A lot of the human ability to prove hard mathematical theorems can be ascribedto a problem-specific problem solving know-how. Such knowledge is intrinsicallyincomplete. In order to prove related problems human mathematicians, however,can go beyond the acquired knowledge by adapting their know-how to new relatedproblems. These two aspects, having rich experience and extending it by need, can besimulated in a proof planning framework: the problem-specific reasoning knowledge isrepresented in form of declarative planning operators, called methods; since these aredeclarative, they can be mechanically adapted to new situations by so-called meta-methods. In this contribution we apply this framework to two prominent proofs intheorem proving, first, we present methods for proving the ground completeness ofbinary resolution, which essentially correspond to key lemmata, and then, we showhow these methods can be reused for the proof of the ground completeness of lockresolution.
Mechanised reasoning systems and computer algebra systems have apparentlydifferent objectives. Their integration is, however, highly desirable, since in manyformal proofs both of the two different tasks, proving and calculating, have to beperformed. Even more importantly, proof and computation are often interwoven andnot easily separable. In the context of producing reliable proofs, the question howto ensure correctness when integrating a computer algebra system into a mechanisedreasoning system is crucial. In this contribution, we discuss the correctness prob-lems that arise from such an integration and advocate an approach in which thecalculations of the computer algebra system are checked at the calculus level of themechanised reasoning system. This can be achieved by adding a verbose mode to thecomputer algebra system which produces high-level protocol information that can beprocessed by an interface to derive proof plans. Such a proof plan in turn can beexpanded to proofs at different levels of abstraction, so the approach is well-suited forproducing a high-level verbalised explication as well as for a low-level machine check-able calculus-level proof. We present an implementation of our ideas and exemplifythem using an automatically solved extended example.
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.
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 concept for an automated theorem prover that employs a searchcontrol based on ideas from several areas of artificial intelligence (AI). The combi-nation of case-based reasoning, several similarity concepts, a cooperation conceptof distributed AI and reactive planning enables a system using our concept tolearn form previous successful proof attempts. In a kind of bootstrapping processeasy problems are used to solve more and more complicated ones.We provide case studies from two domains of interest in pure equationaltheorem proving taken from the TPTP library. These case studies show thatan instantiation of our architecture achieves a high grade of automation andoutperforms state-of-the-art conventional theorem provers.
We propose a specification language for the formalization of data types with par-tial or non-terminating operations as part of a rewrite-based logical frameworkfor inductive theorem proving. The language requires constructors for designat-ing data items and admits positive/negative conditional equations as axioms inspecifications. The (total algebra) semantics for such specifications is based onso-called data models. We present admissibility conditions that guarantee theunique existence of a distinguished data model with properties similar to thoseof the initial model of a usual equational specification. Since admissibility of aspecification requires confluence of the induced rewrite relation, we provide aneffectively testable confluence criterion which does not presuppose termination.
To prove difficult theorems in a mathematical field requires substantial know-ledge of that field. In this paper a frame-based knowledge representation formalismis presented, which supports a conceptual representation and to a large extent guar-antees the consistency of the built-up knowledge bases. We define a semantics ofthe representation by giving a translation into the underlaying logic.
We investigate the usage of so-called inference rights. We point out the prob-lems arising from the inflexibility of existing approaches to heuristically controlthe search of automated deduction systems, and we propose the application ofinference rights that are well-suited for controlling the search more flexibly. More-over, inference rights allow for a mechanism of "partial forgetting" of facts thatis not realizable in the most controlling aproaches. We study theoretical founda-tions of inference rights as well as the integration of inference rights into alreadyexisting inference systems. Furthermore, we present possibilities to control suchmodified inference systems in order to gain efficiency. Finally, we report onexperimental results obtained in the area of condensed detachment.The author was supported by the Deutsche Forschungsgemeinschaft (DFG).
The amount of user interaction is the prime cause of costs in interactiveprogram verification. This paper describes an internal analogy techniquethat reuses subproofs in the verification of state-based specifications. Itidentifies common patterns of subproofs and their justifications in orderto reuse these subproofs; thus significant savings on the number of userinteractions in a verification proof are achievable.
We present an empirical study of mathematical proofs by diagonalization, the aim istheir mechanization based on proof planning techniques. We show that these proofs canbe constructed according to a strategy that (i) finds an indexing relation, (ii) constructsa diagonal element, and (iii) makes the implicit contradiction of the diagonal elementexplicit. Moreover we suggest how diagonal elements can be represented.