Implicit Quantification Made Explicit: How to Interpret Blank Nodes and Universal Variables in Notation3 Logic

Abstract

Since the invention of Notation3 Logic, several years have passed in which the theory has been refined and applied in different reasoning engines like Cwm, EYE, and FuXi. But despite these developments, a clear formal definition of Notation3’s semantics is still missing. This does not only form an obstacle for the formal investigation of that logic and its relations to other formalisms, it has also practical consequences: in many cases the interpretations of the same formula di er between reasoning engines. In this paper we tackle one of the main sources of that problem, namely the uncertainty about implicit quantification. This refers to Notation3’s ability to use bound variables for which the universal or existential quantifiers are not explicitly stated, but implicitly assumed. We provide a tool for clarification through the definition of a core logic for Notation3 that only supports explicit quantification. We specify an attribute grammar which maps Notation3 formulas to that logic according to the different interpretations and thereby define the semantics of Notation3. This grammar is then implemented and used to test the impact of the differences between interpretations on practical cases. Our dataset includes Notation3 implementations from former research projects and test cases developed for the reasoner EYE. We find that 31% of these files are understood differently by different reasoners. We further analyse these cases and categorize them in different classes of which we consider one most harmful: if a file is manually written by a user and no specific built-in predicates are used (13% of our critical files), it is unlikely that this user is aware of possible differences. We therefore argue the need to come to an agreement on implicit quantification, and discuss the different possibilities.

Keywords: N3, Scoping, RDF, Attribute Grammar, Formal Semantics