Agent Based Mathematical Reasoning 1

ion also makes use of a model of the world and can be viewed as a deliberative approach to solve reasoning tasks. As an antithesis to this classical AI paradigm (and in particular to the planning paradigm) an approach has been developed that explicitly does not make use of knowledge representation and complicated deliberations (such as planning the best next step in the proof search). Brooks phrased it as \Intelligence without Reason" [7]. In this approach it is possible to obtain complex, apparently goal directed and intentional behaviour which has no long term internal state and no internal communication. This is referred to as a reactive form of modelling behaviour. Its key notions are: Situatedness: The world is its own best model. Embodiment: The world grounds regress. Intelligence: Intelligence is determined by the dynamics of interaction with the world. Emergence: Intelligence is in the eye of the observer. Although the machine-oriented approaches were not designed in the light of recent work in reactive systems, they can be reinterpreted in this framework. The main aspect is the locality of the search for a solution. For instance, when we consider binary resolution theorem proving, the decision on which two literals to perform a resolution step is often made on the basis of the knowledge of the current proof state only. That is, it does not depend on what has been done previously (of course this view simpli es matters and is not true in a strict sense for all strategies). In particular, there is no overall long term strategy to derive the empty clause. We can view the behaviour of the theorem prover as a reactive process: the world consists of clauses and there is no abstract model of these clauses. The theorem prover acts directly in this world, and the behaviour is determined by the interaction with the world. It is a characteristic of reactiveness that some reactive systems such as Otter normally do not do any backtracking. Furthermore, some reactive systems do complete restarts when the search for a proof is lost in the search space. Such restarts can also be viewed as a typical characteristic of reactive systems. For a detailed discussion of Brooks' approach and its relationship to theorem proving see [9]. Recent years have seen an attempt to reconcile the deliberative and the reactive approaches in single agent architectures [26]. This is partly motivated by looking at the human way of acting and reasoning which can be better explained as a combination of the two cases rather than by any one of them alone. Also, practical issues play an important rôle: in certain cases reactive behaviour is computationally more e cient, while in others reactive behaviour gets stuck. In the latter case deliberative behaviour can sometimes prevent blocking of a reasoning process. 4 3 Agent based mathematical reasoning A weakness of most state of the art reasoning systems is that they usually follow rigid and in exible solution strategies in their search for proofs. Instead, human mathematicians use | depending on their level of expertise | \a colourful mixture of proof strategies" (as Wittgenstein phrases it). In an attempt to prove a mathematical theorem they typically rst try a well known standard technique in the focus of the mathematical theory. If this technique does not lead to the wanted results in a reasonable amount of time, they may doubt that the theorem holds at all and look for a counterexample. If this also fails, they may try again by widening or deepening the proof search. The aim of our approach is to emulate this exible problem solving behaviour of human mathematicians in an agent based reasoning approach. Thus, our system will re ect at least some of the ideas of a sophisticated and experienced problem solver as described by P olya in [23], p. 64: \: : : when he does not succeed in guessing the whole answer, [he] tries to guess some part of the answer, some feature of the solution, some approach to the solution, or some feature of an approach to the solution. Then he seeks to expand his guess, and so he seeks to adapt his guess to the best information he can get at the moment." Agents allow a number of proof search attempts to be executed in parallel. Each agent may try a di erent proof strategy to nd the proof of a conjecture. Hence, a number of di erent proof strategies are used at the same time in the proof search. However, following all the available strategies simultaneously would quickly consume the available system resources consisting of computation time and memory space. In order to prevent this, and furthermore, to guide the proof search we propose to develop and employ a resource management concept in proof search. Resource management is a technique which distributes the available resources amongst the available agents (cf. [28]). Periodically, it assesses the state of the proof search process, evaluates the progress and redistributes the available resources accordingly. Hence, only successful or promising proof attempts will be allowed to continue searching for a proof. This process is repeated until a proof is found, or some other terminating condition is reached. An important aspect will be that in each assessment/evaluation phase the global proof state is updated, that is, promising partial proofs and especially solved subproblems are inserted into the global proof tree. Furthermore, interesting results may be communicated between the agents (for instance, an open subproblem may be passed to a theorem prover that seems to be more appropriate). The resource management mechanism analyses the theorem and decides which agents, i.e., provers, need to be launched and what proportion of the resources needs to be assigned to a particular agent. The mechanism is also responsible for restricting the amount of information exchange between agents, so that not all of the resources are 5 allocated to the communication. Figure 1 demonstrates this agent based proof planning architecture. (Agent Communication/ Exchange of Results) Assesment / Evaluation (Agent Communication/ Exchange of Results) Assesment / Evaluation Exchange of Results) Assesment / Evaluation Assesment / Evaluation (Agent Communication/ Exchange of Results) (Agent Communication/ Completed Proof Theorem Partial Proof Partial Proof Partial Proof Partial Proof PA1 PA2 PA3 PA1 PA4 PA2 PA5 PAk PAm PAn Fig. 1. The reasoning process { iterative allocation of resources to proof agents (PAx) by assessment/evaluation, and the subsequent construction of a proof of a given theorem. 6 Of course, the evaluation of the success of a proof strategy is crucial for determining the amount of resources that is allocated to an agent. This evaluation is based on the contribution that the agent has made in the proof attempt as well as on its prospect of success in the rest of the search. For example, a favourable contribution is a partial problem solution. mega integrates most external systems as glass boxes. That is, it provides mechanisms to map particular results of external systems (e.g., single clauses derived by Spass or Otter) to ND-derivations in mega's central proof data structure (PDS). This feature of mega will bene t our approach in that the evaluation of the contribution of external systems can be based on the examination of the corresponding ND-proofs or proof plans. The future prospect of an agent is estimated with respect to the updated global proof tree and according to the information communicated between the agents. The proposed system should be able to tackle mathematical problems that are currently not automatically solvable by any of the embedded systems alone (nor by any other system). An example of such a problem is described in detail in [2], we brie y summarise it here. The problem states that if there is a partition p of some set, then there is an equivalence relation q whose equivalence classes are exactly the elements of p: 8p partition(p)) (9q equivalence-rel(q) ^ (equivalence-classes(q) = p)) Note that partition, equivalence-classes, and equivalence-rel are derived higherorder concepts de ned in the mega knowledge base of mathematical theories. The search for a proof of this theorem has not been automated yet. We indicate here how the proof might be found within our proposed architecture. First, the initial proof goal is split into three subgoals (e.g., with the help of a proof planner). Namely, from a given partition p we can derive the existence of an equivalence relation q, constituting subgoal (1). For the same equivalence relation q it holds that its equivalence classes are exactly p. The two directions of the set equality give us subgoals (2) and (3). Next, higher-order equality and extensionality reasoning is required. The rst two subgoals (1) and (2) can be solved automatically by the higher-order prover Tps [1]. The last subproblem (3), which requires a fair amount of extensionality reasoning, we expect to be solvable by cooperation between the higher-order extensionality prover Leo [4] and a rst-order automated theorem prover. Leo provides the necessary higher-order extensionality treatment, however, it cannot cope with the large number of rst-order clauses that are generated subsequently. Therefore, this set of clauses could be passed via mega to the rst-order specialist available within our agent society. Our proposed system will be able to organise the sketched cooperation between the integrated systems in a goal oriented way in order to solve such kinds of problems automatically. 7 4 Theorem Proving by a Society of Agents The system we propose will provide a powerful architecture for reasoning systems consisting of a society of specialised reasoning agents. These agents are aware of their own capabilities and partly even of those of the other agents. The knowledge can initially be provided by the user or the implementor of a single agent. However, additional knowledge can be gained by evaluating successful and unsuccessful proof attempts in various mathematical domains as well as by feedback from other agents (for instance, the usefulness of results from some agents can be used in a reinforcement learning approach). Initially, a given mathematical problem is investigated in order to estimate and classify the potential of the solution strategies, i.e., of the agents, available for solving this problem. Depending on the evaluation process an initial resource distribution is computed, in particular, a main strategy line may be manifested. An infrastructure allowing to distribute resources is already provided by the Mathweb architecture [17]. An automatic evaluation module will then be added to the system. The goal is not to remove the human in this process; on the contrary, the agent and resource approach should strongly facilitate the communication between the human and the machine. In fact, human expertise can be incorporated during each of the assessment/evaluation phases. After consuming the available resources, the reasoning agents terminate and investigate whether they have produced useful information or not. For instance, the Otter-agent could look for the shortest derived clauses with assertion clauses as ancestors in order to estimate how close it is to a completed proof. More interesting in our context will be contributions of the Tps system, since it can return partial proofs to the mega-system. These results may be evaluated using adequate criteria like the complexity and the number of the remaining open subproblems. For example, the only open subproblem might be a rst-order goal, whereas the original problem was a higher-order one. Then, the partial proof may be communicated to other systems and the open subgoal can be passed to rst-order provers. Depending on the evaluation of the agents' contributions, a new resource distribution is computed. The starting point for the design of the system we propose here consists of a proof data structure and a proof planning mechanism. The rst prototype of the system can use the existing proof data structure, proof planning components, and proof methods of mega. The system will be extended by implementing a mechanism for knowledge based automatic distribution of subproblems to societies of agents, and an assessment module which will enable an interaction between agents. The agent results can be incorporated directly into mega's partial proofs, enabling the evaluation of usefulness of heterogeneous agents on some uniform 8 level. The information can then be propagated to other agents. However, incase of an unsuccessful proof attempt of the overall system a special back-tracking mechanism needs to be supplied. It has to do book-keeping on theparts of the proofs which have been computed by each agent. Furthermore,the mechanism must be able to subsequently remove both, whole and partialresults of an agent from the overall proof.One of the potential problems, which we foresee, is that increasing theheterogeneity of a system might increase the organisational complexity of thecommunication between the agents. Namely, the greater the variety of thesystems that are integrated, the less there might be a common interest tothe di erent agents, and furthermore, the general viewpoint of the problemsolving process of the overall system might be lost. For instance, some in-termediate result that is of central importance to one prover might not be ofinterest to another prover, because the proof strategies that they use are verydi erent. Hence, establishing the communication between agents might proveto be di cult. As a rst approximation, our approach will be to broadcast theresults of each agent to every other agent in the hope that the results might beuseful to other agents. In further re nements we will look into more sophisti-cated forms of communication which will allow for a more e cient exchangeof information between agents. The possibility to translate into the standardform of mega's partial proofs should help in this task.5 ConclusionWe summarise our proposal by delineating some of the most challenging re-search tasks in this project:(i) The extension of mega's underlying Mathweb-architecture and itsproof data structure by suitable resource distribution, communicationand backtracking facilities. In a rst attempt we want to adapt theblackboard mechanism underlying mega's interactive suggestion mech-anism [5,6] and integrate it with the Mathweb architecture.(ii) The development and realisation of a suitable evaluation criteria; someobvious candidates are the simplicity/complexity of partial proofs, thetheory/logic a subproblem belongs to (e.g., rst-order logic, set theory),and the similarity of open subproblems to already solved problems storedin the database.(iii) The extension of the system, such that it allows a grouping of homo-geneous agents tackling similar kinds of problems into one single meta-agent. For instance, it may be useful to group classical rst-order rea-soners together to form a centre of expertise for classical rst-order logic.Ideally, such centres of expertise may use a mechanism analogous to the9 overall system in order to organise the communication between its sys-tems (sub-agents) and to further distribute the resources they obtain atthe upper level.The systems in a centre of expertise can be evaluated using a ne-grainedevaluation criteria. Evaluation experiments of this kind have been carried outin the past on, for instance, rst-order theorem provers and other homogeneoussystems (cf. [16,27,13]). They proved to be successful and gave positive results.Hence, we could realise a more homogeneous system communication withinthe centre of expertise. Furthermore, the centres of expertise could have adynamic nature, that is, they might remodel themselves di erently for di erentproblem domains or explicitly learn in which areas their particular strengthsand weaknesses are.Related to our proposal is the work on Techs in [13] where no restrictionis imposed on the type of the provers that can be integrated into a system. Acomparison of both architectures might provide some useful insights into thepotential problems as well as the advantages of the approach proposed in thispaper. One di erence between the proposed approach and Techs is that thelatter does not provide techniques to translate selected results of the reasoningagents (e.g., clauses derived by a rst-order theorem prover) into derivationsin a uniform proof data structure, whereas for most systems in our approachthis will be possible. Hence, our evaluation criteria may exploit knowledge ona more abstract level and may relate the contributions of the agents to thecurrent partial proof in the global proof attempt.In conclusion, we propose that a reasoning system with an agent basedarchitecture incorporated into a proof planning framework, as we described inthis paper, will result in improved mechanised reasoning capabilities. Unlike aconventional distributed parallel model of theorem proving, an agent architec-ture provides a paradigm where the communication between agents and themanagement of resources for agents can be realised. The hope is that such asystem will be able to prove theorems that have previously not been provedautomatically.References[1] P. B. Andrews, M. Bishop, S. Issar, D. Nesmith, F. Pfenning, and H. Xi. TPS:A theorem proving system for classical type theory. Journal of AutomatedReasoning, 16(3):321{353, 1996.[2] C. Benzmuller, M. Bishop, and V. Sorge. Integrating tps and mega. Journalof Universal Computer Science, 1999. Special issue on Integrating of DeductionSystem, forthcoming.[3] C. Benzmuller, L. Cheikhrouhou, D. Fehrer, A. Fiedler, X. Huang, M. Kerber,M. Kohlhase, K. Konrad, E. Melis, A. Meier, W. Schaarschmidt, J. Siekmann,10 and V. Sorge. mega: Towards a mathematical assistant. In W. McCune,editor, Proceedings of the 14th Conference on Automated Deduction, pages 252{255. Townsville, Australia, 1997. Springer, LNAI 1249.[4] C. Benzmuller and M. Kohlhase. LEO | a higher-order theorem prover. InC. Kirchner and H. Kirchner, editors, Proceedings of the 15th Conference onAutomated Deduction, pages 139{144, Lindau, Germany, 1998. Springer, LNAI1421.[5] C. Benzmuller and V. Sorge. A blackboard architecture for guiding interactiveproofs. In F. Giunchiglia, editor, Arti cial Intelligence: Methodology, Systemsand Applications, pages 102{114, Sozopol, Bulgaria, 1998. Springer, LNAI 1480.[6] C. Benzmuller and V. Sorge. Critical agents supporting interactive theoremproving. SEKI-Report SR-99-02, Fachbereich Informatik, Universitat desSaarlandes, 1999.[7] R. A. Brooks. Intelligence without reason. Memo No. 1293, MIT, April 1991.[8] A. Bundy. The use of explicit plans to guide inductive proofs. In E. Luskand R. Overbeek, editors, Proceedings of the 9th Conference on AutomatedDeduction, pages 111{120, Argonne, Illinois, USA, 1988. Springer, LNCS 310.[9] A. Bundy. A subsumption architecture for theorem proving. In PhilosophicalTransactions of the Royal Society of London, pages 71{85, Volume 349/1689,October 1994.[10] A. Bundy, F. van Harmelen, C. Horn, and A. Smaill. The OYSTER-CLAM system.In M. Stickel, editor, 10th Conference on Automated Deduction, pages 647{648,1990. Springer, LNAI 449.[11] I. Dahn. Integration of automated and interactive theorem proving in ILF.In W. McCune, editor, Proceedings of the 14th Conference on AutomatedDeduction, pages 57{60. Townsville, Australia, 1997. Springer, LNAI 1249,[12] J. Denzinger. Knowledge-Based Distributed Search Using Teamwork. InProceedings of the First International Conference on Multiagent Systems(ICMAS-95), pages 81{88, 1995.[13] J. Denzinger and I. Dahn. Cooperating theorem provers. In W. Bibel and P. H.Schmitt, editors, Automated Deduction, a Basis for Application { Handbook ofthe German Focus Programme on Automated Deduction, chapter II.14, pages383{416. Kluwer academic publishers, Dordrecht, Netherlands, 1998.[14] J. Denzinger, M. Kronenburg, and S. Schulz. DISCOUNT { a distributed andlearning equational prover. Journal of Automated Reasoning, 18(2):189{198,1997.[15] R. E. Fikes and N. J. Nilsson. STRIPS: A new approach to the application oftheorem proving to problem solving. Arti cial Intelligence, 2:189{208, 1971.11 [16] M. Fisher. An open approach to concurrent theorem proving. In J. Geller,H. Kitano, and C. Suttner, editors, Parallel Processing for Arti cial Intelligence,volume 3. Elsevier/North Holland, 1997.[17] A. Franke, S. Hess, C. Jung, M. Kohlhase, and V. Sorge. Agent-orientedintegration of distributed mathematical services. Journal of UniversalComputer Science, 5(3):156{187, 1999. Special issue on Integration ofDeduction System.[18] X. Huang and A. Fiedler. Proof verbalization in PROVERB. In Proceedings ofthe First International Workshop on Proof Transformation and Presentation,pages 35{36, Schloss Dagstuhl, Germany, 1997.[19] M. Kerber, M. Kohlhase, and V. Sorge. Integrating Computer Algebra IntoProof Planning. Journal of Automated Reasoning, 21(3):327{355, 1998.[20] J. McCarthy and P. Hayes. Some philosophical problems from the standpointof arti cial intelligence. Machine Intelligence, 4:463{502, 1969.[21] W. McCune. Solution of the Robbins problem. Journal of AutomatedReasoning, 19(3):263{276, 1997.[22] E. Melis. A model of analogy-driven proof-plan construction. In C. S.Mellish, editor, Proceedings of the 14th International Joint Conference onArti cial Intelligence (IJCAI95), pages 182{189, Montreal, Canada, 1995.Morgan Kaufmann.[23] G. P olya. Mathematical Discovery. John Wiley & Sons, 3rd edition, 1981.[24] J. Richardson, A. Smaill, and I. Green. System description: proof planning inhigher-order logic with -CLAM. In C. Kirchner and H. Kirchner, editors, 15thConference on Automated Deduction, 1998. Springer, LNAI 1421.[25] J. Siekmann, S. Hess, C. Benzmuller, L. Cheikhrouhou, A. Fiedler, H. Horacek,M. Kohlhase, K. Konrad, A. Meier, E. Melis, M. Pollet, and V. Sorge. Loui:Lovely mega User Interface. Special Issue on Integrated Systems of theJournal of Formal Aspects of Computer Science, 1999. forthcoming.[26] A. Sloman. Architectural requirements for human-like agents { both naturaland arti cial. (What sorts of machines can love?). In K. Dautenhahn, editor,Human Cognition and Social Agent Technology. John Benjamins Publishing,1999.[27] A. Wolf. P-SETHEO: Strategy parallelism in automated theorem proving. InH. de Swart, editor, Proceedings of the International Conference on AutomatedReasoning with Analytic Tableaux and Related Methods (TABLEAUX-98), 1998,page 320. Springer, LNAI 1397.[28] S. Zilberstein. Models of Bounded Rationality. In AAAI Fall Symposium onRational Agency, Cambridge, Massachusetts, November 1995.12