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The Formal Dual of Birkhoffs Completeness Theorem Jesse Hughes jesseh@cs.kun.nl University of Nijmegen The Formal Dual of Birkhoffs Completeness Theorem p.1/26 Outline I. Coequations The Formal Dual of Birkhoffs Completeness

  1. � � ✁ ✁ Outline I. Coequations II. Conditional coequations III. Horn coequations IV. Some co-Birkhoff type theorems (again) V. Birkhoff’s completeness theorem VI. Dualizing deductive closure VII. The operator VIII. The operator IX. The invariance theorem X. Commutativity of , The Formal Dual of Birkhoff’s Completeness Theorem – p.10/26

  2. � � ✁ ✁ Outline I. Coequations II. Conditional coequations III. Horn coequations IV. Some co-Birkhoff type theorems (again) V. Birkhoff’s completeness theorem VI. Dualizing deductive closure VII. The operator VIII. The operator IX. The invariance theorem X. Commutativity of , The Formal Dual of Birkhoff’s Completeness Theorem – p.10/26

  3. Some co-Birkhoff-type theorems Define Th V = { P � � UHC | V | = C P } The Formal Dual of Birkhoff’s Completeness Theorem – p.11/26

  4. Some co-Birkhoff-type theorems Define Th V = { P � � UHC | V | = C P } Imp V = { P ⇒ C Q | V | = C P ⇒ Q } The Formal Dual of Birkhoff’s Completeness Theorem – p.11/26

  5. Some co-Birkhoff-type theorems Define Th V = { P � � UHC | V | = C P } Imp V = { P ⇒ C Q | V | = C P ⇒ Q } C | V | Horn V = Imp V ∪ { P = C P } The Formal Dual of Birkhoff’s Completeness Theorem – p.11/26

  6. Some co-Birkhoff-type theorems Define Th V = { P � � UHC | V | = C P } Imp V = { P ⇒ C Q | V | = C P ⇒ Q } C | V | Horn V = Imp V ∪ { P = C P } Further, let Mod S denote the models of S for S a class of coequations, conditional coequations or Horn coequations. The Formal Dual of Birkhoff’s Completeness Theorem – p.11/26

  7. Some co-Birkhoff-type theorems Theorem (Birkhoff covariety theorem). Mod Th V = SH Σ V Theorem (Quasi-covariety theorem). Mod Imp V = H Σ V Theorem (Horn covariety theorem). Mod Horn V = H Σ + V The Formal Dual of Birkhoff’s Completeness Theorem – p.11/26

  8. Birkhoff’s deduction theorem Fix a set X of variables and let E be a set of equations over X . E is deductively closed just in case E satisfies the following: (i) x = x ∈ E ; (ii) t 1 = t 2 ∈ E ⇒ t 2 = t 1 ∈ E ; (iii) t 1 = t 2 ∈ E and t 2 = t 3 ∈ E ⇒ t 1 = t 3 ∈ E ; 2 ∈ E and f ∈ Σ ⇒ f ( � t 1 ) = f ( � (iv) t i 1 = t i t 2 ) ∈ E ; (v) t 1 = t 2 ∈ E ⇒ t 1 [ t/x ] = t 2 [ t/x ] ∈ E. The Formal Dual of Birkhoff’s Completeness Theorem – p.12/26

  9. Birkhoff’s deduction theorem Fix a set X of variables and let E be a set of equations over X . E is deductively closed just in case E satisfies the following: (i) x = x ∈ E ; (ii) t 1 = t 2 ∈ E ⇒ t 2 = t 1 ∈ E ; (iii) t 1 = t 2 ∈ E and t 2 = t 3 ∈ E ⇒ t 1 = t 3 ∈ E ; 2 ∈ E and f ∈ Σ ⇒ f ( � t 1 ) = f ( � (iv) t i 1 = t i t 2 ) ∈ E ; (v) t 1 = t 2 ∈ E ⇒ t 1 [ t/x ] = t 2 [ t/x ] ∈ E. Items (i) – (iv) ensure that E is a congruence and hence uniquely determines a quotient of FX . The Formal Dual of Birkhoff’s Completeness Theorem – p.12/26

  10. ✂ Birkhoff’s deduction theorem Fix a set X of variables and let E be a set of equations over X . E is deductively closed just in case E satisfies the following: (i) x = x ∈ E ; (ii) t 1 = t 2 ∈ E ⇒ t 2 = t 1 ∈ E ; (iii) t 1 = t 2 ∈ E and t 2 = t 3 ∈ E ⇒ t 1 = t 3 ∈ E ; 2 ∈ E and f ∈ Σ ⇒ f ( � t 1 ) = f ( � (iv) t i 1 = t i t 2 ) ∈ E ; (v) t 1 = t 2 ∈ E ⇒ t 1 [ t/x ] = t 2 [ t/x ] ∈ E. Item (v) ensures that E is a stable -algebra, i.e., closed under substitutions. The Formal Dual of Birkhoff’s Completeness Theorem – p.12/26

  11. Birkhoff’s deduction theorem E is deductively closed just in case E satisfies the following: (i) x = x ∈ E ; (ii) t 1 = t 2 ∈ E ⇒ t 2 = t 1 ∈ E ; (iii) t 1 = t 2 ∈ E and t 2 = t 3 ∈ E ⇒ t 1 = t 3 ∈ E ; 2 ∈ E and f ∈ Σ ⇒ f ( � t 1 ) = f ( � (iv) t i 1 = t i t 2 ) ∈ E ; (v) t 1 = t 2 ∈ E ⇒ t 1 [ t/x ] = t 2 [ t/x ] ∈ E. � Rel ( UFX ) be the closure operation Let Ded : Rel ( UFX ) taking a set E of equations over X to its deductive closure. We can decompose Ded into two closure operators. The Formal Dual of Birkhoff’s Completeness Theorem – p.12/26

  12. Birkhoff’s deduction theorem E is deductively closed just in case E satisfies the following: (i) x = x ∈ E ; (ii) t 1 = t 2 ∈ E ⇒ t 2 = t 1 ∈ E ; (iii) t 1 = t 2 ∈ E and t 2 = t 3 ∈ E ⇒ t 1 = t 3 ∈ E ; 2 ∈ E and f ∈ Σ ⇒ f ( � t 1 ) = f ( � (iv) t i 1 = t i t 2 ) ∈ E ; (v) t 1 = t 2 ∈ E ⇒ t 1 [ t/x ] = t 2 [ t/x ] ∈ E. The first takes E to the congruence it generates. The Formal Dual of Birkhoff’s Completeness Theorem – p.12/26

  13. Birkhoff’s deduction theorem E is deductively closed just in case E satisfies the following: (i) x = x ∈ E ; (ii) t 1 = t 2 ∈ E ⇒ t 2 = t 1 ∈ E ; (iii) t 1 = t 2 ∈ E and t 2 = t 3 ∈ E ⇒ t 1 = t 3 ∈ E ; 2 ∈ E and f ∈ Σ ⇒ f ( � t 1 ) = f ( � (iv) t i 1 = t i t 2 ) ∈ E ; (v) t 1 = t 2 ∈ E ⇒ t 1 [ t/x ] = t 2 [ t/x ] ∈ E. The second closes it under substitution of terms for variables. The Formal Dual of Birkhoff’s Completeness Theorem – p.12/26

  14. � � ✁ ✁ Outline I. Coequations II. Conditional coequations III. Horn coequations IV. Some co-Birkhoff type theorems (again) V. Birkhoff’s completeness theorem VI. Dualizing deductive closure VII. The operator VIII. The operator IX. The invariance theorem X. Commutativity of , The Formal Dual of Birkhoff’s Completeness Theorem – p.13/26

  15. � � ✁ ✁ Outline I. Coequations II. Conditional coequations III. Horn coequations IV. Some co-Birkhoff type theorems (again) V. Birkhoff’s completeness theorem VI. Dualizing deductive closure VII. The operator VIII. The operator IX. The invariance theorem X. Commutativity of , The Formal Dual of Birkhoff’s Completeness Theorem – p.13/26

  16. Dualizing the completeness theorem Theorem (Birkhoff completeness theorem). For any E ∈ Rel ( UFX ) , Th Mod ( E ) = Ded ( E ) The Formal Dual of Birkhoff’s Completeness Theorem – p.14/26

  17. Dualizing the completeness theorem Theorem (Birkhoff completeness theorem). For any E ∈ Rel ( UFX ) , Th Mod ( E ) = Ded ( E ) Compare this to the variety theorem. Theorem (Birkhoff variety theorem). Mod Th V = HSP V The Formal Dual of Birkhoff’s Completeness Theorem – p.14/26

  18. Dualizing the completeness theorem Theorem (Birkhoff completeness theorem). For any E ∈ Rel ( UFX ) , Th Mod ( E ) = Ded ( E ) Th Mod ( E ) satisfies the following fixed point description. • Mod ( E ) | = Th Mod ( E ) ; = E ′ , then E ′ ⊆ Th Mod ( E ) . • If Mod ( E ) | The Formal Dual of Birkhoff’s Completeness Theorem – p.14/26

  19. Dualizing the completeness theorem Th Mod ( E ) satisfies the following fixed point description. • Mod ( E ) | = Th Mod ( E ) ; = E ′ , then E ′ ⊆ Th Mod ( E ) . • If Mod ( E ) | We dualize this fixed point description to yield its coalgebraic analogue. We call the analogue the “generating coequation for Mod ( P ) ”, written Gen Mod ( P ) . The Formal Dual of Birkhoff’s Completeness Theorem – p.14/26

  20. Dualizing the completeness theorem We dualize this fixed point description to yield its coalgebraic analogue. We call the analogue the “generating coequation for Mod ( P ) ”, written Gen Mod ( P ) . Gen Mod ( P ) satisfies the following fixed point description. • Mod ( P ) | = Gen Mod ( E ) ; = P ′ , then Gen Mod ( P ) ⊆ P ′ . • If Mod ( P ) | The Formal Dual of Birkhoff’s Completeness Theorem – p.14/26

  21. Dualizing the completeness theorem We dualize this fixed point description to yield its coalgebraic analogue. We call the analogue the “generating coequation for Mod ( P ) ”, written Gen Mod ( P ) . Gen Mod ( P ) satisfies the following fixed point description. • Mod ( P ) | = Gen Mod ( E ) ; = P ′ , then Gen Mod ( P ) ⊆ P ′ . • If Mod ( P ) | Recall that sets of equations correspond to coequations, so this is an appropriate dualization. The Formal Dual of Birkhoff’s Completeness Theorem – p.14/26

  22. Dualizing the completeness theorem We dualize this fixed point description to yield its coalgebraic analogue. We call the analogue the “generating coequation for Mod ( P ) ”, written Gen Mod ( P ) . Gen Mod ( P ) satisfies the following fixed point description. • Mod ( P ) | = Gen Mod ( E ) ; = P ′ , then Gen Mod ( P ) ⊆ P ′ . • If Mod ( P ) | Recall that sets of equations correspond to coequations, so this is an appropriate dualization. A generating coequation gives a measure of the “coequational commitment” of V . The Formal Dual of Birkhoff’s Completeness Theorem – p.14/26

  23. Dualizing deductive closure Theorem (Birkhoff completeness theorem). For any E ∈ Rel ( UFX ) , Th Mod ( E ) = Ded ( E ) To dualize Ded , we consider again its components. Algebras Coalgebras Projective set of variables X Injective set of colors C Set of equations Coequation E �� UFX P � � UHC q : FX � � � Q, ν � i :[ P ] � � HC The Formal Dual of Birkhoff’s Completeness Theorem – p.15/26

  24. Dualizing deductive closure Theorem (Birkhoff completeness theorem). For any E ∈ Rel ( UFX ) , Th Mod ( E ) = Ded ( E ) To dualize Ded , we consider again its components. Algebras Coalgebras Projective set of variables X Injective set of colors C Set of equations Coequation E �� UFX P � � UHC Congruence generated by E Greatest subcoalgebra in P The Formal Dual of Birkhoff’s Completeness Theorem – p.15/26

  25. Dualizing deductive closure Theorem (Birkhoff completeness theorem). For any E ∈ Rel ( UFX ) , Th Mod ( E ) = Ded ( E ) To dualize Ded , we consider again its components. Algebras Coalgebras Projective set of variables X Injective set of colors C Set of equations Coequation E �� UFX P � � UHC Congruence generated by E Greatest subcoalgebra in P Greatest endo-invariant sub- Closure under substitution object The Formal Dual of Birkhoff’s Completeness Theorem – p.15/26

  26. � � ✁ ✁ Outline I. Coequations II. Conditional coequations III. Horn coequations IV. Some co-Birkhoff type theorems (again) V. Birkhoff’s completeness theorem VI. Dualizing deductive closure VII. The operator VIII. The operator IX. The invariance theorem X. Commutativity of , The Formal Dual of Birkhoff’s Completeness Theorem – p.16/26

  27. � � ✁ ✁ Outline I. Coequations II. Conditional coequations III. Horn coequations IV. Some co-Birkhoff type theorems (again) V. Birkhoff’s completeness theorem VI. Dualizing deductive closure VII. The operator VIII. The operator IX. The invariance theorem X. Commutativity of , The Formal Dual of Birkhoff’s Completeness Theorem – p.16/26

  28. � � � � The modal operator Let P, Q � � A be given. We write P ⊢ Q if there is a map P � Q such that the diagram below commutes. Q P � � � ��� � A The Formal Dual of Birkhoff’s Completeness Theorem – p.17/26

  29. � � � � � The modal operator Let P, Q � � A be given. We write P ⊢ Q if there is a map P � Q such that the diagram below commutes. Q P � � � ��� � A In fact, P � � Q is necessarily an S -morphism. The Formal Dual of Birkhoff’s Completeness Theorem – p.17/26

  30. � � The modal operator � Sub ( UHC ) be the composite U [ − ] . : Sub ( UHC ) Let is a comonad taking a coequation P to In other terms, the largest subcoalgebra � A, α � of HC such that A ≤ P . The Formal Dual of Birkhoff’s Completeness Theorem – p.17/26

  31. � � � � � � � � � � � � The modal operator � Sub ( UHC ) be the composite U [ − ] . : Sub ( UHC ) Let is a comonad taking a coequation P to In other terms, the largest subcoalgebra � A, α � of HC such that A ≤ P . As is well-known, if Γ preserves pullbacks of S -morphisms, then is an S4 operator. (i) If P ⊢ Q then P ⊢ Q ; P ⊢ P ; (ii) P ⊢ P ; (iii) ( P → Q ) ⊢ P → Q ; (iv) The Formal Dual of Birkhoff’s Completeness Theorem – p.17/26

  32. � � � � � � � � � � � The modal operator � Sub ( UHC ) be the composite U [ − ] . : Sub ( UHC ) Let is a comonad taking a coequation P to In other terms, the largest subcoalgebra � A, α � of HC such that A ≤ P . (i) If P ⊢ Q then P ⊢ Q ; P ⊢ P ; (ii) P ⊢ P ; (iii) ( P → Q ) ⊢ P → Q ; (iv) (i) follows from functoriality. The Formal Dual of Birkhoff’s Completeness Theorem – p.17/26

  33. � � � � � � � � � � � The modal operator � Sub ( UHC ) be the composite U [ − ] . : Sub ( UHC ) Let is a comonad taking a coequation P to In other terms, the largest subcoalgebra � A, α � of HC such that A ≤ P . (i) If P ⊢ Q then P ⊢ Q ; P ⊢ P ; (ii) P ⊢ P ; (iii) ( P → Q ) ⊢ P → Q ; (iv) (ii) and (iii) are the counit and comultiplication of the comonad. The Formal Dual of Birkhoff’s Completeness Theorem – p.17/26

  34. � � � � � � � � � � � The modal operator � Sub ( UHC ) be the composite U [ − ] . : Sub ( UHC ) Let is a comonad taking a coequation P to In other terms, the largest subcoalgebra � A, α � of HC such that A ≤ P . (i) If P ⊢ Q then P ⊢ Q ; P ⊢ P ; (ii) P ⊢ P ; (iii) ( P → Q ) ⊢ P → Q ; (iv) (iv) follows from the fact that U : E Γ � E preserves finite meets. The Formal Dual of Birkhoff’s Completeness Theorem – p.17/26

  35. � � � � � � � � � The modal operator (i) If P ⊢ Q then P ⊢ Q ; P ⊢ P ; (ii) P ⊢ P ; (iii) ( P → Q ) ⊢ P → Q ; (iv) Proof. P → Q ⊢ P → Q ( P → Q ) ∧ P ⊢ Q By the counit of adjunction − ∧ P ⊣ P → − . The Formal Dual of Birkhoff’s Completeness Theorem – p.17/26

  36. � � � � � � � � � � � The modal operator (i) If P ⊢ Q then P ⊢ Q ; P ⊢ P ; (ii) P ⊢ P ; (iii) ( P → Q ) ⊢ P → Q ; (iv) Proof. ( P → Q ) ∧ P ⊢ Q (( P → Q ) ∧ P ) ⊢ Q By (i). The Formal Dual of Birkhoff’s Completeness Theorem – p.17/26

  37. � � � � � � � � � � � � � � � The modal operator (i) If P ⊢ Q then P ⊢ Q ; P ⊢ P ; (ii) P ⊢ P ; (iii) ( P → Q ) ⊢ P → Q ; (iv) Proof. (( P → Q ) ∧ P ) ⊢ Q ( P → Q ) ∧ P ⊢ Q Because preserves meets. The Formal Dual of Birkhoff’s Completeness Theorem – p.17/26

  38. � � � � � � � � � � � � � � � The modal operator (i) If P ⊢ Q then P ⊢ Q ; P ⊢ P ; (ii) P ⊢ P ; (iii) ( P → Q ) ⊢ P → Q ; (iv) Proof. ( P → Q ) ∧ P ⊢ Q ( P → Q ) ⊢ P → Q Again, by the adjunction − ∧ P ⊣ P → − . The Formal Dual of Birkhoff’s Completeness Theorem – p.17/26

  39. � � � � Invariant coequations Let f : � A, α � � � B, β � and P � � A be given. We let ∃ f P denote the image of the composite P � � B. � A � � ∃ f P P � B A The Formal Dual of Birkhoff’s Completeness Theorem – p.18/26

  40. Invariant coequations Let P ⊆ UHC . We say that P is endomorphism-invariant just in case, for every “repainting” p : UHC � C, p : HC � HC , we have equivalently, every homomorphism � ∃ � p P ≤ P. The Formal Dual of Birkhoff’s Completeness Theorem – p.18/26

  41. Invariant coequations Let P ⊆ UHC . We say that P is endomorphism-invariant just in case, for every “repainting” p : UHC � C, p : HC � HC , we have equivalently, every homomorphism � ∃ c ∈ UHC ( � p ( c ) = x ∧ P ( c )) ⊢ P ( x ) . The Formal Dual of Birkhoff’s Completeness Theorem – p.18/26

  42. Invariant coequations Let P ⊆ UHC . We say that P is endomorphism-invariant just in case, for every “repainting” p : UHC � C, p : HC � HC , we have equivalently, every homomorphism � ∃ c ∈ UHC ( � p ( c ) = x ∧ P ( c )) ⊢ P ( x ) . In other words, however we repaint HC , the elements of P are again (under this new coloring) elements of P . The Formal Dual of Birkhoff’s Completeness Theorem – p.18/26

  43. Definition of Let P ⊆ UHC . Define I P = { Q ≤ UHC | ∀ p : HC � HC ( ∃ p Q ≤ P ) } . The Formal Dual of Birkhoff’s Completeness Theorem – p.19/26

  44. Definition of Let P ⊆ UHC . Define I P = { Q ≤ UHC | ∀ p : HC � HC ( ∃ p Q ≤ P ) } . That is, I P is the collection of all those coequations Q such that, however we “repaint” UHC , the image of Q still lands in P . The Formal Dual of Birkhoff’s Completeness Theorem – p.19/26

  45. Definition of Let P ⊆ UHC . Define I P = { Q ≤ UHC | ∀ p : HC � HC ( ∃ p Q ≤ P ) } . That is, I P is the collection of all those coequations Q such that, however we “repaint” UHC , the image of Q still lands in P . In particular, if Q ∈ I P , then Q ⊢ P . The Formal Dual of Birkhoff’s Completeness Theorem – p.19/26

  46. ✁ ✁ ✁ Definition of Let P ⊆ UHC . Define I P = { Q ≤ UHC | ∀ p : HC � HC ( ∃ p Q ≤ P ) } . � Sub ( UHC ) by : Sub ( UHC ) We define a functor � P = I P . P is the greatest invariant subobject of UHC con- Then tained in P . The Formal Dual of Birkhoff’s Completeness Theorem – p.19/26

  47. ✁ ✁ ✁ ✁ ✁ Definition of � Sub ( UHC ) by : Sub ( UHC ) We define a functor � I P . P = P satisfies the following: That is, • For all p : HC � HC , ∃ p P ⊢ P . The Formal Dual of Birkhoff’s Completeness Theorem – p.19/26

  48. ✁ ✁ ✁ ✁ ✁ ✁ Definition of � Sub ( UHC ) by : Sub ( UHC ) We define a functor � I P . P = P satisfies the following: That is, • For all p : HC � HC , ∃ p P ⊢ P . • If Q ⊢ P and for all p : HC � HC , ∃ p Q ⊢ Q , then Q ⊢ P . The Formal Dual of Birkhoff’s Completeness Theorem – p.19/26

  49. Example (cont.) The coequation P . The Formal Dual of Birkhoff’s Completeness Theorem – p.20/26

  50. Example (cont.) P is not invariant. The Formal Dual of Birkhoff’s Completeness Theorem – p.20/26

  51. ✁ Example (cont.) P . The coequation The Formal Dual of Birkhoff’s Completeness Theorem – p.20/26

  52. ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ is S4 is an S4 operator. One can show that (i) If P ⊢ Q then P ⊢ Q ; P ⊢ P ; (ii) P ⊢ P ; (iii) ( P → Q ) ⊢ P → Q ; (iv) The Formal Dual of Birkhoff’s Completeness Theorem – p.21/26

  53. ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ is S4 is an S4 operator. One can show that (i) If P ⊢ Q then P ⊢ Q ; P ⊢ P ; (ii) P ⊢ P ; (iii) ( P → Q ) ⊢ P → Q ; (iv) (i) - (iii) follow from the fact that is a comonad, as before. The Formal Dual of Birkhoff’s Completeness Theorem – p.21/26

  54. ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ is S4 is an S4 operator. One can show that (i) If P ⊢ Q then P ⊢ Q ; P ⊢ P ; (ii) P ⊢ P ; (iii) ( P → Q ) ⊢ P → Q ; (iv) (iv) requires an argument that the meet of two invariant co- equations is again invariant. This is not difficult. The Formal Dual of Birkhoff’s Completeness Theorem – p.21/26

  55. � � ✁ ✁ Outline I. Coequations II. Conditional coequations III. Horn coequations IV. Some co-Birkhoff type theorems (again) V. Birkhoff’s completeness theorem VI. Dualizing deductive closure VII. The operator VIII. The operator IX. The invariance theorem X. Commutativity of , The Formal Dual of Birkhoff’s Completeness Theorem – p.22/26

  56. � � ✁ ✁ Outline I. Coequations II. Conditional coequations III. Horn coequations IV. Some co-Birkhoff type theorems (again) V. Birkhoff’s completeness theorem VI. Dualizing deductive closure VII. The operator VIII. The operator IX. The invariance theorem X. Commutativity of , The Formal Dual of Birkhoff’s Completeness Theorem – p.22/26

  57. � The invariance theorem Lemma. � A, α � | = P iff � A, α � | = P . The Formal Dual of Birkhoff’s Completeness Theorem – p.23/26

  58. � ✁ The invariance theorem Lemma. � A, α � | = P iff � A, α � | = P . Lemma. � A, α � | = P iff � A, α � | = P . The Formal Dual of Birkhoff’s Completeness Theorem – p.23/26

  59. � ✁ ✁ The invariance theorem Lemma. � A, α � | = P iff � A, α � | = P . P ] | Lemma. [ = P . The Formal Dual of Birkhoff’s Completeness Theorem – p.23/26

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