Explicitly Typed λμ-Calculus for Polymorphism and Call-by-Value

Soundness and principal contexts for a shallow polymorphic type system based on classical logic

“…Therefore, the polymorphic type-system presented above must almost inevitably be unsound. Furthermore, Fujita has shown that a similar unsoundness arises in the context of an alternative calculus based on classical natural deduction [27]. However, we believe that the source of the unsoundness is actually much clearer in the sequent calculus setting: it is clear that the attempted left propagation of a cut 'past' an occurrence of (∀R) in the left-hand typing derivation is the exact source of the problem.…”

Compositional semantics for composable continuations

“…Indeed, even de Groote [10] and Ong and Stewart's [24] original analysis in callby-name and call-by-value, respectively, relates typed Λµ-calculi to variants of Felleisen's [14] theory of control with a composable form of continuations that do not abort like in Felleisen's theory. Furthermore, Fujita [13] analyzes an alternative call-by-value theory for Λµ with a CPS transformation that actually composes continuations much like the continuation-composing style transformation of the shift operator [8]. In essence, the type systems prevent programs from taking advantage of the extra expressibility that is latently present in the untyped calculi.…”

“…In the approach taken here, the untyped semantics of the Λµv-calculus is already expressive enough to represent the dynamic behavior of delimited control. The difference between classical and delimited control is then a matter of choosing a less expressive, classical type system -like de Groote [10], Ong and Stewart [24], or Fujita [13] or a more expressive, "delimited" one that allows programs like ([μx.x]1) + 2 to be well-typed.…”

Confluency and strong normalizability of call-by-value λμ-calculus

“…The call-by-value systems with control operations have been widely studied: the theory of sequential control [8], the calculus of exception handling A«.,, in [7], the call-by-value Aµ-calculus [9], [10], [13], and so on. For example, in [13], Ong In this paper, we prove the confluency and the strong normalizability of the domain-free callby-value Ap.-calculus for polymorphic types, which was introduced by Fujita in [9]. The results of this paper are applied to the Church-style calculus in a straightforward way, since the domain-free style may be considered as shorthand for the second-order Church-style.…”

“…[a]M N 11.1 (Al does not contain free a). In [9] and [10J, the confluency of the call-by-value Ay-calculus with the µ77-rule was proved for only typable terms by assuming strong normalizability . In this paper, we show the confluency of untyped terms of the call-by-value Ay-calculus including the µr7-rule .…”