In this paper we present the probabilistic typed natural deduction calculus TPTND, designed to reason about and derive trustworthiness properties of probabilistic computational processes, like those underlying current AI applications. Derivability in TPTND is interpreted as the process of extracting $n$ samples of possibly complex outputs with a certain frequency from a given categorical distribution. We formalize trust for such outputs as a form of hypothesis testing on the distance between such frequency and the intended probability. The main advantage of the calculus is to render such notion of trustworthiness checkable. We present a computational semantics for the terms over which we reason and then the semantics of TPTND, where logical operators as well as a Trust operator are defined through introduction and elimination rules. We illustrate structural and metatheoretical properties, with particular focus on the ability to establish under which term evolutions and logical rules applications the notion of trustworhtiness can be preserved.
翻译:检查一个有类型的自然演绎系统中概率计算的可信度
研究摘要:
本文介绍了概率类型自然演绎推理系统(TPTND),旨在推导和证明当前人工智能应用背后的概率计算过程的可信度属性。在TPTND中,可推导性被解释为从给定的分类分布中抽取n个可能复杂的输出样本,并达到一定的频率。我们将这种输出的可信性形式化为基于频率与既定概率之间距离的一种假设测试形式。这个推理系统的主要优势在于可以使这种可信度概念成为可检查的。我们提出了推理对象的计算语义,然后定义了逻辑运算符以及信任运算符的引入和消解规则。我们阐述了结构和元理论性质,特别重点是关于能够建立在哪些术语演变和逻辑规则应用之下,可信度概念能够被保留的能力。