549 research outputs found
PaL Diagrams: A Linear Diagram-Based Visual Language
Linear diagrams have recently been shown to be
more effective than Euler diagrams when used
for set-based reasoning. However, unlike the
growing corpus of knowledge about formal aspects
of Euler and Venn diagrams, there has been no
formalisation of linear diagrams. To fill this
knowledge gap, we present and formalise Point
and Line (PaL) diagrams, an extension of simple
linear diagrams containing points, thus providing
a formal foundation for an effective visual
language.We prove that PaL diagrams are exactly
as expressive as monadic first-order logic with
equality, gaining, as a corollary, an equivalence
with the Euler diagram extension called spider
diagrams. The method of proof provides translations
between PaL diagrams and sentences of monadic
first-order logic
On the Completeness of Spider Diagrams Augmented with Constants
Diagrammatic reasoning can be described formally by a number of diagrammatic logics; spider diagrams are one of these, and are used for expressing logical statements about set membership and containment. Here, existing work on spider diagrams is extended to include constant spiders that represent specific individuals. We give a formal syntax and semantics for the extended diagram language before introducing a collection of reasoning rules encapsulating logical equivalence and logical consequence. We prove that the resulting logic is sound, complete and decidable
A Normal Form for Spider Diagrams of Order
We develop a reasoning system for an Euler diagram based visual logic, called spider diagrams of order. We de- fine a normal form for spider diagrams of order and provide an algorithm, based on the reasoning system, for producing diagrams in our normal form. Normal forms for visual log- ics have been shown to assist in proving completeness of associated reasoning systems. We wish to use the reasoning system to allow future direct comparison of spider diagrams of order and linear temporal logic
Evaluation of the usability of constraint diagrams as a visual modelling language: theoretical and empirical investigations
This research evaluates the constraint diagrams (CD) notation, which is a formal representation for program specification that has some promise to be used by people who are not expert in software design. Multiple methods were adopted in order to provide triangulated evidence of the potential benefits of constraint diagrams compared with other notational systems. Three main approaches were adopted for this research.
The first approach was a semantic and task analysis of the CD notation. This was conducted by the application of the Cognitive Dimensions framework, which was used to examine the relative strengths and weaknesses of constraint diagrams and conventional notations in terms of the perceptive facilitation or impediments of these different representations. From this systematic analysis, we found that CD cognitively reduced the cost of exploratory design, modification, incrementation, searching, and transcription activities with regard to the cognitive dimensions: consistency, visibility, abstraction, closeness of mapping, secondary notation, premature commitment, role-expressiveness, progressive evaluation, diffuseness, provisionality, hidden dependency, viscosity, hard mental operations, and error-proneness.
The second approach was an empirical evaluation of the comprehension of CD compared to natural language (NL) with computer science students. This experiment took the form of a web-based competition in which 33 participants were given instructions and training on either CD or the equivalent NL specification expressions, and then after each example, they responded to three multiple-choice questions requiring the interpretation of expressions in their particular notation. Although the CD group spent more time on the training and had less confidence, they obtained comparable interpretation scores to the NL group and took less time to answer the questions, although they had no prior experience of CD notation.
The third approach was an experiment on the construction of CD. 20 participants were given instructions and training on either CD or the equivalent NL specification expressions, and then after each example, they responded to three questions requiring the construction of expressions in their particular notation. We built an editor to allow the construction of the two notations, which automatically logged their interactions. In general, for constructing program specification, the CD group had more accurate answers, they had spent less time in training, and their returns to the training examples were fewer than those of the NL group.
Overall it was found that CD is understandable, usable, intuitive, and expressive with unambiguous semantic notation
A cognitive exploration of the “non-visual” nature of geometric proofs
Why are Geometric Proofs (Usually) “Non-Visual”? We asked this question as
a way to explore the similarities and differences between diagrams and text (visual
thinking versus language thinking). Traditional text-based proofs are considered
(by many to be) more rigorous than diagrams alone. In this paper we focus on
human perceptual-cognitive characteristics that may encourage textual modes for
proofs because of the ergonomic affordances of text relative to diagrams. We suggest
that visual-spatial perception of physical objects, where an object is perceived
with greater acuity through foveal vision rather than peripheral vision, is similar
to attention navigating a conceptual visual-spatial structure. We suggest that attention
has foveal-like and peripheral-like characteristics and that textual modes
appeal to what we refer to here as foveal-focal attention, an extension of prior
work in focused attention
A diagrammatic calculus of fermionic quantum circuits
We introduce the fermionic ZW calculus, a string-diagrammatic language for
fermionic quantum computing (FQC). After defining a fermionic circuit model, we
present the basic components of the calculus, together with their
interpretation, and show how the main physical gates of interest in FQC can be
represented in our language. We then list our axioms, and derive some
additional equations. We prove that the axioms provide a complete equational
axiomatisation of the monoidal category whose objects are systems of finitely
many local fermionic modes (LFMs), with maps that preserve or reverse the
parity of states, and the tensor product as monoidal product. We achieve this
through a procedure that rewrites any diagram in a normal form. As an example,
we show how the statistics of a fermionic Mach-Zehnder interferometer can be
calculated in the diagrammatic language. We conclude by giving a diagrammatic
treatment of the dual-rail encoding, a standard method in optical quantum
computing used to perform universal quantum computation
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