The Expressiveness of Spider Diagrams

Stapleton, Gem; Howse, John; Taylor, John; Thompson, Simon

doi:10.1093/logcom/14.6.857

Cited by 34 publications

(35 citation statements)

References 14 publications

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“…The language of spider diagrams is also accompanied by inference rules, which results in the logic of spider diagrams. This logic is expressively equivalent to monadic first-order logic with equality (MFOLE) [23]. We also developed a new set of sound inference rules, which represent an extensions of the system in [12] and prove them to be complete -for details, see Section 3.…”

Section: Spider Diagramsmentioning

confidence: 99%

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Speedith: A Reasoner for Spider Diagrams

Urbas

Jamnik

2015

J of Log Lang and Inf

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In this paper, we introduce Speedith which is an interactive diagrammatic theorem prover for the well-known language of spider diagrams. Speedith provides a way to input spider diagrams, transform them via the diagrammatic inference rules, and prove diagrammatic theorems. Speedith's inference rules are sound and complete, extending previous research by including all the classical logic connectives. In addition to being a stand-alone proof system, Speedith is also designed as a program that plugs into existing general purpose theorem provers. This allows for other systems to access diagrammatic reasoning via Speedith, as well as a formal verification of diagrammatic proof steps within standard sentential proof assistants. We describe the general structure of Speedith, the diagrammatic language, the automatic mechanism that draws the diagrams when inference rules are applied on them, and how formal diagrammatic proofs are constructed.

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Section: Spider Diagramsmentioning

confidence: 99%

“…Note that the connectives ←→, −→, and ¬ were excluded from the sound and complete spider diagram logic studied by Stapleton et al [23]. In Fig.…”

Section: Syntaxmentioning

confidence: 99%

Speedith: A Reasoner for Spider Diagrams

Urbas

Jamnik

2015

J of Log Lang and Inf

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“…It has also been shown that spider diagrams are equivalent to MFOL with equality [10]; MFOL [=] extends MFOL by including =, allowing one to assert the distinctness of elements. To establish the expressiveness of spider diagrams, a different approach to that of Shin's for Venn-II was utilized.…”

Section: Introductionmentioning

confidence: 99%

Fragments of Spider Diagrams of Order and Their Relative Expressiveness

Delaney

Taylor

Thompson

2010

Diagrammatic Representation and Inference

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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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“…From [16] we know that spider diagrams without order are expressively equivalent to monadic first order with equality, denoted M F oL [=]. Our extension to spider diagrams adds an order relation to the semantic models.…”

Section: Comparing Classes Of Regular Languages and Spider Diagramsmentioning

confidence: 99%

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

Delaney

Taylor

Thompson

Diagrammatic Representation and Inference

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Abstract. The spider diagram logic forms a fragment of constraint diagram logic and is designed to be primarily used as a diagrammatic software specification tool. Our interest is in using the logical basis of spider diagrams and the existing known equivalences between certain logics, formal language theory classes and some automata to inform the development of diagrammatic logic. Such developments could have many advantages, one of which would be aiding software engineers who are familiar with formal languages and automata to more intuitively understand diagrammatic logics. In this paper we consider relationships between spider diagrams of order (an extension of spider diagrams) and the star-free subset of regular languages. We extend the concept of the language of a spider diagram to encompass languages over arbitrary alphabets. Furthermore, the product of spider diagrams is introduced. This operator is the diagrammatic analogue of language concatenation. We establish that star-free languages are definable by spider diagrams of order equipped with the product operator and, based on this relationship, spider diagrams of order are as expressive as first order monadic logic of order.

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The Expressiveness of Spider Diagrams

Abstract: The version in the Kent Academic Repository may differ from the final published version. Users are advised to check http://kar.kent.ac.uk for the status of the paper. Users should always cite the published version of record.

Cited by 34 publications

References 14 publications

Speedith: A Reasoner for Spider Diagrams

Speedith: A Reasoner for Spider Diagrams

Fragments of Spider Diagrams of Order and Their Relative Expressiveness

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

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