Infinitary intersection types as sequences: A new answer to Klop's problem

Abstract

We provide a type-theoretical characterization of weakly-normalizing terms in an infinitary lambda-calculus. We adapt for this purpose the standard quantitative (with non-idempotent intersections) type assignment system of the lambda-calculus to our infinite calculus.Our work provides a new answer to Klop's HHN-problem, namely, finding out if there is a type system characterizing the hereditary head-normalizing (HHN) lambda-terms. Tatsuta showed that HHN could not be characterized by a finite type system. We prove that an infinitary type system endowed with a validity condition called approximability can achieve it.

Every λ-Term is Meaningful for the Infinitary Relational Model

“…As we will see (see § II-A), relevance disables the argument proving that every term is typable with ρ, the non-idempotent counterpart of the type R considered above. However, while trying to characterize a form of infinitary weak normalization, we noticed in [17] that Ω is also typable R. We recovered soundness by defining a validity criterion, discarding degenerate typing derivations, which was possible by introducing a rigid variant of system R, namely system S. System S has many nice features e.g., tracking (see § II-B).…”

“…Tracking can be retrieved while keeping most of system R 0 's nice features (e.g., syntax-direction) by considering system S, that we introduced in [17]. System S uses sequence types instead of multiset types to represent intersection.…”

“…In an ax-rule concluding with x : (k • T ) ⊢ x : T , the track k is called the axiom track of this axiom rule. We refer to § III and IV of [17] for additional examples and figures for all what concerns the basics of system S, which are presented in this § II. As expected:…”

“…If (a, c) ∈ bisupp(P ), then P (a, c) denotes T P (a, c) e.g., P ex (0•6, ε) = o ′ and P ex (0 • 1, ε) = o, P ex (0 • 1, 1) = o ′′ and P ex (ε, 9) = o ′ . Note that, contrary to [17], we only consider right bipositions. For this article, we think a biposition as type symbol (o ∈ O or →) nested in a given S-derivation P and we often use this heuristic identification implicitly.…”

“…We do not give details (that be found in § IV and Fig. 1 in [17]), but this induces a partial injective function Res b from the right bisupport of P to that of P ′ . The function Res b turns out to be compatible with subjugation:…”