### Abstract

Investigating the expressiveness of a diagrammatic logic provides insight into how its syntactic elements interact at the semantic level. Moreover, it allows for comparisons with other notations. Various expressiveness results for diagrammatic logics are known, such as the theorem that Shin's Venn-II system is equivalent to monadic first order logic. The techniques employed by Shin for Venn-II were adapted to allow the expressiveness of Euler diagrams to be investigated. We consider the expressiveness of spider diagrams of order (SDoO), which extend spider diagrams by including syntax that provides ordering information between elements. Fragments of SDoO are created by systematically removing each aspect of the syntax. We establish the relative expressiveness of the various fragments. In particular, one result establishes that spiders are syntactic sugar in any fragment that contains order, negation and shading. We also show that shading is syntactic sugar in any fragment containing negation and spiders. The existence of syntactic redundancy within the spider diagram of order logic is unsurprising however, we find it interesting that spiders or shading are redundant in fragments of the logic. Further expressiveness results are presented throughout the paper. The techniques we employ may well extend to related notations, such as the Euler/Venn logic of Swoboda et al. and Kent's constraint diagrams.

Original language | English |
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Title of host publication | Proceedings of the 6th international conference on diagrammatic representation and inference |

Place of Publication | Berlin Heidelberg |

Publisher | Springer-Verlag |

Pages | 69-83 |

Number of pages | 15 |

Volume | 6170 |

ISBN (Electronic) | 9783642146008 |

ISBN (Print) | 9783642145995 |

DOIs | |

Publication status | Published - 1 Jan 2010 |

Event | Proceedings of the 6th international conference on diagrammatic representation and inference - Portland, OR, USA, 9-11 August, 2010 Duration: 1 Jan 2010 → … |

### Publication series

Name | Lecture Notes in Computer Science |
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### Conference

Conference | Proceedings of the 6th international conference on diagrammatic representation and inference |
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Period | 1/01/10 → … |

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## Cite this

Delaney, A., Stapleton, G., Taylor, J., & Thompson, S. (2010). Fragments of spider diagrams of order and their relative expressiveness. In

*Proceedings of the 6th international conference on diagrammatic representation and inference*(Vol. 6170, pp. 69-83). (Lecture Notes in Computer Science). Springer-Verlag. https://doi.org/10.1007/978-3-642-14600-8_10