Types by Need

Abstract

A cornerstone of the theory of λ-calculus is that intersection types characterise termination properties. They are a flexible tool that can be adapted to various notions of termination, and that also induces adequate denotational models. Since the seminal work of de Carvalho in 2007, it is known that multi types (i.e. non-idempotent intersection types) refine intersection types with quantitative information and a strong connection to linear logic. Typically, type derivations provide bounds for evaluation lengths, and minimal type derivations provide exact bounds. De Carvalho studied call-by-name evaluation, and Kesner used his system to show the termination equivalence of call-by-need and call-byname. De Carvalho's system, however, cannot provide exact bounds on call-by-need evaluation lengths. In this paper we develop a new multi type system for call-by-need. Our system produces exact bounds and induces a denotational model of callby-need, providing the first tight quantitative semantics of call-by-need.

Intersection types and (positive) almost-sure termination

“…Non-idempotent intersection types have been known since the work by Gardner [Gardner 1994], studied in connection with expansion variables by Carlier et al [Carlier et al 2004], and further analysed in their relation to normalisation by Mairson and Möller-Neergard [Neergaard and Mairson 2004]. The precise correspondence between non-idempotent intersection type system derivations and the number of reduction steps necessary to normalise the underlying term has been first noticed by De Carvalho [de Carvalho 2018], and further refined by Bernadet and Lengrand [Bernadet and Lengrand 2013], and later by Accattoli et al [Accattoli et al 2018], and [Accattoli et al 2019], the latter being a source of inspiration for this work in its reflecting weak notions of reduction inside intersection types. All these contributions, however, deal with deterministic λ-calculi.…”

“…In non-idempotent intersection types, a natural number w can be assigned to any type derivation in such a way that w ⊢ M : [] (where [] is the empty multiset seen as an intersection type) if and only if M can be reduced to normal form in exactly w steps. The resulting system is in Figure 3, and is essentially the one from [Accattoli et al 2019].…”

“…In intersection type systems for the λ-calculus such as those by Lengrand and co-authors [Accattoli et al 2018[Accattoli et al , 2019Bernadet and Lengrand 2013], the weight of any type derivation accounts for how many times redexes can be fired. Roughly speaking, to each λ-abstraction in the type derivation corresponds a β-redex being fired, therefore to measure the runtime of a λ-term, the weight is increased by one at each instance of the λ rule.…”

The Spirit of Node Replication

“…Indeed, not only typability and normalization can be proved to be equivalent, but a measure based on type derivations provides an upper bound to normalizing reduction sequences. This was extensively investigated in different logical/computational frameworks [5,18,20,25,42,47]. However, no quantitative result based on types exists in the literature for the node replication model, including the attempts done for deep inference [30].…”

The (In)Efficiency of interaction

“…They were first considered by themselves by Gardner [1994], and then by de Carvalho [2007,2018]; Kfoury [2000];Neergaard and Mairson [2004]Ða survey is [Bucciarelli et al 2017]. De Carvalho's use of multi types to give bounds to evaluation lengths has also been used in [Accattoli et al 2020c;Accattoli and Guerrieri 2018;Accattoli et al 2019b;Bernadet and Graham-Lengrand 2013;Bucciarelli et al 2020;de Carvalho et al 2011;Kesner and Vial 2020].…”