Automated Generation of Geometric Theorems from Images of Diagrams

We propose an approach to generate geometric theorems from electronic images of diagrams automatically. The approach makes use of techniques of Hough transform to recognize geometric objects and their labels and of numeric verification to mine basic geometric relations. Candidate propositions are generated from the retrieved information by using six strategies and geometric theorems are obtained from the candidates via algebraic computation. Experiments with a preliminary implementation illustrate the effectiveness and efficiency of the proposed approach for generating nontrivial theorems from images of diagrams. This work demonstrates the feasibility of automated discovery of profound geometric knowledge from simple image data and has potential applications in geometric knowledge management and education.Comment: 31 pages. Submitted to Annals of Mathematics and Artificial Intelligence (special issue on Geometric Reasoning

Geometry meets the computer

In a rather short space of time, computers have changed in character from being large numerical devices that could only be communicated with obliquely to small visual devices that allow for much more direct forms of person-machine communication. We have gone from the roomfull to the pocketfull, from paper tape and punched cards to keyboards, mice and touch screens and from strings of binary digits to visual images. All of this has taken not much more than one (human) generation. The IBM Corporation confidently predicted in 1945 that there would never be a market for more than two or three computers in the world, and yet in affluent countries like Australia, there are already many households with more computers than that, depending a bit on how one defines 'computer'

Euler Diagram Transformations

Euler diagrams are a visual language which are used for purposes such as the presentation of set-based data or as the basis of visual logical languages which can be utilised for software specification and reasoning. Such Euler diagram reasoning systems tend to be defined at an abstract level, and the concrete level is simply a visualisation of an abstract model, thereby capturing some subset of the usual boolean logic. The visualisation process tends to be divorced from the data transformation process thereby affecting the user's mental map and reducing the effectiveness of the diagrammatic notation. Furthermore, geometric and topological constraints, called wellformedness conditions, are often placed on the concrete diagrams to try to reduce human comprehension errors, and the effects of these conditions are not modelled in these systems. We view Euler diagrams as a type of graph, where the faces that are present are the key features that convey information and we provide transformations at the dual graph level that correspond to transformations of Euler diagrams, both in terms of editing moves and logical reasoning moves. This original approach gives a correspondence between manipulations of diagrams at an abstract level (such as logical reasoning steps, or simply an update of information) and the manipulation at a concrete level. Thus we facilitate the presentation of diagram changes in a manner that preserves the mental map. The approach will facilitate the realisation of reasoning systems at the concrete level; this has the potential to provide diagrammatic reasoning systems that are inherently different from symbolic logics due to natural geometric constraints. We provide a particular concrete transformation system which preserves the important criteria of planarity and connectivity, which may form part of a framework encompassing multiple concrete systems each adhering to different sets of wellformedness conditions

Two-loop Integral Reduction from Elliptic and Hyperelliptic Curves

We show that for a class of two-loop diagrams, the on-shell part of the integration-by-parts (IBP) relations correspond to exact meromorphic one-forms on algebraic curves. Since it is easy to find such exact meromorphic one-forms from algebraic geometry, this idea provides a new highly efficient algorithm for integral reduction. We demonstrate the power of this method via several complicated two-loop diagrams with internal massive legs. No explicit elliptic or hyperelliptic integral computation is needed for our method.Comment: minor changes: more references adde