arXiv subjects

Bram Westerbaan

Publications and source records attributed to Bram Westerbaan.

3 recordsLinked to original sources

arXiv2026

Quantum combinatorial games

A combinatorial game is a deterministic game with no hidden information played between two opponents such as tic-tac-toe, checkers or chess. In this paper we extend combinatorial games to the quantum setting, by first revisiting and reformulating existing theory of classical combinatorial games. We investigate in which case a quantum opponent has an advantage over a classical one. Surprisingly, our instantiation of Zermelo's classical theorem in the quantum setting shows that the effects of quantum mechanics do not convey an advantage against a classical player that plays a perfect classical strategy. In a more realistic scenario, when the classical player makes mistakes, we show how the quantum opponent can amplify the mistake to increase their chance of winning. Our theory has application beyond the mere playing of board games and can be used as a tool in finite deterministic adversarial models with perfect information.

cs.GT↗

arXiv2017

Statman's Hierarchy Theorem

In the Simply Typed $λ$-calculus Statman investigates the reducibility relation $\leq_{βη}$ between types: for $A,B \in \mathbb{T}^0$, types freely generated using $\rightarrow$ and a single ground type $0$, define $A \leq_{βη} B$ if there exists a $λ$-definable injection from the closed terms of type $A$ into those of type $B$. Unexpectedly, the induced partial order is the (linear) well-ordering (of order type) $ω+ 4$. In the proof a finer relation $\leq_{h}$ is used, where the above injection is required to be a Böhm transformation, and an (a posteriori) coarser relation $\leq_{h^+}$, requiring a finite family of Böhm transformations that is jointly injective. We present this result in a self-contained, syntactic, constructive and simplified manner. En route similar results for $\leq_h$ (order type $ω+ 5$) and $\leq_{h^+}$ (order type $8$) are obtained. Five of the equivalence classes of $\leq_{h^+}$ correspond to canonical term models of Statman, one to the trivial term model collapsing all elements of the same type, and one does not even form a model by the lack of closed terms of many types.

cs.LO↗

arXiv2015

Quotient-Comprehension Chains

Quotients and comprehension are fundamental mathematical constructions that can be described via adjunctions in categorical logic. This paper reveals that quotients and comprehension are related to measurement, not only in quantum logic, but also in probabilistic and classical logic. This relation is presented by a long series of examples, some of them easy, and some also highly non-trivial (esp. for von Neumann algebras). We have not yet identified a unifying theory. Nevertheless, the paper contributes towards such a theory by introducing the new quotient-and-comprehension perspective on measurement instruments, and by describing the examples on which such a theory should be built.

cs.LO↗