2,058 research outputs found
Automating Change of Representation for Proofs in Discrete Mathematics (Extended Version)
Representation determines how we can reason about a specific problem.
Sometimes one representation helps us find a proof more easily than others.
Most current automated reasoning tools focus on reasoning within one
representation. There is, therefore, a need for the development of better tools
to mechanise and automate formal and logically sound changes of representation.
In this paper we look at examples of representational transformations in
discrete mathematics, and show how we have used Isabelle's Transfer tool to
automate the use of these transformations in proofs. We give a brief overview
of a general theory of transformations that we consider appropriate for
thinking about the matter, and we explain how it relates to the Transfer
package. We show our progress towards developing a general tactic that
incorporates the automatic search for representation within the proving
process
The "Artificial Mathematician" Objection: Exploring the (Im)possibility of Automating Mathematical Understanding
Reuben Hersh confided to us that, about forty years ago, the late Paul Cohen predicted to him that at some unspecified point in the future, mathematicians would be replaced by computers. Rather than focus on computers replacing mathematicians, however, our aim is to consider the (im)possibility of human mathematicians being joined by âartificial mathematiciansâ in the proving practiceânot just as a method of inquiry but as a fellow inquirer
Reasoned modelling critics: turning failed proofs into modelling guidance
The activities of formal modelling and reasoning are closely related. But while the rigour of building formal models brings significant benefits, formal reasoning remains a major barrier to the wider acceptance of formalism within design. Here we propose reasoned modelling critics â an approach which aims to abstract away from the complexities of low-level proof obligations, and provide high-level modelling guidance to designers when proofs fail. Inspired by proof planning critics, the technique combines proof-failure analysis with modelling heuristics. Here, we present the details of our proposal, implement them in a prototype and outline future plans
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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
Analysis of Inpainting via Clustered Sparsity and Microlocal Analysis
Recently, compressed sensing techniques in combination with both wavelet and
directional representation systems have been very effectively applied to the
problem of image inpainting. However, a mathematical analysis of these
techniques which reveals the underlying geometrical content is completely
missing. In this paper, we provide the first comprehensive analysis in the
continuum domain utilizing the novel concept of clustered sparsity, which
besides leading to asymptotic error bounds also makes the superior behavior of
directional representation systems over wavelets precise. First, we propose an
abstract model for problems of data recovery and derive error bounds for two
different recovery schemes, namely l_1 minimization and thresholding. Second,
we set up a particular microlocal model for an image governed by edges inspired
by seismic data as well as a particular mask to model the missing data, namely
a linear singularity masked by a horizontal strip. Applying the abstract
estimate in the case of wavelets and of shearlets we prove that -- provided the
size of the missing part is asymptotically to the size of the analyzing
functions -- asymptotically precise inpainting can be obtained for this model.
Finally, we show that shearlets can fill strictly larger gaps than wavelets in
this model.Comment: 49 pages, 9 Figure
Computer simulations, mathematics and economics
Economists lise different kinds of computer simulation. However, there is little attention on the theory of simulation, which is considered either a technology or an extension of mathematical theory or, else, a way of modelling that is alternative to verbal description and mathematical models. The paper suggests a systematisation of the relationship between simulations, mathematics and economics. In particular, it traces the evolution of simulation techniques, comments some of the contributions that deal with their nature, and, finally, illustrates with some examples their influence on economie theory. Keywords: Computer simulation, economie methodology, multi-agent programming techniques.
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