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Ruben Verborgh

Implicit quantification made explicit: How to interpret blank nodes and universal variables in Notation3 Logic

by Dörthe Arndt, Tom Schrijvers, Jos De Roo, and Ruben Verborgh

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 differ 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.

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Published in 2019 in Journal of Web Semantics.

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Cite this article easily using its BibTeX entry:

@article{arndt_jws_2019,
  title = {Implicit quantification made explicit: {How} to interpret blank nodes and universal variables in {Notation3 Logic}},
  author = {Arndt, D\"orthe and Schrijvers, Tom and De Roo, Jos and Verborgh, Ruben},
  journal = {Journal of Web Semantics},
  year = 2019,
  month = oct,
  volume = 58,
  pages = 100501,
  issn = {1570-8268},
  doi = {10.1016/j.websem.2019.04.001},
  url = {https://authors.elsevier.com/c/1ZWg55bAYUaMkH},
}

Alternatively, pick a reference of your choice below:

IEEE
D. Arndt, T. Schrijvers, J. De Roo, and R. Verborgh, “Implicit quantification made explicit: How to interpret blank nodes and universal variables in Notation3 Logic,” Journal of Web Semantics, vol. 58, p. 100501, Oct. 2019.
ACM
Dörthe Arndt, Tom Schrijvers, Jos De Roo, and Ruben Verborgh. 2019. Implicit quantification made explicit: How to interpret blank nodes and universal variables in Notation3 Logic. Journal of Web Semantics 58, (October 2019), 100501.
LNCS
Arndt, D., Schrijvers, T., De Roo, J., Verborgh, R.: Implicit quantification made explicit: How to interpret blank nodes and universal variables in Notation3 Logic. Journal of Web Semantics. 58, 100501 (2019).
APA
Arndt, D., Schrijvers, T., De Roo, J., & Verborgh, R. (2019). Implicit quantification made explicit: How to interpret blank nodes and universal variables in Notation3 Logic. Journal of Web Semantics, 58, 100501.
MLA
Arndt, Dörthe, et al. “Implicit Quantification Made Explicit: How to Interpret Blank Nodes and Universal Variables in Notation3 Logic.” Journal of Web Semantics, vol. 58, Oct. 2019, p. 100501.

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