## Call-by-name and call-by-value in normal modal logic

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Citations: | 1 - 1 self |

### BibTeX

@MISC{Kakutani_call-by-nameand,

author = {Yoshihiko Kakutani},

title = {Call-by-name and call-by-value in normal modal logic},

year = {}

}

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### Abstract

Abstract. This paper provides a call-by-name and a call-by-value calculus, both of which have a Curry-Howard correspondence to the minimal normal logic K. The calculi are extensions of the λµ-calculi, and their semantics are given by CPS transformations into a calculus corresponding to the intuitionistic fragment of K. The duality between call-by-name and call-by-value with modalities is investigated in our calculi. 1

### Citations

921 |
Categories for the working mathematician
- Lane
- 1998
(Show Context)
Citation Context ...✷(σ → τ) → ✷σ → ✷τ. Assuming that this axiom is parametric, the modality is just a monoidal endofunctor with respect to cartesian products. (Fundamental properties of monoidal categories are found in =-=[20]-=-.) Hence, a model of IK is naturally considered a cartesian closed category with a lax monoidal endofunctor with respect to cartesian products. An interpretation is given in the usual manner: a type i... |

518 | Lambda calculi with types
- Barendregt
- 1992
(Show Context)
Citation Context ...er Howard’s work [15]. A Curry-Howard correspondence enables us to study equality on proofs computationally. Though the correspondence can be extended to higher-order and predicate logics as shown in =-=[3]-=-, we investigate only propositional logics in this paper. The aim of this study is to give a proper calculus that have a CurryHoward correspondence to the modal logic K. Through a Curry-Howard corresp... |

443 |
Y.: Modal Logic
- Blackburn, Rijke, et al.
- 2002
(Show Context)
Citation Context ...ations, and are now widely accepted both theoretically and practically. Especially, studies of modal logics by Kripke semantics [18] are quite active and a large number of results exist, for example, =-=[7]-=- is a textbook about such studies. Since Kripke semantics concern only provability, equality on proofs is less studied on modal logics compared with traditional logics. It is well-known that the intui... |

441 |
The formulae-as-types notion of construction
- Howard
- 1980
(Show Context)
Citation Context ...nistic propositional logic exactly corresponds to the simply typed λ-calculus: formulae as types and proofs as terms. Such a correspondence is called a Curry-Howard correspondence after Howard’s work =-=[15]-=-. A Curry-Howard correspondence enables us to study equality on proofs computationally. Though the correspondence can be extended to higher-order and predicate logics as shown in [3], we investigate o... |

440 | Computational lambda-calculus and monads
- Moggi
- 1989
(Show Context)
Citation Context ...tatement of the duality is given in the next section. The call-by-value calculus is an extension of Selinger’s call-by-value version [30] of the λµ-calculus, and hence an extension of the λc-calculus =-=[24]-=-. Definition 14. The call-by-value λµ✷-calculus has the same syntax as the callby-name λµ✷-calculus. The equality of the call-by-value λµ✷-calculus is defined by the transformation [[−]]v given in Tab... |

321 |
λµ-calculus: an algorithmic interpretation of classic natural deduction
- Parigot
- 1992
(Show Context)
Citation Context ...pondence between the classical propositional logic and the λ-calculus with continuations was provided in [13] by Griffin. Parigot has proposed the λµ-calculus as a calculus for the classical logic in =-=[26]-=-. Now, kinds of λµ-calculi exist and some of them are defined by CPS transformations. A CPS transformation was originally introduced in [11], and the relation between call-by-value and CPS semantics w... |

234 | A formulae-as-types notion of control
- Griffin
- 1990
(Show Context)
Citation Context ... since the exponential of the linear logic [12] is a kind of S4 modality.sA Curry-Howard correspondence between the classical propositional logic and the λ-calculus with continuations was provided in =-=[13]-=- by Griffin. Parigot has proposed the λµ-calculus as a calculus for the classical logic in [26]. Now, kinds of λµ-calculi exist and some of them are defined by CPS transformations. A CPS transformatio... |

185 | A modal analysis of staged computation
- Davies, Pfenning
(Show Context)
Citation Context ... correspondence, any type system can be regarded as a logic by forgetting terms. In this sense, modal logics are contributing to practical studies for programming languages, e.g., staged computations =-=[8]-=- and information flow analysis [23]. Since K is known as a minimal modal logic, this paper focuses on K rather than S4. Before defining a calculus for K, we consider the intuitionistic fragment of K, ... |

171 |
Call-by-name, call-by-value, and the lambda calculus
- Plotkin
- 1975
(Show Context)
Citation Context ...t and some of them are defined by CPS transformations. A CPS transformation was originally introduced in [11], and the relation between call-by-value and CPS semantics was first studied by Plotkin in =-=[27]-=-. De Groote defines a CPS transformation on a call-by-name λµ-calculus in [9], but in this paper, we adopt Selinger’s CPS transformation [30], which is an extension of Hofmann and Streicher’s [14]. In... |

140 |
P.J.: Introduction to Higher-Order Categorical Logic
- Lambek, Scott
- 1986
(Show Context)
Citation Context ...lculus for IK is defined as a refinement of Bellin et al.’s calculus [4] rather than Martini and Masini’s[22]. Our calculus is sound and complete for the categorical semantics given in [4]. The study =-=[19]-=- about simply typed λ-calculus and cartesian closed categories is a typical study of categorical semantics. Categorical semantics of modal logics are studied by Bierman and de Paiva, and Bellin et al.... |

102 | Linear logic, *-autonomous categories and cofree coalgebras
- Seely
- 1989
(Show Context)
Citation Context ...al semantics. Categorical semantics of modal logics are studied by Bierman and de Paiva, and Bellin et al. in [6] and [4]. Their semantics are based on studies about semantics of linear logics (e.g., =-=[29]-=- and [5]) since the exponential of the linear logic [12] is a kind of S4 modality.sA Curry-Howard correspondence between the classical propositional logic and the λ-calculus with continuations was pro... |

100 | Lambda calculus schemata
- Fischer
- 1972
(Show Context)
Citation Context ...ed the λµ-calculus as a calculus for the classical logic in [26]. Now, kinds of λµ-calculi exist and some of them are defined by CPS transformations. A CPS transformation was originally introduced in =-=[11]-=-, and the relation between call-by-value and CPS semantics was first studied by Plotkin in [27]. De Groote defines a CPS transformation on a call-by-name λµ-calculus in [9], but in this paper, we adop... |

100 | The Proof Theory and Semantics of Intuitionistic Modal Logic
- Simpson
- 1994
(Show Context)
Citation Context ...e write “¬τ” for “τ → ⊥”. 2 Calculus for Intuitionistic Normal Modal Logic In this section, we study the intuitionistic modal logic IK. Intuitionism of a diamond modality is not trivial, for example, =-=[33]-=- gives an account of it, but this section focuses on the box fragment of IK. We call also this fragment itself IK in this paper. A diamond modality is investigated in a classical logic after the next ... |

99 | What is a categorical model of intuitionistic linear logic
- Bierman
- 1995
(Show Context)
Citation Context ...ics. Categorical semantics of modal logics are studied by Bierman and de Paiva, and Bellin et al. in [6] and [4]. Their semantics are based on studies about semantics of linear logics (e.g., [29] and =-=[5]-=-) since the exponential of the linear logic [12] is a kind of S4 modality.sA Curry-Howard correspondence between the classical propositional logic and the λ-calculus with continuations was provided in... |

89 |
Dual intuitionistic linear logic
- Barber
- 1996
(Show Context)
Citation Context .... In [6], Bierman and de Paiva propose a monad as a model of ✸ in IS4. Our semantics matches their observation. An S4 extension of the dual calculus [34] along the line of dual context calculi (e.g., =-=[2]-=-) is provided in [32] by Shan. Since the λµ-calculus has a bijective correspondence to the dual calculus, the λµ✷calulus remains to be formalized in the dual calculus and to be compared with Shan’s ca... |

83 | Control categories and duality: on the categorical semantics of the lambda-mu calculus
- Selinger
- 2001
(Show Context)
Citation Context ...by-value and CPS semantics was first studied by Plotkin in [27]. De Groote defines a CPS transformation on a call-by-name λµ-calculus in [9], but in this paper, we adopt Selinger’s CPS transformation =-=[30]-=-, which is an extension of Hofmann and Streicher’s [14]. In Section 3, we provide a call-by-name λµ-calculus with a box modality, which has a Curry-Howard correspondence to K, by the CPS semantics int... |

76 | A curry-howard foundation for functional computation with control
- Ong, Stewart
- 1997
(Show Context)
Citation Context ...l-by-name λµ-calculus with a box modality, which has a Curry-Howard correspondence to K, by the CPS semantics into the calculus for IK defined in Section 2. A call-by-value λµ-calculus is provided in =-=[25]-=- by Ong and Stewart. We define a call-by-value calculus for K also as an extension of Selinger’s call-by-value λµ-calculus [30] via the CPS transformation in Section 4. The duality between call-by-nam... |

63 |
Semantical analysis of modal logic I, normal propositional calculi
- Kripke
- 1963
(Show Context)
Citation Context ...duction Modal logics have a long history since logics with strict implications, and are now widely accepted both theoretically and practically. Especially, studies of modal logics by Kripke semantics =-=[18]-=- are quite active and a large number of results exist, for example, [7] is a textbook about such studies. Since Kripke semantics concern only provability, equality on proofs is less studied on modal l... |

46 |
Call-by-value is dual to call-by-name
- Wadler
(Show Context)
Citation Context ...on a programming language with first-class continuations was first formalized by Filinski in [10]. It has been formalized on the λµ-calculi in [30] by Selinger, and reformulated as sequent calculi in =-=[34]-=- by Wadler. In [16], the duality is developed with recursion by the author. In Section 5, we study such duality on the classical modal logic K. In addition, we investigate the logic S4 with the CPS se... |

29 | Declarative Continuations and Categorical Duality
- Filinski
- 1989
(Show Context)
Citation Context ...uality between call-by-name and call-by-value is an important property of the classical logic. The duality on a programming language with first-class continuations was first formalized by Filinski in =-=[10]-=-. It has been formalized on the λµ-calculi in [30] by Selinger, and reformulated as sequent calculi in [34] by Wadler. In [16], the duality is developed with recursion by the author. In Section 5, we ... |

28 | A computational interpretation of modal proofs
- Martini, Masini
- 1996
(Show Context)
Citation Context ...nsider the intuitionistic fragment of K, which is called IK in this paper. In Section 2, the calculus for IK is defined as a refinement of Bellin et al.’s calculus [4] rather than Martini and Masini’s=-=[22]-=-. Our calculus is sound and complete for the categorical semantics given in [4]. The study [19] about simply typed λ-calculus and cartesian closed categories is a typical study of categorical semantic... |

20 |
On the semantics of classical disjunction
- Pym, Ritter
(Show Context)
Citation Context ... intuitionistic case, call-by-name classical disjunctions are not coproducts. (Our formulation of disjunctions is based on Selinger’s [30], but it is possible to define the calculus along the line of =-=[28]-=-.) Instead of case functions and injections, we use the syntax sugar [λx σ1 1 . M1, λx σ2 2 . M2] ≡ λx σ1∨σ2 τ σ1 σ1 σ2 . µb . [b]((λx1 . M1)(µa1 . [b]((λxσ2 2 . M2)(µa2 . [a1, a2]x)))) ιjM ≡ µ(a τ1 1... |

19 | On an intuitionistic modal logic
- Bierman, Paiva
(Show Context)
Citation Context ...ut simply typed λ-calculus and cartesian closed categories is a typical study of categorical semantics. Categorical semantics of modal logics are studied by Bierman and de Paiva, and Bellin et al. in =-=[6]-=- and [4]. Their semantics are based on studies about semantics of linear logics (e.g., [29] and [5]) since the exponential of the linear logic [12] is a kind of S4 modality.sA Curry-Howard corresponde... |

19 |
Constructive modal logics I
- Wijesekera
- 1990
(Show Context)
Citation Context ...n → τ) Γ ⊢ ✷σ1 Γ ⊢ ✷(σ2 → · · · → σn → τ) . Γ ⊢ ✷(σn → τ) Γ ⊢ ✷σn → ✷τ Γ ⊢ ✷σn Γ ⊢ ✷τ . We can also show more directly that our logic corresponds to the sequent calculus formulation of IK proposed in =-=[35]-=-. According to the above encoding, it is not trivial whether an exchange rule commutes with a box operation. Therefore, we distinguish such symmetricity from other axioms although it is common to cons... |

15 | A CPS-Translation of the λµ-Calculus
- Groote
- 1994
(Show Context)
Citation Context ...s originally introduced in [11], and the relation between call-by-value and CPS semantics was first studied by Plotkin in [27]. De Groote defines a CPS transformation on a call-by-name λµ-calculus in =-=[9]-=-, but in this paper, we adopt Selinger’s CPS transformation [30], which is an extension of Hofmann and Streicher’s [14]. In Section 3, we provide a call-by-name λµ-calculus with a box modality, which ... |

14 |
Continuation models are universal for λµ-calculus
- Hofmann, Streicher
- 1997
(Show Context)
Citation Context ... in [27]. De Groote defines a CPS transformation on a call-by-name λµ-calculus in [9], but in this paper, we adopt Selinger’s CPS transformation [30], which is an extension of Hofmann and Streicher’s =-=[14]-=-. In Section 3, we provide a call-by-name λµ-calculus with a box modality, which has a Curry-Howard correspondence to K, by the CPS semantics into the calculus for IK defined in Section 2. A call-by-v... |

14 | Relating categorical semantics for intuitionistic linear logic. Applied Categorical Structures 13(1
- Maietti, Maneggia, et al.
- 2005
(Show Context)
Citation Context ...in addition. Because of the space limitation, we omit a discussion about the category of λ✷-theories along the line of [19]. The notion of equivalence on categories with monoidal endofunctors follows =-=[21]-=-. Definition 4. The internal language of a bicartesian closed category C with a monoidal endofunctor is a λ✷-theory whose type constants consist of objects of C and whose constants consist of morphism... |

10 | Extended curry-howard correspondence for a basic constructive modal logic
- Bellin, Paiva, et al.
- 2001
(Show Context)
Citation Context ...ore defining a calculus for K, we consider the intuitionistic fragment of K, which is called IK in this paper. In Section 2, the calculus for IK is defined as a refinement of Bellin et al.’s calculus =-=[4]-=- rather than Martini and Masini’s[22]. Our calculus is sound and complete for the categorical semantics given in [4]. The study [19] about simply typed λ-calculus and cartesian closed categories is a ... |

9 | Duality between call-by-name recursion and call-by-value iteration
- Kakutani
- 2002
(Show Context)
Citation Context ...nguage with first-class continuations was first formalized by Filinski in [10]. It has been formalized on the λµ-calculi in [30] by Selinger, and reformulated as sequent calculi in [34] by Wadler. In =-=[16]-=-, the duality is developed with recursion by the author. In Section 5, we study such duality on the classical modal logic K. In addition, we investigate the logic S4 with the CPS semantics. It is show... |

5 | A modal foundation for secure information flow
- Miyamoto, Igarashi
- 2004
(Show Context)
Citation Context ...an be regarded as a logic by forgetting terms. In this sense, modal logics are contributing to practical studies for programming languages, e.g., staged computations [8] and information flow analysis =-=[23]-=-. Since K is known as a minimal modal logic, this paper focuses on K rather than S4. Before defining a calculus for K, we consider the intuitionistic fragment of K, which is called IK in this paper. I... |

4 | Some remarks on control categories
- Selinger
- 2003
(Show Context)
Citation Context ... disjunctions is used in the definition of values, but it is possible to introduce [M1, M2] and ιjM as primitive syntax instead of µ(a1, a2). M and [a1, a2]M in the call-by-value calculus as noted in =-=[31]-=-. In our definition, there is a value that has a redex exterior to abstractions, but it is not serious because we are not focusing on reductions. Our notion of values is based on semantical effect-fre... |

2 |
Completeness of modal proofs in first-order predicate logic
- Abe
(Show Context)
Citation Context ...→ Q, −→ L , −→ P 〉 in M{N/x} says that adjacent boxes can be combined into one box. In addition, Abe characterizes the λ✷-calculus by a standard translation into the intuitionistic predicate logic in =-=[1]-=-. Computational meaning of the λ✷-calculus is shown as follows. We consider categorical models of IK along the line of [4]. Because Kripke semantics cover provability but not proofs themselves, they a... |

1 | Calculi for intuitionistic normal modal logic
- Kakutani
- 2007
(Show Context)
Citation Context ... Nn〉 in M : ✷τ the λ-calculus with a box construct. Our calculus, called the λ✷-calculus, is a refinement of Bellin et al.’s calculus given in [4]. The difference from Bellin et al.’s is discussed in =-=[17]-=-. Definition 1. The λ✷-calculus is defined as follows. Types τ and terms M are defined by τ ::= p | τ → τ | ⊤ | τ ∧ τ | ⊥ | τ ∨ τ | ✷τ M ::= α τ | x | λx τ . M | MM | 〈 〉 | 〈M, M〉 | π1M | π2M | [ ]τ |... |

1 | A computastional interpretation of classical S4 modality - Shan - 2005 |