Mateja Jamnik

On Automating Diagrammatic Proofs of Arithmetic Arguments

Theorems in automated theorem proving are usually proved by formal logical proofs. However, there is a subset of problems which humans can prove by the use of geometric operations on diagrams, so called diagrammatic proofs. Insight is often more clearly perceived in these proofs than in the corresponding algebraic proofs; they capture an intuitive notion of truthfulness that humans find easy to see and understand. We are investigating and automating such diagrammatic reasoning about mathematical theorems. Concrete, rather than general diagrams are used to prove particular concrete instances of the universally quantified theorem. The diagrammatic proof is captured by the use of geometric operations on the diagram. These operations are the “inference steps” of the proof. An abstracted schematic proof of the universally quantified theorem is induced from these proof instances. The constructive ω-rule provides the mathematical basis for this step from schematic proofs to theoremhood. In this way we avoid the difficulty of treating a general case in a diagram. One method of confirming that the abstraction of the schematic proof from the proof instances is sound is proving the correctness of schematic proofs in the meta-theory of diagrams. These ideas have been implemented in the system, called Diamond, which is presented here.

Combined reasoning by automated cooperation

Different reasoning systems have different strengths and weaknesses, and often it is useful to combine these systems to gain as much as possible from their strengths and retain as little as possible from their weaknesses. Of particular interest is the integration of first-order and higher-order techniques. First-order reasoning systems, on the one hand, have reached considerable strength in some niches, but in many areas of mathematics they still cannot reliably solve relatively simple problems, for example, when reasoning about sets, relations, or functions. Higher-order reasoning systems, on the other hand, can solve problems of this kind automatically. But the complexity inherent in their calculi prevents them from solving a whole range of problems. However, while many problems cannot be solved by any one system alone, they can be solved by a combination of these systems. We present a general agent-based methodology for integrating different reasoning systems. It provides a generic integration framework which facilitates the cooperation between diverse reasoners, but can also be refined to enable more efficient, specialist integrations. We empirically evaluate its usefulness, effectiveness and efficiency by case studies involving the integration of first-order and higher-order automated theorem provers, computer algebra systems, and model generators.

Speedith: a diagrammatic reasoner for spider diagrams

In this paper, we introduce Speedith which is a diagrammatic theorem prover for the language of spider diagrams. Spider diagrams are a well-known logic for which there is a sound and complete set of inference rules. Speedith provides a way to input diagrams, transform them via the diagrammatic inference rules, and prove diagrammatic theorems. It is designed as a program that plugs into existing general purpose theorem provers. This allows for seamless formal verification of diagrammatic proof steps within established proof assistants such as Isabelle. We describe the general structure of Speedith, the diagrammatic language, the automatic mechanism that draws the diagrams when inference rules are applied on them, and how formal diagrammatic proofs are constructed.

Using Animation in Diagrammatic Theorem Proving

Diagrams have many uses in mathematics, one of the most ambitious of which is as a form of proof. The domain we consider is real analysis, where quantification issues are subtle but crucial. Computers offer new possibilities in diagrammatic reasoning, one of which is animation. Here we develop animated rules as a solution to problems of quantification. We show a simple application of this to constraint diagrams, and also how it can deal with the more complex questions of quantification and generalisation in diagrams that use more specific representations. This allows us to tackle difficult theorems that previously could only be proved algebraically.

Diabelli: a heterogeneous proof system

We present Diabelli, a formal reasoning system that enables users to construct so-called heterogeneous proofs that intermix sentential formulae with diagrams.

Experiments with an Agent-Oriented Reasoning System

This paper discusses experiments with an agent oriented approach to automated and interactive reasoning. The approach combines ideas from two subfields of AI (theorem proving/proof planning and multi-agent systems) and makes use of state of the art distribution techniques to decentralise and spread its reasoning agents over the internet. It particularly supports cooperative proofs between reasoning systems which are strong in different application areas, e.g., higher-order and first-order theorem provers and computer algebra systems.

Can a Higher-Order and a First-Order Theorem Prover Cooperate?

State-of-the-art first-order automated theorem proving systems have reached considerable strength over recent years. However, in many areas of mathematics they are still a long way from reliably proving theorems that would be considered relatively simple by humans. For example, when reasoning about sets, relations, or functions, first-order systems still exhibit serious weaknesses. While it has been shown in the past that higher-order reasoning systems can solve problems of this kind automatically, the complexity inherent in their calculi and their inefficiency in dealing with large numbers of clauses prevent these systems from solving a whole range of problems.

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.

A Framework for Heterogeneous Reasoning in Formal and Informal Domains

Heterogeneous reasoning refers to theorem proving with mixed diagrammatic and sentential languages and inference steps. We introduce a heterogeneous logic that enables a simple and flexible way to extend logics of existing general-purpose theorem provers with representations from entirely different and possibly not formalised domains. We use our heterogeneous logic in a framework that enables integrating different reasoning tools into new heterogeneous reasoning systems. Our implementation of this framework is MixR – we demonstrate its flexibility and extensibility with a few examples.

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.