Mathematical Knowledge Management in               Algebraic Topology                          J´nathan Heras Vicente     ...
Table of Contents   1   Introduction   2   Communication   3   Computation   4   Deduction   5   Conclusions and Further w...
IntroductionTable of Contents   1   Introduction   2   Communication   3   Computation   4   Deduction   5   Conclusions a...
Introduction   ContextContext       Mathematical Knowledge Management (MKM)              Computation              Deductio...
Introduction   ContextContext       Mathematical Knowledge Management (MKM)              Computation              Deductio...
Introduction   ContextContext       Mathematical Knowledge Management (MKM)              Computation              Deductio...
Introduction   Particularization of MKM to Algebraic TopologyParticularization of MKM to Algebraic Topology        Communi...
Introduction   Particularization of MKM to Algebraic TopologyParticularization of MKM to Algebraic Topology        Communi...
Introduction   Particularization of MKM to Algebraic TopologyParticularization of MKM to Algebraic Topology        Communi...
Introduction   Particularization of MKM to Algebraic TopologyParticularization of MKM to Algebraic Topology        Communi...
Introduction   Particularization of MKM to Algebraic TopologyParticularization of MKM to Algebraic Topology        Communi...
Introduction   Particularization of MKM to Algebraic TopologyParticularization of MKM to Algebraic Topology        Communi...
Introduction   Particularization of MKM to Algebraic TopologyParticularization of MKM to Algebraic Topology        Communi...
CommunicationTable of Contents   1   Introduction   2   Communication   3   Computation   4   Deduction   5   Conclusions ...
Communication   MotivationMotivation        Algebraic Topology Expert                                              ⇒ Makes...
Communication   MotivationMotivation        Algebraic Topology Expert                                              ⇒ Makes...
Communication   MotivationMotivation        Algebraic Topology Expert                                              ⇒ Makes...
Communication   MotivationGoal          Develop a system which increases Kenzo accessibility       J´nathan Heras Vicente ...
Communication   MotivationGoal          Develop a system which increases Kenzo accessibility                 Friendly fron...
Communication    Providing a mediated access to KenzoProviding a mediated access to Kenzo                                 ...
Communication    Providing a mediated access to KenzoProviding a mediated access to Kenzo                                 ...
Communication    Providing a mediated access to KenzoProviding a mediated access to Kenzo                                 ...
Communication    Providing a mediated access to KenzoProviding a mediated access to Kenzo                                 ...
Communication                   Providing a mediated access to KenzoProviding a mediated access to Kenzo                  ...
Communication                   Providing a mediated access to KenzoProviding a mediated access to Kenzo                  ...
Communication                   Providing a mediated access to KenzoProviding a mediated access to Kenzo                  ...
Communication   The whole frameworkTowards the whole framework        This architecture provides               Mediated ac...
Communication   The whole frameworkTowards the whole framework        This architecture provides               Mediated ac...
Communication                   The whole frameworkTowards different clients                                               ...
Communication                   The whole frameworkTowards different clients                                               ...
Communication       The whole frameworkTowards different clients                                                           ...
Communication       The whole frameworkTowards Efficiency                                                                 Me...
Communication                   The whole frameworkThe Kenzo framework                                                    ...
Communication   The whole frameworkTowards extensibility   Include new functionality     J´nathan Heras Vicente      o    ...
Communication   The whole frameworkTowards extensibility   Include new functionality        Increase Kenzo functionality  ...
Communication   The whole frameworkTowards extensibility   Include new functionality        Increase Kenzo functionality  ...
Communication   The whole frameworkTowards extensibility   Include new functionality        Increase Kenzo functionality  ...
Communication               The whole frameworkTowards extensibility   Include new functionality        Increase Kenzo fun...
Communication                   The whole frameworkIncluding new Kenzo functionality   <code id="new-constructor">      <d...
Communication   The whole frameworkA distinguished client of our frameworks        A Graphical User Interface implemented ...
Communication   The whole frameworkA distinguished client of our frameworks        A Graphical User Interface implemented ...
Communication   The whole frameworkA distinguished client of our frameworks        A Graphical User Interface implemented ...
Communication   The whole frameworkA distinguished client of our frameworks        A Graphical User Interface implemented ...
Communication                           The whole frameworkA distinguished client of our frameworks           A Graphical ...
Communication   The whole frameworkA customizable GUI        Features               Extensibility               Adaptabili...
Communication   The whole frameworkA customizable GUI        Features               Extensibility               Adaptabili...
Communication          The whole frameworkA customizable GUI        Features               Extensibility               Ada...
Communication   fKenzofKenzo: A tool adaptable to different needs        Beginner student of Algebraic Topology            ...
Communication   fKenzofKenzo: A tool adaptable to different needs        Beginner student of Algebraic Topology            ...
Communication   fKenzofKenzo: A tool adaptable to different needs        Beginner student of Algebraic Topology            ...
Communication   fKenzofKenzo summary       fKenzo    J´nathan Heras Vicente     o                                  Mathema...
Communication   fKenzofKenzo summary       fKenzo              Integral assistant for Algebraic Topology    J´nathan Heras...
Communication   fKenzofKenzo summary       fKenzo              Integral assistant for Algebraic Topology              Cons...
Communication   fKenzofKenzo summary       fKenzo              Integral assistant for Algebraic Topology              Cons...
Communication   fKenzofKenzo summary       fKenzo              Integral assistant for Algebraic Topology              Cons...
Communication   fKenzofKenzo summary       fKenzo              Integral assistant for Algebraic Topology              Cons...
Communication   fKenzofKenzo summary       fKenzo              Integral assistant for Algebraic Topology              Cons...
ComputationTable of Contents   1   Introduction   2   Communication   3   Computation   4   Deduction   5   Conclusions an...
ComputationNew Kenzo modules       Increase Kenzo computational capabilities              Simplicial Complexes            ...
ComputationNew Kenzo modules       Increase Kenzo computational capabilities              Simplicial Complexes            ...
ComputationNew Kenzo modules       Increase Kenzo computational capabilities              Simplicial Complexes            ...
ComputationNew Kenzo modules       Increase Kenzo computational capabilities              Simplicial Complexes            ...
Computation   Effective Homology preliminariesEffective Homology preliminaries: Intuitive Idea   Compute homology groups of ...
Computation   Effective Homology preliminariesEffective Homology preliminaries: Intuitive Idea   Compute homology groups of ...
Computation   Effective Homology preliminariesEffective Homology preliminaries: Intuitive Idea   Compute homology groups of ...
Computation   Effective Homology preliminariesEffective Homology preliminaries: Intuitive Idea   Compute homology groups of ...
Computation   Effective Homology preliminariesEffective Homology preliminaries   Definition   An effective chain complex is a ...
Computation   Effective Homology preliminariesEffective Homology preliminaries   Definition   An effective chain complex is a ...
Computation      Effective Homology preliminariesEffective Homology preliminaries   Definition   A reduction ρ between two ch...
Computation      Effective Homology preliminariesEffective Homology preliminaries   Definition   A reduction ρ between two ch...
Computation       Effective Homology preliminariesEffective Homology preliminaries   Definition   A (strong chain) equivalenc...
Computation       Effective Homology preliminariesEffective Homology preliminaries   Definition   A (strong chain) equivalenc...
Computation       Effective Homology preliminariesEffective Homology preliminaries   Definition   A (strong chain) equivalenc...
Computation   Pushout preliminariesPushout preliminaries   Definition   Let f , g morphisms,        X                 f    ...
Computation   Pushout preliminariesPushout preliminaries   Definition   Let f , g morphisms, the pushout of f , g is an obj...
Computation   Pushout preliminariesPushout preliminaries   Definition   Let f , g morphisms, the pushout of f , g is an obj...
Computation   Pushout preliminariesPushout preliminaries   Definition   Let f , g morphisms, the pushout of f , g is an obj...
Computation        Effective Homology of the PushoutConstruction of the Effective Homology of the Pushout I   Given f : X → ...
Computation        Effective Homology of the PushoutConstruction of the Effective Homology of the Pushout I   Given f : X → ...
Computation        Effective Homology of the PushoutConstruction of the Effective Homology of the Pushout I   Given f : X → ...
Computation      Effective Homology of the PushoutConstruction of the Effective Homology of the Pushout II   Given f : X → Y...
Computation      Effective Homology of the PushoutConstruction of the Effective Homology of the Pushout II   Given f : X → Y...
Computation      Effective Homology of the PushoutConstruction of the Effective Homology of the Pushout II   Given f : X → Y...
Computation      Effective Homology of the PushoutConstruction of the Effective Homology of the Pushout II   Given f : X → Y...
Computation     Effective Homology of the PushoutSketch of the algorithm   Theorem (SES Theorems)   Let                    ...
Computation        Effective Homology of the PushoutSketch of the algorithm   Theorem (SES Theorems)   Let                 ...
Computation             Effective Homology of the PushoutSketch of the algorithm continued  Step 1. From f : X → Y and g : ...
Computation   Effective Homology of the PushoutExample: π∗ (Σ(SL2 (Z)))   SL2 (Z)        Group of 2 × 2 matrices with deter...
Computation   Effective Homology of the PushoutExample: π∗ (Σ(SL2 (Z)))   SL2 (Z)        Group of 2 × 2 matrices with deter...
Computation   Effective Homology of the PushoutExample: π∗ (Σ(SL2 (Z)))   SL2 (Z)        Group of 2 × 2 matrices with deter...
Computation   Effective Homology of the PushoutExample: π∗ (Σ(SL2 (Z)))   Construction of the object K (SL2 (Z, 1))     J´n...
Computation   Effective Homology of the PushoutExample: π∗ (Σ(SL2 (Z)))   Construction of the object K (SL2 (Z, 1))        ...
Computation   Effective Homology of the PushoutExample: π∗ (Σ(SL2 (Z)))   Construction of the object K (SL2 (Z, 1))        ...
Computation   Effective Homology of the PushoutExample: π∗ (Σ(SL2 (Z)))   Construction of the object K (SL2 (Z, 1))        ...
Computation   Effective Homology of the PushoutExample: π∗ (Σ(SL2 (Z)))   Computation of π∗ (Σ(SL2 (Z)))     J´nathan Heras...
Computation   Effective Homology of the PushoutExample: π∗ (Σ(SL2 (Z)))   Computation of π∗ (Σ(SL2 (Z)))        Firstly, we...
Computation   Effective Homology of the PushoutExample: π∗ (Σ(SL2 (Z)))   Computation of π∗ (Σ(SL2 (Z)))        Firstly, we...
Computation   Effective Homology of the PushoutExample: π∗ (Σ(SL2 (Z)))   Computation of π∗ (Σ(SL2 (Z)))        Firstly, we...
Computation   Effective Homology of the PushoutExample: π∗ (Σ(SL2 (Z)))        Computing Homotopy groups continued (Whitehe...
Computation   Effective Homology of the PushoutExample: π∗ (Σ(SL2 (Z)))        Computing Homotopy groups continued (Whitehe...
DeductionTable of Contents   1   Introduction   2   Communication   3   Computation   4   Deduction   5   Conclusions and ...
Deduction   ACL2ACL2       ACL2 (A Computational Logic for an Applicative Common Lisp)   J´nathan Heras Vicente    o      ...
Deduction   ACL2ACL2       ACL2 (A Computational Logic for an Applicative Common Lisp)             Programming Language   ...
Deduction   ACL2ACL2       ACL2 (A Computational Logic for an Applicative Common Lisp)             Programming Language   ...
Deduction   ACL2ACL2       ACL2 (A Computational Logic for an Applicative Common Lisp)             Programming Language   ...
Deduction    GoalGoal   Simplified view of Kenzo way of working       1   Construction of initial spaces       2   Construc...
Deduction    GoalGoal   Simplified view of Kenzo way of working       1   Construction of initial spaces               Cert...
Deduction        Certifying the correctness of Kenzo simplicial setsMathematical context: Simplicial Sets   Definition   A ...
Deduction        Certifying the correctness of Kenzo simplicial setsMathematical context: Simplicial Sets   Definition   A ...
Deduction        Certifying the correctness of Kenzo simplicial setsMathematical context: Simplicial Sets   Definition   A ...
Deduction        Certifying the correctness of Kenzo simplicial setsMathematical context: Simplicial Sets   Definition   A ...
Deduction   Certifying the correctness of Kenzo simplicial setsMathematical context: Example                       c      ...
Deduction   Certifying the correctness of Kenzo simplicial setsMathematical context: Example                       c      ...
Deduction   Certifying the correctness of Kenzo simplicial setsMathematical context: Example                       c      ...
Deduction       Certifying the correctness of Kenzo simplicial setsMathematical context: abstract simplexes   Proposition ...
Deduction       Certifying the correctness of Kenzo simplicial setsMathematical context: abstract simplexes   Proposition ...
Deduction       Certifying the correctness of Kenzo simplicial setsMathematical context: abstract simplexes   Proposition ...
Deduction   Certifying the correctness of Kenzo simplicial setsMathematical context        degeneracy operator            ...
Deduction   Certifying the correctness of Kenzo simplicial setsMathematical context        degeneracy operator            ...
Deduction   Certifying the correctness of Kenzo simplicial setsMathematical context        degeneracy operator            ...
Mathematical Knowledge Management in Algebraic Topology
Mathematical Knowledge Management in Algebraic Topology
Mathematical Knowledge Management in Algebraic Topology
Mathematical Knowledge Management in Algebraic Topology
Mathematical Knowledge Management in Algebraic Topology
Mathematical Knowledge Management in Algebraic Topology
Mathematical Knowledge Management in Algebraic Topology
Mathematical Knowledge Management in Algebraic Topology
Mathematical Knowledge Management in Algebraic Topology
Mathematical Knowledge Management in Algebraic Topology
Mathematical Knowledge Management in Algebraic Topology
Mathematical Knowledge Management in Algebraic Topology
Mathematical Knowledge Management in Algebraic Topology
Mathematical Knowledge Management in Algebraic Topology
Mathematical Knowledge Management in Algebraic Topology
Mathematical Knowledge Management in Algebraic Topology
Mathematical Knowledge Management in Algebraic Topology
Mathematical Knowledge Management in Algebraic Topology
Mathematical Knowledge Management in Algebraic Topology
Mathematical Knowledge Management in Algebraic Topology
Mathematical Knowledge Management in Algebraic Topology
Mathematical Knowledge Management in Algebraic Topology
Mathematical Knowledge Management in Algebraic Topology
Mathematical Knowledge Management in Algebraic Topology
Mathematical Knowledge Management in Algebraic Topology
Mathematical Knowledge Management in Algebraic Topology
Mathematical Knowledge Management in Algebraic Topology
Mathematical Knowledge Management in Algebraic Topology
Mathematical Knowledge Management in Algebraic Topology
Mathematical Knowledge Management in Algebraic Topology
Mathematical Knowledge Management in Algebraic Topology
Mathematical Knowledge Management in Algebraic Topology
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Mathematical Knowledge Management in Algebraic Topology

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Mathematical Knowledge Management in Algebraic Topology

  1. 1. Mathematical Knowledge Management in Algebraic Topology J´nathan Heras Vicente o Supervisors: Dr. Vico Pascual Mart´ ınez-Losa Dr. Julio Rubio Garc´ ıa Department of Mathematics and Computer Science University of La Rioja Spain May 31, 2011J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
  2. 2. Table of Contents 1 Introduction 2 Communication 3 Computation 4 Deduction 5 Conclusions and Further work J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
  3. 3. IntroductionTable of Contents 1 Introduction 2 Communication 3 Computation 4 Deduction 5 Conclusions and Further work J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
  4. 4. Introduction ContextContext Mathematical Knowledge Management (MKM) Computation Deduction Communication J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
  5. 5. Introduction ContextContext Mathematical Knowledge Management (MKM) Computation Deduction Communication Algebraic Topology Mathematical subject which studies topological spaces by algebraic means Applications in Coding theory, Robotics, Digital Image analysis J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
  6. 6. Introduction ContextContext Mathematical Knowledge Management (MKM) Computation Deduction Communication Algebraic Topology Mathematical subject which studies topological spaces by algebraic means Applications in Coding theory, Robotics, Digital Image analysis Kenzo Computer Algebra system devoted to Algebraic Topology developed by F. Sergeraert Common Lisp package Homology groups unreachable by any other means J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
  7. 7. Introduction Particularization of MKM to Algebraic TopologyParticularization of MKM to Algebraic Topology Communication Human beings Other programs J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
  8. 8. Introduction Particularization of MKM to Algebraic TopologyParticularization of MKM to Algebraic Topology Communication Human beings Other programs Computation Kenzo J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
  9. 9. Introduction Particularization of MKM to Algebraic TopologyParticularization of MKM to Algebraic Topology Communication Human beings Other programs Computation Kenzo GAP J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
  10. 10. Introduction Particularization of MKM to Algebraic TopologyParticularization of MKM to Algebraic Topology Communication Human beings Other programs Computation Kenzo GAP New modules J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
  11. 11. Introduction Particularization of MKM to Algebraic TopologyParticularization of MKM to Algebraic Topology Communication Human beings Other programs Computation Kenzo GAP New modules Deduction J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
  12. 12. Introduction Particularization of MKM to Algebraic TopologyParticularization of MKM to Algebraic Topology Communication Human beings Other programs Computation Kenzo GAP New modules Deduction Certification of Kenzo algorithms Isabelle/Hol Coq J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
  13. 13. Introduction Particularization of MKM to Algebraic TopologyParticularization of MKM to Algebraic Topology Communication Human beings Other programs Computation Kenzo GAP New modules Deduction Certification of Kenzo algorithms Isabelle/Hol Coq Certification of Kenzo programs ACL2 J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
  14. 14. CommunicationTable of Contents 1 Introduction 2 Communication 3 Computation 4 Deduction 5 Conclusions and Further work J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
  15. 15. Communication MotivationMotivation Algebraic Topology Expert ⇒ Makes the most of Kenzo Common Lisp Expert J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
  16. 16. Communication MotivationMotivation Algebraic Topology Expert ⇒ Makes the most of Kenzo Common Lisp Expert Non Common Lisp Expert ⇒ Unfriendly front-end J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
  17. 17. Communication MotivationMotivation Algebraic Topology Expert ⇒ Makes the most of Kenzo Common Lisp Expert Non Common Lisp Expert ⇒ Unfriendly front-end Non Algebraic Topology Expert ⇒ Needs guidance J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
  18. 18. Communication MotivationGoal Develop a system which increases Kenzo accessibility J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
  19. 19. Communication MotivationGoal Develop a system which increases Kenzo accessibility Friendly front-end Mediated access to Kenzo J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
  20. 20. Communication Providing a mediated access to KenzoProviding a mediated access to Kenzo Mediated access Client Kenzo J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
  21. 21. Communication Providing a mediated access to KenzoProviding a mediated access to Kenzo Mediated access Client Kenzo Representation of mathematical knowledge   OpenMath Independent of programming language ⇒ XML ⇒ MathML Easily interchangeable New language  J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
  22. 22. Communication Providing a mediated access to KenzoProviding a mediated access to Kenzo Mediated access XML-Kenzo XML-Kenzo Client Kenzo XML-Kenzo construction and computation requests Includes some mathematical knowledge Small Type System: CC, SS, SG, ASG Some restrictions about arguments (S n ) J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
  23. 23. Communication Providing a mediated access to KenzoProviding a mediated access to Kenzo Mediated access XML-Kenzo validator XML-Kenzo XML-Kenzo Client Kenzo XML-Kenzo validator Common Lisp module Receives XML-Kenzo requests → validates them against XML schema definition Type restrictions Mathematical restrictions Kenzo restrictions J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
  24. 24. Communication Providing a mediated access to KenzoProviding a mediated access to Kenzo Mediated access Valid XML-Kenzo Construction Modules XML-Kenzo validator XML-Kenzo XML-Kenzo Client Kenzo Construction modules Process construction requests Check restrictions not included in XML-Kenzo (functional dependencies) 4 modules: CC, SS, SG and ASG J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
  25. 25. Communication Providing a mediated access to KenzoProviding a mediated access to Kenzo Mediated access Construction Modules Valid XML-Kenzo XML-Kenzo validator XML-Kenzo XML-Kenzo Client Kenzo Computation Modules Computation modules Process computation requests Check restrictions not included in XML-Kenzo (reduction degree) 2 modules: Homology and Homotopy J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
  26. 26. Communication Providing a mediated access to KenzoProviding a mediated access to Kenzo Mediated access Construction Modules Valid XML-Kenzo XML-Kenzo validator XML-Kenzo XML-Kenzo Client Processing Kenzo Modules Computation Modules Processing modules Help construction and computation modules 2 modules: Reduction degree and Homotopy assistant J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
  27. 27. Communication The whole frameworkTowards the whole framework This architecture provides Mediated access to Kenzo Friendly front-end but . . . J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
  28. 28. Communication The whole frameworkTowards the whole framework This architecture provides Mediated access to Kenzo Friendly front-end but . . . Desirable features Different clients Efficient Extensible Adaptable to different needs J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
  29. 29. Communication The whole frameworkTowards different clients Mediated access Construction Modules Valid XML-Kenzo XML-Kenzo validator XML-Kenzo XML-Kenzo Client Processing Kenzo Modules Computation Modules XML-Kenzo is ad-hoc J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
  30. 30. Communication The whole frameworkTowards different clients Mediated access Construction Modules Valid XML-Kenzo XML-Kenzo validator XML-Kenzo OpenMath Client Processing Kenzo Modules Computation Modules XML-Kenzo is ad-hoc OpenMath Standard Communication with the outside world J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
  31. 31. Communication The whole frameworkTowards different clients Mediated access Construction Modules Valid XML-Kenzo Client Phrasebook XML-Kenzo validator XML-Kenzo XML-Kenzo OpenMath Client Processing Kenzo Modules Client Computation Modules XML-Kenzo is ad-hoc OpenMath Standard Communication with the outside world Phrasebook J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
  32. 32. Communication The whole frameworkTowards Efficiency Mediated access Construction Modules Valid XML-Kenzo Client Phrasebook XML-Kenzo validator XML-Kenzo XML-Kenzo OpenMath Client Internal Processing Kenzo Memory Modules Client Computation Modules System roughly equivalent to Kenzo Internal memory Memoization technique Store spaces and computations J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
  33. 33. Communication The whole frameworkThe Kenzo framework Framework Microkernel Construction Modules __ Internal External __ Adapter Server Server __ __ __ XML-Kenzo __ XMLValidator __ Kenzo XML-Kenzo __ XML-Kenzo OpenMath __ Internal Processing __ Valid __ __ Clients Memory __ __ __ __ Modules __ Phrasebook __ __ __ Computation Modules Based on well-known patterns and methods J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
  34. 34. Communication The whole frameworkTowards extensibility Include new functionality J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
  35. 35. Communication The whole frameworkTowards extensibility Include new functionality Increase Kenzo functionality J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
  36. 36. Communication The whole frameworkTowards extensibility Include new functionality Increase Kenzo functionality Include new systems Computer Algebra systems Theorem Prover tools J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
  37. 37. Communication The whole frameworkTowards extensibility Include new functionality Increase Kenzo functionality Include new systems Computer Algebra systems Theorem Prover tools Interoperability J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
  38. 38. Communication The whole frameworkTowards extensibility Include new functionality Increase Kenzo functionality Include new systems Computer Algebra systems Theorem Prover tools Interoperability Kenzo framework as a client of a plug-in framework Plug-in Framework Plug-in Manager Client-1 Client-2 __ Module Module OMDoc Plug-in Client __ __ Client-3 Client-4 Plug-in repository Module Module Plug-in Registry J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
  39. 39. Communication The whole frameworkIncluding new Kenzo functionality <code id="new-constructor"> <data format="Kf/internal-server"> new-constructor-is.lisp </data> <data format="Kf/microkernel"> new-constructor-m.lisp </data> <data format="Kf/external-server"> XML-Kenzo.xsd </data> <data format="Kf/adapter"> new-constructor-a.lisp </data> </code> Framework Microkernel Construction Modules __ Internal External __ Adapter Server Server __ __ __ XML-Kenzo __ XMLValidator __ Kenzo XML-Kenzo __ XML-Kenzo OpenMath __ Internal Processing __ Valid __ __ Clients Memory __ __ __ __ Modules __ Phrasebook __ __ __ Computation Modules J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
  40. 40. Communication The whole frameworkA distinguished client of our frameworks A Graphical User Interface implemented in Common Lisp J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
  41. 41. Communication The whole frameworkA distinguished client of our frameworks A Graphical User Interface implemented in Common Lisp Design decisions Functionality (Common Lisp) + Structure (XUL) + Layout (stylesheet) J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
  42. 42. Communication The whole frameworkA distinguished client of our frameworks A Graphical User Interface implemented in Common Lisp Design decisions Functionality (Common Lisp) + Structure (XUL) + Layout (stylesheet) Guided by heuristics J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
  43. 43. Communication The whole frameworkA distinguished client of our frameworks A Graphical User Interface implemented in Common Lisp Design decisions Functionality (Common Lisp) + Structure (XUL) + Layout (stylesheet) Guided by heuristics Design of interaction: Noesis method J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
  44. 44. Communication The whole frameworkA distinguished client of our frameworks A Graphical User Interface implemented in Common Lisp Design decisions Functionality (Common Lisp) + Structure (XUL) + Layout (stylesheet) Guided by heuristics Design of interaction: Noesis method Introduce the dimension,n, of the loop space [Yes] Object Selected in [No] Main tab? Error: Invalid dimension Is the object, X, a Simplicial Set? Select Object X from a list of simplicial sets Is the dimension a [No] [No] natural number? [Yes] Error: Invalid Object [Yes] Construction of the loop space iterated n times of the object X J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
  45. 45. Communication The whole frameworkA customizable GUI Features Extensibility Adaptability J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
  46. 46. Communication The whole frameworkA customizable GUI Features Extensibility Adaptability The Graphical User Interface is client of Kenzo framework client of plug-in framework GUI organized through modules: basic and experimental J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
  47. 47. Communication The whole frameworkA customizable GUI Features Extensibility Adaptability The Graphical User Interface is client of Kenzo framework client of plug-in framework GUI organized through modules: basic and experimental OpenMath fKenzo Kenzo framework GUI fKenzo plug-in Kenzo framework registry plug-in registry OMDoc plug-in Plug-in framework The whole system is called fKenzo J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
  48. 48. Communication fKenzofKenzo: A tool adaptable to different needs Beginner student of Algebraic Topology Load and remove basic modules J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
  49. 49. Communication fKenzofKenzo: A tool adaptable to different needs Beginner student of Algebraic Topology Load and remove basic modules Advanced student or researcher of Algebraic Topology Load and remove basic modules Load and remove experimental modules J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
  50. 50. Communication fKenzofKenzo: A tool adaptable to different needs Beginner student of Algebraic Topology Load and remove basic modules Advanced student or researcher of Algebraic Topology Load and remove basic modules Load and remove experimental modules Digital images GAP ACL2 ... J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
  51. 51. Communication fKenzofKenzo summary fKenzo J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
  52. 52. Communication fKenzofKenzo summary fKenzo Integral assistant for Algebraic Topology J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
  53. 53. Communication fKenzofKenzo summary fKenzo Integral assistant for Algebraic Topology Constrains Kenzo functionality J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
  54. 54. Communication fKenzofKenzo summary fKenzo Integral assistant for Algebraic Topology Constrains Kenzo functionality Provides guidance in the interaction and a friendly front-end J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
  55. 55. Communication fKenzofKenzo summary fKenzo Integral assistant for Algebraic Topology Constrains Kenzo functionality Provides guidance in the interaction and a friendly front-end Extensible J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
  56. 56. Communication fKenzofKenzo summary fKenzo Integral assistant for Algebraic Topology Constrains Kenzo functionality Provides guidance in the interaction and a friendly front-end Extensible Adaptable to different needs J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
  57. 57. Communication fKenzofKenzo summary fKenzo Integral assistant for Algebraic Topology Constrains Kenzo functionality Provides guidance in the interaction and a friendly front-end Extensible Adaptable to different needs Our architecture can be applied in other contexts J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
  58. 58. ComputationTable of Contents 1 Introduction 2 Communication 3 Computation 4 Deduction 5 Conclusions and Further work J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
  59. 59. ComputationNew Kenzo modules Increase Kenzo computational capabilities Simplicial Complexes Digital Images Pushout of Simplicial Sets J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
  60. 60. ComputationNew Kenzo modules Increase Kenzo computational capabilities Simplicial Complexes Digital Images Pushout of Simplicial Sets Integrated in fKenzo J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
  61. 61. ComputationNew Kenzo modules Increase Kenzo computational capabilities Simplicial Complexes Digital Images Pushout of Simplicial Sets Integrated in fKenzo Verified using ACL2 J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
  62. 62. ComputationNew Kenzo modules Increase Kenzo computational capabilities Simplicial Complexes Digital Images Pushout of Simplicial Sets Integrated in fKenzo Verified using ACL2 J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
  63. 63. Computation Effective Homology preliminariesEffective Homology preliminaries: Intuitive Idea Compute homology groups of a space X H∗ (X ) = H∗ (C∗ (X )) J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
  64. 64. Computation Effective Homology preliminariesEffective Homology preliminaries: Intuitive Idea Compute homology groups of a space X H∗ (X ) = H∗ (C∗ (X )) C∗ (X ) has finite nature Differential maps can be expressed as integer matrices Homology groups: Smith Normal Form J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
  65. 65. Computation Effective Homology preliminariesEffective Homology preliminaries: Intuitive Idea Compute homology groups of a space X H∗ (X ) = H∗ (C∗ (X )) C∗ (X ) has finite nature Differential maps can be expressed as integer matrices Homology groups: Smith Normal Form C∗ (X ) has non finite nature Previous methods cannot be applied Effective Homology · Sergeraert’s ideas · Provides real algorithms to compute homology groups · Implemented in the Kenzo system J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
  66. 66. Computation Effective Homology preliminariesEffective Homology preliminaries: Intuitive Idea Compute homology groups of a space X H∗ (X ) = H∗ (C∗ (X )) C∗ (X ) has finite nature (Effective Chain Complex) Differential maps can be expressed as integer matrices Homology groups: Smith Normal Form C∗ (X ) has non finite nature (Locally Effective Chain Complex) Previous methods cannot be applied Effective Homology · Sergeraert’s ideas · Provides real algorithms to compute homology groups · Implemented in the Kenzo system J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
  67. 67. Computation Effective Homology preliminariesEffective Homology preliminaries Definition An effective chain complex is a free chain complex of Z-modules, C∗ = (Cn , dn )n∈N , where each group Cn is finitely generated and an algorithm returns a Z-base in each grade n an algorithm provides the differentials dn Definition A locally effective chain complex is a free chain complex of Z-modules C∗ = (Cn , dn )n∈N where each group Cn can have infinite nature, but there exists an algorithm such that ∀x ∈ Cn , we can compute dn (x) J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
  68. 68. Computation Effective Homology preliminariesEffective Homology preliminaries Definition An effective chain complex is a free chain complex of Z-modules, C∗ = (Cn , dn )n∈N , where each group Cn is finitely generated and an algorithm returns a Z-base in each grade n an algorithm provides the differentials dn differentials dn : Cn → Cn−1 can be expressed as integer matrices possible to compute Ker dn and Im dn+1 possible to compute the homology groups Definition A locally effective chain complex is a free chain complex of Z-modules C∗ = (Cn , dn )n∈N where each group Cn can have infinite nature, but there exists an algorithm such that ∀x ∈ Cn , we can compute dn (x) impossible to compute Ker dn and Im dn+1 possible to perform local computations, differential of a generator J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
  69. 69. Computation Effective Homology preliminariesEffective Homology preliminaries Definition A reduction ρ between two chain complexes C∗ y D∗ (denoted by ρ : C∗ ⇒ D∗ ) is a triple ρ = (f , g , h) h f +D C∗ k ∗ g satisfying the following relations 1) fg = IdD∗ ; 2) dC h + hdC = IdC∗ −gf ; 3) fh = 0; hg = 0; hh = 0. J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
  70. 70. Computation Effective Homology preliminariesEffective Homology preliminaries Definition A reduction ρ between two chain complexes C∗ y D∗ (denoted by ρ : C∗ ⇒ D∗ ) is a triple ρ = (f , g , h) h f +D C∗ k ∗ g satisfying the following relations 1) fg = IdD∗ ; 2) dC h + hdC = IdC∗ −gf ; 3) fh = 0; hg = 0; hh = 0. Theorem If C∗ ⇒ D∗ , then C∗ ∼ D∗ ⊕ A∗ , with A∗ acyclic, which implies that = Hn (C∗ ) ∼ Hn (D∗ ) for all n. = J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
  71. 71. Computation Effective Homology preliminariesEffective Homology preliminaries Definition A (strong chain) equivalence ε between C∗ and D∗ , ε : C∗ ⇐ D∗ , is a triple ⇒ ε = (B∗ , ρ, ρ ) where B∗ is a chain complex, ρ : B∗ ⇒ C∗ and ρ : B∗ ⇒ D∗ . 42 B∗ 30 s{ #+ 14 s{ #+ 21 C∗ D∗ 10 15 J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
  72. 72. Computation Effective Homology preliminariesEffective Homology preliminaries Definition A (strong chain) equivalence ε between C∗ and D∗ , ε : C∗ ⇐ D∗ , is a triple ⇒ ε = (B∗ , ρ, ρ ) where B∗ is a chain complex, ρ : B∗ ⇒ C∗ and ρ : B∗ ⇒ D∗ . 42 B∗ 30 s{ #+ 14 s{ #+ 21 C∗ D∗ 10 15 Definition An object with effective homology is a quadruple (X , C∗ (X ), HC∗ , ε) where X is a locally effective object C∗ (X ) is a (locally effective) chain complex canonically associated with X , which allows the study of the homological nature of X HC∗ is an effective chain complex ε is a equivalence ε : C∗ (X ) ⇐ HC∗ ⇒ J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
  73. 73. Computation Effective Homology preliminariesEffective Homology preliminaries Definition A (strong chain) equivalence ε between C∗ and D∗ , ε : C∗ ⇐ D∗ , is a triple ⇒ ε = (B∗ , ρ, ρ ) where B∗ is a chain complex, ρ : B∗ ⇒ C∗ and ρ : B∗ ⇒ D∗ . 42 B∗ 30 s{ #+ 14 s{ #+ 21 C∗ D∗ 10 15 Definition An object with effective homology is a quadruple (X , C∗ (X ), HC∗ , ε) where X is a locally effective object C∗ (X ) is a (locally effective) chain complex canonically associated with X , which allows the study of the homological nature of X HC∗ is an effective chain complex ε is a equivalence ε : C∗ (X ) ⇐ HC∗ ⇒ Theorem Let an object with effective homology (X , C∗ (X ), HC∗ , ε) then Hn (X ) ∼ Hn (HC∗ ) for = all n. J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
  74. 74. Computation Pushout preliminariesPushout preliminaries Definition Let f , g morphisms, X f /Y g f Z g / P(f ,g ) f ∃!p g !, Q J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
  75. 75. Computation Pushout preliminariesPushout preliminaries Definition Let f , g morphisms, the pushout of f , g is an object P(f ,g ) for which the diagram X f /Y g commutes f Z g / P(f ,g ) f ∃!p g !, Q J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
  76. 76. Computation Pushout preliminariesPushout preliminaries Definition Let f , g morphisms, the pushout of f , g is an object P(f ,g ) for which the diagram X f /Y g commutes f respects the universal Z g / P(f ,g ) f property ∃!p g !, Q J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
  77. 77. Computation Pushout preliminariesPushout preliminaries Definition Let f , g morphisms, the pushout of f , g is an object P(f ,g ) for which the diagram X f /Y g commutes f respects the universal Z g / P(f ,g ) f property ∃!p g !, Q Standard Construction P(f ,g ) ∼ (Y (X × I ) = Z )/ ∼ where I is the unit interval for every x ∈ X , ∼ (x, 0) ∼ f (x) ∈ Y (x, 1) ∼ g (x) ∈ Z J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
  78. 78. Computation Effective Homology of the PushoutConstruction of the Effective Homology of the Pushout I Given f : X → Y and g : X → Z simplicial morphisms where X , Y and Z are simplicial sets X f /Y g f Z g / P(f ,g ) J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
  79. 79. Computation Effective Homology of the PushoutConstruction of the Effective Homology of the Pushout I Given f : X → Y and g : X → Z simplicial morphisms where X , Y and Z are simplicial sets X f /Y g f Z g / P(f ,g ) J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
  80. 80. Computation Effective Homology of the PushoutConstruction of the Effective Homology of the Pushout I Given f : X → Y and g : X → Z simplicial morphisms where X , Y and Z are simplicial sets X f /Y g f Z g / P(f ,g ) Algorithm (Standard Construction) Input: two simplicial morphisms f : X → Y and g : X → Z where X , Y and Z are simplicial sets Output: the pushout P(f ,g ) , a simplicial set J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
  81. 81. Computation Effective Homology of the PushoutConstruction of the Effective Homology of the Pushout II Given f : X → Y and g : X → Z simplicial morphisms where X , Y and Z are simplicial sets with effective homology (X , C∗ (X ), HX∗ , εX ) f / (Y , C∗ (Y ), HY∗ , εY ) g (Z , C∗ (Z ), HZ∗ , εZ ) / (P(f ,g ) , C∗ (P(f ,g ) ), HP∗ , εP ) J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
  82. 82. Computation Effective Homology of the PushoutConstruction of the Effective Homology of the Pushout II Given f : X → Y and g : X → Z simplicial morphisms where X , Y and Z are simplicial sets with effective homology (X , C∗ (X ), HX∗ , εX ) f / (Y , C∗ (Y ), HY∗ , εY ) g (Z , C∗ (Z ), HZ∗ , εZ ) / (P(f ,g ) , C∗ (P(f ,g ) ), HP∗ , εP ) J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
  83. 83. Computation Effective Homology of the PushoutConstruction of the Effective Homology of the Pushout II Given f : X → Y and g : X → Z simplicial morphisms where X , Y and Z are simplicial sets with effective homology (X , C∗ (X ), HX∗ , εX ) f / (Y , C∗ (Y ), HY∗ , εY ) g (Z , C∗ (Z ), HZ∗ , εZ ) / (P(f ,g ) , C∗ (P(f ,g ) ), HP∗ , εP ) J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
  84. 84. Computation Effective Homology of the PushoutConstruction of the Effective Homology of the Pushout II Given f : X → Y and g : X → Z simplicial morphisms where X , Y and Z are simplicial sets with effective homology (X , C∗ (X ), HX∗ , εX ) f / (Y , C∗ (Y ), HY∗ , εY ) g (Z , C∗ (Z ), HZ∗ , εZ ) / (P(f ,g ) , C∗ (P(f ,g ) ), HP∗ , εP ) Algorithm (joint work with F. Sergeraert) Input: two simplicial morphisms f : X → Y and g : X → Z where X , Y and Z are simplicial sets with effective homology Output: the effective homology version of P(f ,g ) , that is, an equivalence C∗ (P(f ,g ) ) ⇐ HP∗ , where HP∗ is an effective chain complex ⇒ J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
  85. 85. Computation Effective Homology of the PushoutSketch of the algorithm Theorem (SES Theorems) Let 0 o 0 A∗ o σ / B∗ o ρ / C∗ o 0 j i be an effective short exact sequence of chain complexes. Then three general algorithms are available SES1 : (B∗,EH , C∗,EH ) → A∗,EH SES2 : (A∗,EH , C∗,EH ) → B∗,EH SES3 : (A∗,EH , B∗,EH ) → C∗,EH producing the effective homology of one chain complex when the effective homology of both others is given. J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
  86. 86. Computation Effective Homology of the PushoutSketch of the algorithm Theorem (SES Theorems) Let 0 o 0 A∗ o σ / B∗ o ρ / C∗ o 0 j i be an effective short exact sequence of chain complexes. Then three general algorithms are available SES1 : (B∗,EH , C∗,EH ) → A∗,EH SES2 : (A∗,EH , C∗,EH ) → B∗,EH SES3 : (A∗,EH , B∗,EH ) → C∗,EH producing the effective homology of one chain complex when the effective homology of both others is given. 0 o M∗ o σ / C∗ P o ρ / C∗ Y ⊕ C∗ Z o 0 j i M∗ is the chain complex associated with X × ∆1 but with the simplexes of X × (0) and X × (1) cancelled J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
  87. 87. Computation Effective Homology of the PushoutSketch of the algorithm continued Step 1. From f : X → Y and g : X → Z simplicial morphisms, P(f ,g ) and its associated chain complex C∗ P are constructed Step 2. The effective homology of M∗ is constructed · Define 0 o M∗ o σ2 / C∗ (X × ∆1 ) o / ρ2 C∗ (X × (0)) ⊕ C∗ (X × (1)) o 0 j2 i2 · Construct the chain complex M∗ · Build the effective homology of C∗ (X × ∆1 ) · Construct the effective homology of C∗ (X × (0)) ⊕ C∗ (X × (1)) · Construct the effective homology of Cone(i2) · Construct the reduction M∗⇐ Cone(i2) ⇐ · Obtain the effective homology of M∗ applying case SES1 Step 3. The effective homology of C∗ X ⊕ C∗ Y is constructed Step 4. The effective homology of the pushout P(f ,g ) is constructed · Define 0 o M∗ o σ / C∗ P o ρ / C∗ Y ⊕ C∗ Z o 0 j i · Construct the effective homology of (C∗ Y ⊕ C∗ Z )[1] · Define the morphism shift : C∗ Y ⊕ C∗ Z → (C∗ Y ⊕ C∗ Z )[1] · Define the chain complex morphism χ : M∗ → (C∗ Y ⊕ C∗ Z )[1] · Construct the effective homology of Cone(χ) · Obtain the effective homology of P(f ,g ) applying case SES2 J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
  88. 88. Computation Effective Homology of the PushoutExample: π∗ (Σ(SL2 (Z))) SL2 (Z) Group of 2 × 2 matrices with determinant 1 over Z J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
  89. 89. Computation Effective Homology of the PushoutExample: π∗ (Σ(SL2 (Z))) SL2 (Z) Group of 2 × 2 matrices with determinant 1 over Z Isomorphic to Z4 ∗Z2 Z6 J. P. Serre. Trees. Springer-Verlag, 1980 J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
  90. 90. Computation Effective Homology of the PushoutExample: π∗ (Σ(SL2 (Z))) SL2 (Z) Group of 2 × 2 matrices with determinant 1 over Z Isomorphic to Z4 ∗Z2 Z6 J. P. Serre. Trees. Springer-Verlag, 1980 i1 K (Z2 , 1) / K (Z4 , 1) i2 K (Z6 , 1) / K (SL2 (Z), 1) K. S. Brown. Cohomology of Groups. Springer-Verlag, 1982 J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
  91. 91. Computation Effective Homology of the PushoutExample: π∗ (Σ(SL2 (Z))) Construction of the object K (SL2 (Z, 1)) J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
  92. 92. Computation Effective Homology of the PushoutExample: π∗ (Σ(SL2 (Z))) Construction of the object K (SL2 (Z, 1)) Firstly, we define K (Z2 , 1), K (Z4 , 1) and K (Z6 , 1) (setf kz2 (k-zp-1 2)) [K2 Abelian-Simplicial-Group] (setf kz4 (k-zp-1 4)) [K15 Abelian-Simplicial-Group] (setf kz6 (k-zp-1 6)) [K28 Abelian-Simplicial-Group] J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
  93. 93. Computation Effective Homology of the PushoutExample: π∗ (Σ(SL2 (Z))) Construction of the object K (SL2 (Z, 1)) Firstly, we define K (Z2 , 1), K (Z4 , 1) and K (Z6 , 1) (setf kz2 (k-zp-1 2)) [K2 Abelian-Simplicial-Group] (setf kz4 (k-zp-1 4)) [K15 Abelian-Simplicial-Group] (setf kz6 (k-zp-1 6)) [K28 Abelian-Simplicial-Group] Subsequently, we define the two simplicial morphisms i1 and i2 (setf i1 (kzps-incl 2 4)) [K40 Simplicial-Morphism K2 - K15] (setf i2 (kzps-incl 2 6)) [K41 Simplicial-Morphism K2 - K28] J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
  94. 94. Computation Effective Homology of the PushoutExample: π∗ (Σ(SL2 (Z))) Construction of the object K (SL2 (Z, 1)) Firstly, we define K (Z2 , 1), K (Z4 , 1) and K (Z6 , 1) (setf kz2 (k-zp-1 2)) [K2 Abelian-Simplicial-Group] (setf kz4 (k-zp-1 4)) [K15 Abelian-Simplicial-Group] (setf kz6 (k-zp-1 6)) [K28 Abelian-Simplicial-Group] Subsequently, we define the two simplicial morphisms i1 and i2 (setf i1 (kzps-incl 2 4)) [K40 Simplicial-Morphism K2 - K15] (setf i2 (kzps-incl 2 6)) [K41 Simplicial-Morphism K2 - K28] Finally, we construct the pushout of i1 and i2 (K (SL2 (Z), 1)) (setf ksl2z (pushout i1 i2)) [K52 Simplicial-Set] J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
  95. 95. Computation Effective Homology of the PushoutExample: π∗ (Σ(SL2 (Z))) Computation of π∗ (Σ(SL2 (Z))) J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
  96. 96. Computation Effective Homology of the PushoutExample: π∗ (Σ(SL2 (Z))) Computation of π∗ (Σ(SL2 (Z))) Firstly, we define Σ(K (SL2 (Z), 1)) (setf sksl2z (suspension ksl2z)) [K62 Simplicial-Set] J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
  97. 97. Computation Effective Homology of the PushoutExample: π∗ (Σ(SL2 (Z))) Computation of π∗ (Σ(SL2 (Z))) Firstly, we define Σ(K (SL2 (Z), 1)) (setf sksl2z (suspension ksl2z)) [K62 Simplicial-Set] Computing Homotopy groups (Hurewicz theorem) (homology sksl2z 1 3) Homology in dimension 1: Homology in dimension 2: Component Z/12Z J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
  98. 98. Computation Effective Homology of the PushoutExample: π∗ (Σ(SL2 (Z))) Computation of π∗ (Σ(SL2 (Z))) Firstly, we define Σ(K (SL2 (Z), 1)) (setf sksl2z (suspension ksl2z)) [K62 Simplicial-Set] Computing Homotopy groups (Hurewicz theorem) (homology sksl2z 1 3) Homology in dimension 1: Homology in dimension 2: Component Z/12Z π1 (Σ(SL2 (Z))) = 0, π2 (Σ(SL2 (Z))) = Z/12Z J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
  99. 99. Computation Effective Homology of the PushoutExample: π∗ (Σ(SL2 (Z))) Computing Homotopy groups continued (Whitehead tower) (setf tau (zp-whitehead 12 sksl2z (chml-clss sksl2z 2))) [K91 Fibration K62 - K79] (setf x (fibration-total tau)) [K97 Simplicial-Set] (homology x 3) Homology in dimension 3: Component Z/12Z (setf tau2 (zp-whitehead 12 x (chml-clss x 3))) [K228 Fibration K97 - K214] (setf x2 (fibration-total tau2)) [K234 Simplicial-Set] (homology x2 4) Homology in dimension 4: Component Z/12Z Component Z/2Z ... J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
  100. 100. Computation Effective Homology of the PushoutExample: π∗ (Σ(SL2 (Z))) Computing Homotopy groups continued (Whitehead tower) (setf tau (zp-whitehead 12 sksl2z (chml-clss sksl2z 2))) [K91 Fibration K62 - K79] (setf x (fibration-total tau)) [K97 Simplicial-Set] (homology x 3) Homology in dimension 3: Component Z/12Z (setf tau2 (zp-whitehead 12 x (chml-clss x 3))) [K228 Fibration K97 - K214] (setf x2 (fibration-total tau2)) [K234 Simplicial-Set] (homology x2 4) Homology in dimension 4: Component Z/12Z Component Z/2Z ... π3 (Σ(SL2 (Z))) = Z/12Z, π4 (Σ(SL2 (Z))) = Z/12Z ⊕ Z/2Z, . . . J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
  101. 101. DeductionTable of Contents 1 Introduction 2 Communication 3 Computation 4 Deduction 5 Conclusions and Further work J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
  102. 102. Deduction ACL2ACL2 ACL2 (A Computational Logic for an Applicative Common Lisp) J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
  103. 103. Deduction ACL2ACL2 ACL2 (A Computational Logic for an Applicative Common Lisp) Programming Language First-Order Logic Theorem Prover J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
  104. 104. Deduction ACL2ACL2 ACL2 (A Computational Logic for an Applicative Common Lisp) Programming Language First-Order Logic Theorem Prover Proof techniques Simplification Induction “The Method” J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
  105. 105. Deduction ACL2ACL2 ACL2 (A Computational Logic for an Applicative Common Lisp) Programming Language First-Order Logic Theorem Prover Proof techniques Simplification Induction “The Method” Encapsulate principle Simulation of Higher-Order Logic J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
  106. 106. Deduction GoalGoal Simplified view of Kenzo way of working 1 Construction of initial spaces 2 Construction of new spaces from other ones 3 Computation of homology groups J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
  107. 107. Deduction GoalGoal Simplified view of Kenzo way of working 1 Construction of initial spaces Certification of the correctness of Kenzo constructors of simplicial sets 2 Construction of new spaces from other ones Easy Perturbation Lemma Composition of reductions Cone Equivalence Theorem ... 3 Computation of homology groups .............................................................................................................. (sphere 3) [K1 Simplicial-Set] .............................................................................................................. J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
  108. 108. Deduction Certifying the correctness of Kenzo simplicial setsMathematical context: Simplicial Sets Definition A simplicial set K , is a union K = K q , where the K q are disjoints sets, together with functions q≥0 q ∂i : K q → K q−1 , q 0, i = 0, . . . , q, q ηi : K q → K q+1 , q ≥ 0, i = 0, . . . , q, subject to the relations q−1 q q−1 q (1) ∂i ∂j = ∂j−1 ∂i if i j, q+1 q q+1 q (2) ηi ηj = ηj+1 ηi if i ≤ j, q+1 q q−1 q (3) ∂i ηj = ηj−1 ∂i if i j, q+1 q q+1 q (4) ∂i ηi = identity = ∂i+1 ηi , q+1 q q−1 q (5) ∂i ηj = ηj ∂i−1 if i j + 1, J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
  109. 109. Deduction Certifying the correctness of Kenzo simplicial setsMathematical context: Simplicial Sets Definition A simplicial set K , is a union K = K q , where the K q are disjoints sets, together with functions q≥0 q ∂i : K q → K q−1 , q 0, i = 0, . . . , q, q ηi : K q → K q+1 , q ≥ 0, i = 0, . . . , q, subject to the relations q−1 q q−1 q (1) ∂i ∂j = ∂j−1 ∂i if i j, q+1 q q+1 q (2) ηi ηj = ηj+1 ηi if i ≤ j, q+1 q q−1 q (3) ∂i ηj = ηj−1 ∂i if i j, q+1 q q+1 q (4) ∂i ηi = identity = ∂i+1 ηi , q+1 q q−1 q (5) ∂i ηj = ηj ∂i−1 if i j + 1, The elements of K q are called q-simplexes J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
  110. 110. Deduction Certifying the correctness of Kenzo simplicial setsMathematical context: Simplicial Sets Definition A simplicial set K , is a union K = K q , where the K q are disjoints sets, together with functions q≥0 q ∂i : K q → K q−1 , q 0, i = 0, . . . , q, q ηi : K q → K q+1 , q ≥ 0, i = 0, . . . , q, subject to the relations q−1 q q−1 q (1) ∂i ∂j = ∂j−1 ∂i if i j, q+1 q q+1 q (2) ηi ηj = ηj+1 ηi if i ≤ j, q+1 q q−1 q (3) ∂i ηj = ηj−1 ∂i if i j, q+1 q q+1 q (4) ∂i ηi = identity = ∂i+1 ηi , q+1 q q−1 q (5) ∂i ηj = ηj ∂i−1 if i j + 1, The elements of K q are called q-simplexes q−1 A q-simplex x is degenerate if x = ηi y for some simplex y ∈ K q−1 J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
  111. 111. Deduction Certifying the correctness of Kenzo simplicial setsMathematical context: Simplicial Sets Definition A simplicial set K , is a union K = K q , where the K q are disjoints sets, together with functions q≥0 q ∂i : K q → K q−1 , q 0, i = 0, . . . , q, q ηi : K q → K q+1 , q ≥ 0, i = 0, . . . , q, subject to the relations q−1 q q−1 q (1) ∂i ∂j = ∂j−1 ∂i if i j, q+1 q q+1 q (2) ηi ηj = ηj+1 ηi if i ≤ j, q+1 q q−1 q (3) ∂i ηj = ηj−1 ∂i if i j, q+1 q q+1 q (4) ∂i ηi = identity = ∂i+1 ηi , q+1 q q−1 q (5) ∂i ηj = ηj ∂i−1 if i j + 1, The elements of K q are called q-simplexes q−1 A q-simplex x is degenerate if x = ηi y for some simplex y ∈ K q−1 Otherwise x is called non-degenerate (or geometric) J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
  112. 112. Deduction Certifying the correctness of Kenzo simplicial setsMathematical context: Example c 0-simplexes: vertices: (a), (b), (c), (d) non-degenerate 1-simplexes: edges: (a b),(a c),(a d),(b c),(b d),(c d) non-degenerate 2-simplexes: d (filled) triangles: (a b c),(a b d),(a c d),(b c d) non-degenerate 3-simplexes: a b (filled) tetrahedron: (a b c d) J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
  113. 113. Deduction Certifying the correctness of Kenzo simplicial setsMathematical context: Example c 0-simplexes: vertices: (a), (b), (c), (d) non-degenerate 1-simplexes: edges: (a b),(a c),(a d),(b c),(b d),(c d) non-degenerate 2-simplexes: d (filled) triangles: (a b c),(a b d),(a c d),(b c d) non-degenerate 3-simplexes: a b (filled) tetrahedron: (a b c d)   (b c) if i = 0 face: ∂i (a b c) = (a c) if i = 1 (a b) if i = 2  J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
  114. 114. Deduction Certifying the correctness of Kenzo simplicial setsMathematical context: Example c 0-simplexes: vertices: (a), (b), (c), (d) non-degenerate 1-simplexes: edges: (a b),(a c),(a d),(b c),(b d),(c d) non-degenerate 2-simplexes: d (filled) triangles: (a b c),(a b d),(a c d),(b c d) non-degenerate 3-simplexes: a b (filled) tetrahedron: (a b c d)   (b c) if i = 0 face: ∂i (a b c) = (a c) if i = 1 (a b) if i = 2     (a a b c) if i = 0  degeneracy: ηi (a b c) = (a b b c) if i = 1 non-geometrical meaning (a b c c) if i = 2   J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
  115. 115. Deduction Certifying the correctness of Kenzo simplicial setsMathematical context: abstract simplexes Proposition Let K be a simplicial set. Any n-simplex x ∈ K n can be expressed in a unique way as a (possibly) iterated degeneracy of a non-degenerate simplex y in the following way x = ηj . . . ηj1 y k r with y ∈ K , k = n − r ≥ 0, and 0 ≤ j1 · · · jk n. J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
  116. 116. Deduction Certifying the correctness of Kenzo simplicial setsMathematical context: abstract simplexes Proposition Let K be a simplicial set. Any n-simplex x ∈ K n can be expressed in a unique way as a (possibly) iterated degeneracy of a non-degenerate simplex y in the following way x = ηj . . . ηj1 y k r with y ∈ K , k = n − r ≥ 0, and 0 ≤ j1 · · · jk n. abstract simplex dgop is a strictly decreasing sequence of degeneracy maps (dgop gmsm) := gmsm is a geometric simplex J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
  117. 117. Deduction Certifying the correctness of Kenzo simplicial setsMathematical context: abstract simplexes Proposition Let K be a simplicial set. Any n-simplex x ∈ K n can be expressed in a unique way as a (possibly) iterated degeneracy of a non-degenerate simplex y in the following way x = ηj . . . ηj1 y k r with y ∈ K , k = n − r ≥ 0, and 0 ≤ j1 · · · jk n. abstract simplex dgop is a strictly decreasing sequence of degeneracy maps (dgop gmsm) := gmsm is a geometric simplex Examples simplex abstract simplex non-degenerate (a b) (∅ (a b)) degenerate (a a b c) (η0 (a b c)) J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
  118. 118. Deduction Certifying the correctness of Kenzo simplicial setsMathematical context degeneracy operator ηiq (dgop gmsm) := (ηiq ◦ dgop gmsm) J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
  119. 119. Deduction Certifying the correctness of Kenzo simplicial setsMathematical context degeneracy operator ηiq (dgop gmsm) := (ηiq ◦ dgop gmsm) face operator (∂iq ◦ dgop gmsm) if ηi ∈ dgop ∨ ηi−1 ∈ dgop ∂iq (dgop gmsm) := (∂iq ◦ dgop ∂k gmsm) r otherwise; where r = q − {number of degeneracies in dgop} and k = i − {number of degeneracies in dgop with index lower than i} J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
  120. 120. Deduction Certifying the correctness of Kenzo simplicial setsMathematical context degeneracy operator ηiq (dgop gmsm) := (ηiq ◦ dgop gmsm) face operator (∂iq ◦ dgop gmsm) if ηi ∈ dgop ∨ ηi−1 ∈ dgop ∂iq (dgop gmsm) := (∂iq ◦ dgop ∂k gmsm) r otherwise; where r = q − {number of degeneracies in dgop} and k = i − {number of degeneracies in dgop with index lower than i} invariant operator (dgop gmsm) ∈ K q (length dgop) q gmsm ∈ K r where r = q − (length dgop) index of the first degeneracy in dgop is lower than q J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology

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