301 research outputs found
Lightweight Formal Verification in Classroom Instruction of Reasoning about Functional Code
In college courses dealing with material that requires mathematical rigor, the adoption of a machine-readable representation for formal arguments can be advantageous. Students can focus on a specific collection of constructs that are represented consistently. Examples and counterexamples can be evaluated. Assignments can be assembled and checked with the help of an automated formal reasoning system. However, usability and accessibility do not have a high priority and are not addressed sufficiently well in the design of many existing machine-readable representations and corresponding formal reasoning systems. In earlier work [Lap09], we attempt to address this broad problem by proposing several specific design criteria organized around the notion of a natural context: the sphere of awareness a working human user maintains of the relevant constructs, arguments, experiences, and background materials necessary to accomplish the task at hand. We report on our attempt to evaluate our proposed design criteria by deploying within the classroom a lightweight formal verification system designed according to these criteria. The lightweight formal verification system was used within the instruction of a common application of formal reasoning: proving by induction formal propositions about functional code. We present all of the formal reasoning examples and assignments considered during this deployment, most of which are drawn directly from an introductory text on functional programming. We demonstrate how the design of the system improves the effectiveness and understandability of the examples, and how it aids in the instruction of basic formal reasoning techniques. We make brief remarks about the practical and administrative implications of the system’s design from the perspectives of the student, the instructor, and the grader
3-filiform Leibniz algebras of maximum length
This work completes the study of the solvable Leibniz algebras, more precisely, it completes
the classi cation of the 3- liform Leibniz algebras of maximum length [4]. Moreover, due to
the good structure of the algebras of maximum length, we also tackle some of their cohomological
properties. Our main tools are the previous result of Cabezas and Pastor [3], the construction of
appropriate homogeneous basis in the considered connected gradation and the computational support
provided by the two programs implemented in the software Mathematica.Ministerio de EconomÃa y Competitividad MTM2013–43687–
regularity of solutions of degenerate fully non-linear elliptic equations
In the present paper, a class of fully non-linear elliptic equations are
considered, which are degenerate as the gradient becomes small. H\"older
estimates obtained by the first author (2011) are combined with new Lipschitz
estimates obtained through the Ishii-Lions method in order to get
estimates for solutions of these equations.Comment: Submitte
The Surprise Examination Paradox and the Second Incompleteness Theorem
We give a new proof for Godel's second incompleteness theorem, based on
Kolmogorov complexity, Chaitin's incompleteness theorem, and an argument that
resembles the surprise examination paradox. We then go the other way around and
suggest that the second incompleteness theorem gives a possible resolution of
the surprise examination paradox. Roughly speaking, we argue that the flaw in
the derivation of the paradox is that it contains a hidden assumption that one
can prove the consistency of the mathematical theory in which the derivation is
done; which is impossible by the second incompleteness theorem.Comment: 8 page
A discrete Schrodinger spectral problem and associated evolution equations
A recently proposed discrete version of the Schrodinger spectral problem is
considered. The whole hierarchy of differential-difference nonlinear evolution
equations associated to this spectral problem is derived. It is shown that a
discrete version of the KdV, sine-Gordon and Liouville equations are included
and that the so called `inverse' class in the hierarchy is local. The whole
class of related Darboux and Backlund transformations is also exhibited.Comment: 14 pages, LaTeX2
- …