论文标题
高阶因果理论是BV-Logic的模型
Higher-order causal theories are models of BV-logic
论文作者
论文摘要
CAUS [ - ]构造采取了紧凑的基本过程的封闭类别，并产生了遵守某些信号传导/因果关系约束的高阶过程的 * - 自主类别，如所得类别中的类型系统所决定。本文研究了基本类别C满足其他属性的实例，该属性在CAUS [C]上产生了仿射线性结构，并且内部逻辑基本上更丰富。虽然原始结构仅提供乘法线性逻辑，但在这里我们还获得了添加剂和非共同的，自偶联的顺序产物，该产品产生了Guglielmi的BV逻辑模型。此外，我们将顺序产物的自然解释视为“ c可以信号对b，但反之亦然”，该解释位于非信号张量与完全信号（即无约束）par之间的预期。 C固定C的正数矩阵恢复了由Blute，Panangaden和Slavnov确定的概率相干空间的BV类别结构，仅限于归一化图。另一方面，固定完全正面地图的类别提供了一个全新的BV模型，该模型由高阶量子通道组成，其中包括量子和无限性因果结构研究中的最新工作。
The Caus[-] construction takes a compact closed category of basic processes and yields a *-autonomous category of higher-order processes obeying certain signalling/causality constraints, as dictated by the type system in the resulting category. This paper looks at instances where the base category C satisfies additional properties yielding an affine-linear structure on Caus[C] and a substantially richer internal logic. While the original construction only gave multiplicative linear logic, here we additionally obtain additives and a non-commutative, self-dual sequential product yielding a model of Guglielmi's BV logic. Furthermore, we obtain a natural interpretation for the sequential product as "A can signal to B, but not vice-versa", which sits as expected between the non-signalling tensor and the fully-signalling (i.e. unconstrained) par. Fixing matrices of positive numbers for C recovers the BV category structure of probabilistic coherence spaces identified by Blute, Panangaden, and Slavnov, restricted to normalised maps. On the other hand, fixing the category of completely positive maps gives an entirely new model of BV consisting of higher order quantum channels, encompassing recent work in the study of quantum and indefinite causal structures.
