## Term Assignment for Intuitionistic Linear Logic (1992)

Citations: | 57 - 9 self |

### BibTeX

@TECHREPORT{Benton92termassignment,

author = {Nick Benton and Gavin Bierman and Valeria Paiva and Martin Hyland},

title = {Term Assignment for Intuitionistic Linear Logic},

institution = {},

year = {1992}

}

### Years of Citing Articles

### OpenURL

### Abstract

In this paper we consider the problem of deriving a term assignment system for Girard's Intuitionistic Linear Logic for both the sequent calculus and natural deduction proof systems. Our system differs from previous calculi (e.g. that of Abramsky) and has two important properties which they lack. These are the substitution property (the set of valid deductions is closed under substitution) and subject reduction (reduction on terms is well-typed). We define a simple (but more general than previous proposals) categorical model for Intuitionistic Linear Logic and show how this can be used to derive the term assignment system. We also consider term reduction arising from cut-elimination in the sequent calculus and normalisation in natural deduction. We explore the relationship between these, as well as with the equations which follow from our categorical model.

### Citations

754 |
An Introduction to Modal Logic
- Hughes, Cresswell
- 1968
(Show Context)
Citation Context ...ical operator !, is introduced which allows a formula to be used as many times as required (including zero). This operator is, in some ways, similar to the modal necessity operator 2 from Modal Logic =-=[16]-=-. We shall follow Girard's original presentation [11], and give the rules for Intuitionistic Linear Logic in a sequent calculus system. The logic is given in Figure 1. Identity A ` A ?; A;B; ` C Exch... |

669 | Light linear logic
- Girard
- 1998
(Show Context)
Citation Context ...een given names by other authors. ! L1 is called Weakening , ! L2 Contraction, ! L3 Dereliction and (! R ) Promotion 1 . We shall use these terms throughout this paper. 1 Girard, Scedrov and Scott =-=[13]-=- prefer to call this rule Storage. 4 In the Promotion rule, !? means that every formula in the set ? is modal, in other words, if ? is the set fA 1 ; A 2 ; : : : A n g, then !? denotes the set f!A 1 ;... |

469 |
The formulae-as-types notion of construction
- Howard
- 1980
(Show Context)
Citation Context ...system. 2. By considering a linear natural deduction system. Using this system we can construct the rules for the linear logic connectives. We can then apply the so-called Curry-Howard Correspondence =-=[15]-=- to derive a term assignment system. These two approaches produce equivalent term assignment systems. However, when we come to consider equality (reduction) of terms, matters are more subtle. As ever ... |

449 |
Introduction to Higher Order Categorical Logic
- Lambek, Scott
- 1986
(Show Context)
Citation Context ...onal language with variables.) Here we carry out this programme for Intuitionistic Linear Logic. The reader may wish to compare our discussion with the treatment of the -calculus in Lambek and Scott =-=[19]-=-. Since we are dealing with sequents ? ` A, in principle we should deal with multicategories. However it simplies things to assume at once that the multicategorical structure is represented by a tens... |

393 | Basic concepts of enriched category theory
- Kelly
- 2005
(Show Context)
Citation Context ...l functors and monoidal natural transformations. (The basic notions are again due to Eilenberg and Kelly [7], and are spelt out in detail at the end of the paper. The reader may wish to consult Kelly =-=[20]-=- for further information on category theory in the enriched setting.) The equations corresponding to the standard presentation of the notion of a monoidal comonad are quite messy to write down in term... |

290 | Computational interpretation of linear logic
- Abramsky
- 1990
(Show Context)
Citation Context ...es an operation taking an arrow ? A ! B to an arrow ?!A ! B, an appropriate syntax is ?; x : A ` e : B Dereliction ?; z :!A ` let z be !x in e : B 8 and indeed this is the syntax given by Abramsky =-=[1]-=-. With this formulation naturality in B gives rise to an equation f [let z be !x in e=y] = let z be !x in f [e=y] However it is a consequence of naturality that our operation is determined by its eec... |

226 |
Closed categories
- Eilenberg, Kelly
- 1966
(Show Context)
Citation Context ...a morphism m I : I !!I (note that this morphism is the nullary form of the natural transformation) and making a standard collection of diagrams commute. (The denition is given in Eilenberg and Kelly =-=[7]-=-. For the convenience of the reader we display the relevent diagrams in Appendix A.) We have appropriate candidates for the maps m A;B and m I in the interpretations of the proofs: A ` A Dereliction !... |

69 |
Semantics of weakening and contraction
- Jacobs
- 1992
(Show Context)
Citation Context ...ar to require further study. Certainly the work of Zucker [31] and Pottinger [24] need to be considered in this new linear framework. Many variants of Intuitionistic Linear Logic have been proposed =-=[18, 3, 17, 14]-=-. Clearly these need to be considered in the light of this work. Details of term calculi and various resource logics will be discussed in [4]. It has been postulated that computation of Intuitionist... |

64 |
Bounded linear logic: A modular approach to polynomial time computability
- Girard, Scedrov, et al.
- 1992
(Show Context)
Citation Context ...z 0 i in g[promote z i for x i in f=y; promote z 0 i for x i in f=y 0 ] (27) 48 Note that the three cases of cut elimination above involving Promotion are only considered by Girard, Scedrov and Scott =-=[14]-=- when the context (!?) is empty. If the context is nonempty these are called irreducible cuts. The principal cuts correspond to thes-reductions in natural deduction. Hence the reductions that we have ... |

41 |
Linear logic and lazy computation
- Lafont
- 1987
(Show Context)
Citation Context ... to be used as many times as required (including zero). This operator is, in some ways, similar to the modal necessity operator 2 from Modal Logic [16]. We shall follow Girard's original presentation =-=[11]-=-, and give the rules for Intuitionistic Linear Logic in a sequent calculus system. The logic is given in Figure 1. Identity A ` A ?; A;B; ` C Exchange ?; B;A; ` C ? ` B B; ` C Cut ?; ` C ? ` A (I ... |

23 |
The Dialectica categories, in
- Paiva
- 1989
(Show Context)
Citation Context ...of theory of Intuitionistic Linear Logic. Much work has been done on providing such (categorical) models of Intuitionistic Linear Logic. Here we shall just mention the work of Seely [27] and de Paiva =-=[5, 6]-=-. This section is self-contained and the reader need not be familiar with the above. With a view to understanding what is involved here, let us consider the traditional analysis of the proof theory of... |

15 |
Coalgebras and cartesian categories
- Fox
- 1976
(Show Context)
Citation Context ...natural isomorphisms I = !I !A !B = !(A&B) 4. If, on the other hand, C ! has equalizers of core exive pairs of arrows then C ! is cartesian closed. This theorem, which in essence goes back to Fox =-=[8]-=-, is the basis for the Girard translation of intuitionistic logic into Intuitionistic Linear Logic. In the usual formulation this translation is based on 3, that is on the natural isomorphisms introdu... |

9 |
Type systems, linearity and functional languages
- Bierman
- 1992
(Show Context)
Citation Context ...ar to require further study. Certainly the work of Zucker [31] and Pottinger [24] need to be considered in this new linear framework. Many variants of Intuitionistic Linear Logic have been proposed =-=[18, 3, 17, 14]-=-. Clearly these need to be considered in the light of this work. Details of term calculi and various resource logics will be discussed in [4]. It has been postulated that computation of Intuitionist... |

5 |
Axiomatic systems, deduction and implication
- Avron
- 1991
(Show Context)
Citation Context ...ing for !A 1 , the deduction 3 Projections are only dened for the additive connectives. 4 The fundamental importance of closure under substitution for a given logical system is well known; see Avron =-=[2]-=- and Gabbay [9] for example. 14 C!A 1 C ( E ) !A 1 we get the following deduction C!A 1 C ( E ) !A 1 !A n B Promotion !B which is no longer a valid deduction (the assumptions are no... |

4 |
Full intuitionistic linear logic
- Hyland, Paiva
- 1992
(Show Context)
Citation Context ...ar to require further study. Certainly the work of Zucker [31] and Pottinger [24] need to be considered in this new linear framework. Many variants of Intuitionistic Linear Logic have been proposed =-=[18, 3, 17, 14]-=-. Clearly these need to be considered in the light of this work. Details of term calculi and various resource logics will be discussed in [4]. It has been postulated that computation of Intuitionist... |

1 |
Resource Logics and Functional Programming
- Bierman
- 1993
(Show Context)
Citation Context ...o reason about a program (but which a compiler makes little or no use of) and those other cases which are used extensively in the compilation process. Further discussions of this point will appear in =-=[4]-=-. 10 x : A ` x : A ? ` e : A ; x : A ` f : B Cut ?; ` f [e=x] : B ? ` e : A ; x : B ` f : C ( L ) ?; g : AB; ` f [(ge)=x] : C ?; x : A ` e : B ( R ) ? ` x:e : AB ? ` e : A (I L ) ?; x : I... |