Providing Hints, Next Steps and Feedback in a Tutoring System for Structural Induction

Structural induction is a proof technique that is widely used to prove statements about discrete structures. Students find it hard to construct inductive proofs, and when learning to construct such proofs, receiving feedback is important. In this paper we discuss the design of a tutoring system, LogInd, that helps students with constructing stepwise inductive proofs by providing hints, next steps and feedback. As far as we know, this is the first tutoring system for structural induction with this functionality. We explain how we use a strategy to construct proofs for a restricted class of problems. This strategy can also be used to complete partial student solutions, and hence to provide hints or next steps. We use constraints to provide feedback. A pilot evaluation with a small group of students shows that LogInd indeed can give hints and next steps in almost all cases.Comment: In Proceedings ThEdu'19, arXiv:2002.1189

A Vernacular for Coherent Logic

We propose a simple, yet expressive proof representation from which proofs for different proof assistants can easily be generated. The representation uses only a few inference rules and is based on a frag- ment of first-order logic called coherent logic. Coherent logic has been recognized by a number of researchers as a suitable logic for many ev- eryday mathematical developments. The proposed proof representation is accompanied by a corresponding XML format and by a suite of XSL transformations for generating formal proofs for Isabelle/Isar and Coq, as well as proofs expressed in a natural language form (formatted in LATEX or in HTML). Also, our automated theorem prover for coherent logic exports proofs in the proposed XML format. All tools are publicly available, along with a set of sample theorems.Comment: CICM 2014 - Conferences on Intelligent Computer Mathematics (2014

Construction and evaluation of a gold standard syntax for formal logic formulas and systems

Classical logic plays a significant role in computer science where formal proofs eventually make their way into a student’s curriculum via discrete mathematics, philosophy logic, or some other medium. We traditionally see propositional logic in Boolean algebra, conditional statements, program and data structure definitions and invariants, and much more. In fact, everyday language is easily expressible in first-order logic. Accordingly, a solid understanding of classical logic is paramount. Natural deduction, as the name suggests, is a method of reasoning about an argument using natural intuition, and as a result, it appears quite frequently as a topic of study in introductory logic courses. Due to its relevance, natural deduction intelligent tutors and solvers are widespread on the internet and in the classroom to improve the pedagogical appeal of logic. In this thesis, we present and solve two questions. The first is a proposed research question wherein we evaluate the efficacy of publicly-available natural deduction tutors/solvers with the prospect of determining what inherently defines understandability and difficulty in natural deduction proofs. In the second question, we investigate the problem of unnecessary intermingling of logic syntaxes. With this, we propose a gold standard language for zeroth and first-order logic with the goal and hope of tutor/proof systems adapting to said language to ease the currently laborious task of system evaluation and comparison

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.

ProverX: rewriting and extending prover9

O propósito principal deste projecto é tornar o demonstrador automático de teoremas Prover9 programável e, por conseguinte, extensível. Este propósito foi conseguido acrescentando um interpretador de Python, uma linha de comandos e uma biblioteca de módulos, objectos e funções escritos em Python para interagir com ficheiros de Prover9 e Mace4. Foi também criada uma “interface” gráfica de utilizador (GUI) sob a forma de uma aplicação web para trazer aos utilizadores um meio mais eficiente e rápido de trabalhar com demonstrações automáticas de teoremas. A nova biblioteca de “scripting” oferece aos utilizadores novas funcionalidades tais como correr várias sessões simultâneas de Prover9 parando automaticamente quando uma demonstração (ou um contraexemplo) é encontrada, elaborar estratégias para aumentar a velocidade com que as demonstrações são encontradas ou diminuir o tamanho das mesmas. Outro módulo permite interagir com o sistema de álgebra GAP. Sobre esta biblioteca, muitas outras funcionalidades podem ser facilmente acrescentadas pois o objectivo principal é dar aos utilizadores a capacidade de acrescentar novas funcionalidades ao Prover9. Resumindo, o objectivo deste projecto é oferecer à comunidade matemática um ambiente integrado para trabalhar com demonstração automática de teoremas.The primary purpose of this project is to extend Prover9 with a scripting language. This was achieved by adding a Python interpreter, an interactive command line and a special scripting library to interact with Prover9 and Mace4 files. A user interface in the form of a web application was also created to help users achieve a more rapid and efficient way of working with automated theorem proving. The new scripting library offers utilities that allows a user to run several Prover9 sessions concurrently and to create strategies for increasing the effectiveness of the proof search or to search for shorter proofs. Another module allows to interact with the algebra system GAP. Based on the library, many more functionalities can be easily added, as the main goal is to give users the ability to extend the functionality of Prover9 the way they see fit. In conclusion, the aim of this project is to offer to the mathematical community an integrated environment for working with automated reasonin