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....

The team work method is a concept for distributing automated theorem provers and so to activate several experts to work on a given problem. We have implemented this for pure equational logic using the unfailing Knuth-Bendix completion procedure as basic prover. In this paper we present three classes of experts working in a goal oriented fashion. In general, goal oriented experts perform their job "unfair" and so are often unable to solve a given problem alone. However, as a team member in the team work method they perform highly efficient, even in comparison with such respected provers as Otter 3.0 or REVEAL, as we demonstrate by examples, some of which can only be proved using team work. The reason for these achievements results from the fact that the team work method forces the experts to compete for a while and then to cooperate by exchanging their best results. This allows one to collect "good" intermediate results and to forget "useless" ones. Completion based proof methods are fr...

Problems stemming from the study of logic calculi in connection with an inference rule called "condensed detachment" are widely acknowledged as prominent test sets for automated deduction systems and their search guiding heuristics. It is in the light of these problems that we demonstrate the power of heuristics that make use of past proof experience with numerous experiments. We present two such heuristics. The first heuristic attempts to re-enact a proof of a proof problem found in the past in a flexible way in order to find a proof of a similar problem. The second heuristic employs "features" in connection with past proof experience to prune the search space. Both these heuristics not only allow for substantial speed-ups, but also make it possible to prove problems that were out of reach when using so-called basic heuristics. Moreover, a combination of these two heuristics can further increase performance. We compare our results with the results the creators of Otter obtained with t...

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...

: In this report we give an overview of the development of our new Waldmeister prover for equational theories. We elaborate a systematic stepwise design process, starting with the inference system for unfailing Knuth--Bendix completion and ending up with an implementation which avoids the main diseases today's provers suffer from: overindulgence in time and space. Our design process is based on a logical three--level system model consisting of basic operations for inference step execution, aggregated inference machine, and overall control strategy. Careful analysis of the inference system for unfailing completion has revealed the crucial points responsible for time and space consumption. For the low level of our model, we introduce specialized data structures and algorithms speeding up the running system and cutting it down in size --- both by one order of magnitude compared with standard techniques. Flexible control of the mid--level aggregation inside the resulting prover is made po...

Gaining efficiency in completion-based theorem proving requires improvements on three levels: fast inference step execution, careful aggregation into an inference machine, and sophisticated control strategies, all that combined with space saving representation of derived facts.

this paper we first introduce the TEAMWORK method and the DISCOUNT

Proof reconstruction is the operation of extracting the computed proof from the trace of a theorem-proving run. We study the problem of proof reconstruction in distributed theorem proving: because of the distributed nature of the derivation and especially because of deletions of clauses by contraction, it may happen that a deductive process generates the empty clause, but does not have all the necessary information to reconstruct the proof. We analyze this problem and we present a method for distributed theorem proving, called Modified Clause-Diffusion, which guarantees that the deductive process that generates the empty clause will be able to reconstruct the distributed proof. This result is obtained without imposing a centralized control on the deductive processes or resorting to a round of post-processing with ad hoc communication. We prove that Modified Clause-Diffusion is fair (hence complete) and guarantees proof reconstruction. First we define a set of conditions, next we prove that they are sufficient for proof reconstruction, then we show that Modified Clause-Diffusion satisfies them. Fairness is proved in the same way, which has the advantage that the sufficient conditions provide a treatment of the problem relevant for distributed theorem proving in general. 1.

. The Distributed Larch Prover, DLP, is a distributed and parallel version of LP, an interactive prover. DLP helps users find proofs by creating and managing many proof attempts that run in parallel. Parallel attempts may cooperate by working on different subgoals, and they may compete by using different inference methods to prove the same goal. DLP runs on a network of workstations. 1 Introduction The Distributed Larch Prover, DLP, is an experiment in parallelizing LP, the Larch Prover. LP is an interactive, rewrite-rule based reasoning system for proving formulas in first-order, multi-sorted logic by first-order reasoning and induction [4]. DLP runs on a network of workstations. DLP uses a novel approach for exploiting parallelism. The user of DLP is encouraged to launch many parallel attempts to prove conjectures. Some attempts compete by using different inference methods, such as proof-by-cases and proofby -induction, to try to prove the same goal, while other attempts cooperate...