Robert Furber

From Kleisli Categories to Commutative C*-Algebras: Probabilistic Gelfand Duality

C *-algebras form rather general and rich mathematical structures that can be studied with different morphisms (preserving multiplication, or not), and with different properties (commutative, or not). These various options can be used to incorporate various styles of computation (set-theoretic, probabilistic, quantum) inside categories of C *-algebras. This paper concentrates on the commutative case and shows that there are functors from several Kleisli categories, of monads that are relevant to model probabilistic computations, to categories of C *-algebras. This yields a new probabilistic version of Gelfand duality, involving the “Radon” monad on the category of compact Hausdorff spaces. We also show that a commutative C *-algebra is isomorphic to the space of convex continuous functionals from its state space to the complex numbers. This allows us to obtain an appropriately commuting state-and-effect triangle for commutative C *-algebras.

Towards a Categorical Account of Conditional Probability

QPL 2015 : 12th International Workshop on Quantum Physics and Logic, 13-17 July 2015, Oxford, United Kingdom

Riesz Modal logic for Markov processes

We investigate a modal logic for expressing properties of Markov processes whose semantics is real-valued, rather than Boolean, and based on the mathematical theory of Riesz spaces. We use the duality theory of Riesz spaces to provide a connection between Markov processes and the logic. This takes the form of a duality between the category of coalgebras of the Radon monad (modeling Markov processes) and the category of a new class of algebras (algebraizing the logic) which we call modal Riesz spaces. As a result, we obtain a sound and complete axiomatization of the Riesz Modal logic.

Unrestricted stone duality for Markov processes

Stone duality relates logic, in the form of Boolean algebra, to spaces. Stone-type dualities abound in computer science and have been of great use in understanding the relationship between computational models and the languages used to reason about them. Recent work on probabilistic processes has established a Stone-type duality for a restricted class of Markov processes. The dual category was a new notion—Aumann algebras—which are Boolean algebras equipped with countable family of modalities indexed by rational probabilities. In this article we consider an alternative definition of Aumann algebra that leads to dual adjunction for Markov processes that is a duality for many measurable spaces occurring in practice. This extends a duality for measurable spaces due to Sikorski. In particular, we do not require that the probabilistic modalities preserve a distinguished base of clopen sets, nor that morphisms of Markov processes do so. The extra generality allows us to give a perspicuous definition of event bisimulation on Aumann algebras.

Boolean-Valued Semantics for the Stochastic λ-Calculus

The ordinary untyped λ-calculus has a λ-theoretic model proposed in two related forms by Scott and Plotkin in the 1970s. Recently Scott showed how to introduce probability by extending these models with random variables. However, to reason about correctness and to add further features, it is useful to reinterpret the construction in a higher-order Boolean-valued model involving a measure algebra. We develop the semantics of an extended stochastic λ-calculus suitable for modeling a simple higher-order probabilistic programming language. We exhibit a number of key equations satisfied by the terms of our language. The terms are interpreted using a continuation-style semantics with an additional argument, an infinite sequence of coin tosses, which serves as a source of randomness. We also introduce a fixpoint operator as a new syntactic construct, as β-reduction turns out not to be sound for unrestricted terms. Finally, we develop a new notion of equality between terms interpreted in a measure algebra, allowing one to reason about terms that may not be equal almost everywhere. This provides a new framework and reasoning principles for probabilistic programs and their higher-order properties.

Infinite-dimensionality in quantum foundations: W*-algebras as presheaves over matrix algebras

In this paper, W*-algebras are presented as canonical colimits of diagrams of matrix algebras and completely positive maps. In other words, matrix algebras are dense in W*-algebras.

Unordered Tuples in Quantum Computation

It is well known that the C*-algebra of an ordered pair of qubits is M_2 (x) M_2. What about unordered pairs? We show in detail that M_3 (+) C is the C*-algebra of an unordered pair of qubits. Then we use Schur-Weyl duality to characterize the C*-algebra of an unordered n-tuple of d-level quantum systems. Using some further elementary representation theory and number theory, we characterize the quantum cycles. We finish with a characterization of the von Neumann algebra for unordered words.