Integrating Computer Algebra into Proof Planning

• Authors: Manfred Kerber;Michael Kohlhase;Volker Sorge
• Venue: Journal of Automated Reasoning
• Year: 1998

Mechanized reasoning systems and computer algebra systems have differentobjectives. Their integration is highly desirable, since formal proofs ofteninvolve both of the two different tasks proving and calculating. Even moreimportant, proof and computation are often interwoven and not easilyseparable.In this article we advocate an integration of computer algebra intomechanized reasoning systems at the proof plan level. This approach allowsus to view the computer algebra algorithms as methods, that is, declarativerepresentations of the problem-solving knowledge specific to a certainmathematical domain. Automation can be achieved in many cases by searchingfor a hierarchic proof plan at the method level by using suitabledomain-specific control knowledge about the mathematical algorithms. Inother words, the uniform framework of proof planning allows us to solve alarge class of problems that are not automatically solvable by separatesystems.Our approach also gives an answer to the correctness problems inherent insuch an integration. We advocate an approach where the computer algebrasystem produces high-level protocol information that can be processed by aninterface to derive proof plans. Such a proof plan in turn can be expandedto proofs at different levels of abstraction, so the approach is well suitedfor producing a high-level verbalized explication as well as for alow-level, machine-checkable, calculus-level proof.We present an implementation of our ideas and exemplify them using anautomatically solved example.Changes in the criterion of ‘rigor of the proof’ engendermajor revolutions in mathematics. H. Poincaré, 1905