Galois Embedding from Polymorphic Types into Existential Types

Relational Parametricity and Control

“…In tthe literature, the second-order λ-calculus (λ2) is often taken as the target of the CPS translation for λµ2. Fujita observed that it actually suffices to consider a fragment of λ2 with negations, conjunctions and existential types as a target [7]. In this paper we follow this insight.…”

“…The CPS translation. We present a call-by-name CPS translation which can be considered as an extension of that introduced by Streicher [16,39,36] (rather than the translations by Plotkin [29], Parigot [27] or Fujita [7] which introduce extra negations and do not respect extensionality).…”

“…Parametric CPS semantics. We first consider the semantics of λµ2 given by a CPS-translation into a fragment of λ2 -that of the second-order existential types ∃X.τ , conjunction types τ 1 ∧ τ 2 , and arrow types τ → R into a distinguished type R (this choice of the target calculus is due to a recent work of Fujita [7]). The translation (−) • sends a type variable X to X, arrow type σ 1 → σ 2 to (σ • 1 → R) ∧ σ • 2 , and the universal type ∀X.σ to ∃X.σ • -while a term M : σ is sent to [[M ]] : σ • → R. It can be considered as a natural extension of Streicher's call-by-name CPS translation [36,39,43].…”

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“…no (2) no (2) no (3) DFno (4) no no (3) Figure 1: Decidability of TC, STI, TI and INH This paper is an extended version of the paper [9] by the same authors, particularly focusing on the decision problems in the calculi with multiple-quantifier rules and the type-free-style calculi.…”

“…The computational meaning of the second-order existential quantifiers has also been actively studied since the work of [8] on the abstract data types. More recently, [2] and [6] pointed out that calculi with negation, conjunction, and existence are suitable for target calculi of the continuation-passing-style (CPS) translations of polymorphic typed calculi.…”