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

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

Making mathematics on paper : constructing representations of stories about related linear functions

This dissertation takes up the problem of applied quantitative inference as a central question for cognitive science, asking what must happen during problem solving for people to obtain a meaningful and effective representation of the problem. The core of the dissertation reports exploratory empirical studies that seek to answer the descriptive question of how quantitative inferences are generated, pursued, and evaluated by problem solvers with different mathematical backgrounds. These are framed against a controversy, described in Chapter 2, over the theoretical and empirical validity of current cognitive science accounts of problems, solutions, knowledge, and competent human activity outside of laboratory or school settings.Chapter 3 describes a written protocol study of algebra story problem solving among advanced undergraduates in computer science. A relatively open-ended interpretive framework for "problem-solving episodes" is developed and applied to their written solution attempts. The resulting description of problem-solving activities gives a surprising image of competence among an important occupational target for standard mathematics instruction.Chapter 4 follows these results into detailed verbal problem-solving interviews with algebra students and teachers. These provide a comparison across settings and levels of competence for the same set of problems. The results corroborate similar generative activities outside the standard formalism of algebra across levels of competence. Notable among these nonalgebraic problem-solving activities are "model-based reasoning tactics," in which people reason about quantitative relations in terms of the dimensional structure of functional relations described in the problem. These tactics support different activities within surrounding solution attempts and usually describe "states" in the problem's situational structure.Chapter 5 holds these activities accountable to local combinations of notation and quantity, reinterpreting results for model-based reasoning in an ecological analysis of material designs for constructing and evaluating quantitative inferences. This analysis brings forward important relations between what material designs afford problem solvers and the complexity of episodic structure observed in their solution attempts. The dissertation closes with a reappraisal of the relationship between knowledge, person, and setting and, I will argue, puts us on a more promising track for a descriptively adequate theoretical account of constructing mathematical representations that support applied quantitative inference

A Graphical Language for Proof Strategies

Complex automated proof strategies are often difficult to extract, visualise, modify, and debug. Traditional tactic languages, often based on stack-based goal propagation, make it easy to write proofs that obscure the flow of goals between tactics and are fragile to minor changes in input, proof structure or changes to tactics themselves. Here, we address this by introducing a graphical language called PSGraph for writing proof strategies. Strategies are constructed visually by "wiring together" collections of tactics and evaluated by propagating goal nodes through the diagram via graph rewriting. Tactic nodes can have many output wires, and use a filtering procedure based on goal-types (predicates describing the features of a goal) to decide where best to send newly-generated sub-goals. In addition to making the flow of goal information explicit, the graphical language can fulfil the role of many tacticals using visual idioms like branching, merging, and feedback loops. We argue that this language enables development of more robust proof strategies and provide several examples, along with a prototype implementation in Isabelle

Personalised correction, feedback, and guidance in an automated tutoring system for skills training

In addition to knowledge, in various domains skills are equally important. Active learning and training are effective forms of education. We present an automated skills training system for a database programming environment that promotes procedural knowledge acquisition and skills training. The system provides support features such as correction of solutions, feedback and personalised guidance, similar to interactions with a human tutor. Specifically, we address synchronous feedback and guidance based on personalised assessment. Each of these features is automated and includes a level of personalisation and adaptation. At the core of the system is a pattern-based error classification and correction component that analyses student input

Implementing a Business Process Management System Using ADEPT: A Real-World Case Study

This article describes how the agent-based design of ADEPT (advanced decision environment for processed tasks) and implementation philosophy was used to prototype a business process management system for a real-world application. The application illustrated is based on the British Telecom (BT) business process of providing a quote to a customer for installing a network to deliver a specified type of telecommunication service. Particular emphasis is placed upon the techniques developed for specifying services, allowing heterogeneous information models to interoperate, allowing rich and flexible interagent negotiation to occur, and on the issues related to interfacing agent-based systems and humans. This article builds upon the companion article (Applied Artificial Intelligence Vol.14, no 2, pgs. 145-189) that provides details of the rationale and design of the ADEPT technology deployed in this application

Expertise and intuition: A tale of three theories

Several authors have hailed intuition as one of the defining features of expertise. In particular, while disagreeing on almost anything that touches on human cognition and artificial intelligence, Hubert Dreyfus and Herbert Simon agreed on this point. However, the highly influential theories of intuition they proposed differed in major ways, especially with respect to the role given to search and as to whether intuition is holistic or analytic. Both theories suffer from empirical weaknesses. In this paper, we show how, with some additions, a recent theory of expert memory (the template theory) offers a coherent and wide-ranging explanation of intuition in expert behaviour. It is shown that the theory accounts for the key features of intuition: it explains the rapid onset of intuition and its perceptual nature, provides mechanisms for learning, incorporates processes showing how perception is linked to action and emotion, and how experts capture the entirety of a situation. In doing so, the new theory addresses the issues problematic for Dreyfus’s and Simon’s theories. Implications for research and practice are discussed

Expertise in chess

This chapter provides an overview of research into chess expertise. After an historical background and a brief description of the game and the rating system, it discusses the information processes enabling players to choose good moves, and in particular the trade-offs between knowledge and search. Other topics include blindfold chess, talent, and the role of deliberate practice and tournament experience

Architectural type and the discourse of urbanism

Introduction to special issue, entitled Architectural type and the discourse of urbanism