1,957 research outputs found
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
Tactical diagrammatic reasoning
Although automated reasoning with diagrams has been possible for some years,
tools for diagrammatic reasoning are generally much less sophisticated than
their sentential cousins. The tasks of exploring levels of automation and
abstraction in the construction of proofs and of providing explanations of
solutions expressed in the proofs remain to be addressed. In this paper we take
an interactive proof assistant for Euler diagrams, Speedith, and add tactics to
its reasoning engine, providing a level of automation in the construction of
proofs. By adding tactics to Speedith's repertoire of inferences, we ease the
interaction between the user and the system and capture a higher level
explanation of the essence of the proof. We analysed the design options for
tactics by using metrics which relate to human readability, such as the number
of inferences and the amount of clutter present in diagrams. Thus, in contrast
to the normal case with sentential tactics, our tactics are designed to not
only prove the theorem, but also to support explanation
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Considerations in Representation Selection for Problem Solving: A Review
Choosing how to represent knowledge effectively is a long-standing open problem. Cognitive science has shed light on the taxonomisation of representational systems from the perspective of cognitive processes, but a similar analysis is absent from the perspective of problem solving, where the representations are employed. In this paper we review how representation choices are made for solving problems in the context of theorem proving from three perspectives: cognition, heterogeneity, and computational demands. We contrast the different factors that are most important for each perspective in the context of problem solving to produce a list of considerations for developers of problem solving tools regarding representations that are appropriate for particular users and effective for specific problem domains
Accessible reasoning with diagrams: From cognition to automation
High-tech systems are ubiquitous and often safety and se- curity critical: reasoning about their correctness is paramount. Thus, precise modelling and formal reasoning are necessary in order to convey knowledge unambiguously and accurately. Whilst mathematical mod- elling adds great rigour, it is opaque to many stakeholders which leads to errors in data handling, delays in product release, for example. This is a major motivation for the development of diagrammatic approaches to formalisation and reasoning about models of knowledge. In this paper, we present an interactive theorem prover, called iCon, for a highly expressive diagrammatic logic that is capable of modelling OWL 2 ontologies and, thus, has practical relevance. Significantly, this work is the first to design diagrammatic inference rules using insights into what humans find accessible. Specifically, we conducted an experiment about relative cognitive benefits of primitive (small step) and derived (big step) inferences, and use the results to guide the implementation of inference rules in iCon
The ZX-calculus is complete for stabilizer quantum mechanics
The ZX-calculus is a graphical calculus for reasoning about quantum systems
and processes. It is known to be universal for pure state qubit quantum
mechanics, meaning any pure state, unitary operation and post-selected pure
projective measurement can be expressed in the ZX-calculus. The calculus is
also sound, i.e. any equality that can be derived graphically can also be
derived using matrix mechanics. Here, we show that the ZX-calculus is complete
for pure qubit stabilizer quantum mechanics, meaning any equality that can be
derived using matrices can also be derived pictorially. The proof relies on
bringing diagrams into a normal form based on graph states and local Clifford
operations.Comment: 26 page
Dynamic Euler Diagram Drawing
In this paper we describe a method to lay out a graph enhanced Euler diagram so that it looks similar to a previously drawn graph enhanced Euler diagram. This task is non-trivial when the underlying structures of the diagrams differ. In particular, if a structural change is made to an existing drawn diagram, our work enables the presentation of the new diagram with minor disruption to the user's mental map. As the new diagram can be generated from an abstract representation, its initial embedding may be very different from that of the original. We have developed comparison measures for Euler diagrams, integrated into a multicriteria optimizer, and applied a force model for associated graphs that attempts to move nodes towards their positions in the original layout. To further enhance the usability of the system, the transition between diagrams can be animated
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