School of Computing

# Fragments of Spider Diagrams of Order and their Relative Expressiveness

## Aidan Delaney, Gem Stapleton, John Taylor, and Simon Thompson

In Ashok K. Goel, Mateja Jamnik, and N. Hari Narayanan, editors, Diagrammatic Representation and Inference 6th International Conference, Diagrams 2010, volume 6170 of Lecture Notes in Computer Science, pages 182-196. Springer, August 2010 [doi].

### 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.

### Bibtex Record

```@inproceedings{3034,
author = {Aidan Delaney and Gem Stapleton and John Taylor and Simon Thompson},
title = {{Fragments of Spider Diagrams of Order and their Relative Expressiveness}},
month = {August},
year = {2010},
pages = {182-196},
keywords = {determinacy analysis, Craig interpolants},
note = {},
doi = {10.1007/978-3-642-14600-8},
url = {http://www.cs.kent.ac.uk/pubs/2010/3034},
publication_type = {inproceedings},
submission_id = {2982_1281951859},
booktitle = {Diagrammatic Representation and Inference 6th International Conference, Diagrams 2010},
editor = {Ashok K. Goel and Mateja Jamnik and N. Hari Narayanan},
volume = {6170},
series = {Lecture Notes in Computer Science},
publisher = {Springer},
refereed = {yes},
}
```

School of Computing, University of Kent, Canterbury, Kent, CT2 7NF