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Integrating computer algebra into proof planning
 Journal of Automated Reasoning
, 1998
"... Abstract. Mechanised reasoning systems and computer algebra systems have different objectives. Their integration is highly desirable, since formal proofs often involve both of the two di erent tasks, proving and calculating. Even more importantly, proof and computation are often interwoven and not e ..."
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Cited by 41 (26 self)
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Abstract. Mechanised reasoning systems and computer algebra systems have different objectives. Their integration is highly desirable, since formal proofs often involve both of the two di erent tasks, proving and calculating. Even more importantly, proof and computation are often interwoven and not easily separable. In this contribution we advocate an integration of computer algebra into mechanised reasoning systems at the proof plan level. This approach allows to view the computer algebra algorithms as methods, that is, declarative representations of the problem solving knowledge speci c to a certain mathematical domain. Automation can be achieved in many cases bysearching for a hierarchic proof plan at the methodlevel using suitable domainspeci c control knowledge about the mathematical algorithms. In other words, the uniform framework of proof planning allows to solve a large class of problems that are not automatically solvable by separate systems. Our approach also gives an answer to the correctness problems inherent insuch an integration. We advocate an approach where the computer algebra system produces highlevel protocol information that can be processed by aninterface to derive proof plans. Such a proof plan in turn can be expanded to proofs at di erent levels of abstraction, so the approach iswellsuited for producing a highlevel verbalised explication as well as for a lowlevel machine checkable calculuslevel proof. We present an implementation of our ideas and exemplify them using an automatically solved example. Changes in the criterion of `rigour of the proof ' engender major revolutions in mathematics.
PROVERB  A System Explaining MachineFound Proofs
 IN PROC. OF 16TH ANNUAL CONFERENCE OF THE COGNITIVE SCIENCE SOCIETY
, 1994
"... This paper outlines an implemented system called PROVERB that explains machinefound natural deduction proofs in natural language. Different from earlier works, we pursue a reconstructive approach. Based on the observation that natural deduction proofs are at a too low level of abstraction compared ..."
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Cited by 10 (3 self)
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This paper outlines an implemented system called PROVERB that explains machinefound natural deduction proofs in natural language. Different from earlier works, we pursue a reconstructive approach. Based on the observation that natural deduction proofs are at a too low level of abstraction compared with proofs found in mathematical textbooks, we define first the concept of socalled assertion level inference rules. Derivations justified by these rules can intuitively be understood as the application of a definition or a theorem. Then an algorithm is introduced that abstracts machinefound ND proofs using the assertion level inference rules. Abstracted proofs are then verbalized into natural language by a presentation module. The most significant feature of the presentation module is that it combines standard hierarchical text planning and techniques that locally organize argumentative texts based on the derivation relation under the guidance of a focus mechanism. The behavior of the s...
Adapting Methods to Novel Tasks in Proof Planning
 KI94: ADVANCES IN ARTIFICIAL INTELLIGENCE  PROCEEDINGS OF KI94, 18TH GERMAN ANNUAL CONFERENCE ON ARTIFICIAL INTELLIGENCE
, 1994
"... In this paper we generalize the notion of method for proof planning. While we adopt the general structure of methods introduced by Alan Bundy, we make an essential advancement in that we strictly separate the declarative knowledge from the procedural knowledge. This change of paradigm not only leads ..."
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Cited by 8 (8 self)
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In this paper we generalize the notion of method for proof planning. While we adopt the general structure of methods introduced by Alan Bundy, we make an essential advancement in that we strictly separate the declarative knowledge from the procedural knowledge. This change of paradigm not only leads to representations easier to understand, it also enables modeling the important activity of formulating metamethods, that is, operators that adapt the declarative part of existing methods to suit novel situations. Thus this change of representation leads to a considerably strengthened planning mechanism. After presenting our declarative approach towards methods we describe the basic proof planning process with these. Then we define the notion of metamethod, provide an overview of practical examples and illustrate how metamethods can be integrated into the planning process.
Adapting the Diagonalization Method by Reformulations
, 1995
"... ion, Reformulation, and Approximation, SARA95, Ville d'Esterel, Canada, 1995, forthcoming. Adapting the Diagonalization Method by Reformulations Xiaorong Huang Manfred Kerber Lassaad Cheikhrouhou Fachbereich Informatik Universitat des Saarlandes D66041 Saarbrucken Germany fhuangkerberlas ..."
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Cited by 3 (1 self)
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ion, Reformulation, and Approximation, SARA95, Ville d'Esterel, Canada, 1995, forthcoming. Adapting the Diagonalization Method by Reformulations Xiaorong Huang Manfred Kerber Lassaad Cheikhrouhou Fachbereich Informatik Universitat des Saarlandes D66041 Saarbrucken Germany fhuangkerberlassaadg@cs.unisb.de Abstract Extending the planbased paradigm for automated theorem proving, we developed in previous work a declarative approach towards representing methods in a proof planning framework to support their mechanical modification. This paper presents a detailed study of a class of particular methods, embodying variations of a mathematical technique called diagonalization. The purpose of this paper is mainly twofold. First we demonstrate that typical mathematical methods can be represented in our framework in a natural way. Second we illustrate our philosophy of proof planning: besides planning with a fixed repertoire of methods, metamethods create new methods by modifying e...
Proving Ground Completeness of Resolution by Proof Planning
, 1997
"... A lot of the human ability to prove hard mathematical theorems can be ascribed to a problemspecific problem solving knowhow. Such knowledge is intrinsically incomplete. In order to prove related problems human mathematicians, however, can go beyond the acquired knowledge by adapting their knowhow ..."
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Cited by 2 (1 self)
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A lot of the human ability to prove hard mathematical theorems can be ascribed to a problemspecific problem solving knowhow. Such knowledge is intrinsically incomplete. In order to prove related problems human mathematicians, however, can go beyond the acquired knowledge by adapting their knowhow to new related problems. These two aspects, having rich experience and extending it by need, can be simulated in a proof planning framework: the problemspecific reasoning knowledge is represented in form of declarative planning operators, called methods; since these are declarative, they can be mechanically adapted to new situations by socalled metamethods. In this contribution we apply this framework to two prominent proofs in theorem proving, first, we present methods for proving the ground completeness of binary resolution, which essentially correspond to key lemmata, and then, we show how these methods can be reused for the proof of the ground completeness of lock resolution.
An Implementation of Distributed Mathematical Services
, 1998
"... Realworld applications of theorem proving require open and modern software environments that enable modularization, distribution, interoperability, networking, and coordination. This paper describes the DMS architecture for automated theorem proving that connects a widerange of mathematical servi ..."
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Cited by 1 (0 self)
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Realworld applications of theorem proving require open and modern software environments that enable modularization, distribution, interoperability, networking, and coordination. This paper describes the DMS architecture for automated theorem proving that connects a widerange of mathematical services by a common, mathematical software bus. It also presents an implementation, OzDMS, of the architecture in the Oz programming language. OzDMS provides the functionality to turn existing theorem proving systems and tools into mathematical services that are homogeneously integrated into a networked proof development environment. The environment thus gains the services from these particular modules, but each module in turn gains from using the features of other, pluggedin components. 1 Introduction The work reported in this paper originates in the effort to develop a practical mathematical assistant system that integrates external deductive components. The\Omega megasystem [BCF + 97...
A Declarative Language for Formulating Methods
, 1994
"... ral Content: A procedure interpreting the declarative content. In this contribution it is sufficient to consider the tactic part (consisting of the declarative and procedural content) as a procedure that produces new proof lines. It is separated in a declarative and a procedural part in order to ena ..."
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ral Content: A procedure interpreting the declarative content. In this contribution it is sufficient to consider the tactic part (consisting of the declarative and procedural content) as a procedure that produces new proof lines. It is separated in a declarative and a procedural part in order to enable an automated adaption of methods. In the following we will concentrate on the specification (the first four components), which forms that part of a method that is considered in the planning process. For further details on planning see [3]. The declarative part of the specification is structured in three items: while the premises and the conclusions are kept simple, namely they are just patterns for proof lines, more sophisticated criteria for the applicability of the method must be expressed in the constraints. In order to be effective in a proof planning environment, the specification language has to respect the followin