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Using Open Mathematical Documents to interfaceComputer Algebra and Proof Assistant systems

Jonathan Heras Vico Pascual Julio Rubio

Departamento de Matematicas y ComputacionUniversidad de La Rioja

8th International Conference on Mathematical Knowledge Management

July 11, 2009

J. Heras, V. Pascual and J. Rubio Interfacing Computer Algebra and Proof Assistant systems 1/20

Table of Contents

1 Introduction

2 Specifying with OMDoc Documents

3 Conclusions and Further Work

Introduction

Table of Contents

1 Introduction

Introduction

Kenzo:

Symbolic Computation System devoted to Algebraic TopologyHomology groups unreachable by any other means

Increase the reliability of Kenzo

Integration of Kenzo with ACL2

Necessary:

ComputationRepresentation of the Mathematical KnowledgeDeduction

Introduction

Kenzo:

Goal:Increase the reliability of Kenzo

Necessary:

Introduction

Kenzo:

Goal:Increase the reliability of Kenzo

Necessary:

Introduction

Computation

Kenzo + mediated access

Figure: Simplified diagram of the architecture

Introduction

Representation

OpenMath: XML standard

Kenzo works with the main mathematical structures used in Simplicial Algebraic Topology

A CD without axioms for each Mathematical Structure was developed

Introduction

Representation

Introduction

Representation

Introduction

Deduction

ACL2 (A Computational Logic for an Applicative CommonLisp)

Programming LanguageFirst-Order LogicTheorem Prover

Proof techniques:

SimplificationInduction“The Method”

Encapsulate: to the constrained introduction of new functions

SignaturesPropertiesWitness

Introduction

Deduction

Proof techniques:

Introduction

Deduction

Proof techniques:

Introduction

Deduction

Proof techniques:

Introduction

Gathering all the pieces

A Graphical User Interface to gather all the pieces wasdeveloped

Customizable by means of an OMDoc Document RepositoryOpenMath → OMDoc due to OMDoc tools

Specifying with OMDoc Documents

Table of Contents

1 Introduction

OMDoc Documents

OMDoc format:mathematical documents + knowledge encapsulate in themthree levels of information:

formulæmathematical statementsmathematical theories

Sub-languages:

OMDoc Documents

Sub-languages:

Figure: OMDoc Sub-Languages and ModulesJ. Heras, V. Pascual and J. Rubio Interfacing Computer Algebra and Proof Assistant systems 10/20

OMDoc Documents

Sub-languages:

Figure: OMDoc Sub-Languages usedJ. Heras, V. Pascual and J. Rubio Interfacing Computer Algebra and Proof Assistant systems 10/20

OMDoc Documents

Sub-languages:

5 kinds of OMDoc documents:

Definition of Mathematical StructuresLogic to interact with KenzoPresentation for the GUIACL2 interpreterACL2 presentation for the GUI

Definition of Mathematical Structures

Define the mathematical structures of Kenzo

Sub-language:

OMDoc Content Dictionaries Sub-language

OMDoc Content Dictionaries for each mathematical structure:

SignaturePropertiesExample

OpenMath CDs ⊂ OMDoc CDs

Sub-language:

Logic to interact with Kenzo

Kenzo keeps on growing

Include new functionalityExtensibility of the system

Sub-language:

MathWeb sub-language (EXT module)

<code> tagCommon Lisp codecode for different applications

Functionality for each mathematical structure defined

Sub-language:MathWeb sub-language (EXT module)

Presentation for the GUI

Goal:Define the structure of our GUI

Sub-language:

MathWeb sub-language

<OMForeign> tagXUL

Mozilla’s XML-based user interface languageTo build feature rich cross-platforms

Sub-language:MathWeb sub-language

<OMForeign> tagXUL

Mozilla’s XML-based user interface languageTo build feature rich cross-platforms

<OMForeign> tagXUL

XUL:Mozilla’s XML-based user interface languageTo build feature rich cross-platforms

ACL2 Interpreter

Integrate Kenzo with ACL2By means of OMDoc CDs

Sub-language:

MathWeb sub-language (EXT module)

<code> tagInterpreter from OMDoc CDs to ACL2 encapsulates

OMDoc CDs ACL2 EncapsulatesSignatures → SignaturesProperties → PropertiesExamples → Witness

ACL2 Interpreter

ACL2 presentation for the GUI

Integrate the GUI with ACL2

Sub-language:

MathWeb sub-language

<OMForeign> tagXUL

Glue all the parts of a mathematical structure

Sub-language:

Basic OMDoc

<omgroup> tag

</omgroup>

</omdoc>

Sub-language:Basic OMDoc

<omgroup> tag

</omgroup>

</omdoc>

Sub-language:Basic OMDoc

<omgroup> tag

</omgroup>

</omdoc>

Workflow

Loading Simplicial Sets and ACL2 in our GUI:

Figure: Workflow diagram

Workflow

Conclusions and Further Work

Table of Contents

1 Introduction

Conclusions

Conclusions:An OMDoc Documents Repository has been developedSame OMDoc sub-language to reach different goalsIntegration of representation, computation and deduction inthe same system

Further Work:

Implement interesting interactions between ACL2 and KenzoIntegration with other Symbolic Computation Systems

Further Work

Further Work:Implement interesting interactions between ACL2 and KenzoIntegration with other Symbolic Computation Systems

Further Work

Further Work:Implement interesting interactions between ACL2 and KenzoIntegration with other Symbolic Computation Systems

The End Thank you for your attention

Using Open Mathematical Documents to interfaceComputer Algebra and Proof Assistant systems

Jonathan Heras Vico Pascual Julio Rubio

Departamento de Matematicas y ComputacionUniversidad de La Rioja

8th International Conference on Mathematical Knowledge Management

July 11, 2009