Spider Diagrams of Order and a Hierarchy of Star-Free Regular Languages

Delaney, Aidan; Taylor, John; Thompson, Simon

doi:10.1007/978-3-540-87730-1_18

“…Research on the language was carried out mainly (but not only) at the Visual Modelling Group (VMG) at the University of Brighton, to which the authors belong. Other logics based on spider diagrams are spider diagrams of order [11], constraint diagrams [28,43,14], generalized constraint diagrams [41] and concept diagrams [42].…”

Section: History and Overviewmentioning

confidence: 99%

The Semiotics of Spider Diagrams

Burton

¹

,

Howse

²

2017

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Abstract. Spider diagrams are based on Euler and Venn/Peirce diagrams, forming a system which is as expressive as monadic first order logic with equality. Rather than being primarily intended for logicians, spider diagrams were developed at the end of the 1990s in the context of visual modelling and software specification. We examine the original goals of the designers, the ways in which the notation has evolved and its connection with the philosophical origins of the logical diagrams of Euler, Venn and Peirce on which spider diagrams are based. Using Peirce's concepts and classification of signs, we analyse the ways in which different sign types are exploited in the notation. Our hope is that this analysis may be of interest beyond those readers particularly interested in spider diagrams, and act as a case study in deconstructing a simple visual logic. Along the way, we discuss the need for a deeper semiotic engagement in visual modelling.

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“…al. [11]). An interpretation is a triple I = (U, < , Ψ ) where U is called the universal set and Ψ : C → PU is a function that assigns a subset of U to each contour label and < is a strict total order on U .…”

Section: Spider Diagrams Of Ordermentioning

confidence: 99%

“…al. [11]). Spider diagrams of order, D 1 and D 2 , are semantically equivalent provided they have exactly the same models.…”

Section: Spider Diagrams Of Ordermentioning

confidence: 99%

Diagrammatic Representation and Inference

2010

Lecture Notes in Computer Science

4

0

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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.

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“…The proof strategy involved a model theoretic analysis of the closure properties of the model sets for the formulas of the language. In the case of spider diagrams of order, socalled becasue they provide ordering constraints on elements, it has been shown that they are equivalent to MFOL of Order [11]; MFOL[<] extends MFOL by including <, which is interpreted as a strict total order. MFOL [<] is strictly more expressive than MFOL[=] which, in turn, is strictly more expressive than MFOL.…”

Section: Introductionmentioning

confidence: 99%

Fragments of Spider Diagrams of Order and Their Relative Expressiveness

Delaney

¹

,

Taylor

²

,

Thompson

³

2010

Diagrammatic Representation and Inference

Self Cite

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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.

show abstract

Spider Diagrams of Order and a Hierarchy of Star-Free Regular Languages

Cited by 9 publications

References 13 publications

The Semiotics of Spider Diagrams

The Semiotics of Spider Diagrams

Diagrammatic Representation and Inference

Fragments of Spider Diagrams of Order and Their Relative Expressiveness

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