Using Open Mathematical Documents to interface Computer ... · ... Using Open Mathematical Documents

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

Spain

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

Introduction

Kenzo:

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

Goal:

Increase the reliability of Kenzo

Integration of Kenzo with ACL2

Necessary:

ComputationRepresentation of the Mathematical KnowledgeDeduction


Introduction

Introduction

Kenzo:


Goal:Increase the reliability of Kenzo


Necessary:



Introduction

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)

ACL2

Programming LanguageFirst-Order LogicTheorem Prover

Proof techniques:

SimplificationInduction“The Method”

Encapsulate: to the constrained introduction of new functions

SignaturesPropertiesWitness


Introduction

Deduction


ACL2


Proof techniques:





Introduction

Deduction


ACL2


Proof techniques:





Introduction

Deduction


ACL2


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

Goal:

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




Goal:


Sub-language:








Goal:


Sub-language:








Goal:


Sub-language:







Logic to interact with Kenzo

Kenzo keeps on growing

Goal:

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





Goal:


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







Goal:







Presentation for the GUI

Goal:Define the structure of our GUI

Sub-language:

MathWeb sub-language

<OMForeign> tagXUL

XUL:

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





Sub-language:MathWeb sub-language

<OMForeign> tagXUL

XUL:

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

Goal:

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

Goal:







ACL2 Interpreter

Goal:







ACL2 presentation for the GUI

Goal:

Integrate the GUI with ACL2

Sub-language:

MathWeb sub-language

<OMForeign> tagXUL




Goal:



<OMForeign> tagXUL




Goal:



<OMForeign> tagXUL




Goal:



<OMForeign> tagXUL




Goal:

Glue all the parts of a mathematical structure

Sub-language:

Basic OMDoc

<omgroup> tag

<omdoc id="simplicial-sets.omdoc">

...

<omgroup type="sequence">

<ref xref="simplicial-sets-logic"/>

<ref xref="simplicial-sets-presentation"/>

<ref xref="simplicial-sets-conceptual-model"/>

</omgroup>

...

</omdoc>




Goal:


Sub-language:Basic OMDoc

<omgroup> tag


...





</omgroup>

...

</omdoc>




Goal:


Sub-language:Basic OMDoc

<omgroup> tag


...





</omgroup>

...

</omdoc>



Workflow

Loading Simplicial Sets and ACL2 in our GUI:

Figure: Workflow diagram



Workflow





Workflow





Workflow





Workflow





Workflow





Workflow





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

Spain

8th International Conference on Mathematical Knowledge Management

July 11, 2009
