表达逻辑中句子的概率
Probabilities on Sentences in an Expressive Logic
摘要：
关于不确定性知识的自动推理有着广泛的应用。开发这种系统的一个困难是缺乏逻辑和概率的完全令人满意的整合。我们直接解决这个问题。高阶逻辑等表达性语言非常适合表示和推理结构化知识。不确定性知识可以通过使用分级概率而不是二进制真值来建模。本文研究的主要技术问题是：给定一组句子，每个句子都有一定的概率为真，其他句子的概率应该是多少？一个自然的愿望清单，除其他外，是概率分布（i）与知识库一致，（ii）允许一致的推理过程，特别是（iii）在概率为0和1的限制下简化为演绎逻辑，（四）允许（贝叶斯）归纳推理和（v）学习的限制，特别是（vi）允许确认普遍量化的假设/句子。我们将这个愿望清单转化为先验概率的技术要求，并表明满足我们所有标准的概率存在。我们还给出了明确的建设和几个一般的概率特征，满足一些或所有的标准和各种（反）的例子。我们还推导出必要和充分的条件，扩展信念的许多句子，以合适的概率在所有的句子，特别是最少的教条主义或偏见。最后，我们如何开发的理论可能会被使用和近似的自主推理代理的简要展望。我们的理论是一个全球一致的和经验上令人满意的统一概率和逻辑的一步。
英文摘要：
Automated reasoning about uncertain knowledge has many applications. One difficulty when developing such systems is the lack of a completely satisfactory integration of logic and probability. We address this problem directly. Expressive languages like higher-order logic are ideally suited for representing and reasoning about structured knowledge. Uncertain knowledge can be modeled by using graded probabilities rather than binary truth-values. The main technical problem studied in this paper is the following: Given a set of sentences, each having some probability of being true, what probability should be ascribed to other (query) sentences? A natural wish-list, among others, is that the probability distribution (i) is consistent with the knowledge base, (ii) allows for a consistent inference procedure and in particular (iii) reduces to deductive logic in the limit of probabilities being 0 and 1, (iv) allows (Bayesian) inductive reasoning and (v) learning in the limit and in particular (vi) allows confirmation of universally quantified hypotheses/sentences. We translate this wish-list into technical requirements for a prior probability and show that probabilities satisfying all our criteria exist. We also give explicit constructions and several general characterizations of probabilities that satisfy some or all of the criteria and various (counter) examples. We also derive necessary and sufficient conditions for extending beliefs about finitely many sentences to suitable probabilities over all sentences, and in particular least dogmatic or least biased ones. We conclude with a brief outlook on how the developed theory might be used and approximated in autonomous reasoning agents. Our theory is a step towards a globally consistent and empirically satisfactory unification of probability and logic.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
