Alexandra Mendes

Logic training through algorithmic problem solving

Although much of mathematics is algorithmic in nature, the skills needed to formulate and solve algorithmic problems do not form an integral part of mathematics education. In particular, logic, which is central to algorithm development, is rarely taught explicitly at preuniversity level, under the justification that it is implicit in mathematics and therefore does not need to be taught as an independent topic. This paper argues in the opposite direction, describing a one-week workshop done at the University of Minho, in Portugal, whose goal was to introduce to high-school students calculational principles and techniques of algorithmic problem solving supported by calculational logic. The workshop resorted to recreational problems to convey the principles and to software tools, the Alloy Analyzer and Netlogo, to animate models.

Which Mathematics for the Information Society

MathIS is a new project that aims to reinvigorate secondary-school mathematics by exploiting insights of the dynamics of algorithmic problem solving. This paper describes the main ideas that underpin the project. In summary, we propose a central role for formal logic, the development of a calculational style of reasoning, the emphasis on the algorithmic nature of mathematics, and the promotion of self-discovery by the students. These ideas are discussed and the case is made, through a number of examples that show the teaching style that we want to introduce, for their relevance in shaping mathematics training for the years to come. In our opinion, the education of software engineers that work effectively with formal methods and mathematical abstractions should start before university and would benefit from the ideas discussed here.

Structure Editing of Handwritten Mathematics: Improving the Computer Support for the Calculational Method

We present a structure editor that aims to facilitate the presentation and manipulation of handwritten mathematical expressions. The editor is oriented to the calculational mathematics involved in algorithmic problem solving and it provides features that allow reliable structure manipulation of mathematical formulae, as well as flexible and interactive presentations. We describe some of its most important features, including the use of gestures to manipulate algebraic formulae, the structured selection of expressions, definition and redefinition of operators in runtime, gestures editor, and handwritten templates. The editor is made available in the form of a C# class library which can be easily used to extend existing tools. For example, we have extended Classroom Presenter, a tool for ink-based teaching presentations and classroom interaction. We have tested and evaluated the editor with target users. The results obtained seem to indicate that the software is usable, suitable for its purpose and a valuable contribution to teaching and learning algorithmic problem solving.

The magic of algorithm design and analysis: teaching algorithmic skills using magic card tricks

We describe our experience using magic card tricks to teach algorithmic skills to first-year Computer Science undergraduates. We illustrate our approach with a detailed discussion on a card trick that is typically presented as a test to the psychic abilities of an audience. We use the trick to discuss concepts like problem decomposition, pre- and post-conditions, and invariants. We discuss pedagogical issues and analyse feedback collected from students. The feedback has been very positive and encouraging.

Work in progress - structure editing of handwritten mathematics

This project aims to develop a pen-based software tool that will assist in the process of doing mathematics by providing structured manipulation of handwritten mathematical expressions. The tool will be used to support the teaching of the dynamics of problem solving in a way that combines the advantages of the traditional blackboard style of teaching with the flexibility and accuracy of computer software. It will provide not only a simpler way to input mathematics - by allowing the recognition of handwritten mathematics - but also enhance studentspsila understanding of the calculational techniques and facilitate the process of doing mathematics - by providing structure editing. Some of the most important features of this tool are the accurate selection and copy of expressions, the automatic application of algebraic rules and the use of gestures to apply them, and also the combined writing of mathematics and text. These features will have a major impact on writing, doing, and presenting mathematics. This project includes the required technical developments and also the application and testing of the tool in concrete situations, namely in mathematics and computing science courses.

A calculational approach to path-based properties of the Eisenstein–Stern and Stern–Brocot trees via matrix algebra

Abstract This paper proposes a calculational approach to prove properties of two well-known binary trees used to enumerate the rational numbers: the Stern–Brocot tree and the Eisenstein–Stern tree (also known as Calkin–Wilf tree). The calculational style of reasoning is enabled by a matrix formulation that is well-suited to naturally formulate path-based properties, since it provides a natural way to refer to paths in the trees. Three new properties are presented. First, we show that nodes with palindromic paths contain the same rational in both the Stern–Brocot and Eisenstein–Stern trees. Second, we show how certain numerators and denominators in these trees can be written as the sum of two squares x 2 and y 2 , with the rational x y appearing in specific paths. Finally, we show how we can construct Sierpinskis triangle from these trees of rationals.

Towards Verified Handwritten Calculational Proofs

Despite great advances in computer-assisted proof systems, writing formal proofs using a traditional computer is still challenging due to mouse-and-keyboard interaction. This leads to scientists often resorting to pen and paper to write their proofs. However, when handwriting a proof, there is no formal guarantee that the proof is correct. In this paper we address this issue and present the initial steps towards a system that allows users to handwrite proofs using a pen-based device and that communicates with an external theorem prover to support the users throughout the proof writing process. We focus on calculational proofs, whereby a theorem is proved by a chain of formulae, each transformed in some way into the next. We present the implementation of a proof-of-concept prototype that can formally verify handwritten calculational proofs without the need to learn the specific syntax of theorem provers.

Certified Password Quality

We propose the use of modern proof assistants to specify, implement, and verify password quality checkers. We use the proof assistant Coq, focusing on Linux PAM, a widely-used implementation of pluggable authentication modules for Linux. We show how password quality policies can be expressed in Coq and how to use Coq’s code extraction features to automatically encode these policies as PAM modules that can readily be used by any Linux system.

ReqCap: Hierarchical Requirements Modeling and Test Generation for Industrial Control Systems

This paper presents ReqCap, an implementation of a new method that articulates hierarchical requirements modeling and test generation to assist in the process of capturing requirements for PLC-based control systems. ReqCap is based on a semi-formal graphical model that supports hierarchical modeling, thus enabling compositional specifications. The tool supports automated generation of test cases according to different coverage criteria. It can also import requirements directly from ReqIF files and automatically generate Sequential Function Charts (SFCs).We use a real-world case study to show how ReqCap can be used to model realistic system requirements. We show how the automated generation of SFCs and test cases can support engineers (and clients) in visualizing and reviewing requirements. Moreover, all the tests listed in the original test document of the case study are also generated automatically by ReqCap, demonstrating that the tool can be used to effectively capture requirements and generate valid and useful test cases.