Probabilistic Monads, Domains and Classical Information

Michael Mislove
(Tulane University)

Shannon's classical information theory uses probability theory to analyze channels as mechanisms for information flow. In this paper, we generalize results of Martin, Allwein and Moskowitz for binary channels to show how some more modern tools - probabilistic monads and domain theory in particular - can be used to model classical channels. As initiated Martin, et al., the point of departure is to consider the family of channels with fixed inputs and outputs, rather than trying to analyze channels one at a time. The results show that domain theory has a role to play in the capacity of channels; in particular, the (n x n)-stochastic matrices, which are the classical channels having the same sized input as output, admit a quotient compact ordered space which is a domain, and the capacity map factors through this quotient via a Scott-continuous map that measures the quotient domain. We also comment on how some of our results relate to recent discoveries about quantum channels and free affine monoids.

In Elham Kashefi, Jean Krivine and Femke van Raamsdonk: Proceedings 7th International Workshop on Developments of Computational Methods (DCM 2011), Zurich, Switzerland, 3rd July 2011, Electronic Proceedings in Theoretical Computer Science 88, pp. 87–100.
Published: 30th July 2012.

ArXived at: http://dx.doi.org/10.4204/EPTCS.88.8 bibtex PDF
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