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One of the aims of Implicit Computational Complexity is the design ofprogramming languages with bounded computational complexity; indeed,guaranteeing and certifying a limited resources usage is of central importancefor various aspects of computer science. One of the more promising approachesto this aim is based on the use of lambda-calculus as paradigmatic programminglanguage and the design of type assignment systems for lambda-terms, wheretypes guarantee both the functional correctness and the complexity bound. Herewe propose a system of stratified types, inspired by intersection types, whereintersection is a non-associative operator. The system, called STR, is correctand complete for polynomial time computations; moreover, all the stronglynormalizing terms are typed in it, thus increasing the typing power withrespect to the previous proposals. Moreover, STR enjoys a stronger expressivitywith respect to the previous system STA, since it allows to type a restrictedversion of iteration.

A type assignment for lambda-calculus complete both for FPTIME and strong normalization