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

• Preprint: Some No-Go Results in Quantum Domain Theory
  In the first part we show that superoperators between finite-dimensional C*-algebras cannot be approximated from a countable set by using directed suprema. Then we show that the unit interval of a C*-algebra is a continuous dcpo iff the C*-algebra is a product of finite-dimensional matrix algebras (i.e. it is hereditarily atomic in the sense of Kornell). Supersedes Continuous Dcpos in Quantum Computing.
  Slides from the talk at QPL 2022.
• Preprint: A Probability Monad on Measure Spaces
  We show that the category of commutative W*-algebras with positive unital normal maps is dual to the Kleisli category of a monad on the category of (compact complete strictly localizable) measure spaces. This is based on the probabilistic Gelfand duality between commutative unital C*-algebras with positive unital maps and the Kleisli category of the Radon monad on compact Hausdorff spaces.
  Slides from the talk at ACT 2022.
  Slides from the invited talk at SYCO 9

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. One these required an update to my thesis.
• 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

• Conference Paper: 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

2017

• Conference Paper: Infinite-Dimensionality in Quantum Foundations: W*-algebras as Presheaves over Matrix Algebras with Mathys Rennela and Sam Staton, QPL 2016, published in EPTCS 236, 2017, pp. 161-173.
  This paper relates W*-algebras to presheaves on the category of finite-dimensional matrix algebras with completely positive maps.
• Conference Paper: Unrestricted Stone Duality for Markov Processes with Dexter Kozen, Kim Larsen, Radu Mardare and Prakash Panangaden, LICS 2017
  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
• Conference Paper: Riesz Modal Logic for Markov Processes with Radu Mardare and Matteo Mio, LICS 2017
  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

2016

• Journal Paper: 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.

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

• 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)
  
• Categorical Duality in Probability and Quantum Foundations (Second Version)
  An updated version of my PhD thesis, where I removed a local convexity assumption in the main text needed for an example of a base-norm space, and added some new appendices, including one on the expectation monad in the absence of the axiom of choice.
• The submitted version of my PhD thesis (2016)
  This contains the original chapter 5 on the duality between commutative W*-algebras and compact complete strictly localizable measure spaces. Be warned that I have not altered it in any way, such as to fix any typos and minor errors or change the terminology to that which I now prefer.
  At the time I wrote it, I did have a proof that every nullset-reflecting map from a compact localizable measure space to a strictly localizable measure space is normal (in the sense of Section 5.5) if the size of the decomposition of the strictly localizable space was smaller than the first measurable cardinal, i.e. I had a proof that eliminated the problem of atomlessly measurable cardinals in Counterexample 5.5.8, but not atomic ones. I only found out about Fremlin's 451Q (c), which implies every nullset-reflecting map from a compact localizable measure space to a strictly localizable measure space is normal, between when I submitted my thesis and when I resubmitted it without Chapter 5.
  I did not include my proof of the atomlessly-measurable case in my original thesis submission above in a futile attempt not to annoy Klaas Landsman.