### Abstract

This paper introduces a hierarchy of Euler and Venn diagrammatic reasoning systems in terms of their expressive powers in topological-relation-based formalization. At the bottom of the hierarchy is the Euler diagrammatic system introduced in Mineshima-Okada-Sato-Takemura [13, 12], which is expressive enough to characterize syllogistic reasoning in terms of unification and deletion rules. At the top of the hierarchy is a Venn diagrammatic system such as Swoboda-Allwein's Euler/Venn diagrammatic system [23]. In order to understand the hierarchy uniformly, we introduce an algebraic structure, which also provides another description of our unification rule of Euler diagrams. We prove that each system S' of the hierarchy is conservative over any lower system S with respect to validity-in the sense that S' is an extension of S, and the semantic consequence relations of S and S' are equivalent for diagrams of S. Furthermore, we prove that a region-based Venn diagrammatic system is conservative over our topological-relation-based Euler diagrammatic system with respect to provability.

Original language | English |
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Title of host publication | CEUR Workshop Proceedings |

Pages | 37-61 |

Number of pages | 25 |

Volume | 510 |

Publication status | Published - 2009 |

Event | 2nd International Workshop on Visual Languages and Logic, VLL 2009 - As Part of the 2009 IEEE Symposium on Visual Languages and Human Centric Computing, VL/HCC 2009 - Corvallis, OR, United States Duration: 2009 Sep 20 → 2009 Sep 20 |

### Other

Other | 2nd International Workshop on Visual Languages and Logic, VLL 2009 - As Part of the 2009 IEEE Symposium on Visual Languages and Human Centric Computing, VL/HCC 2009 |
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Country | United States |

City | Corvallis, OR |

Period | 09/9/20 → 09/9/20 |

### Fingerprint

### ASJC Scopus subject areas

- Computer Science(all)

### Cite this

*CEUR Workshop Proceedings*(Vol. 510, pp. 37-61)

**Conservativity for a hierarchy of Euler and Venn reasoning systems.** / Mineshima, Koji; Okada, Mitsuhiro; Takemura, Ryo.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*CEUR Workshop Proceedings.*vol. 510, pp. 37-61, 2nd International Workshop on Visual Languages and Logic, VLL 2009 - As Part of the 2009 IEEE Symposium on Visual Languages and Human Centric Computing, VL/HCC 2009, Corvallis, OR, United States, 09/9/20.

}

TY - GEN

T1 - Conservativity for a hierarchy of Euler and Venn reasoning systems

AU - Mineshima, Koji

AU - Okada, Mitsuhiro

AU - Takemura, Ryo

PY - 2009

Y1 - 2009

N2 - This paper introduces a hierarchy of Euler and Venn diagrammatic reasoning systems in terms of their expressive powers in topological-relation-based formalization. At the bottom of the hierarchy is the Euler diagrammatic system introduced in Mineshima-Okada-Sato-Takemura [13, 12], which is expressive enough to characterize syllogistic reasoning in terms of unification and deletion rules. At the top of the hierarchy is a Venn diagrammatic system such as Swoboda-Allwein's Euler/Venn diagrammatic system [23]. In order to understand the hierarchy uniformly, we introduce an algebraic structure, which also provides another description of our unification rule of Euler diagrams. We prove that each system S' of the hierarchy is conservative over any lower system S with respect to validity-in the sense that S' is an extension of S, and the semantic consequence relations of S and S' are equivalent for diagrams of S. Furthermore, we prove that a region-based Venn diagrammatic system is conservative over our topological-relation-based Euler diagrammatic system with respect to provability.

AB - This paper introduces a hierarchy of Euler and Venn diagrammatic reasoning systems in terms of their expressive powers in topological-relation-based formalization. At the bottom of the hierarchy is the Euler diagrammatic system introduced in Mineshima-Okada-Sato-Takemura [13, 12], which is expressive enough to characterize syllogistic reasoning in terms of unification and deletion rules. At the top of the hierarchy is a Venn diagrammatic system such as Swoboda-Allwein's Euler/Venn diagrammatic system [23]. In order to understand the hierarchy uniformly, we introduce an algebraic structure, which also provides another description of our unification rule of Euler diagrams. We prove that each system S' of the hierarchy is conservative over any lower system S with respect to validity-in the sense that S' is an extension of S, and the semantic consequence relations of S and S' are equivalent for diagrams of S. Furthermore, we prove that a region-based Venn diagrammatic system is conservative over our topological-relation-based Euler diagrammatic system with respect to provability.

UR - http://www.scopus.com/inward/record.url?scp=77955831897&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=77955831897&partnerID=8YFLogxK

M3 - Conference contribution

AN - SCOPUS:77955831897

VL - 510

SP - 37

EP - 61

BT - CEUR Workshop Proceedings

ER -