## Managing automatically formed mathematical theories (2006)

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Venue: | In Proceedings of the 5th International Conference on Mathematical Knowledge Management |

Citations: | 4 - 3 self |

### BibTeX

@INPROCEEDINGS{Colton06managingautomatically,

author = {Simon Colton and Pedro Torres and Paul Cairns and Volker Sorge},

title = {Managing automatically formed mathematical theories},

booktitle = {In Proceedings of the 5th International Conference on Mathematical Knowledge Management},

year = {2006}

}

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### Abstract

Abstract. The HR system forms scientific theories, and has found particularly successful application in domains of pure mathematics. Starting with only the axioms of an algebraic system, HR can generate dozens of example algebras, hundreds of concepts and thousands of conjectures, many of which have first order proofs. Given the overwhelming amount of knowledge produced, we have provided HR with sophisticated tools for handling this data. We present here the first full description of these management tools. Moreover, we describe how careful analysis of the theories produced by HR – which is enabled by the management tools – has led us to make interesting discoveries in algebraic domains. We demonstrate this with some illustrative results from HR’s theories about an algebra of one axiom. The results fueled further developments, and led us to discover and prove a fundamental theorem about this domain. 1

### Citations

182 |
Proofs and refutations : The logic of mathematical discovery
- Lakatos
- 1976
(Show Context)
Citation Context ...ture. More recent approaches to theory formation include further application to graph theory [19] and the approach by Pease et. al [18] to modeling the philosophy of mathematics championed by Lakatos =-=[14]-=-. Our contribution to mathematical theory formation has been to develop a novel descriptive machine learning algorithm, known as automated theory formation [3], and to apply this to discovery tasks in... |

138 |
The TPTP Problem Library: CNF Release v1.2.1
- Sutcliffe, Suttner
- 1998
(Show Context)
Citation Context ...tuents directly, e.g., a text search facility in the concepts screen. HR can present concept definitions and conjecture statements in plain text, in Otter format, in Prolog format, and in TPTP format =-=[22]-=-. It also has an ability to simplify concept definitions, e.g., it would simplify the Otter-style definition of commutative groups from §2 above to: 4.[a] : ∀ b, c ∈ G (b∗c = c∗d). In addition, HR use... |

98 |
Automated Theory Formation in Pure Mathematics
- Colton
- 2002
(Show Context)
Citation Context ...hy of mathematics championed by Lakatos [14]. Our contribution to mathematical theory formation has been to develop a novel descriptive machine learning algorithm, known as automated theory formation =-=[3]-=-, and to apply this to discovery tasks in domains of pure mathematics, such as group theory, graph theory and number theory. Our first implementation of this technique (in the HR1 program) was written... |

97 | A Davis-Putnam program and its application to finite first-order model search: Quasigroup existence problems
- McCune
- 1994
(Show Context)
Citation Context ...and from Horn clause implicates, it extracts prime implicates, where no proper subset of the premises implies the goal. In addition, HR uses the Otter theorem prover [16] and the Mace model generator =-=[17]-=- to prove/disprove conjectures. Otter therefore introduces a third type of theory constituent, namely proofs, and Mace introduces a fourth, namely new objects of interest (e.g., groups) which act as a... |

66 | On the notion of interestingness in automated mathematical discovery
- Colton, Bundy, et al.
(Show Context)
Citation Context ..., and the novelty of a concept calculates the reciprocal of the number of times the categorisation of examples afforded by that concept is also the categorisation for another concept. As discussed in =-=[6]-=-, while the measures of interestingness drive HR’s heuristic search, they are also useful for sorting and pruning the concepts it produces. In particular, we often find that concepts which score highl... |

64 |
On two conjectures of Graffiti
- Fajtlowicz, Waller
- 1986
(Show Context)
Citation Context ... and non-associative algebras [12]. Particular attention has been paid to graph theory, with Epstein’s GT program [8] providing a generic model for theory formation, and Fajtlowicz’s Graffiti program =-=[9]-=- producing many conjectures, the proofs/disproofs of which have led to publication in the mathematical literature. More recent approaches to theory formation include further application to graph theor... |

58 |
AM: Discovery in Mathematics as Heuristic Search
- Davis, Lenat
- 1982
(Show Context)
Citation Context ... Intelligence researchers for nearly 40 years. This fascination began with Lenat’s inspirational – but ultimately flawed 1 – approach to mathematical concept formation via the AM and Eurisko programs =-=[15]-=-, which formed concepts in set and number theory. Following these early attempts, methods for theory formation in particular domains were implemented, e.g., plane geometry [2], number systems (such as... |

46 |
The OTTER user’s guide
- McCune
- 1990
(Show Context)
Citation Context ... premises implies a single goal), and from Horn clause implicates, it extracts prime implicates, where no proper subset of the premises implies the goal. In addition, HR uses the Otter theorem prover =-=[16]-=- and the Mace model generator [17] to prove/disprove conjectures. Otter therefore introduces a third type of theory constituent, namely proofs, and Mace introduces a fourth, namely new objects of inte... |

26 | Automatic identification of mathematical concepts
- Colton, Bundy, et al.
- 2000
(Show Context)
Citation Context ...ce a new one. In mode m2, the initial set of concepts is supplied by the user, whereas in mode m1, the concepts are extracted from the axioms supplied. The production rules are described in detail in =-=[5]-=- and chapter 6 of [3]. To give a flavour of how they operate and the concepts they produce, we describe how HR can construct the concept of commutative groups. It begins by constructing the concept of... |

21 | Automatic generation of classification theorems for finite algebras
- Colton, Meier, et al.
- 2004
(Show Context)
Citation Context ...any size) produce star algebras (proof omitted). Note that we used Mace to disprove the hypothesis that such star algebras are the only idempotent ones. We used the classification system described in =-=[7]-=- to prove that idempotent star algebras for which all the non-diagonal entries are the same element (not necessarily zero) form an isomorphism class for sizes 6, 7 and 8. This gave us good empirical e... |

20 | The HR program for theorem generation
- Colton
- 2002
(Show Context)
Citation Context ...mely proofs, and Mace introduces a fourth, namely new objects of interest (e.g., groups) which act as a counterexample to a non-theorem. HR’s conjecture making functionality is described in detail in =-=[4]-=-. As an illustration, in mode m1, HR is given no example groups, so it first forms the conjecture that there are actually no groups. Mace disproves this by supplying the trivial group as a counterexam... |

18 |
On the discovery of mathematical theorems
- Epstein
- 1987
(Show Context)
Citation Context ...re implemented, e.g., plane geometry [2], number systems (such as Conway numbers) [21] and non-associative algebras [12]. Particular attention has been paid to graph theory, with Epstein’s GT program =-=[8]-=- providing a generic model for theory formation, and Fajtlowicz’s Graffiti program [9] producing many conjectures, the proofs/disproofs of which have led to publication in the mathematical literature.... |

18 |
A Course in Group Theory
- Humphreys
- 1996
(Show Context)
Citation Context ...anagement tools available in HR, and we propose improvements for future implementations of automated theory formation tools. We will use group theory – a well known algebraic system with one operator =-=[11]-=- – as a running example. The operator in groups is usually denoted ∗, and this multiplies pairs of elements, x and y of a set, G, to produce a third element, x∗y in such a way that: (i) ∗ is associati... |

14 |
Automatic theorem generation in plane geometry
- Bagai, Shanbhogue, et al.
(Show Context)
Citation Context ...AM and Eurisko programs [15], which formed concepts in set and number theory. Following these early attempts, methods for theory formation in particular domains were implemented, e.g., plane geometry =-=[2]-=-, number systems (such as Conway numbers) [21] and non-associative algebras [12]. Particular attention has been paid to graph theory, with Epstein’s GT program [8] providing a generic model for theory... |

13 |
DOT and NEATO user’s guide
- North, Koutsofios
- 1996
(Show Context)
Citation Context ...lso has an ability to simplify concept definitions, e.g., it would simplify the Otter-style definition of commutative groups from §2 above to: 4.[a] : ∀ b, c ∈ G (b∗c = c∗d). In addition, HR uses Dot =-=[13]-=- to present graphical construction histories of concepts and conjectures. To illustrate usage of these screens, after the theory formation in session S described in §2 above, we looked at the conjectu... |

10 |
AM: A case study in methodology
- Ritchie, Hanna
- 1984
(Show Context)
Citation Context ...asks in domains of pure mathematics, such as group theory, graph theory and number theory. Our first implementation of this technique (in the HR1 program) was written in Prolog and allowed 1 See [1], =-=[20]-=- and chapter 13 of [3] for criticisms of this work.sus to investigate various concept formation and conjecture making mechanisms at a fundamental level. Our second implementation (in the HR2 program) ... |

8 |
A model of Lakatos’s philosophy of mathematics
- Pease, Colton, et al.
- 2004
(Show Context)
Citation Context ...disproofs of which have led to publication in the mathematical literature. More recent approaches to theory formation include further application to graph theory [19] and the approach by Pease et. al =-=[18]-=- to modeling the philosophy of mathematics championed by Lakatos [14]. Our contribution to mathematical theory formation has been to develop a novel descriptive machine learning algorithm, known as au... |

6 |
IL: An Artificial Intelligence approach to theory formation in mathematics. Doctoral dissertation
- Sims
- 1990
(Show Context)
Citation Context ...ncepts in set and number theory. Following these early attempts, methods for theory formation in particular domains were implemented, e.g., plane geometry [2], number systems (such as Conway numbers) =-=[21]-=- and non-associative algebras [12]. Particular attention has been paid to graph theory, with Epstein’s GT program [8] providing a generic model for theory formation, and Fajtlowicz’s Graffiti program ... |

5 | Experimenting with the identity (xy)z = y(zx
- HENTZEL, JACOBS, et al.
- 1993
(Show Context)
Citation Context ...wn below. Definition. An algebra is said to be k-nice if the product of any k elements is the same regardless of bracketing or the order of the elements in the product. (This definition is taken from =-=[10]-=-). For instance, commutative, associative algebras are 3-nice, because multiplying a triple of elements in any way always produces the same product. Definition. An algebra is said to be redundant if t... |

4 | Automatic theory formation in graph theory
- Pistori, Wainer
- 1999
(Show Context)
Citation Context ...roducing many conjectures, the proofs/disproofs of which have led to publication in the mathematical literature. More recent approaches to theory formation include further application to graph theory =-=[19]-=- and the approach by Pease et. al [18] to modeling the philosophy of mathematics championed by Lakatos [14]. Our contribution to mathematical theory formation has been to develop a novel descriptive m... |

4 |
Applying model generation to concept formation
- Torres, Colton
- 2006
(Show Context)
Citation Context ...raphrased, this states that, in star algebras, any element which is a left identity for another element is also a right identity for that element. We also used the first order generic production rule =-=[23]-=- to specify closure under multiplication as interesting to study. Using this, HR conjectured – and Otter proved – that the elements which have a left identity are closed under multiplication, i.e., th... |

1 |
A critical evaluation of Lenat’s AM program
- Anderson
- 1989
(Show Context)
Citation Context ...ery tasks in domains of pure mathematics, such as group theory, graph theory and number theory. Our first implementation of this technique (in the HR1 program) was written in Prolog and allowed 1 See =-=[1]-=-, [20] and chapter 13 of [3] for criticisms of this work.sus to investigate various concept formation and conjecture making mechanisms at a fundamental level. Our second implementation (in the HR2 pro... |

1 | The Albert non-associative algebra system: a progress report
- Jacobs
- 1994
(Show Context)
Citation Context ...ollowing these early attempts, methods for theory formation in particular domains were implemented, e.g., plane geometry [2], number systems (such as Conway numbers) [21] and non-associative algebras =-=[12]-=-. Particular attention has been paid to graph theory, with Epstein’s GT program [8] providing a generic model for theory formation, and Fajtlowicz’s Graffiti program [9] producing many conjectures, th... |