Two types of diagrammatic inference systems: Natural deduction style and resolution style

Koji Mineshima, Mitsuhiro Okada, Ryo Takemura

Research output: Chapter in Book/Report/Conference proceedingConference contribution

8 Citations (Scopus)

Abstract

Since the 1990s, reasoning with Venn and Euler diagrams has been studied from mathematical and logical viewpoints. The standard approach to a formalization is a "region-based" approach, where a diagram is defined as a set of regions. An alternative is a "relation-based" approach, where a diagram is defined in terms of topological relations (inclusion and exclusion) between circles and points. We compare these two approaches from a proof-theoretical point of view. In general, diagrams correspond to formulas in symbolic logic, and diagram manipulations correspond to applications of inference rules in a certain logical system. From this perspective, we demonstrate the following correspondences. On the one hand, a diagram construed as a set of regions corresponds to a disjunctive normal form formula and the inference system based on such diagrams corresponds to a resolution calculus. On the other hand, a diagram construed as a set of topological relations corresponds to an implicational formula and the inference system based on such diagrams corresponds to a natural deduction system. Based on these correspondences, we discuss advantages and disadvantages of each framework.

Original languageEnglish
Title of host publicationDiagrammatic Representation and Inference - 6th International Conference, Diagrams 2010, Proceedings
Pages99-114
Number of pages16
DOIs
Publication statusPublished - 2010
Event6th International Conference on the Theory and Application of Diagrams, Diagrams 2010 - Portland, OR, United States
Duration: 2010 Aug 92010 Aug 11

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume6170 LNAI
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349

Other

Other6th International Conference on the Theory and Application of Diagrams, Diagrams 2010
CountryUnited States
CityPortland, OR
Period10/8/910/8/11

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Computer Science(all)

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    Mineshima, K., Okada, M., & Takemura, R. (2010). Two types of diagrammatic inference systems: Natural deduction style and resolution style. In Diagrammatic Representation and Inference - 6th International Conference, Diagrams 2010, Proceedings (pp. 99-114). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 6170 LNAI). https://doi.org/10.1007/978-3-642-14600-8_12