Noncommutative coherence spaces for full linear logic
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We propose a model of full propositional linear logic based on topological vector spaces. The essential feature is that we consider partially ordered vector spaces, or, rather, positive cones in such spaces. Thus we introduce a category whose objects are dual pairs of normed cones satisfying certain specific completeness properties, such as existence of norm-bounded monotone suprema, and whose morphisms are bounded (adjointable) positive maps. Norms allow us distinct interpretation of dual additive connectives as product and coproduct; in this sense the model is nondegenerate. Also, unlike the familiar case of probabilistic coherence spaces, there is no reference or need for preferred basis; in this sense the model is invariant. Probabilistic coherence spaces form a full subcategory. whose objects, seen as posets, are lattices. Thus we get a model fitting in the tradition of interpreting linear logic in linear algebraic setting, which arguably is free from the drawbacks of its predecessors. We choose the somewhat show-offy title "noncommutative coherence spaces", hinting of noncommutative geometry, because the passage from probabilistic coherence spaces with their preferred bases to general partially ordered cones seems analogous to the passage from commutative algebras of functions on topological spaces to general, not necessarily commutative algebras. Also, natural non-lattice examples of our spaces come from self-adjoint parts of noncommutative operator algebras. Probabilistic coherence spaces then appear as a "commutative" subcategory.
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