承诺多态的平凡性
Triviality of promise polymorphisms
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中文总结 AI 辅助
本文研究承诺多态的平凡性，将泛代数中多态平凡性的相关结果，分别推广到含两个谓词的P、Q场景和P⊆Q的承诺场景。
中文摘要 AI 辅助
给定两个m元谓词P、Q，一个n元多态是函数组(f₁,…,fₘ)，使得当x⁽¹⁾,…,x⁽ⁿ⁾∈P时，有(f₁(y₁),…,fₘ(yₘ))∈Q，其中yᵢ=(x⁽¹⁾ᵢ,…,x⁽ᵐ⁾ᵢ)。这推广了泛代数中的常用定义，该定义中P=Q且f₁=…=fₘ。在早期工作中，我们研究单个谓词的所有多态何时是“平凡的”：要么全部依赖于一个公共坐标，要么构成该谓词的一个“证书”。我们表明，只需对2元多态甚至1元多态检查该条件，模一个明确的阻碍列表即可。本文中，我们将第一个结果推广到P、Q场景，针对一种放宽的证书概念；还将第二个结果推广到承诺场景，其中P、Q在同一字母表上取值且P⊆Q。
英文摘要
Given two $m$-ary predicates $P,Q$, an $n$-ary polymorphism is a tuple $(f_1,\dots,f_m)$ of functions such that $x^{(1)},\dots,x^{(n)} \in P$ implies $(f_1(y_1),\dots,f_m(y_m)) \in Q$, where $y_i = (x^{(1)}_i,\dots,x^{(m)}_i)$. This generalizes the usual definition in universal algebra, in which $P = Q$ and $f_1 = \cdots = f_m$. In earlier work, we studied when all polymorphisms of a single predicate are "trivial": either all depend on a single coordinate (common to all of them), or they constitute a "certificate" for the predicate. We showed that it suffices to check this condition for $2$-ary polymorphisms, and even for $1$-ary polymorphisms, modulo an explicit list of obstructions. In this paper we generalize the first result to the $P,Q$ setting, for a relaxed notion of certificate. We also generalize the second result in the promise setting, in which $P,Q$ range over the same alphabets and $P \subseteq Q$.
