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 eectiveness 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.

Real-world applications of automated theorem proving require modern software environments that enable modularisation, networked inter-operability, robustness, and scalability. These requirements are met by the Agent-Oriented Programming paradigm of Distributed Artificial Intelligence. We argue that a reasonable framework for automated theorem proving in the large regards typical mathematical services as autonomous agents that provide internal functionality to the outside and that, in turn, are able to access a variety of existing external services. This article describes...

We report on an experiment in exploring properties of residue classes over the integers with the combined effort of a multi-strategy proof planner and two computer algebra systems. An exploration module classifies a given set and a given operation in terms of the algebraic structure they form. It then calls the proof planner to prove or refute simple properties of the operation. Moreover, we use different proof planning strategies to implement various proving techniques: from naive testing of all possible cases to elaborate techniques of equational reasoning and reduction to known cases.

One of the main applications of computational techniques to pure mathematics has been the use of computer algebra systems to perform calculations which mathematicians cannot perform by hand.

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.

The invention of suitable concepts to characterise mathematical structures is one of the most challenging tasks for both human mathematicians and automated theorem provers alike. We present an approach where automatic concept formation is used to guide non-isomorphism proofs in the residue class domain. The main idea behind the proof is to automatically identify discriminants for two given structures to show that they are not isomorphic. Suitable discriminants are generated by a theory formation system; the overall proof is constructed by a proof planner with the additional support of traditional automated theorem provers and a computer algebra system.

We discuss a pragmatic approach to integrate computer algebra into proof planning. It is based on the idea to separate computation and verification and can thereby exploit the fact that many elaborate symbolic computations are trivially checked. In proof planning the separation is realized by using a powerful computer algebra system during the planning process to do non-trivial symbolic computations. Results of these computations are checked during the refinement of a proof plan to a calculus level proof using a small, self-implemented, system that gives us protocol information on its calculation. This protocol can be easily expanded into a checkable low-level calculus proof ensuring the correctness of the computation. We demonstrate our approach with the concrete implementation in the Omega system.

This paper describes a the integration of computer algebra systems and constraint solvers into proof planners. It shows how efficient external reasoners can be employed in proof planning and how the shortcuts of the external reasoners can be expanded to verifiable natural deduction proofs in the proof planning framework. It illustrates the integration and cooperation of the external reasoners with an example from proof planning limit theorems.

Proof planning is the application of Artificial Intelligence planning techniques to prove mathematical theorems. While exploring the domain of the residue classes over the integers with the multi-strategy proof planner Multi we found a class of hard problems on which proof planning showed a remarkable high degree of variance. On problems of the same complexity we either succeeded very quickly with short proofs or the proof planning process took significantly longer and resulted in a large proof. Recent work in Artificial Intelligence points out that the unpredictability in the running time of heuristic search procedures can often be explained by the phenomenon of heavy-tailed cost distributions. Because of the non-standard nature of these heavy-tailed cost distributions the controled introduction of randomization into the search procedures and quick restarts of the randomized procedure can eliminate heavy-tailed behavior and can take advantage of short runs. In this report,...

In this paper, we describe an architecture for resource guided concurrent mechanised deduction which is motivated by some findings in cognitive science. Its benefits are illustrated by comparing it with traditional proof search techniques. In particular, we introduce the notion of focused search and show that a reasoning system can be built as the cooperative collection of concurrently acting specialised problem solvers. These reasoners typically perform well in a particular problem domain. The system architecture that we describe assesses the subgoals of a theorem and distributes them to the specialised solvers that look the most promising. Furthermore it allocates resources (above all computation time and memory) to the specialised reasoners. This technique is referred to as resource management. Each reasoner terminates its search for a solution of a given subgoal when the solution is found or when it runs out of its assigned resources. We argue that the effect of resource ma...