论文标题
扩展BELNAP 2：深入的双重类别
Expanding Belnap 2: the dual category in depth
论文作者
论文摘要
N.D. Belnap在1977年的一篇题为“计算机应该如何思考”的论文中引入了Bilattices，它为同时建模知识和真理提供了代数工具。优先考虑的默认双重双歧杆不仅包括Belnap的四个值，用于“ true”（$ t $），``false''（$ f $），``矛盾''（$ \ top $）和``no Information''（$ \ bot $），还包括同时建模知识和真理的默认值的索引家族。优先考虑的默认双拉在包括人工智能在内的许多领域都有应用。 在我们的同伴论文中，我们引入了一个新的优先级默认双性恋家庭，$ \ mathbf j_n $，$ n \ inω$，其中$ \ mathbf j_0 $是Belnap的开创性示例。我们给出了由$ \ mathbf j_n $生成的品种$ \ mathcal v_n $的二元性，而双重类别的对象$ \ mathcal x_n $是多组的拓扑结构。 在这里，我们深入研究双重类别。我们给出类别$ \ Mathcal x_n $的公理化，并表明它与单分类拓扑结构的类别$ \ Mathcal y_n $同构。 $ \ Mathcal y_n $的对象是Priestley空间，并具有连续回缩，其中该顺序具有自然排名。我们展示了如何通过$ \ Mathcal y_n $中的dual构建$ \ mathcal v_n $中基础有限的分布晶格的Priestley双重二元；作为一个应用程序，我们表明免费代数$ \ Mathbf f _ {\ Mathcal v_n}（1）$的大小由多项式$ n $ of of $ 6 $给出。
Bilattices, which provide an algebraic tool for simultaneously modelling knowledge and truth, were introduced by N.D. Belnap in a 1977 paper entitled 'How a computer should think'. Prioritised default bilattices include not only Belnap's four values, for `true' ($t$), `false'($f$), `contradiction' ($\top$) and `no information' ($\bot$), but also indexed families of default values for simultaneously modelling degrees of knowledge and truth. Prioritised default bilattices have applications in a number of areas including artificial intelligence. In our companion paper, we introduced a new family of prioritised default bilattices, $\mathbf J_n$, for $n \in ω$, with $\mathbf J_0$ being Belnap's seminal example. We gave a duality for the variety $\mathcal V_n$ generated by $\mathbf J_n$, with the objects of the dual category $\mathcal X_n$ being multi-sorted topological structures. Here we study the dual category in depth. We give an axiomatisation of the category $\mathcal X_n$ and show that it is isomorphic to a category $\mathcal Y_n$ of single-sorted topological structures. The objects of $\mathcal Y_n$ are Priestley spaces endowed with a continuous retraction in which the order has a natural ranking. We show how to construct the Priestley dual of the underlying bounded distributive lattice of an algebra in $\mathcal V_n$ via its dual in $\mathcal Y_n$; as an application we show that the size of the free algebra $\mathbf F_{\mathcal V_n}(1)$ is given by a polynomial in $n$ of degree $6$.
