Robert Furber
My interests are in probability, logic, quantum, and category theory, especially monads. I also work in domain theory and Boolean-valued models.
Publications, Preprints, etc.
Conference papers are listed in their year of final publication. Conference talks without a proceedings paper are listed in the year of the talk.
2026
- Conference Paper:Interpreting Lambda Calculus in Domain-Valued Random Variables with Radu Mardare, Prakash Panangaden and Dana Scott.
We show how to interpret untyped lambda calculus in the set of P(N)-valued random variables using Boolean-valued models of set theory, and give a proof, using only the notions of lambda calculus, that there are two sets of integers such that neither can be mapped to the other by a lambda-definable function. Previously, this was proved using recursion theory (there are two sets of integers that are not many-one comparable).
2023
- Abstract: Commutative W*-algebras as a Markov Category (Extended Abstract)
We show that the probability monad on measure spaces is commutative. We do this is by duality, showing that the comonad on commutative W*-algebras is cocommutative. This requires us to characterize normal positive unital maps out of a colimit of commutative W*-algebras, which we do by introducing the notion of a positive operator-valued measure with "truly continuous marginals". In passing, we show that the product of [0,1] with Lebesgue measure, as intepreted in the category of measure spaces, is isomorphic to the coproduct of continuum-many copies of itself, so cannot be expressed using sigma-finite measures.
2022
2021
- Preprint: Interpreting Lambda Calculus in Domain-Valued Random Variables with Radu Mardare, Prakash Panangaden and Dana Scott.
We show how to interpret untyped lambda calculus in the set of P(N)-valued random variables using Boolean-valued models of set theory, and give a lambda calculus proof that there are incomparable many-one degrees as an example application. See 2026 for the conference version.
2020
- Conference Paper: Scott Continuity in Generalized Probabilistic Theories, published in EPTCS 318, pp. 66-84. Originally a talk at QPL 2019.
In this paper, I construct counterexamples to various generalizations of the use of Scott continuity in W*-algebras to the setting of base-norm and order-unit spaces. In particular, one cannot recover the predual of an order-unit space (if it has one) using Scott continuous states.
Using these constructions, and some classical counterexamples from functional analysis, I was able to produce several other counterexamples.
- Journal Paper: Probabilistic Logics Based on Riesz Spaces with Radu Mardare and Matteo Mio. Published in Logical Methods in Computer Science Volume 16, Issue 1.
This is an extended version of earlier results, some joint work, and some by Matteo Mio alone, for Riesz modal logic, a logic, based on Riesz spaces, for reasoning about continuous Markov chains on compact Hausdorff spaces. This logic stands in relation to such Markov chains just as Boolean modal logic does to Stone coalgebras.
2019
- Conference Paper: Categorical Equivalences from State-Effect Adjunctions, talk at QPL 2018, published in EPTCS 287, 2019, pp. 107-126.
In an earlier paper, Bart Jacobs defined a dual adjunction between effect algebras and abstract convex sets. This paper characterizes the subcategories on which this dual adjunction is a contravariant equivalence. I then outline how to get two more adjunctions and dualities using the theory of Smith base-norm and Smith order-unit spaces, like in my PhD thesis. In an appendix I characterize the effect modules/convex effect algebras for which effect algebra morphisms are automatically effect module homomorphisms, and give counterexamples showing that the result is the best possible.
- Preprint: Continuous Dcpos in Quantum Computing
In this paper, I show that if the unit interval of a directed-complete C*-algebra A is a continuous dcpo, then A is a product of finite-dimensional matrix algebras. Combined with previous results due to Selinger, this characterizes the directed-complete C*-algebras with continuous unit interval. I also show that if the unit interval of A has a countable base (as a dcpo) then A is isomorphic to the algebra of bounded functions on a countable set, and is therefore commutative.
2018
2017
2016
2015
- Journal Paper: From Kleisli categories to commutative C*-algebras: Probabilistic Gelfand Duality with Bart Jacobs, published in Logical Methods in Computer Science, 2015, Volume 11, Issue 2.
The Radon monad is a kind of Giry monad (though predating Giry's paper) that assigns a compact Hausdorff space to its space of Radon measures. In this paper, we show that the Kleisli category of the Radon monad is equivalent to the category of commutative C*-algebras, under the functor that assigns a compact Hausdorff space X to its C*-algebra of complex-valued functions C(X).
An earlier version as a conference paper from CALCO 2013 was published by Springer in the proceedings LNCS 8089, pages 141-157.
- Conference Paper: Towards a Categorical Account of Conditional Probability with Bart Jacobs, originally for QPL 2013, published in EPTCS 195, pp. 179-195.
This paper gives a definition of conditional probability that can be applied in both the Kleisli category of the distribution monad and the category of C*-algebras (with positive unital maps). As an example, we use the Elitzur-Vaidman "bomb tester".
- Conference Paper: Unordered Tuples in Quantum Computation with Bas Westerbaan, QPL 2015, published in EPTCS 195, pp. 196-207.
This paper is on how to realize certain quotient types (unordered tuples and necklaces) in C*-algebraic quantum theory.
Theses