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.
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).
This paper provides a description of PLATIN. With PLATIN we present an imple-mented system for planning inductive theorem proofs in equational theories that arebased on rewrite methods. We provide a survey of the underlying architecture ofPLATIN and then concentrate on details and experiences of the current implementa-tion.
We present a cooperation concept for automated theorem provers that isbased on a periodical interchange of selected results between several incarnationsof a prover. These incarnations differ from each other in the search heuristic theyemploy for guiding the search of the prover. Depending on the strengths' andweaknesses of these heuristics different knowledge and different communicationstructures are used for selecting the results to interchange.Our concept is easy to implement and can easily be integrated into alreadyexisting theorem provers. Moreover, the resulting cooperation allows the dis-tributed system to find proofs much faster than single heuristics working alone.We substantiate these claims by two case studies: experiments with the DiCoDesystem that is based on the condensed detachment rule and experiments with theSPASS system, a prover for first order logic with equality based on the super-position calculus. Both case studies show the improvements by our cooperationconcept.