Results 1  10
of
40
On the Notion of Interestingness in Automated Mathematical Discovery
 International Journal of Human Computer Studies
, 2000
"... We survey ve mathematical discovery programs by looking in detail at the discovery processes they illustrate and the success they've had. We focus on how they estimate the interestingness of concepts and conjectures and extract some common notions about interestingness in automated mathema ..."
Abstract

Cited by 66 (25 self)
 Add to MetaCart
We survey ve mathematical discovery programs by looking in detail at the discovery processes they illustrate and the success they've had. We focus on how they estimate the interestingness of concepts and conjectures and extract some common notions about interestingness in automated mathematical discovery. We detail how empirical evidence is used to give plausibility to conjectures, and the dierent ways in which a result can be thought of as novel. We also look at the ways in which the programs assess how surprising and complex a conjecture statement is, and the dierent ways in which the applicability of a concept or conjecture is used. Finally, we note how a user can set tasks for the program to achieve and how this aects the calculation of interestingness. We conclude with some hints on the use of interestingness measures for future developers of discovery programs in mathematics.
Constraint Generation via Automated Theory Formation
, 2001
"... Introduction Adding constraints to a basic CSP model can signi cantly reduce search, e.g. for Golomb rulers [6]. The generation process is usually performed by hand, although some recent work has focused on automatically generating symmetry breaking constraints [4] and (less so) on generating impl ..."
Abstract

Cited by 29 (18 self)
 Add to MetaCart
Introduction Adding constraints to a basic CSP model can signi cantly reduce search, e.g. for Golomb rulers [6]. The generation process is usually performed by hand, although some recent work has focused on automatically generating symmetry breaking constraints [4] and (less so) on generating implied constraints [5]. We describe an approach to generating implied, symmetry breaking and specialisation constraints and apply this technique to quasigroup construction [10]. Given a problem class parameterised by size, we use a basic model to solve small instances with the Choco constraint programming language [7]. We then give these solutions to the HR automated theory formation program [1] which detects implied constraints (proved to follow from the speci cations) and induced constraints (true of a subset of solutions). Interpreting HR's results to reformulate the model can lead to a reduction in search on larger instances. It is often more ecient to run HR, interpret the results and so
Automatic Invention of Integer Sequences
, 2000
"... We report on the application of the HR program (Colton, Bundy, & Walsh 1999) to the problem of automatically inventing integer sequences. Seventeen sequences invented by HR are interesting enough to have been accepted into the Encyclopedia of Integer Sequences (Sloane 2000) and all were sup ..."
Abstract

Cited by 28 (16 self)
 Add to MetaCart
We report on the application of the HR program (Colton, Bundy, & Walsh 1999) to the problem of automatically inventing integer sequences. Seventeen sequences invented by HR are interesting enough to have been accepted into the Encyclopedia of Integer Sequences (Sloane 2000) and all were supplied with interesting conjectures about their nature, also discovered by HR. By extending HR, we have enabled it to perform a two stage process of invention and investigation. This involves generating both the definition and terms of a new sequence, relating it to sequences already in the Encyclopedia and pruning the output to help identify the most surprising and interesting results.
Automatic identification of mathematical concepts
 In Proceedings of the 17th International Conference on Machine Learning
, 2000
"... ..."
Evaluating Machine Creativity
 IN WORKSHOP ON CREATIVE SYSTEMS, 4TH INTERNATIONAL CONFERENCE ON CASE BASED REASONING
, 2001
"... We consider aspects pertinent to evaluating creativity to be input, output and the process by which the output is achieved. These issues ..."
Abstract

Cited by 23 (3 self)
 Add to MetaCart
We consider aspects pertinent to evaluating creativity to be input, output and the process by which the output is achieved. These issues
The Effect of Input Knowledge on Creativity
, 2001
"... Recently, many programs have been written to perform tasks which are usually regarded as requiring creativity in humans. We can ..."
Abstract

Cited by 14 (7 self)
 Add to MetaCart
Recently, many programs have been written to perform tasks which are usually regarded as requiring creativity in humans. We can
Agent based cooperative theory formation in pure mathematics
 In Proceedings of the AISB00 Symposium on Creative & Cultural Aspects and Applications of AI & Cognitive Science
, 2000
"... The HR program, Colton et al. (1999), performs theory formation in domains of pure mathematics. Given only minimal information about a domain, it invents concepts, make conjectures, proves theorems and finds counterexamples to false conjectures. We present here a multiagent version of HR which may ..."
Abstract

Cited by 12 (7 self)
 Add to MetaCart
The HR program, Colton et al. (1999), performs theory formation in domains of pure mathematics. Given only minimal information about a domain, it invents concepts, make conjectures, proves theorems and finds counterexamples to false conjectures. We present here a multiagent version of HR which may provide a model for how individual mathematicians perform separate investigations but communicate their results to the mathematical community, learning from others as they do. We detail the exhaustive categorisation problem to which we have applied a multiagent approach. 1
Automatic Generation of Benchmark Problems for Automated Theorem Proving Systems
 In: Proceedings of the Seventh AI and Maths Symposium
, 2002
"... Automated Theorem Proving (ATP) researchers who always use the same problems for testing their systems, run the risk of producing systems that can solve only those problems, and are weak on new problems or applications. Furthermore, as the stateoftheart in ATP progresses, existing test problems b ..."
Abstract

Cited by 10 (6 self)
 Add to MetaCart
Automated Theorem Proving (ATP) researchers who always use the same problems for testing their systems, run the risk of producing systems that can solve only those problems, and are weak on new problems or applications. Furthermore, as the stateoftheart in ATP progresses, existing test problems become too easy, and testing on them provides little useful information. It is thus important to regularly nd new and harder problems for testing ATP systems. HR is a program that performs automated theory formation in mathematical domains, such as group theory, quasigroup theory, and ring theory. Given the axioms of the domain...
Experiments in metatheory formation
 In Proceedings of the AISB’01 Symposium on Artificial Intelligence and Creativity in Arts and Science
, 2001
"... ..."
Cooperating reasoning processes: More than just the sum of their parts. Research Excellence Award Acceptance Speech at IJCAI07
, 2007
"... Using the achievements of my research group over the last 30+ years, I provide evidence to support the following hypothesis: By complementing each other, cooperating reasoning process can achieve much more than they could if they only acted individually. Most of the work of my group has been on pro ..."
Abstract

Cited by 8 (0 self)
 Add to MetaCart
Using the achievements of my research group over the last 30+ years, I provide evidence to support the following hypothesis: By complementing each other, cooperating reasoning process can achieve much more than they could if they only acted individually. Most of the work of my group has been on processes for mathematical reasoning and its applications, e.g. to formal methods. The reasoning processes we have studied include: Proof Search: by metalevel inference, proof planning, abstraction, analogy, symmetry, and reasoning with diagrams. Representation Discovery, Formation and Evolution: by analysing, diagnosing and repairing failed proof and planning attempts, forming and repairing new concepts and conjectures, and forming logical representations of informally stated problems. ¤I would like to thank the many colleagues with whom I have worked over the last 30+ years on the research reported in this paper: