenomenon is erasure: in the diagram above, the step T erases the step R ′ 1 , resulting in no copies of R ′ 1 . The third phenomenon is creation: in the diagram above, the step R 2 creates the step T , meaning that T is not a residual of a step that existed prior to executing R 2 ; that is, T has no ancestor.These three interaction phenomena, especially duplication and erasure, are intimately related with the management of resources. In this work, we aim to explore the hypothesis that having an explicit representation of resource management may provide insight on the structure of derivation spaces.There are many existing λ-calculi that deal with resource management explicitly [6,15,20,21], most of which draw inspiration from Girard's Linear Logic [18]. Recently, calculi endowed with non-idempotent intersection type systems, have received some attention [14,5,7,8,19,34,22]. These type systems are able to statically capture non-trivial dynamic properties of terms, particularly normalization, while at the same time being amenable to elementary proof techniques by induction. Intersection types were originally proposed by Coppo and Dezani-Ciancaglini [11] to study termination in the λ-calculus. They are characterized by the presence of an intersection type constructor A ∩ B. Non-idempotent intersection type systems are distinguished from their usual idempotent counterparts by the fact that intersection is not declared to be idempotent, i.e. A and A ∩ A are not equivalent types. Rather, intersection behaves like a multiplicative connective in linear logic. Arguments to functions are typed many times, typically once per each time that the argument will be used. Non-idempotent intersection types were originally formulated by Gardner [17], and later reintroduced by de Carvalho [9].I...

Hybrid Intersection Types for PCF

“…Proofs. Some proofs in the paper have been ommited, but they can be found in the extended version [5].…”