HR: A System for Machine Discovery in Finite Algebras

We describe the HR concept formation program which invents mathematical definitions and conjectures in finite algebras such as group theory and ring theory. We give the methods behind and the reasons for the concept formation in HR, an evaluation of its performance in its training domain, group theory, and a look at HR in domains other than group theory

Advancements in Research Mathematics through AI: A Framework for Conjecturing

This paper introduces a general framework for computer-based conjecture generation, particularly those conjectures that mathematicians might deem substantial and elegant. We describe our approach and demonstrate its effectiveness by providing examples of its application in producing publishable research and unexpected mathematical insights. We anticipate that our discussion of computer-assisted mathematical conjecturing will catalyze further research into this area and encourage the development of more advanced techniques than the ones presented herein

Purposive discovery of operations

The Generate, Prune & Prove (GPP) methodology for discovering definitions of mathematical operators is introduced. GPP is a task within the IL exploration discovery system. We developed GPP for use in the discovery of mathematical operators with a wider class of representations than was possible with the previous methods by Lenat and by Shen. GPP utilizes the purpose for which an operator is created to prune the possible definitions. The relevant search spaces are immense and there exists insufficient information for a complete evaluation of the purpose constraint, so it is necessary to perform a partial evaluation of the purpose (i.e., pruning) constraint. The constraint is first transformed so that it is operational with respect to the partial information, and then it is applied to examples in order to test the generated candidates for an operator's definition. In the GPP process, once a candidate definition survives this empirical prune, it is passed on to a theorem prover for formal verification. We describe the application of this methodology to the (re)discovery of the definition of multiplication for Conway numbers, a discovery which is difficult for human mathematicians. We successfully model this discovery process utilizing information which was reasonably available at the time of Conway's original discovery. As part of this discovery process, we reduce the size of the search space from a computationally intractable size to 3468 elements

Automated conjecturing III : property-relations conjectures

Discovery in mathematics is a prototypical intelligent behavior, and an early and continuing goal of artificial intelligence research. We present a heuristic for producing mathematical conjectures of a certain typical form and demonstrate its utility. Our program conjectures relations that hold between properties of objects (property-relation conjectures). These objects can be of a wide variety of types. The statements are true for all objects known to the program, and are the simplest statements which are true of all these objects. The examples here include new conjectures for the hamiltonicity of a graph, a well-studied property of graphs. While our motivation and experiments have been to produce mathematical conjectures-and to contribute to mathematical research-other kinds of interesting property-relation conjectures can be imagined, and this research may be more generally applicable to the development of intelligent machinery

On the Notion of Interestingness in Automated Mathematical Discovery

Deciding whether something is interesting or not is of central importance in automated mathematical discovery, as it helps determine both the search space and search strategy for finding and evaluating concepts and conjectures

A Classification Approach for Automated Reasoning Systems--A Case Study in Graph Theory

Reasoning systems which create classifications of structured objects face the problem of how object descriptions can be used to reflect their components as well as relations among these components. Current reasoning systems on graph theory do not adequately provide models to discover complex relations among mathematical concepts (eg: relations involving subgraphs) mainly due to the inability to solve this problem. This thesis presents an approach to construct a knowledge-based system, GC (Graph Classification), which overcomes this difficulty in performing automated reasoning in graph theory. We describe graph concepts based on an attribute called Linear Recursive Constructivity (LRC). LRC defines classes by an algebraic formula supported by background knowledge of graph types. We use subsumption checking on decomposed algebraic expressions of graph classes as a major proof method. The search is guided by case-split-based inferencing. Using the approach GC has generated proofs for many theorems such as any two distinct cycles (closed paths) having a common edge e contain a cycle not traversing e , if cycle C1 contains edges e1, e2, and cycle C2 contains edges e2, e3, then there exists a cycle that contains e1 and e3 and the union of a tree and a path is a tree if they have only a single common vertex. The main contributions of this thesis are: (1) Development of a classification-based knowledge representation and a reasoning approach for graph concepts, thus providing a simple model for structured mathematical objects. (2) Development of an algebraic theory for simplifying and decomposing graph concepts. (3) Development of a proof search and a case-splitting technique with the guidance of graph type knowledge. (4) Development of a proving mechanism that can be generate constructive proofs by manipulating only simple linear formalization of theorems