Omega-MKRP: A Proof Development Environment

This report presents the main ideas underlyingtheOmegaGamma mkrp-system, an environmentfor the development of mathematical proofs. The motivation for the development ofthis system comes from our extensive experience with traditional first-order theoremprovers and aims to overcome some of their shortcomings. After comparing the benefitsand drawbacks of existing systems, we propose a system architecture that combinesthe positive features of different types of theorem-proving systems, most notably theadvantages of human-oriented systems based on methods (our version of tactics) andthe deductive strength of traditional automated theorem provers.In OmegaGamma mkrp a user first states a problem to be solved in a typed and sorted higher-order language (called POST ) and then applies natural deduction inference rules inorder to prove it. He can also insert a mathematical fact from an integrated data-base into the current partial proof, he can apply a domain-specific problem-solvingmethod, or he can call an integrated automated theorem prover to solve a subprob-lem. The user can also pass the control to a planning component that supports andpartially automates his long-range planning of a proof. Toward the important goal ofuser-friendliness, machine-generated proofs are transformed in several steps into muchshorter, better-structured proofs that are finally translated into natural language.This work was supported by the Deutsche Forschungsgemeinschaft, SFB 314 (D2, D3)

[1]  Dieter Hutter,et al.  The Karlsruhe Induction Theorem Proving System , 1986, CADE.

[2]  Xiaorong Huang.  An Explanatory Framework for Human Theorem Proving , 1992, GWAI.

[3]  Paul Bernays,et al.  A System of Axiomatic Set Theory , 1976 .

[4]  Jörg H. Siekmann,et al.  KEIM: A Toolkit for Automated Deduction , 1994, CADE.

[5]  Alan Bundy.  A Science of Reasoning: Extended Abstract , 1990, CADE.

[6]  Michael Kohlhase,et al.  Unification in Order-Sorted Type Theory , 1992, LPAR.

[7]  Alan Bundy,et al.  The Use of Explicit Plans to Guide Inductive Proofs , 1988, CADE.

[8]  Peter Deussen,et al.  Halbgruppen und Automaten , 1971, Heidelberger Taschenbücher.

[9]  Dominique Pastre,et al.  MUSCADET: An Automatic Theorem Proving System Using Knowledge and Metaknowledge in Mathematics , 1989, Artif. Intell..

[10]  G. Gentzen.  Untersuchungen über das logische Schließen. I , 1935 .

[11]  Rance Cleaveland,et al.  Implementing mathematics with the Nuprl proof development system , 1986 .

[12]  Peter B. Andrews.  An introduction to mathematical logic and type theory - to truth through proof , 1986, Computer science and applied mathematics.

[13]  Alan Bundy,et al.  The Computer Modelling of Mathematical Reasoning , 1983 .

[14]  Norbert Eisinger.  Completeness, confluence, and related properties of clause graph resolution , 1988 .

[15]  Alan Bundy,et al.  -mkrp: a Proof Development Environment , 1994 .

[16]  Larry Wos,et al.  Automated Reasoning: Introduction and Applications , 1984 .

[17]  X. Huang.  Reference Choices in Mathematical Proofs , 1990, ECAI.

[18]  I. G. BONNER CLAPPISON.  Editor , 1960, The Electric Power Engineering Handbook - Five Volume Set.

[19]  Lawrence C. Paulson,et al.  Isabelle: The Next 700 Theorem Provers , 2000, ArXiv.

[20]  Christoph Lingenfelder,et al.  Structuring Computer Generated Proofs , 1989, IJCAI.

[21]  Alonzo Church,et al.  A formulation of the simple theory of types , 1940, Journal of Symbolic Logic.

[22]  Manfred Kerber.  How to Prove Higher Order Theorems in First Order Logic , 1991, IJCAI.

[23]  Manfred Kerber.  On the representation of mathematical concepts and their translation into first-order logic , 1992 .

[24]  J. Neumann,et al.  Die Axiomatisierung der Mengenlehre , 1928 .

[25]  G. B. M..  Principia Mathematica , 1911, Nature.

[26]  E. Zermelo.  Untersuchungen über die Grundlagen der Mengenlehre. I , 1908 .

[27]  C. Torrance.  Review: Kurt Gödel, The Consistency of the Axiom of Choice and of the Generalized Continuum-Hypothesis with the Axioms of Set Theory , 1941 .

[28]  Xiaorong Huang,et al.  Human oriented proof presentation - a reconstructive approach , 1996, DISKI.

[29]  Jörg H. Siekmann,et al.  The Markgraf Karl Refutation Procedure , 1980, IJCAI.

[30]  Gottlob Frege,et al.  Begriffsschrift, eine der arithmetischen nachgebildete Formelsprache des reinen Denkens , 1879 .

[31]  G. Gentzen.  Untersuchungen über das logische Schließen. II , 1935 .

[32]  F. Dick.  A survey of the project Automath , 1980 .

[33]  Robert S. Boyer,et al.  Computational Logic , 1990, ESPRIT Basic Research Series.

[34]  Allen Newell,et al.  Empirical explorations with the logic theory machine: a case study in heuristics , 1995 .

[35]  A. Fraenkel,et al.  Zu den Grundlagen der Cantor-Zermeloschen Mengenlehre , 1922 .