The Use of Proof Planning for Cooperative Theorem Proving

AbstractWe describebarnacle: a co-operative interface to theclaminductive theorem proving system. For the foreseeable future, there will be theorems which cannot be proved completely automatically, so the ability to allow human intervention is desirable; for this intervention to be productive the problem of orienting the user in the proof attempt must be overcome. There are many semi-automatic theorem provers: we call our style of theorem provingco-operative, in that the skills of both human and automaton are used each to their best advantage, and used together may find a proof where other methods fail. The co-operative nature of thebarnacleinterface is made possible by the proof planning technique underpinningclam. Our claim is that proof planning makes new kinds of user interaction possible.Proof planning is a technique for guiding the search for a proof in automatic theorem proving. Common patterns of reasoning in proofs are identified and represented computationally as proof plans, which can then be used to guide the search for proofs of new conjectures. We have harnessed the explanatory power of proof planning to enable the user to understand where the automatic prover got to and why it is stuck. A user can analyse the failed proof in terms ofclam's specification language, and hence override the prover to force or prevent the application of a tactic, or discover a proof patch. This patch might be to apply further rules or tactics to bridge the gap between the effects of previous tactics and the preconditions needed by a currently inapplicable tactic

Formalizing graphical notations

The thesis describes research into graphical notations for software engineering, with a principal interest in ways of formalizing them. The research seeks to provide a theoretical basis that will help in designing both notations and the software tools that process them. The work starts from a survey of literature on notation, followed by a review of techniques for formal description and for computational handling of notations. The survey concentrates on collecting views of the benefits and the problems attending notation use in software development; the review covers picture description languages, grammars and tools such as generic editors and visual programming environments. The main problem of notation is found to be a lack of any coherent, rigorous description methods. The current approaches to this problem are analysed as lacking in consensus on syntax specification and also lacking a clear focus on a defined concept of notated expression. To address these deficiencies, the thesis embarks upon an exploration of serniotic, linguistic and logical theory; this culminates in a proposed formalization of serniosis in notations, using categorial model theory as a mathematical foundation. An argument about the structure of sign systems leads to an analysis of notation into a layered system of tractable theories, spanning the gap between expressive pictorial medium and subject domain. This notion of 'tectonic' theory aims to treat both diagrams and formulae together. The research gives details of how syntactic structure can be sketched in a mathematical sense, with examples applying to software development diagrams, offering a new solution to the problem of notation specification. Based on these methods, the thesis discusses directions for resolving the harder problems of supporting notation design, processing and computer-aided generic editing. A number of future research areas are thereby opened up. For practical trial of the ideas, the work proceeds to the development and partial implementation of a system to aid the design of notations and editors. Finally the thesis is evaluated as a contribution to theory in an area which has not attracted a standard approach

Intelligent Combination of Structural Analysis Algorithms: Application to Mathematical Expression Recognition

Structural analysis is an important step in many document based recognition problem. Structural analysis is performed to associate elements in a document and assign meaning to their association. Handwritten mathematical expression recognition is one such problem which has been studied and researched for long. Many techniques have been researched to build a system that produce high performance mathematical expression recognition. We have presented a novel method to combine multiple structural recognition algorithms in which the combined result shows better performance than each individual recognition algorithms. In our experiment we have applied our method to combine multiple mathematical expression recognition parsers called DRACULAE. We have used Graph Transformation Network (GTN) which is a network of function based systems in which each system takes graphs as input, apply function and produces a graph as output. GTN is used to combine multiple DRACULAE parsers and its parameter are tuned using gradient based learning. It has been shown that such a combination method can be used to accentuate the strength of individual algorithms in combination to produce better combination result which higher recognition performance. In our experiment we were able to obtain a highest recognition rate of 74% as compared to best recognition result of 70% from individual DRACULAE parsers. Our experiment also resulted into a maximum of 20% reduction of parent recognition errors and maximum 37% reduction in relation recognition errors between symbols in expressions

Table recognition in mathematical documents

While a number of techniques have been developed for table recognition in ordinary text documents, when dealing with tables in mathematical documents these techniques are often ineffective as tables containing mathematical structures can differ quite significantly from ordinary text tables. In fact, it is even difficult to clearly distinguish table recognition in mathematics from layout analysis of mathematical formulas. Again, it is not straight forward to adapt general layout analysis techniques for mathematical formulas. However, a reliable understanding of formula layout is often a necessary prerequisite to further semantic interpretation of the represented formulae. In this thesis, we present the necessary preprocessing steps towards a table recognition technique that specialises on tables in mathematical documents. It is based on our novel robust line recognition technique for mathematical expressions, which is fully independent of understanding the content or specialist fonts of expressions. We also present a graph representation for complex mathematical table structures. A set of rewriting rules applied to the graph allows for reliable re-composition of cells in order to identify several valid table interpretations. We demonstrate the effectiveness of our technique by applying them to a set of mathematical tables from standard text book that has been manually ground-truthed