Abstract. This paper discusses experiments with an agent oriented approach to automated and interactive reasoning. The approach combines ideas from two subfields of AI (theorem proving/proof planning and multi-agent systems) and makes use of state of the art distribution techniques to decentralise and spread its reasoning agents over the internet. It particularly supports cooperative proofs between reasoning systems which are strong in different application areas, e.g., higher-order and first-order theorem provers and computer algebra systems. 1

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.

Introduction We present recent developments within the equational theorem prover Waldmeister, an implementation of unfailing Knuth-Bendix completion [BDP89] with re nements towards ordered completion. The new developments rely on a novel organization of the underlying saturation-based proof procedure into a system architecture. As is known, the saturation process tends to quickly ll the memory available unless preventive measures are employed. To overcome this problem, our new \Waldmeister loop" features a highly compact representation of the search state, exploiting its inherent structure. The implementation just being available, the cost and the bene ts of the concept now can exactly be measured. Indeed are our expectations met concerning the drastic cut-down of memory usage with only moderate overhead of time. In addition it has turned out that the revealed structure of the search state paves the way to an easily implemented parallelization of the prover with modest communicati

this paper, we have extended our analysis to the impact of the parallelization approaches on the control of search. We observed that approaches with parallelism at the term level may replace the search plan by low-level data-driven forms of concurrency, or produce strategy-compliant parallelizations. It seems that the potential problem is a loss of control for the former, and an excess of control for the latter. Data-driven concurrency may be appropriate for ground computations that are guaranteed to converge (e.g., computing a congruence closure for ground completion), but may represent a counter-productive loss of control in general theorem proving, where the essence, from a practical point of view, is not saturation (i.e., do all the steps, with the order being a secondary issue), but effective search (i.e., find a good order to do the steps in order to avoid doing them all). Strategy-compliant parallelizations, on the other hand, may be too conservative: they avoid the risk of mixing search with parallelism, but they renounce using parallelism to try to generate better searches.

Key concerns in the development of more powerful ATP systems are to provide breadth of coverage – an ability to solve a large range of problems, and to provide greater depth of coverage – an ability to solve more difficult problems, within the same resource limits. This work describes the design and implementation of CSSCPA, a compositional competitioncooperation parallel ATP System. CSSCPA combines existing high performance ATP systems in a framework that allows them to work independently, but also allows communication of intermediate results. The performance data shows that CSSPCA has high breadth and depth of coverage. 1

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Mathematical Services . . . . . . . . . . . . . . . . . . . 10 2.3.2 Autonomy and Decentralization . . . . . . . . . . . . . . . . . . . 11 2.3.3 Coordination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 3 Distributed Articial Intelligence 12 3.1 Agent-Oriented Programming . . . . . . . . . . . . . . . . . . . . . . . . 12 3.2 The Knowledge Query and Manipulation Language . . . . . . . . . . . . 13 3.3 Coordination in Multi-Agent Systems . . . . . . . . . . . . . . . . . . . 13 4 Agent Technology for Distributed Mathematical Reasoning 15 4.1 MathWeb Agents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 4.2 Communication between MathWeb agents . . . . . . . . . . . . . . . . . 18 4.2.1 Technical Issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 4.2.2 Characterization of Reasoning Capabilities . . . . . . . . . . . . 18 4.2.3 Context in Mathematical Communication . . . . . . . . . . . . . 19 4.3 Coordination of MathWeb Agents . . . . . . . . . . . . . . . . . . . . . 20 5 Summary and Work Plan 22 5.1 Work Plan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 1

Negotiation over resources and multi-agent planning are important issues in multi-agent systems research. It has been demonstrated [19] how symbolic negotiation and distributed planning together could be formalised as distributed Linear Logic (LL) theorem proving. LL has been chosen mainly because of its expressive power for representation of resources and its computation-oriented nature. This paper extends the previous work by taking advantage of a richer fragment of LL and introducing two sorts of nondeterministic choices into negotiation. This allows agents to reason and negotiate under certain degree of uncertainty.

Linear logic presents a unified framework for describing and reasoning about stateful systems. Because of its view of hypotheses as resources, it supports such phenomena as concurrency, external and internal choice, and state transitions that are common in such domains as protocol verification, concurrent computation, process calculi and games. It accomplishes this unifying view by providing logical connectives whose behaviour is closely tied to the collection of resources, which is free of structural phenomena such as weakening (allowing more resources than necessary) or contraction (using a resource more than once). The usual (non-linear) logic is embedded in this substructural framework by means of an exponential modal operator. The interaction of the rules for multiplicative, additive and exponential connectives gives rise to a wide and expressive array of behaviours. Various approaches have been taken to produce automated reasoning systems for fragments of linear logic, usually in the form of logic programming engines; but, due to the lack of the full complement of linear connectives, uses of such systems have an idiomatic commitment, for example as serializations or in continuation-passing-style. This thesis addresses the need for automated reasoning for the complete set of operators for first order intuitionistic linear logic (i.e., ⊗, 1, ❜, &, ⊤, ⊕, 0,!, ∀, ∃), which removes the need for such idiomatic constructions and allows direct logical expression. The particular theorem proving technique used is the inverse method, which performs forward

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