TY - GEN

T1 - Two types of diagrammatic inference systems

T2 - 6th International Conference on the Theory and Application of Diagrams, Diagrams 2010

AU - Mineshima, Koji

AU - Okada, Mitsuhiro

AU - Takemura, Ryo

N1 - Copyright:
Copyright 2010 Elsevier B.V., All rights reserved.

PY - 2010

Y1 - 2010

N2 - 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.

AB - 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.

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U2 - 10.1007/978-3-642-14600-8_12

DO - 10.1007/978-3-642-14600-8_12

M3 - Conference contribution

AN - SCOPUS:77955806284

SN - 364214599X

SN - 9783642145995

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 99

EP - 114

BT - Diagrammatic Representation and Inference - 6th International Conference, Diagrams 2010, Proceedings

Y2 - 9 August 2010 through 11 August 2010

ER -