The DISCOUNT system is a distributed equational theorem prover based on the teamwork method for knowledge-based distribution. It uses an extended version of unfailing Knuth-Bendix completion that is able to deal with arbitrarily quantified goals. DISCOUNT features many different control strategies that cooperate using the teamwork approach. Competition between multiple strategies, combined with reactive planning, results in an adaptation of the whole system to given problems, and thus in a very high degree of independence from user interaction. Teamwork also provides a suitable framework for the use of control strategies based on learning from previous proof experiences. One of these strategies forms the core of the expert global learn, which is capable of learning from successful proofs of several problems. This expert, running sequentially, was one of the entrants in the competition (DISCOUNT/GL), while a distributed DISCOUNT system running on two workstations was another entrant....

. We present two learning inference control heuristics for equational deduction. Based on data about facts that contributed to previous proofs, evaluation functions learn to select equations that are likely to be of use in new situations. The first evaluation function works by symbolic retrieval of generalized patterns from a knowledge base, the second function compiles the knowledge into abstract term evaluation trees. We analyze the performance of the two heuristics on a set of examples and demonstrate their usefulness. We also show that these strategies are well suited for cooperation in the framework of the knowledge based distribution method teamwork. 1 Introduction The last years have seen a steady increase in the power of automatic theorem provers. There are a couple of reasons for this trend. The most significant ones are hardware improvements, refined inference engines, and stronger guiding heuristics. However, despite these advances, theorem provers still cannot ri...

: Distributed models for deduction allow for more powerful proof systems, but also lead to new problems. In particular, the analysis of the deduction process becomes harder, as a number of largely independent agents may contribute to the proof. In a system including cooperating agents timing considerations can lead to further problems. In this paper we first introduce the TEAMWORK method and the DISCOUNT system for distributed equational reasoning. We point out the difficulties in obtaining a detailed representation of the proofs generated by a distributed system with completely distributed memory and present our solution for the TEAMWORK approach. Using this solution we are able to explain some of the speedups DISCOUNT was able to obtain in distributed mode. Finally we show how the machine-friendly representation of an equational proof can be transformed into a proof easily readable by humans. Keywords: Distributed deduction, equational reasoning, proof representation, TEAM...

This paper presents a new way to use planning in automated theorem proving by means of distribution. To overcome the problem that often subtasks for a proof problem can not be detected a priori (which prevents the use of the known planning and distribution techniques) we use a team of experts that work independently with different heuristics on the problem. After a certain amount of time referees judge their results using the impact of the results on the behaviour of the expert and a supervisor combines the selected results to a new starting point. This supervisor also selects the experts that can work on the problem in the next round. This selection is a reactive planning task. We outline which information the supervisor can use to fulfill this task and how this information is processed to result in a plan or to revise a plan. We also show that the use of planning for the assignment of experts to the team allows the system to solve many different examples in an acceptable time with th...

We present a method for learning heuristics employed by an automated prover to control its inference machine. The hub of the method is the adaptation of the parameters of a heuristic. Adaptation is accomplished by a genetic algorithm. The necessary guidance during the learning process is provided by a proof problem and a proof of it found in the past. The objective of learning consists in finding a parameter configuration that avoids redundant effort w.r.t. this problem and the particular proof of it. A heuristic learned (adapted) this way can then be applied profitably when searching for a proof of a similar problem. So, our method can be used to train a proof heuristic for a class of similar problems. A number of experiments (with an automated prover for purely equational logic) show that adapted heuristics are not only able to speed up enormously the search for the proof learned during adaptation. They also reduce redundancies in the search for proofs of similar theorems. This not o...

. The combination of logical and symbolic computation systems has recently emerged from prototype extensions of stand-alone systems to the study of environments allowing interaction among several systems. Communication and cooperation mechanisms of systems performing any kind of mathematical service enable to study and solve new classes of problems and to perform efficient computation by distributed specialized packages. The classification of communication and cooperation methods for logical and symbolic computation systems given in this paper provides and surveys different methodologies for combining mathematical services and their characteristics, capabilities, requirements, and differences. The methods are illustrated by recent well-known examples. We separate the classification into communication and cooperation methods. The former includes all aspects of the physical connection, the flow of mathematical information, the communication language(s) and its encoding, encryption, and ...

this paper we first introduce the TEAMWORK method and the DISCOUNT

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.

We present TWlib, a library to support the knowledge-based distribution of certain search processes. The distribution concept, called teamwork, is intended for such search processes that use a set of facts as representation of the state and addition and removal of facts as transition between states. Teamwork has proven to be quite successful in different applications by generating synergetic effects. Teamwork employs four types of components. Experts and specialists work independently on solving the search problem. Cooperation is achieved by periodically judging the new generated facts by referees and generating a new start state out of the best facts. A supervisor controls the whole system and can adapt it to the given problem by exchanging bad experts and specialists. TWlib provides communication functions (including a secure broadcast) and control skeletons for the teamwork specific flow of control. 1

Abstract. Leo-II, a resolution based theorem prover for classical higherorder logic, is currently being developed in a one year research project at the University of Cambridge, UK, with support from Saarland University, Germany. We report on the current stage of development of Leo-II. In particular, we sketch some main aspects of Leo-II’s automated proof search procedure, discuss its cooperation with first-order specialist provers, show that Leo-II is also an interactive proof assistant, and explain its shared term data structure and its term indexing mechanism. 1