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Proximity-based unification theory

机译：基于邻近的统一理论

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Similarity-based Logic Programming has been proposed to enhance the Logic Programming paradigm with similarity relations, in order to represent and manage vague or imprecise information. A similarity relation is a reflexive, symmetric, transitive fuzzy binary relation, extending the standard notion of equivalence relation with the purpose of weakening the concept of equality. As is reported, similarity relations have significant limitations, due to the transitivity restriction, that can lead to representing fuzzy information incorrectly in some situations. Recently we have proposed the use of proximity relations (i.e., reflexive, symmetric, fuzzy binary relations) in the context of a logic programming system, called Bousi~Prolog, as a way of solving this problem and generalizing later approaches exclusively based on similarity relations. However, a naive combination of proximity relations and the existing unification algorithms may cause incompleteness problems. Hence, in this paper, we introduce an accurate definition of proximity between expressions (terms or atomic formulas) and a new unification algorithm able to manage proximity relations properly. The so-called weak unification algorithm is an extension of Martelli and Montanari's unification algorithm supported by the new notion of proximity. As major relevant properties, we prove that the weak unification algorithm terminates and it is sound (i.e., it produces a weak unifier of two expressions, if they are unifiable) and complete (i.e., it is able to compute a weak most general unifier for two unifiable expressions).
机译：为了表示和管理模糊或不精确的信息，已经提出了基于相似度的逻辑编程来增强具有相似关系的逻辑编程范式。相似关系是自反，对称，可传递的模糊二元关系，它扩展了等价关系的标准概念，目的是削弱等价概念。如所报道的，由于可传递性限制，相似关系具有显着的限制，在某些情况下可能导致错误地表示模糊信息。最近，我们提出在称为Bousi〜Prolog的逻辑编程系统中使用邻近关系（即自反，对称，模糊二进制关系），作为解决此问题的方法，并且仅基于相似关系来推广以后的方法。但是，接近关系与现有统一算法的简单组合可能会导致不完整问题。因此，在本文中，我们介绍了表达式（术语或原子公式）之间的接近度的准确定义以及一种能够正确管理接近度关系的新统一算法。所谓的弱统一算法是Martelli和Montanari统一算法的扩展，并由新的邻近概念支持。作为主要的相关属性，我们证明了弱统一算法会终止并且是健全的（即，如果两个表达式都是不可统一的，它会产生两个表达式的弱统一体）和完整的（即，它能够为两个无法确定的表达式）。

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