## There's No Substitute for Linear Logic (1991)

Citations: | 24 - 1 self |

### BibTeX

@TECHREPORT{Wadler91there'sno,

author = {Philip Wadler},

title = {There's No Substitute for Linear Logic},

institution = {},

year = {1991}

}

### Years of Citing Articles

### OpenURL

### Abstract

Surprisingly, there is not a good fit between a syntax for linear logic in the style of Abramsky, and a semantics in the style of Seely. Notably, the Substitution Lemma is valid if and only if !A and !!A are isomorphic in a canonical way. An alternative syntax is proposed, that has striking parallels to Moggi's language for monads. In the old syntax, some terms look like the identity that should not, and vice versa; the new syntax eliminates this awkwardness. 1 Introduction This paper has two purposes: to show that linear logic has no substitute, and to propose one. The first part presents a standard syntax and semantics for linear logic, and notes some resulting difficulties. The linear logic is that of Girard [Gir87]. The syntax is based on lambda terms, following in the footsteps of Abramsky [Abr90]: the four rules associated with the `of course' type, Weakening, Contraction, Dereliction, and Promotion, are each represented by a separate term form. The semantics is based on categor...

### Citations

440 | Computational lambda-calculus and monads
- Moggi
- 1989
(Show Context)
Citation Context ...here is no need to indicate these rules explicitly in a term form. There is also an alternate syntax for Dereliction and Promotion. The new syntax has striking parallels to Moggi's language of monads =-=[Mog89]-=-. Roughly speaking, monads : Moggi :: comonads : Girard : The constructs are not exactly dual: Moggi's language requires a monad with a tensorial strength, whereas Girard's language requires a comonad... |

400 | Deforestation: Transforming programs to eliminate trees
- Wadler
- 1990
(Show Context)
Citation Context ...: a term looks like its semantics should be the identity only when its semantics is the identity. Intriguingly, similar syntax has arisen in other areas: a program transformation method due to Wadler =-=[Wad88]-=-, and a model of graph reduction due to Peyton Jones and Salkild [PS89]. This suggests new possible applications of linear logic, which we are avidly pursuing. 1.1 Background and organisation. The syn... |

281 | Computational interpretations of linear logic
- Abramsky
- 1993
(Show Context)
Citation Context ...d syntax and semantics for linear logic, and notes some resulting difficulties. The linear logic is that of Girard [Gir87]. The syntax is based on lambda terms, following in the footsteps of Abramsky =-=[Abr90]-=-: the four rules associated with the `of course' type, Weakening, Contraction, Dereliction, and Promotion, are each represented by a separate term form. The semantics is based on category theory, foll... |

134 | types can change the world!, in
- Wadler, Linear
- 1990
(Show Context)
Citation Context ...form (let x ( a in b) to indicate that term a is `unblazed' and thus not subject to further transformation, and the similarity between `unblazed' types and `of course' types has already been noted in =-=[Wad90]-=-. In work on compiling functional languages, Peyton Jones and Salkild [PS89] have used the form (let x ( a in b) to indicate that x should be bound to a closure representing a, and the similarity of P... |

115 |
Autonomous Categories
- Barr
- 1979
(Show Context)
Citation Context ..., and diagonal : A ! A \Theta A is the diagonal map. The above definition is derived from Seely [See89]. Usually, Seely's definition is thought of as extending Barr's notion of a -autonomous category =-=[Bar79]-=-, but we have no need of a dualising object since we do not deal with negation. A categorical model is obtained by associating with each type variable an object in C, inducing a map from types to obje... |

102 | Linear logic, *-autonomous categories and cofree coalgebras
- Seely
- 1989
(Show Context)
Citation Context ... the `of course' type, Weakening, Contraction, Dereliction, and Promotion, are each represented by a separate term form. The semantics is based on category theory, following in the footsteps of Seely =-=[See89]-=-: Weakening and Contraction are modelled by a comonoid, while Dereliction and Promotion are modelled by a comonad. Surprisingly, when you combine a syntax like Abramsky's with a semantics like Seely's... |

91 |
The linear abstract machine
- Lafont
- 1988
(Show Context)
Citation Context ...gests new possible applications of linear logic, which we are avidly pursuing. 1.1 Background and organisation. The syntactic approach of Abramsky has a long history, including as predecessors Lafont =-=[Laf88]-=- and Holmstrom [Hol88], and as successors Chirimar, Gurnter, and Riecke [CGR91]. The categorical approach of Seely also has a long history, with a notable successor being the work of Filinski [Fil92],... |

88 | Is there a use for linear logic - Wadler - 1991 |

71 |
Single-threaded polymorphic lambda calculus
- Guzman, Hudak
- 1990
(Show Context)
Citation Context ...ing the work of Filinski [Fil92], which reconciles the categoric approach with operational intuition. Linear logic has been applied to control updates in a functional language by by Guzm'an and Hudak =-=[GH90]-=- and Wadler [Wad90, Wad91], and in an imperative language by O'Hearn [O'He91]. The remainder of this paper is organised as follows. Section 2 gives the syntax and semantics of a fragment of intuitioni... |

46 |
The spineless tagless g-machine
- Jones, Salkild, et al.
- 1989
(Show Context)
Citation Context ...emantics is the identity. Intriguingly, similar syntax has arisen in other areas: a program transformation method due to Wadler [Wad88], and a model of graph reduction due to Peyton Jones and Salkild =-=[PS89]-=-. This suggests new possible applications of linear logic, which we are avidly pursuing. 1.1 Background and organisation. The syntactic approach of Abramsky has a long history, including as predecesso... |

34 |
Inheritance and explicit coercion
- Breazu-Tannen, Coquand, et al.
- 1991
(Show Context)
Citation Context ...we must verify that all possible derivations of a judgement assign it the same semantics. This property is called coherence, and its importance was noted by Breazu-Tannen, Coquand, Gunter and Scedrov =-=[BCGS91]-=-. In our case, two derivations of a judgement can differ only in their use of the Exchange rule, since uses of all other rules are encoded in the term, and the fact that \Omega is symmetric and monoid... |

30 | Linear continuations
- Filinski
- 1992
(Show Context)
Citation Context ...nt condition for The Substitution Lemma to holds is that !A and !!A be isomorphic in a canonical way. This explains the occurrence of similar restrictions in the work of O'Hearn [O'He91] and Filinski =-=[Fil92]-=-. Author's address: Department of Computing Science, University of Glasgow, G12 8QQ, Scotland. Electronic mail: wadler@dcs.glasgow.ac.uk. Phone: +44 41 330 4966. Fax: +44 41 330 4913. 1 The second par... |

19 |
A linear functional language
- Holmström
- 1988
(Show Context)
Citation Context ...lications of linear logic, which we are avidly pursuing. 1.1 Background and organisation. The syntactic approach of Abramsky has a long history, including as predecessors Lafont [Laf88] and Holmstrom =-=[Hol88]-=-, and as successors Chirimar, Gurnter, and Riecke [CGR91]. The categorical approach of Seely also has a long history, with a notable successor being the work of Filinski [Fil92], which reconciles the ... |

17 | Linear logic and interference control - O’Hearn - 1991 |