Abstract
In this paper we define several notions of term expansion, used to define terms with less sharing, but with the same computational properties of terms typable in an intersection type system. Expansion relates terms typed by associative, commutative and idempotent intersections with terms typed in the Curry type system and the relevant type system; terms typed by non-idempotent intersections with terms typed in the affine and linear type systems; and terms typed by non-idempotent and non-commutative intersections with terms typed in an ordered type system. © 2022, The Author(s), under exclusive license to Springer Nature Switzerland AG.

Abstract
We show that recent approaches to quantitative analysis based on non-idempotent typing systems can be extended to programming languages with effects. In particular, we consider two cases: the weak open call-by-name (CBN) and call-by-value (CBV) variants of the $\lambda$ -calculus, equipped with operations to write and read from a global state. In order to capture quantitative information with respect to time and space for both CBN and CBV, we design for each of them a quantitative type system based on a (tight) multi-type system. One key observation of this work is how CBN and CBV influence the composition of state types. That is, each type system is developed by taking into account how each language manages the global state: in CBN, the composition of state types is almost straightforward, since function application does not require evaluation of its argument; in CBV, however, the interaction between functions and arguments makes the composition of state types more subtle since only values can be passed as actual arguments. The main contribution of this paper is the design of type systems capturing quantitative information about effectful CBN and CBV programming languages. Indeed, we develop type systems that are qualitatively and quantitatively sound and complete.

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