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Register a new userHypernormalisation, linear exponential monads and the Giry tricocycloid
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Added by
Richard Garner
Publication date
2018
fields
Informatics Engineering
and research's language is
English
Authors
Richard Garner
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We provide new categorical perspectives on Jacobs notion of hypernormalisation of sub-probability distributions. In particular, we show that a suitable general framework for notions of hypernormalisation is that of a symmetric monoidal category endowed with a linear exponential monad---a notion arising in the categorical semantics of Girards linear logic. We show that Jacobs original notion of hypernormalisation arises in this way from the finitely supported probability measure monad on the category of sets, which can be seen as a linear exponential monad with respect to a monoidal structure on sets arising from a quantum-algebraic object which we term the Giry tricocycloid. We give many other examples of hypernormalisation arising from other linear exponential monads.
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Monads can be interpreted as encoding formal expressions, or formal operations in the sense of universal algebra. We give a construction which formalizes the idea of evaluating an expression partially: for example, 2+3 can be obtained as a partial evaluation of 2+2+1. This construction can be given for any monad, and it is linked to the famous bar construction, of which it gives an operational interpretation: the bar construction induces a simplicial set, and its 1-cells are partial evaluations. We study the properties of partial evaluations for general monads. We prove that whenever the monad is weakly cartesian, partial evaluations can be composed via the usual Kan filler property of simplicial sets, of which we give an interpretation in terms of substitution of terms. In terms of rewritings, partial evaluations give an abstract reduction system which is reflexive, confluent, and transitive whenever the monad is weakly cartesian. For the case of probability monads, partial evaluations correspond to what probabilists call conditional expectation of random variables. This manuscript is part of a work in progress on a general rewriting interpretation of the bar construction.
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