## Linear Logic, Totality and Full Completeness (1994)

Venue: | In Proceedings of LiCS `94 |

Citations: | 12 - 2 self |

### BibTeX

@INPROCEEDINGS{Loader94linearlogic,,

author = {Ralph Loader},

title = {Linear Logic, Totality and Full Completeness},

booktitle = {In Proceedings of LiCS `94},

year = {1994},

pages = {292--298},

publisher = {Press}

}

### OpenURL

### Abstract

I give a `totality space' model for linear logic [4], derived by taking an abstract view of computations on a datatype. The model has similarities with both the coherence space model and game-theoretic models [1, 5], but is based upon a notion of total object. Using this model, I prove a full completeness result, along the lines of the results for game theoretic models in [1] and [5]. In other words, I show that the mapping of proofs to their interpretations (here collections of total objects uniform for a given functor) in the model is a surjection. 1 Introduction We shall give a model of linear logic by formalising a particular view of what an abstract datatype is. Consider a datatype A. There are objects s of type A, and programs t that accept an argument of type A. Taking any such s and t, we may execute the program on the data, and obtain a particular computation---the trace of the execution of the program. We shall consider only this facet of datatypes. For a given data (or pro...

### Citations

619 | Light linear logic
- Girard
- 1998
(Show Context)
Citation Context ...or some s 2 A? , whenever x; y 2 jAj. For any x; y 2 jAj, the set fx; yg is either consistent or co-consistent (i.e., consistent in either A or A ? ). then the reader familiar with the original paper =-=[4]-=- on Linear Logic will note that A is a coherence space in a natural way. The multiplicative connectives on totality spaces agree with those for coherence spaces, where appropriate. Some examples of to... |

209 | Games and full completeness for multiplicative linear logic
- Abramsky, Jagadeesan
- 1992
(Show Context)
Citation Context ...`totality space' model for linear logic [4], derived by taking an abstract view of computations on a datatype. The model has similarities with both the coherence space model and game-theoretic models =-=[1, 5]-=-, but is based upon a notion of total object. Using this model, I prove a full completeness result, along the lines of the results for game theoretic models in [1] and [5]. In other words, I show that... |

102 | Linear logic, *-autonomous categories and cofree coalgebras
- Seely
- 1989
(Show Context)
Citation Context ...verified equations, shows that we have a -autonomous category Tot whose objects are totality spaces, with Tot(A; B) = (A (B)? : This gives a model of the multiplicative fragment of linear logic---see =-=[9]-=-. Noting that 1 = ?, the mix rule is also modeled. On the level of the underlying sets, the interpretation is the same as that of the coherence space semantics. As mentioned in the introduction, this ... |

59 |
Lambda definability in the full type hierarchy
- Plotkin
- 1980
(Show Context)
Citation Context ...e with a natural numbers type. The term `full completeness' was first used in [1] for the result there, although an earlier non-trivial result of this kind, for the simply typed -calculus, appears in =-=[8]-=-. 2 Totality Spaces and Linear Operators Definition 1 A totality space A = (jAj; A? ; A? ) consists of a set jAj together with sets A? ; A? ae PjAj such that: T1 A? is the set of those s ae jAj such t... |

44 |
L.: The structure of multiplicatives. Archive for Mathematical Logic 28
- Danos, Regnier
- 1989
(Show Context)
Citation Context ...e of the Y i to the index of the other end of the axiom link. Thus we must have, for i = 1 : : : m, OE(OE(i)) = i;si =sOE(i) ; i i 6= i OE(i) : (1) So thatsis a proof-net, the Danos-Regnier condition =-=[2]-=- must be satisfied also. As in the coherence space semantics, the interpretation [[]](sA) of any cutfree proof-netsgiven by an involution OE is \Phi (x 1 : : : xm ) 2 jF (sA)j fi fi x i = x OE(i) for ... |

20 |
The System F of Variable Types
- Girard
- 1986
(Show Context)
Citation Context ...e on coherence spaces. The extension to higher order calculi seems to require the totality structure to be given on top of some sort of domain or coherence space structure, as is done for system F in =-=[3]-=-. 4 Full Completeness By the interpretation given in the previous section, any proof of a formula F (X 1 ; : : : ; Xn ) of linear logic gives a total object of F (A 1 ; : : : ; An ) for any totality s... |

5 |
Interpreting higher computations as types with totality
- Kristiansen, Normann
- 1992
(Show Context)
Citation Context ... a counter-strategy (co-total object) to produce a result (their intersection). Similar models, for intuitionistic logics and type theories, have been considered by L. Kristiansen and D. Normann; see =-=[6]-=- and the Ph.D thesis of the first author of that paper. We shall show that, for this model, the correspondance between logic and model is very close, by proving that every object of the appropriate ty... |

3 |
Full completeness for multiplicative linear logic without the mix-rule, electronically distributed manuscript
- Hyland, Ong
- 1993
(Show Context)
Citation Context ...`totality space' model for linear logic [4], derived by taking an abstract view of computations on a datatype. The model has similarities with both the coherence space model and game-theoretic models =-=[1, 5]-=-, but is based upon a notion of total object. Using this model, I prove a full completeness result, along the lines of the results for game theoretic models in [1] and [5]. In other words, I show that... |

2 |
Models of linear logic and inductive datatypes, unpublished manuscript. A very preliminary version of this was presented at the ' Esprit BRA Types workshop in
- Loader
- 1993
(Show Context)
Citation Context ...are programs, so it should be of no surprise that we obtain a model of linear logic. The spaces being used here to represent datatypes are called totality spaces, following an analysis of totality in =-=[7]-=-, where spaces such as these arose. The totality spaces are related to the game-theoretic semantics recently studied by Abramsky et. al. [1, 5], retaining only the fact that a strategy (total object) ... |