Robert Furber

My interests are in probability, logic, quantum, and category theory, especially monads.

Publications, Preprints, etc.

Papers are grouped by year of final publication.

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.

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.
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.

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.

Categorical Equivalences from State-Effect Adjunctions, published in EPTCS 287, 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.
Boolean-valued Semantics for Stochastic Lambda-Calculus with Giorgio Bacci, Dexter Kozen, Radu Mardare, Prakash Panangaden and Dana Scott
Proceedings of the Thirty-Third Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), pp. 669-678. DOI: 10.1145/3209108.3209175

Infinite-Dimensionality in Quantum Foundations: W*-algebras as Presheaves over Matrix Algebras with Mathys Rennela and Sam Staton, published in EPTCS 236, pp. 161-173.
This paper relates W*-algebras to presheaves on the category of finite-dimensional matrix algebras with completely positive maps.
Unrestricted Stone Duality for Markov Processes with Dexter Kozen, Kim Larsen, Radu Mardare and Prakash Panangaden.
This paper defines a duality for Markov processes extending Sikorski's duality for measurable spaces. The version above includes an appendix with the proofs omitted from the proceedings version.
2017 32nd Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). DOI: 10.1109/LICS.2017.8005152
Riesz Modal Logic for Markov Processes with Radu Mardare and Matteo Mio.
This paper makes a connection between Riesz spaces and probabilistic modal logics for Markov processes on compact Hausdorff spaces.
2017 32nd Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). DOI: 10.1109/LICS.2017.8005091

The Expectation Monad in Quantum Foundations with Bart Jacobs and Jorik Mandemaker, published in Information and Computation, 250, pages 87-114.
The original version of this paper was by Jacobs and Mandemaker only, in the proceedings of QPL 2011.

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 was published by Springer in the proceedings LNCS 8089, pages 141-157.
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".
Unordered Tuples in Quantum Computation with Bas Westerbaan, 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.

Quantum Entanglement and Algebraic Group Actions
My Master's thesis, supervised by Bob Coecke
Categorical Duality in Probability and Quantum Foundations
My PhD thesis, supervised by Bart Jacobs (includes corrections to the printed version)