Bounding normalization time through intersection types

Intersection types were originally introduced as idempotent, i.e., modulo the equivalence σ ∧σ = σ . In fact, they have been used essentially for semantic purposes, for building filter models for λ -calculus, where the interpretation of types as properties of terms induces naturally the idempotence property. Recently it has been observed that, when dropping idempotency, intersection types can be used for reasoning about the complexity of β -reduction. Some results have been already obtained along this line. Terui [15] designed a system assigning non-idempotent intersection types to λ -calculus, which can type all and only the strongly normalizing terms, and such that the size of any derivation with subject M is bigger than the size of every term in the β -reduction sequence from M to its normal form. This property can be used for computing a bound of every normalizing β -reduction sequence starting from M. A more precise result in this direction has been obtained by Lengrand [2], who gave a precise measure of the number of β -reduction steps. Namely he designed a type assignment system, where intersection is considered without idempotency, and defined the notions of measure of derivation and of principal derivation for a given term. Then he proved that the measure of a principal derivation of a type for a normalizing term M corresponds to the maximal length of a normalizing β -reduction sequence for M. In this line, we go one step forward, and use intersection types without neither idempotence nor associativity to express the functional dependence of the length of a normalizing β -reduction sequence from a term M on the size of M itself . In order to obtain such a result, we take inspiration from the system STA of Gaboardi and Ronchi Della Rocca [6], in its turn inspired by the Soft Linear Logic of Lafont [9], which characterizes the polynomial time computations. The resulting system allow us to give a bound on the number of steps necessary to reduce a normalizing term M to its normal form, in the form |M|d+1, where |M| is the size of the term, and d is a measure depending on the type derivation for it (the depth). Since for every normalizing term there is a type derivation with minimal depth, this bound does not depend on a particular derivation. A preliminary type assignment of this kind has been described in [1].