A normal form for spider diagrams of order

Delaney, Aidan; Taylor, John; Thompson, Simon

doi:10.1016/j.jvlc.2014.10.029

Cited by 2 publications

(7 citation statements)

References 14 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Our interest is in the relationship between an extension of spider diagrams called spider diagrams of order and regular languages. This paper builds on our previous work [4,5] and provides a proof that star-free regular languages are definable in spider diagrams of order, when augmented with a product operator. Star-free languages may be described by regular expressions without the use of the Kleene star, a fact from which the name of the language class derives [13].…”

Section: Introductionmentioning

confidence: 95%

“…This section provides an overview of the syntax and semantics of spider diagrams of order, originally presented in [5] which in turn extends [11].…”

Section: Syntax and Semantics Of Spider Diagrams Of Ordermentioning

confidence: 99%

“…The use of this function allows us to consider the language of a diagram over an arbitrary alphabet. Previous work [5] considered a much more restricted set of alphabets. In the examples in figure 2 we assume C = {P, Q} and we assign the alphabet Σ = {a, b, c, d} via a function called lettermap in the manner depicted in figure 2(a) i.e.…”

Section: The Language Of a Spider Diagram Of Ordermentioning

confidence: 99%

“…We show, in this paper, that the logic of spider diagram of order describes which correspond to well-known subsets of star-free languages. We have previously shown that spider diagrams (without order) describe (sub)sets of regular languages that are incomparable with well-known hierarchies such as the Straubin-Thérin or dot-depth hierarchies [5]. Conversely regular languages have helped to inform the development of spider diagrams.…”

Section: Introductionmentioning

confidence: 99%

“…In section 3 we define the language of a spider diagram of order, generalising work in [5]. The product of spider diagrams is introduced in section 4.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

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

Delaney

Taylor

Thompson

Diagrammatic Representation and Inference

Self Cite

View full text Add to dashboard Cite

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.

show abstract

Section: Introductionmentioning

confidence: 95%

“…This section provides an overview of the syntax and semantics of spider diagrams of order, originally presented in [5] which in turn extends [11].…”

Section: Syntax and Semantics Of Spider Diagrams Of Ordermentioning

confidence: 99%

Section: The Language Of a Spider Diagram Of Ordermentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…In section 3 we define the language of a spider diagram of order, generalising work in [5]. The product of spider diagrams is introduced in section 4.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

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

Delaney

Taylor

Thompson

Diagrammatic Representation and Inference

Self Cite

View full text Add to dashboard Cite

show abstract

On the expressiveness of spider diagrams and commutative star-free regular languages

Delaney

Taylor

Thompson

2013

Journal of Visual Languages & Computing

Self Cite

View full text Add to dashboard Cite

Spider diagrams provide a visual logic to express relations between sets and their elements, extending the expressiveness of Venn diagrams. Sound and complete inference systems for spider diagrams have been developed and it is known that they are equivalent in expressive power to Monadic First-Order Logic with equality, MFOL[=]. In this paper, we further characterize their expressiveness by articulating a link between them and formal languages. First, we establish that spider diagrams define precisely the languages that are finite unions of languages of the form K⊔⊔Γ⁎ where K is a finite commutative language and Γ is a finite set of letters. We note that it was previously established that spider diagrams define commutative star-free languages. As a corollary K⊔⊔Γ⁎ all languages of the form are commutative star-free languages. We further demonstrate that every commutative star-free language is also such a finite union. In summary, we establish that spider diagrams define precisely: (a) languages definable in MFOL[=], (b) the commutative star-free regular languages, and (c) finite unions of the form K⊔⊔Γ⁎ as just described

show abstract

A normal form for spider diagrams of order

Cited by 2 publications

References 14 publications

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

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

On the expressiveness of spider diagrams and commutative star-free regular languages

Contact Info

Product

Resources

About