## Normal Forms and Cut-Free Proofs as Natural Transformations (1992)

Venue: | in : Logic From Computer Science, Mathematical Science Research Institute Publications 21 |

Citations: | 12 - 4 self |

### BibTeX

@INPROCEEDINGS{Girard92normalforms,

author = {Jean-Yves Girard and Andre Scedrov and Philip J. Scott},

title = {Normal Forms and Cut-Free Proofs as Natural Transformations},

booktitle = {in : Logic From Computer Science, Mathematical Science Research Institute Publications 21},

year = {1992},

pages = {217--241},

publisher = {Springer-Verlag}

}

### Years of Citing Articles

### OpenURL

### Abstract

What equations can we guarantee that simple functional programs must satisfy, irrespective of their obvious defining equations? Equivalently, what non-trivial identifications must hold between lambda terms, thought-of as encoding appropriate natural deduction proofs ? We show that the usual syntax guarantees that certain naturality equations from category theory are necessarily provable. At the same time, our categorical approach addresses an equational meaning of cut-elimination and asymmetrical interpretations of cut-free proofs. This viewpoint is connected to Reynolds' relational interpretation of parametricity ([27], [2]), and to the Kelly-Lambek-Mac LaneMints approach to coherence problems in category theory. 1 Introduction In the past several years, there has been renewed interest and research into the interconnections of proof theory, typed lambda calculus (as a functional programming paradigm) and category theory. Some of these connections can be surprisingly subtle. Here we a...

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Citation Context ...on can be restated in cartesian closed categories (= ccc's). Recall that simply typed lambda calculi correspond to ccc's and that in any ccc morphisms A ! B uniquely correspond to morphisms 1 ! A ) B =-=[23, 19]-=- . We shall blur this latter distinction and abuse the notation accordingly. In any ccc, then, the above equation says that for any morphism f : A ! B the following diagram commutes. 2 B 2B B 2B - r B... |

339 | Theorems for free
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Citation Context ... f) o mAA (u) ; 8 In particular, letting u = 1A , we obtain (2b) mAB (f) = (f 2 f) o mAA (1 A ) The equations (2a) and (2b), which together express equation 2 of Section 1, were first noted by Wadler =-=[32]-=-. Remark 2.4 The equations (y) and (yy) are instances of natural transformations, as follows. From general considerations, dinatural transformations m : F 0! G, where F; G : (C ffi ) 2 C 0! C essentia... |

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Citation Context ... calculus (e.g. [12]) in which types are (implicitly universally quantified) schema, closed under type substitution. The language is closely related to the "Core-ML" language of implicit pol=-=ymorphism [24]-=-. 5 4. If t :s1 2s2 is t =! t 1 ; t 2 ? and kt i k : koe 1 k 2 1 1 1 2 koe k k 0! k i k for i = 1; 2 , then ktkA =! kt 1 kA; kt 2 kA ?. 5. If t :si is 5 i (t 0 ) ( i = 1; 2 ) where kt 0 k : koe 1 k2 1... |

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49 |
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Citation Context ...heorems. If we specialize C to be the ccc freely generated by L, that is, the syntactical term model of typed 2 Indeed, we actually obtain a fibred ccc or hyperdoctrine, ignoring the indexed adjoints =-=[25, 28] 6 lambda calculus w-=-ith type variables, qua ccc (cf. [19],p. 77) then "equality" means "provable equality" in L . In this setting, it follows that the syntactic families ktk above are provably dinatur... |

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Citation Context ...n by the notion of relator) is discussed in Abramsky, et al [1]. 3 Gentzen Sequents and Natural Deduction In what follows, we sketch the translation of Gentzen sequents into natural deduction calculi =-=[12, 31, 33, 6, 22]-=- which motivate our presentation. 3.1 Natural Deduction and Sequent Calculus A deduction D of formula A (to be defined below) is a certain type of finite tree whose vertices are labelled by formulae a... |

46 |
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Citation Context ... variable x of type oe , are then interpreted as certain multivariant (= dinatural ) transformations between (the interpretations of) the types. The resultant calculus has many interesting properties =-=[2, 7]-=-, despite the fact that in general dinatural transformations do not compose. Previous studies [2, 3, 8] emphasized semantical (compositional) models for this calculus; in this paper we prove that the ... |

45 |
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Citation Context ...existence of a factorization. Diagrams of this shape arise as Reynolds' invariance conditions in his approach to parametricity [27], [2], pp.53-55, as well as in Statman's theory of logical relations =-=[30]-=-, and in Wadler's universal Horn conditions which motivated this discussion. We emphasize, however, that the invariance conditions given by diagrams of this shape have deep roots in the left rules of ... |

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Citation Context ...nt of coherence in categories, discusses this very situation in detail for a general calculus of transformations, while special cases of the general problem were already resolved in Eilenberg & Kelly =-=[5]-=-. Cut elimination theorems were successfully applied to coherence questions by Lambek [15, 16, 17, 18] and Mints [22]. Of course, for the simple types of this paper, normalization or cut elimination p... |

38 |
Categorical semantics for higher order polymorphic lambda calculus
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Citation Context ...heorems. If we specialize C to be the ccc freely generated by L, that is, the syntactical term model of typed 2 Indeed, we actually obtain a fibred ccc or hyperdoctrine, ignoring the indexed adjoints =-=[25, 28] 6 lambda calculus w-=-ith type variables, qua ccc (cf. [19],p. 77) then "equality" means "provable equality" in L . In this setting, it follows that the syntactic families ktk above are provably dinatur... |

29 |
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Citation Context ...ts on a preliminary version of this work. 1 ffl Natural Transformations from category theory. ffl Parametricity from the foundations of polymorphism. Familiar work of Curry, Howard, Lambek and others =-=[12, 15, 17]-=- has shown how we may consider constructive proofs as programs. For example, Gentzen's intuitionistic sequents A 1 ; : : : ; A k ` B may be interpreted as functional programs mapping k inputs of types... |

21 | The System F of Variable Types - Girard - 1986 |

18 | A logic program for transforming sequent proofs to natural deduc- tion proofs
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(Show Context)
Citation Context ...n by the notion of relator) is discussed in Abramsky, et al [1]. 3 Gentzen Sequents and Natural Deduction In what follows, we sketch the translation of Gentzen sequents into natural deduction calculi =-=[12, 31, 33, 6, 22]-=- which motivate our presentation. 3.1 Natural Deduction and Sequent Calculus A deduction D of formula A (to be defined below) is a certain type of finite tree whose vertices are labelled by formulae a... |

18 |
Scott,Typed Lambda Models and Cartesian Closed Categories
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- 1989
(Show Context)
Citation Context ...on can be restated in cartesian closed categories (= ccc's). Recall that simply typed lambda calculi correspond to ccc's and that in any ccc morphisms A ! B uniquely correspond to morphisms 1 ! A ) B =-=[23, 19]-=- . We shall blur this latter distinction and abuse the notation accordingly. In any ccc, then, the above equation says that for any morphism f : A ! B the following diagram commutes. 2 B 2B B 2B - r B... |

10 |
Closed categories and the theory of proofs
- Mints
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(Show Context)
Citation Context ...n by the notion of relator) is discussed in Abramsky, et al [1]. 3 Gentzen Sequents and Natural Deduction In what follows, we sketch the translation of Gentzen sequents into natural deduction calculi =-=[12, 31, 33, 6, 22]-=- which motivate our presentation. 3.1 Natural Deduction and Sequent Calculus A deduction D of formula A (to be defined below) is a certain type of finite tree whose vertices are labelled by formulae a... |

9 | Why commutative diagrams coincide with equivalent proofs - Lane - 1982 |

3 |
Logic Without Structural Rules
- Lambek
- 1990
(Show Context)
Citation Context ...uction proofs (cf [12]) is by now quite familiar, a similar process applied to Gentzen sequent calculus appears less so, despite the related work of Lambek connected to categorical coherence theorems =-=[17, 18]-=-. One motivation of this term assignment is to think of an intuitionist sequent 0 ` B (where 0 = fA 1 ; : : : ; A k g) as an input/output device, accepting k inputs of types A 1 ; : : : ; A k and retu... |

3 |
Parametric Polymorphism, in: Information Processing `83
- Types
- 1983
(Show Context)
Citation Context ...l approach addresses an equational meaning of cut-elimination and asymmetrical interpretations of cut-free proofs. This viewpoint is connected to Reynolds' relational interpretation of parametricity (=-=[27]-=-, [2]), and to the Kelly-Lambek-Mac LaneMints approach to coherence problems in category theory. 1 Introduction In the past several years, there has been renewed interest and research into the interco... |

2 |
Normal functors, power series, and -calculus, Ann. Pure and Applied Logic 37
- Girard
- 1986
(Show Context)
Citation Context ...he absence of cut, the symmetry of the situation is broken: we may obtain asymmetrical interpretations of the sequents [12]. Let us give an example to illustrate the power of the method. In the paper =-=[9]-=- one associates to any lambda term an infinite series P a n ~ e n where the coefficients a n are cardinal numbers. The task is to prove that any normalisable lambda term is weakly finite , i.e. all a ... |

2 |
Many-variable functorial calculus.I
- Kelly
(Show Context)
Citation Context ...rical statement: strongly finite = weakly finite. It is precisely the problem of linking positive and negative slots in type expressions which prevents, in general, the composing of dinaturals. Kelly =-=[13]-=-, in his abstract treatment of coherence in categories, discusses this very situation in detail for a general calculus of transformations, while special cases of the general problem were already resol... |