Xiaorong Huang

Omega: Towards a Mathematical Assistant

Ωmega is a mixed-initiative system with the ultimate purpose of supporting theorem proving in main-stream mathematics and mathematics education. The current system consists of a proof planner and an integrated collection of tools for formulating problems, proving subproblems, and proof presentation.

Reconstruction Proofs at the Assertion Level

Most automated theorem provers suffer from the problem that they can produce proofs only in formalisms difficult to understand even for experienced mathematicians. Effort has been made to reconstruct natural deduction (ND) proofs from such machine generated proofs. Although the single steps in ND proofs are easy to understand, the entire proof is usually at a low level of abstraction, containing too many tedious steps. To obtain proofs similar to those found in mathematical textbooks, we propose a new formalism, called ND style proofs at the assertion level, where derivations are mostly justified by the application of a definition or a theorem. After characterizing the structure of compound ND proof segments allowing assertion level justification, we show that the same derivations can be achieved by domain-specific inference rules as well. Furthermore, these rules can be represented compactly in a tree structure. Finally, we describe a system called PROVERB, which substantially shortens ND proofs by abstracting them to the assertion level and then transforms them into natural language.

Presenting Machine-Found Proofs

This paper outlines an implemented system named PROVERB that transforms and abstracts machine-found proofs to natural deduction style proofs at an adequate level of abstraction and then verbalizes them in natural language. The abstracted proofs, originally employed only as an intermediate representation, also prove to be useful for proof planning and proving by analogy.

Ω-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)

KEIM: A Toolkit for Automated Deduction

KEIM is a collection of software modules, written in Common Lisp with CLOS, designed to be used in the implementation of automated reasoning systems. KEIM is intended to be used by those who want to build or use deduction systems (such as resolution theorem provers) without having to write the entire framework. KEIM is also suitable for embedding a reasoning component into another Common Lisp program. It offers a range of datatypes implementing a logical language of type theory (higher order logic), in which first order logic can be easily embedded. KEIMs datatypes and algorithms include: types; terms (symbols, applications, abstractions); unification and substitutions; proofs, including resolution and natural deduction styles.

Adaptation of declaratively represented methods in proof planning

The reasoning power of human-oriented plan-based reasoning systems is primarily derived from their domain-specific problem solving knowledge. Such knowledge is, however, intrinsically incomplete. In order to model the human ability of adapting existing methods to new situations we present in this work a declarative approach for representing methods, which can be adapted by so-called meta-methods. Since the computational success of this approach relies on the existence of general and strong meta-methods, we describe several meta-methods of general interest in detail by presenting the problem solving process of two familiar classes of mathematical problems. These examples should illustrate our philosophy of proof planning as well: besides planning with a pre-defined repertory of methods, the repertory of methods evolves with experience in that new ones are created by meta-methods that modify existing ones.

Adapting Methods to Novel Tasks in Proof Planning

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 meta-methods, 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.

Die Beweisentwicklungsumgebung\(\Omega\) -Mkrp

Zusammenfassung. Die Beweisentwicklungsumgebung

Proof verbalization as an application of NLG

\Omega

Paraphrasing and Aggregating Argumentative Texts Using Text Structure

-Mkrpsoll Mathematiker bei einer ihrer Haupttätigkeiten, nämlich dem Beweisen mathematischer Theoreme unterstützen. Diese Unterstützung muß so komfortabel sein, daß die rechnergestützte Suche nach formalen Beweisen leichter und insbesondere weniger aufwendig ist, als ohne das System. Dazu muß die verwendete Objektsprache ausdrucksstark sein, man muß die Möglichkeit haben, abstrakt über Beweispläne zu reden, die gefundenen Beweise müssen in einer am Menschen orientierte Form präsentiert werden und vor allem muß eine effiziente Unterstützung beim Füllen von Beweislücken zur Verfügung stehen. Das im folgenden vorgestellte