We present an approach to learning cooperative behavior of agents. Our ap-proach is based on classifying situations with the help of the nearest-neighborrule. In this context, learning amounts to evolving a set of good prototypical sit-uations. With each prototypical situation an action is associated that should beexecuted in that situation. A set of prototypical situation/action pairs togetherwith the nearest-neighbor rule represent the behavior of an agent.We demonstrate the utility of our approach in the light of variants of thewell-known pursuit game. To this end, we present a classification of variantsof the pursuit game, and we report on the results of our approach obtained forvariants regarding several aspects of the classification. A first implementationof our approach that utilizes a genetic algorithm to conduct the search for a setof suitable prototypical situation/action pairs was able to handle many differentvariants.
The team work method is a concept for distributing automated theoremprovers and so to activate several experts to work on a given problem. We haveimplemented this for pure equational logic using the unfailing KnuthADBendixcompletion procedure as basic prover. In this paper we present three classes ofexperts working in a goal oriented fashion. In general, goal oriented experts perADform 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 effiADcient, 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 teamwork.The reason for these achievements results from the fact that the team workmethod forces the experts to compete for a while and then to cooperate by exADchanging their best results. This allows one to collect "good" intermediate resultsand to forget "useless" ones. Completion based proof methods are frequently reADgarded to have the disadvantage of being not goal oriented. We believe that ourapproach overcomes this disadvantage to a large extend.
We present an overview of various learning techniques used in automated theorem provers. We characterize the main problems arising in this context and classify the solutions to these problems from published approaches. We analyze the suitability of several combinations of solutions for different approaches to theorem proving and place these combinations in a spectrum ranging from provers using very specialized learning approaches to optimally adapt to a small class of proof problems, to provers that learn more general kinds of knowledge, resulting in systems that are less efficient in special cases but show improved performance for a wide range of problems. Finally, we suggest combinations of solutions for various proof philosophies.
We present first steps towards fully automated deduction that merely requiresthe user to submit proof problems and pick up results. Essentially, this necessi-tates the automation of the crucial step in the use of a deduction system, namelychoosing and configuring an appropriate search-guiding heuristic. Furthermore,we motivate why learning capabilities are pivotal for satisfactory performance.The infrastructure for automating both the selection of a heuristic and integra-tion of learning are provided in form of an environment embedding the "core"deduction system.We have conducted a case study in connection with a deduction system basedon condensed detachment. Our experiments with a fully automated deductionsystem 'AutoCoDe' have produced remarkable results. We substantiate Au-toCoDe's encouraging achievements with a comparison with the renowned the-orem prover Otter. AutoCoDe outperforms Otter even when assuming veryfavorable conditions for Otter.
Evolving Combinators
(1996)
One of the many abilities that distinguish a mathematician from an auto-mated deduction system is to be able to offer appropriate expressions based onintuition and experience that are substituted for existentially quantified variablesso as to simplify the problem at hand substantially. We propose to simulate thisability with a technique called genetic programming for use in automated deduc-tion. We apply this approach to problems of combinatory logic. Our experimen-tal results show that the approach is viable and actually produces very promisingresults. A comparison with the renowned theorem prover Otter underlines theachievements.This work was supported by the Deutsche Forschungsgemeinschaft (DFG).
We present a method for making use of past proof experience called flexiblere-enactment (FR). FR is actually a search-guiding heuristic that uses past proofexperience to create a search bias. Given a proof P of a problem solved previouslythat is assumed to be similar to the current problem A, FR searches for P andin the "neighborhood" of P in order to find a proof of A.This heuristic use of past experience has certain advantages that make FRquite profitable and give it a wide range of applicability. Experimental studiessubstantiate and illustrate this claim.This work was supported by the Deutsche Forschungsgemeinschaft (DFG).
A method for efficiently handling associativity and commutativity (AC) in implementations of (equational) theorem provers without incorporating AC as an underlying theory will be presented. The key of substantial efficiency gains resides in a more suitable representation of permutation-equations (such as f(x,f(y,z))=f(y,f(z,x)) for instance). By representing these permutation-equations through permutations in the mathematical sense (i.e. bijective func- tions :{1,..,n} {1,..,n}), and by applying adapted and specialized inference rules, we can cope more appropriately with the fact that permutation-equations are playing a particular role. Moreover, a number of restrictions concerning application and generation of permuta- tion-equations can be found that would not be possible in this extent when treating permu- tation-equations just like any other equation. Thus, further improvements in efficiency can be achieved.
We are going to present two methods that allow to exploit previous expe-rience in the area of automated deduction. The first method adapts (learns)the parameters of a heuristic employed for controlling the application of infer-ence rules in order to find a known proof with as little redundant search effortas possible. Adaptation is accomplished by a genetic algorithm. A heuristiclearned that way can then be profitably used to solve similar problems. Thesecond method attempts to re-enact a known proof in a flexible manner in orderto solve an unknown problem whose proof is believed to lie in (close) vicinity.The experimental results obtained with an equational theorem prover show thatthese methods not only allow for impressive speed-ups, but also make it possibleto handle problems that were out of reach before.
We present an approach to automating the selection of search-guiding heuris-tics that control the search conducted by a problem solver. The approach centerson representing problems with feature vectors that are vectors of numerical val-ues. Thus, similarity between problems can be determined by using a distancemeasure on feature vectors. Given a database of problems, each problem beingassociated with the heuristic that was used to solve it, heuristics to be employedto solve a novel problem are suggested in correspondence with the similaritybetween the novel problem and problems of the database.Our approach is strongly connected with instance-based learning and nearest-neighbor classification and therefore possesses incremental learning capabilities.In experimental studies it has proven to be a viable tool for achieving the finaland crucial missing piece of automation of problem solving - namely selecting anappropriate search-guiding heuristic - in a flexible way.This work was supported by the Deutsche Forschungsgemeinschaft (DFG).
We present a method for learning heuristics employed by an automated proverto control its inference machine. The hub of the method is the adaptation of theparameters of a heuristic. Adaptation is accomplished by a genetic algorithm.The necessary guidance during the learning process is provided by a proof prob-lem and a proof of it found in the past. The objective of learning consists infinding a parameter configuration that avoids redundant effort w.r.t. this prob-lem and the particular proof of it. A heuristic learned (adapted) this way canthen 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 equationallogic) show that adapted heuristics are not only able to speed up enormously thesearch for the proof learned during adaptation. They also reduce redundancies inthe search for proofs of similar theorems. This not only results in finding proofsfaster, but also enables the prover to prove theorems it could not handle before.