## On Diagram Tokens and Types (2002)

Venue: | In: Proceedings of Diagrams 2002, LNAI 2317 |

Citations: | 19 - 5 self |

### BibTeX

@INPROCEEDINGS{Howse02ondiagram,

author = {John Howse and O Molina and Sun-joo Shin and John Taylor},

title = {On Diagram Tokens and Types},

booktitle = {In: Proceedings of Diagrams 2002, LNAI 2317},

year = {2002},

pages = {146--160},

publisher = {Springer-Verlag}

}

### OpenURL

### Abstract

Abstract. Rejecting the temptation to make up a list of necessary and sufficient conditions for diagrammatic and sentential systems, we present an important distinction which arises from sentential and diagrammatic features of systems. Importantly, the distinction we will explore in the paper lies at a meta-level. That is, we argue for a major difference in metatheory between diagrammatic and sentential systems, by showing the necessity of a more fine-grained syntax for a diagrammatic system than for a sentential system. Unlike with sentential systems, a diagrammatic system requires two levels of syntax—token and type. Token-syntax is about particular diagrams instantiated on some physical medium, and type-syntax provides a formal definition with which a concrete representation of a diagram must comply. While these two levels of syntax are closely related, the domains of type-syntax and token-syntax are distinct from each other. Euler diagrams are chosen as a case study to illustrate the following major points of the paper: (i) What kinds of diagrammatic features (as opposed to sentential features) require two different levels of syntax? (ii) What is the relation between these two levels of syntax? (iii) What is the advantage of having a two-tiered syntax? 1

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Citation Context ...ntersection, respectively. In thisspaper we have added shading to the notation to increase its expressiveness. Euler diagrams form the basis of more expressive diagrammatic notations such as Higraphs =-=[7]-=-, constraint diagrams [3] and some of the notations of UML [9]. This paper is mainly concerned with diagrammatic notations based on Euler diagrams; this is a substantial and important collection in it... |

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Citation Context ...the syntax is not fine-grained enough, then we would have to accept a syntax that does not reflect non-trivial visual features of a system. We revive a time-honored distinction between type and token =-=[4, 10]-=- and suggest a two-tiered syntax for diagrammatic systems. Hence, the diagrams in Fig. 1 are different tokens, but of the same type. But, in the case of diagrams, differences in tokens are not negligi... |

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Citation Context ...of regions, and ˆ R∗ = 2 ˆ Z ∗ be the set of shaded regions. 3. ˆ L is the set of contour labels. The bijection ˆ ℓ : Ĉ ∪ { ˆ β} → ˆ L returns the label of a contour or the boundary rectangle. Hammer =-=[6]-=- defines an Euler diagram as ‘any finite number of closed curves drawn on the page in any arrangement’. This is, of course, a very liberal definition. We have chosen to ban some possible Euler diagram... |

29 |
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Citation Context ...system simply allows the manipulation between these two tokens. When a type-token distinction is implemented in syntax, the syntactic operation between these two diagrams comes almost free. Shimojima =-=[12]-=- uses term ‘free ride’ to refer to an inference in which the conclusion seems to be read off almost automatically from the representation of premises. For further discussion, see [5]. Consider again t... |

24 |
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Citation Context ...another. Visual differences in these three tokens are hard to ignore, but at the same time we know that these three diagrams are of the same kind in an important sense. In the case of the Venn system =-=[13]-=-, the location of contours does not have a representing import. Therefore, we would like to say that these diagrams are the “same” in some sense. On the other hand, we do not want say that the Euler d... |

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Citation Context ...ed by Scotto [11] as a flaw in Shin’s Venn I system [13]. His solution involves combining the original diagram with the diagram composed just of a boundary contour. From a construction proved by More =-=[8]-=-, we can construct a Venn diagram with at least one contour whose erasure would result in a well-formed diagram. Scotto’s solution rearranges the contours so that the contour to be removed is this wel... |

12 | Towards a formalization of constraint diagrams
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Citation Context .... In thisspaper we have added shading to the notation to increase its expressiveness. Euler diagrams form the basis of more expressive diagrammatic notations such as Higraphs [7], constraint diagrams =-=[3]-=- and some of the notations of UML [9]. This paper is mainly concerned with diagrammatic notations based on Euler diagrams; this is a substantial and important collection in its own right as the above ... |

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Citation Context ... free. Shimojima [12] uses term ‘free ride’ to refer to an inference in which the conclusion seems to be read off almost automatically from the representation of premises. For further discussion, see =-=[5]-=-. Consider again the diagrams in Fig. 6. Type-equivalence in a syntactic level gives us a free pass for a commutation between these two diagrams without bringing in transformation rules (except the tr... |

9 |
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Citation Context ... more serious problem can occur. In Fig. 9 removing a contour results in a non-well-formed diagram because we get disconnected zones in the resulting diagram. This problem was first noticed by Scotto =-=[11]-=- as a flaw in Shin’s Venn I system [13]. His solution involves combining the original diagram with the diagram composed just of a boundary contour. From a construction proved by More [8], we can const... |

5 |
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Citation Context ...tax. After presenting a case study of the two-level syntax of the Euler system in the next section, we will return to this issue in §4. 3 Type-syntax and token-syntax of Euler diagrams Euler diagrams =-=[1]-=- illustrate relations between sets. This notation uses topological properties of enclosure, exclusion and intersection to represent the settheoretic notions of subset, disjoint sets, and intersection,... |

3 |
specification, version 1.3. Available from www.omg.org
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Citation Context ... to the notation to increase its expressiveness. Euler diagrams form the basis of more expressive diagrammatic notations such as Higraphs [7], constraint diagrams [3] and some of the notations of UML =-=[9]-=-. This paper is mainly concerned with diagrammatic notations based on Euler diagrams; this is a substantial and important collection in its own right as the above list indicates. We now give a concise... |

1 |
Generating Euler Diagrams. Accepted for Diagrams 2002
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Citation Context ...opment process. In order for this to happen we need to be able to develop and implement an algorithm that takes a diagram-type and instantiates it as a diagram-token. Work is already underway on this =-=[2]-=-. Acknowledgements We would like to thank Jean Flower and the anonymous referees for their very helpful comments. Authors Howse and Taylor were partially supported by UK EPSRC grant GR/R63516. Referen... |