Martin Pollet

Proof Development with OMEGA

The Ωmega proof development system [2] is the core of several related and well integrated research projects of the Ωmega research group.

Proof Development with Ωmega: The Irrationality of \sqrt 2

The well-known theorem asserting the irrationality of \(\sqrt 2\) was proposed as a case study for a comparison of fifteen (interactive) theorem proving systems [Wiedijk, 2002]. This represents an important shift of emphasis in the field of automated deduction away from the somehow artificial problems of the past back to real mathematical challenges.

Comparing approaches to the exploration of the domain of residue classes

We report on a case study on combining proof planning with computer algebra systems. We construct proofs for basic algebraic properties of residue classes as well as for isomorphisms between residue classes using different proof techniques, which are implemented as strategies in a multi-strategy proof planner. The search space of the proof planner can be drastically reduced by employing computations of two computer algebra systems during the planning process. To test the effectiveness of our approach we carried out a large number of experiments and also compared it with some alternative approaches. In particular, we experimented with substituting computer algebra by model generation and by proving theorems with a first-order equational theorem prover instead of a proof planner.

LΩUI: Lovely ΩMEGA User Interface

Abstract. The capabilities of a automated theorem provers interface are essential for the effective use of (interactive) proof systems. LΩUI is the multi-modal interface that combines several features: a graphical display of information in a proof graph, a selective term browser with hypertext facilities, proof and proof plan presentation in natural language, and an editor for adding and maintaining the knowledge base. LΩUI is realized in an agent-based client-server architecture and implemented in the concurrent constraint programming language Oz.

Intuitive and Formal Representations: The Case of Matrices

A major obstacle for bridging the gap between textbook mathematics and formalising it on a computer is the problem how to adequately capture the intuition inherent in the mathematical notation when formalising mathematical concepts. While logic is an excellent tool to represent certain mathematical concepts it often fails to retain all the information implicitly given in the representation of some mathematical objects. In this paper we concern ourselves with matrices, whose representation can be particularly rich in implicit information. We analyse different types of matrices and present a mechanism that can represent them very close to their textbook style appearance and captures the information contained in this representation but that nevertheless allows for their compilation into a formal logical framework. This firstly allows for a more human-oriented interface and secondly enables efficient reasoning with matrices.

Certifying Solutions to Permutation Group Problems

We describe the integration of permutation group algorithms with proof planning. We consider eight basic questions arising in computational permutation group theory, for which our code provides both answers and a set of certificates enabling a user, or an intelligent software system, to provide a full proof of correctness of the answer. To guarantee correctness we use proof planning techniques, which construct proofs in a human-oriented reasoning style. This gives the human mathematician the necessary insight into the computed solution, as well as making it feasible to check the solution for relatively large groups.

LOUI: Lovely OMEGA User Interface

Abstract. The capabilities of a automated theorem provers interface are essential for the effective use of (interactive) proof systems. LΩUI is the multi-modal interface that combines several features: a graphical display of information in a proof graph, a selective term browser with hypertext facilities, proof and proof plan presentation in natural language, and an editor for adding and maintaining the knowledge base. LΩUI is realized in an agent-based client-server architecture and implemented in the concurrent constraint programming language Oz.

Automatic Learning of Proof Methods in Proof Planning

In this paper we present an approach to automated learning within mathematical reasoning systems. In particular, the approach enables proof planning systems to automatically learn new proof methods from well-chosen examples of proofs which use a similar reasoning pattern to prove related theorems. Our approach consists of an abstract representation for methods and a machine learning technique which can learn methods using this representation formalism. We present an implementation of the approach within the mega proof planning system, which we call Learnma tic. We also present the results of the experiments that we ran on this implementation in order to evaluate if and how it improves the power of proof planning systems.

Classifying Isomorphic Residue Classes

We report on a case study on combining proof planning with computer algebra systems. We construct proofs for basic algebraic properties of residue classes as well as for isomorphisms between residue classes using different proving techniques, which are implemented as strategies in a multi-strategy proof planner. We show how these techniques help to successfully derive proofs in our domain and explain how the search space of the proof planner can be drastically reduced by employing computations of two computer algebra systems during the planning process. Moreover, we discuss the results of experiments we conducted which give evidence that with the help of the computer algebra systems the planner is able to solve problems for which it would fail to create a proof otherwise.

On the Comparison of Proof Planning Systems: , Ωmega and IsaPlanner

We present a framework for describing proof planners. This framework is based around a decomposition of proof planners into planning states, proof language, proof plans, proof methods, proof revision, proof control and planning algorithms. We use this framework to motivate the comparison of three recent proof planning systems, @lCLaM, @Wmega and IsaPlanner, and demonstrate how the framework allows us to discuss and illustrate both their similarities and differences in a consistent fashion. This analysis reveals that proof control and the use of contextual information in planning states are key areas in need of further investigation.