Journal of Symbolic Logic

The Shuffle Hopf Algebra and Noncommutative Full Completeness

R. F. Blute and P. J. Scott

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We present a full completeness theorem for the multiplicative fragment of a variant of noncommutative linear logic, Yetter's cyclic linear logic (CyLL). The semantics is obtained by interpreting proofs as dinatural transformations on a category of topological vector spaces, these transformations being equivariant under certain actions of a noncocommutative Hopf algebra called the shuffie algebra. Multiplicative sequents are assigned a vector space of such dinaturals, and we show that this space has as a basis the denotations of cut-free proofs in CyLL + MIX. This can be viewed as a fully faithful representation of a free *-autonomous category, canonically enriched over vector spaces. This paper is a natural extension of the authors' previous work, "Linear Lauchli Semantics", where a similar theorem is obtained for the commutative logic MLL + MIX. In that paper, we interpret proofs as dinaturals which are invariant under certain actions of the additive group of integers. Here we also present a simplification of that work by showing that the invariance criterion is actually a consequence of dinaturality. The passage from groups to Hopf algebras in this paper corresponds to the passage from commutative to noncommutative logic. However, in our noncommutative setting, one must still keep the invariance condition on dinaturals.

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J. Symbolic Logic, Volume 63, Issue 4 (1998), 1413-1436.

First available in Project Euclid: 6 July 2007

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Blute, R. F.; Scott, P. J. The Shuffle Hopf Algebra and Noncommutative Full Completeness. J. Symbolic Logic 63 (1998), no. 4, 1413--1436.

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