## Strong Normalization of Polymorphic Calculus for Delimited Continuations

### BibTeX

@MISC{Kameyama_strongnormalization,

author = {Yukiyoshi Kameyama and Kenichi Asai},

title = {Strong Normalization of Polymorphic Calculus for Delimited Continuations},

year = {}

}

### OpenURL

### Abstract

Abstract. The notion of delimited continuations has been proved useful in various areas of computer programming such as partial evaluation, mobile computing, and web transaction. In our previous work, we proposed polymorphic calculi with control operators for delimited continuations. This paper presents a proof of strong normalization (SN) of these calculi based on a refined (i.e. administrative redex-free) CPS translation.

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Citation Context ... : τ is derivable. Theorem 2 (Progress). If ⊢p 〈e〉 : τ is derivable, then 〈e〉 can be reduced. By Theorems 1 and 2, we can conclude that our type system is sound (strong type soundness in the sense of =-=[18]-=-). Theorem 3 (Principal Type and Type Inference). In λ s/r let , principal type exists, and we can construct a sound and complete type inference algorithm as an extension of HindleyMilner’s algorithm.... |

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Citation Context ...or continuations, which has been intensively studied from theory to implementation and practical applications. Type-theoretic studies on “call/cc” have revealed that it corresponds to classical logic =-=[10]-=-. Delimited continuation is a similar notion to (unlimited) continuation, but it represents part of the rest of the computation rather than whole rest of the computation. Since Danvy and Filinski have... |

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Citation Context ...ression, or a conditional. Note that we omit the fixpoint operator from the calculus in [4]. 3 Filinski did not give a type system explicitly, but his well-known implementation of “shift” and “reset” =-=[8]-=- specifies a certain type system. 2(λx.e)v � e[v/x] 〈v〉 � v 〈F [Sk.e]〉 � 〈let k = λx.〈F [x]〉 in e〉 let x = v in e � e[v/x] if true then e1 else e2 � e1 if false then e1 else e2 � e2 Fig. 2. Reduction... |

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Citation Context ... represents part of the rest of the computation rather than whole rest of the computation. Since Danvy and Filinski have proposed the control operators “shift” and “reset” for delimited continuations =-=[6]-=-, they have been proved useful in various applications such as backtracking [6], A-normalization in direct style [2], let-insertion in partial evaluation [17], type-safe “printf” [3], and web transact... |

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Citation Context ...r delimited continuations comes from the fact that there exists a simple, compositional CPS translation. Danvy and Filinski gave the precise semantics of shift and reset in terms of a CPS translation =-=[6, 7]-=-, and based on their translation, various theoretical results as well as applications using shift and reset have been proposed (see, for instance, [12]). In this section, we present an extension of Da... |

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Citation Context ...e2 else e3 expression Fig. 1. Syntax of λ s/r let . – A typed calculus with “call/cc” is strongly normalizing, while that with exception in Standard ML is not strongly normalizing (see, for instance, =-=[11]-=-). – The calculus with “shift” and “reset” under Filinski’s typing 3 is not normalizing, while, under Danvy and Filinski’s type system, it is strongly normalizing [1]. – A typed calculus with “control... |

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Citation Context ... shift and reset in terms of a CPS translation [6, 7], and based on their translation, various theoretical results as well as applications using shift and reset have been proposed (see, for instance, =-=[12]-=-). In this section, we present an extension of Danvy and Filinski’s CPS translation, namely, a CPS translation from λ s/r let to a pure polymorphic lambda calculus λlet. We call this CPS translation a... |

24 |
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Citation Context ... normalizing (see, for instance, [11]). – The calculus with “shift” and “reset” under Filinski’s typing 3 is not normalizing, while, under Danvy and Filinski’s type system, it is strongly normalizing =-=[1]-=-. – A typed calculus with “control” and “prompt”, the other control operators for delimited continuations, is not normalizing [13]. Similarly, a typed calculus with the control operator cupto is not n... |

20 |
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Citation Context ...t” and “reset” for delimited continuations [6], they have been proved useful in various applications such as backtracking [6], A-normalization in direct style [2], let-insertion in partial evaluation =-=[17]-=-, type-safe “printf” [3], and web transaction [15]. This paper investigates the type structure of “shift” and “reset”, and in particular, proves strong normalization of a polymorphic calculus for them... |

19 | Polymorphic delimited continuations
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Citation Context ...], and web transaction [15]. This paper investigates the type structure of “shift” and “reset”, and in particular, proves strong normalization of a polymorphic calculus for them. In our previous work =-=[4]-=-, we have introduced a polymorphic type system for “shift” and “reset”, and proved a number of properties such as type soundness (subject reduction and progress). The strong normalization property, ho... |

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Citation Context ...ted continuations [6], they have been proved useful in various applications such as backtracking [6], A-normalization in direct style [2], let-insertion in partial evaluation [17], type-safe “printf” =-=[3]-=-, and web transaction [15]. This paper investigates the type structure of “shift” and “reset”, and in particular, proves strong normalization of a polymorphic calculus for them. In our previous work [... |

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Citation Context ...have proposed the control operators “shift” and “reset” for delimited continuations [6], they have been proved useful in various applications such as backtracking [6], A-normalization in direct style =-=[2]-=-, let-insertion in partial evaluation [17], type-safe “printf” [3], and web transaction [15]. This paper investigates the type structure of “shift” and “reset”, and in particular, proves strong normal... |

3 | Typed dynamic control operators for delimited continuations
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(Show Context)
Citation Context ...le, under Danvy and Filinski’s type system, it is strongly normalizing [1]. – A typed calculus with “control” and “prompt”, the other control operators for delimited continuations, is not normalizing =-=[13]-=-. Similarly, a typed calculus with the control operator cupto is not normalizing [14]. Hence, we can say SN for the calculi with control operators is a non-trivial issue. This paper solves the problem... |

1 |
Simply Typed Lambda-Calculus with a Typed-Prompt Delimited Control is not Strongly Normalizing
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(Show Context)
Citation Context ... calculus with “control” and “prompt”, the other control operators for delimited continuations, is not normalizing [13]. Similarly, a typed calculus with the control operator cupto is not normalizing =-=[14]-=-. Hence, we can say SN for the calculi with control operators is a non-trivial issue. This paper solves the problem for the case of “shift” and “reset” under the polymorphic type system. The rest of t... |

1 |
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(Show Context)
Citation Context ...ey have been proved useful in various applications such as backtracking [6], A-normalization in direct style [2], let-insertion in partial evaluation [17], type-safe “printf” [3], and web transaction =-=[15]-=-. This paper investigates the type structure of “shift” and “reset”, and in particular, proves strong normalization of a polymorphic calculus for them. In our previous work [4], we have introduced a p... |