## Chu Spaces as a Semantic Bridge Between Linear Logic and Mathematics (1998)

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Venue: | Theoretical Computer Science |

Citations: | 12 - 2 self |

### BibTeX

@ARTICLE{Pratt98chuspaces,

author = {Vaughan Pratt},

title = {Chu Spaces as a Semantic Bridge Between Linear Logic and Mathematics},

journal = {Theoretical Computer Science},

year = {1998},

volume = {294},

pages = {439--471}

}

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### Abstract

The motivating role of linear logic is as a "logic behind logic." We propose a sibling role for it as a logic of transformational mathematics via the self-dual category of Chu spaces, a generalization of topological spaces. These create a bridge between linear logic and mathematics by soundly interpreting linear logic while fully and concretely embedding a comprehensive range of concrete categories of mathematics. Our main goal is to treat each end of this bridge in expository detail. In addition we introduce the dialectic lambda-calculus, and show that dinaturality semantics is not fully complete for the Chu interpretation of linear logic. 1 Introduction Linear logic was introduced by J.-Y. Girard as a "logic behind logic." It separates logical reasoning into a core linear part in which formulas are merely moved around, and an auxiliary nonlinear part in which formulas may be deleted and copied. The core, multiplicative linear logic (MLL), is a substructural logic whose basic connect...

### Citations

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296 |
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(Show Context)
Citation Context ...logic: phase semantics and coherent spaces [Gir87], Hilbert spaces, and more recently Banach spaces. Phase semantics has truth-valued entailment and resembles Birkhoff and von Neumann’s quantum logic =-=[BvN36]-=-, while the other three have the set-valued entailment characteristic of categorical logic: A ⊢ B as the set of morphisms from A to B. A number of other models have also been proposed. Inspired by Bla... |

240 | Modelling concurrency with partial orders - Pratt - 1986 |

215 | Games and full completeness for multiplicative linear logic
- Abramsky, Jagadeesan
- 1994
(Show Context)
Citation Context ...onnected and containing a cycle. bijection between syntactic and semantic criteria for theoremhood called full completeness, the term coined by Abramsky and Jagadeesan for their game semantics of MLL =-=[AJ92]-=- but equally applicable to switching semantics. Here the bijection is identification: the valid linking that each sound linking is paired with is itself. The sound linkings of A constitute abstract pr... |

145 |
The structure of multiplicatives
- Danos, Regnier
- 1989
(Show Context)
Citation Context ...s we shall be working exclusively with cut-free proofs. There exist both syntactic and semantic characterizations of theorems in terms of linkings, which Danos and Regnier have shown to be equivalent =-=[DR89]-=-. For the syntactic characterization, every MLL derivation of a theorem A determines a proof structure as follows. The proof structure determined by an instance of T matches P ⊥ i and Pi in each conju... |

144 |
Introduction to Higher-Order Categorical Logic. Number 7
- Lambek, Scott
- 1986
(Show Context)
Citation Context ... mathematical proofs and computations. On the other, in the role we propose for it here, it expresses the transformational structure of universal mathematics, extending higher order categorical logic =-=[LS86]-=- from its previous narrow preoccupation with cartesian closed categories, where it has the form of intuitionistic logic, to the broader universe of “the rest of mathematics,” where it takes on the sha... |

137 | A game semantics for linear logic
- Blass
- 1992
(Show Context)
Citation Context ...hile the other three have the set-valued entailment characteristic of categorical logic: A ` B as the set of morphisms from A to B. A number of other models have also been proposed. Inspired by Blass =-=[Bla92]-=-, Abramsky and Jagadeesan [AJ94] have interpreted linear logic over sequential games, further studied by Hyland and Ong [HO93]. Barr has proposed fuzzy relations as a model [Bar96], while Blute [Blu96... |

103 | Towards a geometry of interaction - Girard - 1989 |

88 |
autonomous categories and linear logic
- Barr
- 1991
(Show Context)
Citation Context ...d fuzzy relations as a model [Bar96], while Blute [Blu96] has taken Hopf algebras as an interpretation of noncommutative linear logic. Chu spaces, the model we treat here, were first proposed by Barr =-=[Bar91]-=- and Lafont and Streicher [LS91]. Generalizing an idea of Mackey [Mac45], Barr defined general V -enriched Chu spaces, whose carrier, cocarrier, and alphabet k are objects of a symmetric monoidal clos... |

61 |
Lambda-de in the full type hierarchy
- Plotkin
- 1980
(Show Context)
Citation Context ...h as higher order intuitionistic logic, where a strengthening of dinaturality is required for full completeness, e.g. logical transformations as defined in terms of certain kinds of logical relations =-=[Plo80]-=-. We have recently showed, with H. Devarajan, D. Hughes, and G. Plotkin [DHPP99], that strengthening dinaturality to binary logicality rescues this situation for Chu spaces. Contrasting with this situ... |

55 |
The logic of quantum mechanics
- Birkho¤, Neumann
- 1936
(Show Context)
Citation Context ...logic: phase semantics and coherent spaces [Gir87], Hilbert spaces, and more recently Banach spaces. Phase semantics has truth-valued entailment and resembles Birkhoff and von Neumann's quantum logic =-=[BvN36]-=-, while the other three have the set-valued entailment characteristic of categorical logic: A ` B as the set of morphisms from A to B. A number of other models have also been proposed. Inspired by Bla... |

55 | Origins of the calculus of binary relations - Pratt - 1992 |

49 |
Games semantics for linear logic
- Lafont, Streicher
- 1991
(Show Context)
Citation Context ...96], while Blute [Blu96] has taken Hopf algebras as an interpretation of noncommutative linear logic. Chu spaces, the model we treat here, were first proposed by Barr [Bar91] and Lafont and Streicher =-=[LS91]-=-. Generalizing an idea of Mackey [Mac45], Barr defined general V -enriched Chu spaces, whose carrier, cocarrier, and alphabet k are objects of a symmetric monoidal closed category V , forming the cate... |

46 | Autonomous categories, volume 752 - Barr - 1979 |

42 |
Abramsky and Radha Jagadeesan. Games and full completeness for multiplicative linear logic
- Samson
- 1994
(Show Context)
Citation Context ...-valued entailment characteristic of categorical logic: A ` B as the set of morphisms from A to B. A number of other models have also been proposed. Inspired by Blass [Bla92], Abramsky and Jagadeesan =-=[AJ94]-=- have interpreted linear logic over sequential games, further studied by Hyland and Ong [HO93]. Barr has proposed fuzzy relations as a model [Bar96], while Blute [Blu96] has taken Hopf algebras as an ... |

40 | Fair games and full completeness for multiplicative linear logic without the mix-rule
- Hyland, Ong
- 1993
(Show Context)
Citation Context ...o B. A number of other models have also been proposed. Inspired by Blass [Bla92], Abramsky and Jagadeesan [AJ94] have interpreted linear logic over sequential games, further studied by Hyland and Ong =-=[HO93]-=-. Barr has proposed fuzzy relations as a model [Bar96], while Blute [Blu96] has taken Hopf algebras as an interpretation of noncommutative linear logic. Chu spaces, the model we treat here, were first... |

32 | The Stone gamut: A coordinatization of mathematics
- Pratt
- 1995
(Show Context)
Citation Context ...d poset-like structures, whereas linear logic is that of the larger class of all structures, ranging from the extreme discreteness of sets to the extreme coherence of complete atomic Boolean algebras =-=[Pra95]-=-. These two roles, proof theory and transformational mathematics, are not necessarily best served by the same denotational semantics. In particular Girard has argued the need for a denotational semant... |

29 | Temporal structures - Casley, Crew, et al. - 1991 |

27 | A dialectica-like model of linear logic - Paiva - 1989 |

23 | The Dialectica categories, in - Paiva - 1989 |

21 | Full completeness of the multiplicative linear logic of chu spaces
- Devarajan, Hughes, et al.
- 1999
(Show Context)
Citation Context ...s required for full completeness, e.g. logical transformations as defined in terms of certain kinds of logical relations [Plo80]. We have recently showed, with H. Devarajan, D. Hughes, and G. Plotkin =-=[DHPP99]-=-, that strengthening dinaturality to binary logicality rescues this situation for Chu spaces. Contrasting with this situation, A. Tan has shown [Tan97] that MLL with MIX is fully complete for dinatura... |

19 | Linear Lauchli semantics - Blute, Scott - 1996 |

17 |
Full completeness for models of linear logic
- Tan
- 1997
(Show Context)
Citation Context ...d, with H. Devarajan, D. Hughes, and G. Plotkin [DHPP99], that strengthening dinaturality to binary logicality rescues this situation for Chu spaces. Contrasting with this situation, A. Tan has shown =-=[Tan97]-=- that MLL with MIX is fully complete for dinatural transformations in Girard’s ∗-autonomous category Coh of coherence spaces. We also introduce in this section the dialectic λ-calculus, as a novel way... |

15 | How comprehensive is the category of semigroups - Hedrlín, Lambeck - 1969 |

11 |
Non-commutative intuitionistic linear propositional logic’, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik 36:297–318
- Abrusci
- 1990
(Show Context)
Citation Context ...eral (in S1 they connected complementary pairs). Such a formula is in the language of S2. We further claim that this formula is a theorem of S2. To see this proceed 6 For noncommutative l inear logic =-=[Abr90]-=- the De Morgan laws also reverse order; here we leave the order unchanged so as to preserve the exact structure of all formulas. by induction of the length of proofs in S1. For the basis case, transla... |

9 |
Infinite dimensional vector spaces
- Mackey
- 1945
(Show Context)
Citation Context ... algebras as an interpretation of noncommutative linear logic. Chu spaces, the model we treat here, were first proposed by Barr [Bar91] and Lafont and Streicher [LS91]. Generalizing an idea of Mackey =-=[Mac45]-=-, Barr defined general V -enriched Chu spaces, whose carrier, cocarrier, and alphabet k are objects of a symmetric monoidal closed category V , forming the category Chu(V, k) studied by Barr’s student... |

8 | The shuffle hopf algebra and noncommutative full completeness - Blute, Scott - 1998 |

6 |
Fuzzy models of linear logic
- Barr
- 1996
(Show Context)
Citation Context .... Inspired by Blass [Bla92], Abramsky and Jagadeesan [AJ94] have interpreted linear logic over sequential games, further studied by Hyland and Ong [HO93]. Barr has proposed fuzzy relations as a model =-=[Bar96]-=-, while Blute [Blu96] has taken Hopf algebras as an interpretation of noncommutative linear logic. Chu spaces, the model we treat here, were first proposed by Barr [Bar91] and Lafont and Streicher [LS... |

6 | Towards full completeness of the linear logic of Chu spaces. Mathematical foundations of programming semantics
- Pratt
- 1997
(Show Context)
Citation Context ...Morphisms between functors are usually taken to be natural transformations. However linear logic contains functors of mixed variance 1 such as A−◦A, for which mere naturality is not enough. Elsewhere =-=[Pra97]-=- we have shown that when the morphisms are taken to be ordinary dinatural transformations, as done by Blute and Scott [BS96a, BS96b] for their Lauchli semantics of linear logic, then Girard’s MIX-free... |

5 | Universal categories - Trnková - 1966 |

4 |
Hopf algebras and linear logic (with appendix by
- Blute
- 1996
(Show Context)
Citation Context ...Bla92], Abramsky and Jagadeesan [AJ94] have interpreted linear logic over sequential games, further studied by Hyland and Ong [HO93]. Barr has proposed fuzzy relations as a model [Bar96], while Blute =-=[Blu96]-=- has taken Hopf algebras as an interpretation of noncommutative linear logic. Chu spaces, the model we treat here, were first proposed by Barr [Bar91] and Lafont and Streicher [LS91]. Generalizing an ... |

3 |
Dinatural numbers
- Robert, Román
- 1998
(Show Context)
Citation Context ...his weakness of dinaturality has already been observed in other contexts, e.g. Set as a model of intuitionistic logic where A → A ⊢ A → A also has spurious dinaturals as pointed out by Paré and Román =-=[PR98]-=-. Constructive intuitionistic logic has as the proofs from A → A (the set of all functions on the set A) to itself just the Church numerals, those functions sending f : A → A to f n : A → A for some f... |