## A Noncommutative Full Completeness Theorem (Extended Abstract) (1996)

Venue: | Elsevier Science B.V |

Citations: | 1 - 1 self |

### BibTeX

@INPROCEEDINGS{Blute96anoncommutative,

author = {R. F. Blute and P. J. Scott},

title = {A Noncommutative Full Completeness Theorem (Extended Abstract)},

booktitle = {Elsevier Science B.V},

year = {1996}

}

### OpenURL

### Abstract

) R.F. Blute 1 P.J. Scott 1 Dept. of Mathematics University of Ottawa Ottawa, Ontario K1N 6N5 CANADA E. N. T. C. S. Elsevier Science B. V. Abstract We present a full completeness theorem for the multiplicative fragment of a variant of noncommutative linear logic known as cyclic linear logic (CyLL), first defined by Yetter. The semantics is obtained by considering dinatural transformations on a category of topological vector spaces which are invariant under certain actions of a noncocommutative Hopf algebra, called the shuffle algebra. Multiplicative sequents are assigned a vector space of such dinaturals, and we show that the space has the denotations of cut-free proofs in CyLL+MIX as a basis. This work is a natural extension of the authors' previous work, "Linear Lauchli Semantics", where a similar theorem is obtained for the commutative logic. In that paper, we consider dinaturals which are invariant under certain actions of the additive group of integers. The passage from group...

### Citations

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Citation Context ... thus far are inadequate for the consideration of MALL in that product and coproduct are isomorphic, i.e. RT VEC has all finite biproducts. This problem is avoided by considering normed vector spaces =-=[29]-=-, p. 96. We define a category BAN 1 whose objects are Banach spaces, i.e. complete normed vector spaces, and whose morphisms are linear maps of norm less than or equal to 1. This is a symmetric monoid... |

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Citation Context ...oalgebras We assume that the reader is familiar with the notion of Hopf algebra. In this section we give a quick summary of the representation theory of Hopf algebras. For suitable introductions, see =-=[1,28,19]-=-. Definition 3.1 Given a Hopf algebra H, a module over H is a vector space V , equipped with a k-linear map 2 , called an H-action ae: H\Omega V !V such that: 2 We will assume throughout this paper th... |

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Citation Context ...n combinatorics [20,25], as such Hopf algebras provide an algebriac framework for the study of generating functions. Connections to combinatorics are further established via Joyal's notion of species =-=[21]-=-, a functorial framework for analyzing generating functions. Species were then generalized and given a Hopf-algebraic interpretation by Schmitt in [26]. Thus the representation theory of such structur... |

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Citation Context ...oalgebras We assume that the reader is familiar with the notion of Hopf algebra. In this section we give a quick summary of the representation theory of Hopf algebras. For suitable introductions, see =-=[1,28,19]-=-. Definition 3.1 Given a Hopf algebra H, a module over H is a vector space V , equipped with a k-linear map 2 , called an H-action ae: H\Omega V !V such that: 2 We will assume throughout this paper th... |

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Citation Context ...pology utilized here to obtain a -autonomous category [6]. The resulting structures are closely related to the mixed topologies of [27,32]. Of course, one can also apply the Chu construction to BAN 1 =-=[7]-=-. In so do9 Blute, Scott ing, we obtain a -autonomous category of topological vector spaces in which products and coproducts no longer coincide. Explicitly, if V; W 2 BAN 1 , then we have the followin... |

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Citation Context ...cular variant of linear logic that we will work with is the cyclic linear logic (CyLL) of Yetter [31]. This variant is obtained by adding the cyclic exchange rule to the fully noncommutative logic of =-=[3]-=-. The corresponding version of proof net is also described in [31]. This theory has subsequently been used substantially by Retor'e in his work on linguistics [24]. The Hopf algebra which provides our... |

83 |
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Citation Context ...rengthens this analogy, and suggests a general theory which we hope to explore in the future. The particular variant of linear logic that we will work with is the cyclic linear logic (CyLL) of Yetter =-=[31]-=-. This variant is obtained by adding the cyclic exchange rule to the fully noncommutative logic of [3]. The corresponding version of proof net is also described in [31]. This theory has subsequently b... |

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Citation Context ...a, and show that it is generated by the denotations of proofs in the cyclic fragment. Nonsymmetric monoidal categories which arise from Hopf algebras have recently become important in quantum physics =-=[23]-=-. Since linear logic is a natural vehicle for describing free monoidal categories [10], then modifying the structural rule of exchange should be the logical analogue of the quantizaBlute, Scott tion p... |

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Citation Context ...tipode is bijective. In the case of a cocommutative Hopf algebra, the two internal HOM's are equal. To obtain a -autonomous category of vector spaces, we add a topological structure, due to Lefschetz =-=[22]. Th-=-e categorical structure was worked out by Barr in [5]. Definition 3.5 Let V be a vector space. A topology, �� , on V is linear if it satisfies the following three properties: ffl Addition and scal... |

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Citation Context ...a rotation, then we will not need any explicit representation of the cyclic exchange rule [31,24]. It is possible to represent nets by an inductive procedure analogous to that of the commutative case =-=[15,13,14]-=-. 3 Hopf algebras and Representations 3.1 Algebras and Coalgebras We assume that the reader is familiar with the notion of Hopf algebra. In this section we give a quick summary of the representation t... |

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Citation Context ...ibed in [31]. This theory has subsequently been used substantially by Retor'e in his work on linguistics [24]. The Hopf algebra which provides our semantics is an example of the incidence algebras of =-=[20,25]-=-. It is also refered to as a shuffle algebra in [9], which is the name we have chosen to use. Given a sequent in linear logic, we assign a vector space of dinaturals which are uniform with respect to ... |

22 |
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Citation Context ...analogue of the quantizaBlute, Scott tion process discussed in these references. This suggests for example a logical interpretation of theorems such as the various Tannaka-Krein theorems described in =-=[23,30]-=-. The particular Hopf algebra chosen is of independent interest in several fields. In the theory of distributed and concurrent computation, an important notion is that of interleaving or merging of in... |

19 | Linear Lauchli semantics
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Citation Context ...ive to noncommutative logic. This is an extended abstract. A full version of this paper, with complete proofs, is in preparation. 1 Introduction This paper is a continuation of a program initiated in =-=[12]-=-, where a linear version of Lauchli's semantics for intuitionistic logic is presented. In that paper, 1 Research of each author has been partially supported by operating grants from the Natural Scienc... |

12 | Normal forms and cutfree proofs as natural transformations
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Citation Context ..., i.e. S 2 = id, then RT MOD(H) is a cyclic -autonomous category. Blute, Scott 4 Linear Lauchli Semantics We now review the results of [12]. (Appropriate references for the theory of dinaturality are =-=[4,17,10]-=-). In that paper, a full completeness theorem is established for MLL +MIX via the notion of a uniform dinatural. Definition 4.1 Let F and F 0 be definable functors on RT VEC. A dinatural transformatio... |

11 |
Hopf algebras and linear logic
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Citation Context ...y of (linear combinations of) denotations of proofs. It was observed at the end of that paper that this semantics might be expanded to noncommutative logics by replacing groups with Hopf algebras. In =-=[11]-=-, the representation theory of Hopf algebras is presented as a unifying framework for the analysis of a number of variants of linear logic. By varying the Hopf structure, one obtains models of the com... |

8 | Hopf algebras of combinatorial structures
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Citation Context ...urther established via Joyal's notion of species [21], a functorial framework for analyzing generating functions. Species were then generalized and given a Hopf-algebraic interpretation by Schmitt in =-=[26]-=-. Thus the representation theory of such structures should have important consequences for both these subjects. An overview of the applications of Hopf algebras to various branches of mathematics is g... |

6 |
Lambek’s categorical proof theory and Läuchli’s abstract realizability
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(Show Context)
Citation Context ... F ` F 0 . (Here Z is the additive group of integers.) Before obtaining a full completeness theorem, we first obtained a traditional completeness theorem, which is clearly analogous to the results of =-=[18]-=-. Theorem 4.2 (Completeness) Let M ` N be a balanced binary sequent. If the unique cut-free proof structure associated to M ` N is not a proof net for the theory MLL+MIX, then Z \Gamma Dinat(M; N) is ... |

6 |
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Citation Context ...ion theory of such structures should have important consequences for both these subjects. An overview of the applications of Hopf algebras to various branches of mathematics is given by Hazewinkel in =-=[19]-=-. 2 Cyclic Linear Logic Yetter proposes cyclic linear logic (CyLL) in [31]. He presents a posetal semantics, which he calls Girard quantales and presents a completeness theorem, similar to the phase s... |

5 |
Games and Full Completeness for Multiplicative Linear
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Citation Context ...inaturals uniform. We then show that this vector space has as basis the denotations of cut-free proofs of the sequent in the theory MLL+MIX. Thus we obtain a full completeness theorem in the sense of =-=[2]-=-: our semantics consists entirely of (linear combinations of) denotations of proofs. It was observed at the end of that paper that this semantics might be expanded to noncommutative logics by replacin... |

5 |
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Citation Context ...bstantially by Retor'e in his work on linguistics [24]. The Hopf algebra which provides our semantics is an example of the incidence algebras of [20,25]. It is also refered to as a shuffle algebra in =-=[9]-=-, which is the name we have chosen to use. Given a sequent in linear logic, we assign a vector space of dinaturals which are uniform with respect to this Hopf algebra, and show that it is generated by... |

3 |
Duality of Vector Spaces, Cahiers de Top. et Geom
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Citation Context ...lgebra, the two internal HOM's are equal. To obtain a -autonomous category of vector spaces, we add a topological structure, due to Lefschetz [22]. The categorical structure was worked out by Barr in =-=[5]. De-=-finition 3.5 Let V be a vector space. A topology, �� , on V is linear if it satisfies the following three properties: ffl Addition and scalar multiplication are continuous, when the field k is giv... |

3 |
Des R'eseaux De D'emonstration Pour La Linguistique, manuscript
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- 1996
(Show Context)
Citation Context ... to the fully noncommutative logic of [3]. The corresponding version of proof net is also described in [31]. This theory has subsequently been used substantially by Retor'e in his work on linguistics =-=[24]-=-. The Hopf algebra which provides our semantics is an example of the incidence algebras of [20,25]. It is also refered to as a shuffle algebra in [9], which is the name we have chosen to use. Given a ... |

3 |
spaces with mixed topology
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(Show Context)
Citation Context ... tensor [29]. One can then add a structure analogous to the linear topology utilized here to obtain a -autonomous category [6]. The resulting structures are closely related to the mixed topologies of =-=[27,32]-=-. Of course, one can also apply the Chu construction to BAN 1 [7]. In so do9 Blute, Scott ing, we obtain a -autonomous category of topological vector spaces in which products and coproducts no longer ... |

2 |
Duality of Banach Spaces, Cahiers de Top. et G'eom
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(Show Context)
Citation Context ...sed category, when the tensor product is taken to be the completed projective tensor [29]. One can then add a structure analogous to the linear topology utilized here to obtain a -autonomous category =-=[6]-=-. The resulting structures are closely related to the mixed topologies of [27,32]. Of course, one can also apply the Chu construction to BAN 1 [7]. In so do9 Blute, Scott ing, we obtain a -autonomous ... |

2 |
Coherent Banach Spaces, preprint
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- 1996
(Show Context)
Citation Context ... correspond to the ` 1 and ` 1 norms respectively. Given our previous work, this seems a promising candidate for a full completeness theorem for MALL. Girard, in a recent series of talks and preprint =-=[16]-=-, has proposed the notion of a coherent Banach space in which the additive structure is modeled as above, and the exponentials are modeled via the notion of analytic functions on a Banach space. He al... |

1 |
Appendix to [11
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- 1994
(Show Context)
Citation Context ... T VEC is a symmetric monoidal closed category, when V \Gammaffi W is defined to be the vector space of linear continuous maps, topologized with the topology of pointwise convergence. (It is shown in =-=[8]-=- that the forgetful functorsT VEC!VEC is tensor-preserving.) Lefschetz proves that the embedding V !V ?? is always a bijection, but need not be an isomorphism. We then have: Theorem 3.6 (Barr) RT VEC,... |