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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

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SLIDE 1

The Formal Dual of Birkhoff’s Completeness Theorem

Jesse Hughes

jesseh@cs.kun.nl

University of Nijmegen

The Formal Dual of Birkhoff’s Completeness Theorem – p.1/26

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SLIDE 2

Outline

  • I. Coequations

The Formal Dual of Birkhoff’s Completeness Theorem – p.2/26

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SLIDE 3

Outline

  • I. Coequations
  • II. Conditional coequations

The Formal Dual of Birkhoff’s Completeness Theorem – p.2/26

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SLIDE 4

Outline

  • I. Coequations
  • II. Conditional coequations
  • III. Horn coequations

The Formal Dual of Birkhoff’s Completeness Theorem – p.2/26

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SLIDE 5

Outline

  • I. Coequations
  • II. Conditional coequations
  • III. Horn coequations
  • IV. Some co-Birkhoff type theorems (again)

The Formal Dual of Birkhoff’s Completeness Theorem – p.2/26

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SLIDE 6

Outline

  • I. Coequations
  • II. Conditional coequations
  • III. Horn coequations
  • IV. Some co-Birkhoff type theorems (again)
  • V. Birkhoff’s completeness theorem

The Formal Dual of Birkhoff’s Completeness Theorem – p.2/26

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SLIDE 7

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

The Formal Dual of Birkhoff’s Completeness Theorem – p.2/26

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SLIDE 8

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
  • perator

The Formal Dual of Birkhoff’s Completeness Theorem – p.2/26

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SLIDE 9

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
  • perator
  • VIII. The
✁
  • perator

The Formal Dual of Birkhoff’s Completeness Theorem – p.2/26

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SLIDE 10

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
  • perator
  • VIII. The
✁
  • perator
  • IX. The invariance theorem

The Formal Dual of Birkhoff’s Completeness Theorem – p.2/26

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SLIDE 11

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
  • perator
  • VIII. The
✁
  • perator
  • IX. The invariance theorem
  • X. Commutativity of
  • ,
✁

The Formal Dual of Birkhoff’s Completeness Theorem – p.2/26

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SLIDE 12

Coequations

Let U ⊣ H and C ∈ C be injective with respect to S-morphisms. A coequation over C is an S-morphism P UHC in C.

The Formal Dual of Birkhoff’s Completeness Theorem – p.3/26

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SLIDE 13

Coequations

Let U ⊣ H and C ∈ C be injective with respect to S-morphisms. A coequation over C is an S-morphism P UHC in C. We say A, α | =C P just in case for every homomorphism p:A, α

HC, we have Im(p) ≤ P.

A

∀p ∃

  • UHC

P

  • The Formal Dual of Birkhoff’s Completeness Theorem – p.3/26
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SLIDE 14

Coequations

Let U ⊣ H and C ∈ C be injective with respect to S-morphisms. A coequation over C is an S-morphism P UHC in C. We say A, α | =C P just in case for every homomorphism p:A, α

HC, we have Im(p) ≤ P.

A, α

∀p ∃

  • HC

[P]

  • Here, [P] is the largest subcoalgebra of HC contained in

P.

The Formal Dual of Birkhoff’s Completeness Theorem – p.3/26

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SLIDE 15

Coequations

Let U ⊣ H and C ∈ C be injective with respect to S-morphisms. A coequation over C is an S-morphism P UHC in C. We say A, α | =C P just in case for every homomorphism p:A, α

HC, we have Im(p) ≤ P.

Thus, A, α | =C P iff A, α ∈ Proj([P]), i.e., Hom(A, α, HC) ∼ = Hom(A, α, [P]).

The Formal Dual of Birkhoff’s Completeness Theorem – p.3/26

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SLIDE 16

Example

The cofree coalgebra H2

The Formal Dual of Birkhoff’s Completeness Theorem – p.4/26

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SLIDE 17

Example

A coequation.

The Formal Dual of Birkhoff’s Completeness Theorem – p.4/26

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SLIDE 18

Example

This coalgebra satisfies P.

The Formal Dual of Birkhoff’s Completeness Theorem – p.4/26

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SLIDE 19

Example

Under any coloring, the elements of the coalgebra map to elements of P.

The Formal Dual of Birkhoff’s Completeness Theorem – p.4/26

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SLIDE 20

Example

This coalgebra doesn’t satisfy P.

The Formal Dual of Birkhoff’s Completeness Theorem – p.4/26

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SLIDE 21

Example

If we paint the circle red, it isn’t mapped to an element of P.

The Formal Dual of Birkhoff’s Completeness Theorem – p.4/26

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SLIDE 22

Comparing coequations and equations

Algebras Coalgebras Projective set of variables X Injective set of colors C

The Formal Dual of Birkhoff’s Completeness Theorem – p.5/26

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SLIDE 23

Comparing coequations and equations

Algebras Coalgebras Projective set of variables X Injective set of colors C Set of equations Coequation

The Formal Dual of Birkhoff’s Completeness Theorem – p.5/26

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SLIDE 24

Comparing coequations and equations

Algebras Coalgebras Projective set of variables X Injective set of colors C Set of equations Coequation E

UFX

P

UHC

The Formal Dual of Birkhoff’s Completeness Theorem – p.5/26

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SLIDE 25

Comparing coequations and equations

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.5/26

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SLIDE 26

Comparing coequations and equations

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 | = as q-injective | = as i-projective

The Formal Dual of Birkhoff’s Completeness Theorem – p.5/26

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SLIDE 27

Conditional coequations

Let P, Q ≤ UHC. We write A, α | =C P ⇒ Q just in case, for every p:A, α

HC such that Im(p) ≤ P, we have Im(p) ≤ Q.

The Formal Dual of Birkhoff’s Completeness Theorem – p.6/26

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SLIDE 28

Conditional coequations

Let P, Q ≤ UHC. We write A, α | =C P ⇒ Q just in case, for every p:A, α

HC such that Im(p) ≤ P, we have Im(p) ≤ Q.

A

p

  • ∃
  • UHC

P

  • ⇒

A

p

  • ∃
  • UHC

Q

  • The Formal Dual of Birkhoff’s Completeness Theorem – p.6/26
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SLIDE 29

Conditional coequations

Let P, Q ≤ UHC. We write A, α | =C P ⇒ Q just in case, for every p:A, α

HC such that Im(p) ≤ P, we have Im(p) ≤ Q.

A, α | = P ⇒ Q just in case every homomorphism A, α

[P] factors through [Q], i.e.,

Hom(A, α, [P]) ∼ = Hom(A, α, [Q]).

The Formal Dual of Birkhoff’s Completeness Theorem – p.6/26

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SLIDE 30

Example

Recall our coequation P.

The Formal Dual of Birkhoff’s Completeness Theorem – p.7/26

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SLIDE 31

Example

Let Q be the coequation above.

The Formal Dual of Birkhoff’s Completeness Theorem – p.7/26

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SLIDE 32

Example

P Q

And consider the “conditional coequation” P ⇒ Q.

The Formal Dual of Birkhoff’s Completeness Theorem – p.7/26

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SLIDE 33

Example

Q P

This coalgebra satisfies P ⇒ Q.

The Formal Dual of Birkhoff’s Completeness Theorem – p.7/26

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Example

Q P

However we paint it so that it factors through P, it also factors through Q.

The Formal Dual of Birkhoff’s Completeness Theorem – p.7/26

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SLIDE 35

Example

Q P

(It also satisfies Q ⇒ P.)

The Formal Dual of Birkhoff’s Completeness Theorem – p.7/26

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SLIDE 36

Dualizing negations

Let P ≤ UHC. We write A, α | =C P just in case for every p:A

C, it is

not the case Im( p) ≤ P.

The Formal Dual of Birkhoff’s Completeness Theorem – p.8/26

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SLIDE 37

Dualizing negations

Let P ≤ UHC. We write A, α | =C P just in case for every p:A

C, it is

not the case Im( p) ≤ P. Equivalently, there is no homomorphism A, α

[P], i.e.,

Hom(A, α, [P]) = ∅.

The Formal Dual of Birkhoff’s Completeness Theorem – p.8/26

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SLIDE 38

Dualizing negations

Let P ≤ UHC. We write A, α | =C P just in case for every p:A

C, it is

not the case Im( p) ≤ P. Equivalently, there is no homomorphism A, α

[P], i.e.,

Hom(A, α, [P]) = ∅. No matter how we paint A, there is some element a ∈ A that doesn’t land in P.

The Formal Dual of Birkhoff’s Completeness Theorem – p.8/26

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SLIDE 39

Dualizing negations

Let P ≤ UHC. We write A, α | =C P just in case for every p:A

C, it is

not the case Im( p) ≤ P. No matter how we paint A, there is some element a ∈ A that doesn’t land in P. Note: This does not mean that A, α | = ¬P! “Something in A does not land in P,” is not the same as, “Everything in A does not land in P.”

The Formal Dual of Birkhoff’s Completeness Theorem – p.8/26

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SLIDE 40

Example

The coalgebra on the left satisfies P.

The Formal Dual of Birkhoff’s Completeness Theorem – p.9/26

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SLIDE 41

Example

No matter how we paint it, the square does not land in P

The Formal Dual of Birkhoff’s Completeness Theorem – p.9/26

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SLIDE 42

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
  • perator
  • VIII. The
✁
  • perator
  • IX. The invariance theorem
  • X. Commutativity of
  • ,
✁

The Formal Dual of Birkhoff’s Completeness Theorem – p.10/26

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SLIDE 43

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
  • perator
  • VIII. The
✁
  • perator
  • IX. The invariance theorem
  • X. Commutativity of
  • ,
✁

The Formal Dual of Birkhoff’s Completeness Theorem – p.10/26

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SLIDE 44

Some co-Birkhoff-type theorems

Define Th V = {P

UHC | V |

=C P}

The Formal Dual of Birkhoff’s Completeness Theorem – p.11/26

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SLIDE 45

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

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SLIDE 46

Some co-Birkhoff-type theorems

Define Th V = {P

UHC | V |

=C P} Imp V = {P ⇒C Q | V | =C P ⇒ Q} Horn V = Imp V ∪ {P

C | V |

=C P}

The Formal Dual of Birkhoff’s Completeness Theorem – p.11/26

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SLIDE 47

Some co-Birkhoff-type theorems

Define Th V = {P

UHC | V |

=C P} Imp V = {P ⇒C Q | V | =C P ⇒ Q} Horn V = Imp V ∪ {P

C | V |

=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

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SLIDE 48

Some co-Birkhoff-type theorems

Theorem (Birkhoff covariety theorem). Mod Th V = SHΣV Theorem (Quasi-covariety theorem). Mod Imp V = HΣV Theorem (Horn covariety theorem). Mod Horn V = HΣ+V

The Formal Dual of Birkhoff’s Completeness Theorem – p.11/26

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SLIDE 49

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) t1 = t2 ∈ E ⇒ t2 = t1 ∈ E; (iii) t1 = t2 ∈ E and t2 = t3 ∈ E ⇒ t1 = t3 ∈ E; (iv) ti

1 = ti 2 ∈ E and f ∈ Σ⇒f(

t1) = f( t2) ∈ E; (v) t1 = t2 ∈ E ⇒ t1[t/x] = t2[t/x] ∈ E.

The Formal Dual of Birkhoff’s Completeness Theorem – p.12/26

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SLIDE 50

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) t1 = t2 ∈ E ⇒ t2 = t1 ∈ E; (iii) t1 = t2 ∈ E and t2 = t3 ∈ E ⇒ t1 = t3 ∈ E; (iv) ti

1 = ti 2 ∈ E and f ∈ Σ⇒f(

t1) = f( t2) ∈ E; (v) t1 = t2 ∈ E ⇒ t1[t/x] = t2[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

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SLIDE 51

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) t1 = t2 ∈ E ⇒ t2 = t1 ∈ E; (iii) t1 = t2 ∈ E and t2 = t3 ∈ E ⇒ t1 = t3 ∈ E; (iv) ti

1 = ti 2 ∈ E and f ∈ Σ⇒f(

t1) = f( t2) ∈ E; (v) t1 = t2 ∈ E ⇒ t1[t/x] = t2[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

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SLIDE 52

Birkhoff’s deduction theorem

E is deductively closed just in case E satisfies the following: (i) x = x ∈ E; (ii) t1 = t2 ∈ E ⇒ t2 = t1 ∈ E; (iii) t1 = t2 ∈ E and t2 = t3 ∈ E ⇒ t1 = t3 ∈ E; (iv) ti

1 = ti 2 ∈ E and f ∈ Σ⇒f(

t1) = f( t2) ∈ E; (v) t1 = t2 ∈ E ⇒ t1[t/x] = t2[t/x] ∈ E. Let Ded:Rel(UFX)

Rel(UFX) be the closure operation

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

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SLIDE 53

Birkhoff’s deduction theorem

E is deductively closed just in case E satisfies the following: (i) x = x ∈ E; (ii) t1 = t2 ∈ E ⇒ t2 = t1 ∈ E; (iii) t1 = t2 ∈ E and t2 = t3 ∈ E ⇒ t1 = t3 ∈ E; (iv) ti

1 = ti 2 ∈ E and f ∈ Σ⇒f(

t1) = f( t2) ∈ E; (v) t1 = t2 ∈ E ⇒ t1[t/x] = t2[t/x] ∈ E. The first takes E to the congruence it generates.

The Formal Dual of Birkhoff’s Completeness Theorem – p.12/26

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SLIDE 54

Birkhoff’s deduction theorem

E is deductively closed just in case E satisfies the following: (i) x = x ∈ E; (ii) t1 = t2 ∈ E ⇒ t2 = t1 ∈ E; (iii) t1 = t2 ∈ E and t2 = t3 ∈ E ⇒ t1 = t3 ∈ E; (iv) ti

1 = ti 2 ∈ E and f ∈ Σ⇒f(

t1) = f( t2) ∈ E; (v) t1 = t2 ∈ E ⇒ t1[t/x] = t2[t/x] ∈ E. The second closes it under substitution of terms for variables.

The Formal Dual of Birkhoff’s Completeness Theorem – p.12/26

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SLIDE 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
  • perator
  • VIII. The
✁
  • perator
  • IX. The invariance theorem
  • X. Commutativity of
  • ,
✁

The Formal Dual of Birkhoff’s Completeness Theorem – p.13/26

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SLIDE 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
  • perator
  • VIII. The
✁
  • perator
  • IX. The invariance theorem
  • X. Commutativity of
  • ,
✁

The Formal Dual of Birkhoff’s Completeness Theorem – p.13/26

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SLIDE 57

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

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SLIDE 58

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 = HSPV

The Formal Dual of Birkhoff’s Completeness Theorem – p.14/26

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SLIDE 59

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);

  • If Mod(E) |

= E′, then E′ ⊆ Th Mod(E).

The Formal Dual of Birkhoff’s Completeness Theorem – p.14/26

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SLIDE 60

Dualizing the completeness theorem

Th Mod(E) satisfies the following fixed point description.

  • Mod(E) |

= Th Mod(E);

  • If Mod(E) |

= E′, then E′ ⊆ Th 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

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SLIDE 61

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);

  • If Mod(P) |

= P ′, then Gen Mod(P) ⊆ P ′.

The Formal Dual of Birkhoff’s Completeness Theorem – p.14/26

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SLIDE 62

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);

  • If Mod(P) |

= P ′, then Gen Mod(P) ⊆ 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

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SLIDE 63

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);

  • If Mod(P) |

= P ′, then Gen Mod(P) ⊆ 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

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SLIDE 64

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

slide-65
SLIDE 65

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

slide-66
SLIDE 66

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 Closure under substitution Greatest endo-invariant sub-

  • bject

The Formal Dual of Birkhoff’s Completeness Theorem – p.15/26

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SLIDE 67

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
  • perator
  • VIII. The
✁
  • perator
  • IX. The invariance theorem
  • X. Commutativity of
  • ,
✁

The Formal Dual of Birkhoff’s Completeness Theorem – p.16/26

slide-68
SLIDE 68

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
  • perator
  • VIII. The
✁
  • perator
  • IX. The invariance theorem
  • X. Commutativity of
  • ,
✁

The Formal Dual of Birkhoff’s Completeness Theorem – p.16/26

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SLIDE 69

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.

P Q A

  • The Formal Dual of Birkhoff’s Completeness Theorem – p.17/26
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SLIDE 70

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.

P Q A

  • In fact, P Q is necessarily an S-morphism.

The Formal Dual of Birkhoff’s Completeness Theorem – p.17/26

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SLIDE 71

The modal operator

Let

  • :Sub(UHC)

Sub(UHC) be the composite U[−].

In other terms,

  • is a comonad taking a coequation P to

the largest subcoalgebra A, α of HC such that A ≤ P.

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slide-72
SLIDE 72

The modal operator

Let

  • :Sub(UHC)

Sub(UHC) be the composite U[−].

In other terms,

  • is a comonad taking a coequation P to

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;

(ii)

  • P ⊢ P;

(iii)

  • P ⊢
  • P;

(iv)

  • (P → Q) ⊢
  • P →
  • Q;

The Formal Dual of Birkhoff’s Completeness Theorem – p.17/26

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SLIDE 73

The modal operator

Let

  • :Sub(UHC)

Sub(UHC) be the composite U[−].

In other terms,

  • is a comonad taking a coequation P to

the largest subcoalgebra A, α of HC such that A ≤ P. (i) If P ⊢ Q then

  • P ⊢
  • Q;

(ii)

  • P ⊢ P;

(iii)

  • P ⊢
  • P;

(iv)

  • (P → Q) ⊢
  • P →
  • Q;

(i) follows from functoriality.

The Formal Dual of Birkhoff’s Completeness Theorem – p.17/26

slide-74
SLIDE 74

The modal operator

Let

  • :Sub(UHC)

Sub(UHC) be the composite U[−].

In other terms,

  • is a comonad taking a coequation P to

the largest subcoalgebra A, α of HC such that A ≤ P. (i) If P ⊢ Q then

  • P ⊢
  • Q;

(ii)

  • P ⊢ P;

(iii)

  • P ⊢
  • P;

(iv)

  • (P → Q) ⊢
  • P →
  • Q;

(ii) and (iii) are the counit and comultiplication of the comonad.

The Formal Dual of Birkhoff’s Completeness Theorem – p.17/26

slide-75
SLIDE 75

The modal operator

Let

  • :Sub(UHC)

Sub(UHC) be the composite U[−].

In other terms,

  • is a comonad taking a coequation P to

the largest subcoalgebra A, α of HC such that A ≤ P. (i) If P ⊢ Q then

  • P ⊢
  • Q;

(ii)

  • P ⊢ P;

(iii)

  • P ⊢
  • P;

(iv)

  • (P → Q) ⊢
  • P →
  • Q;

(iv) follows from the fact that U :EΓ

E preserves finite

meets.

The Formal Dual of Birkhoff’s Completeness Theorem – p.17/26

slide-76
SLIDE 76

The modal operator

(i) If P ⊢ Q then

  • P ⊢
  • Q;

(ii)

  • P ⊢ P;

(iii)

  • P ⊢
  • P;

(iv)

  • (P → Q) ⊢
  • P →
  • Q;

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

slide-77
SLIDE 77

The modal operator

(i) If P ⊢ Q then

  • P ⊢
  • Q;

(ii)

  • P ⊢ P;

(iii)

  • P ⊢
  • P;

(iv)

  • (P → Q) ⊢
  • P →
  • Q;

Proof. (P → Q) ∧ P ⊢ Q

  • ((P → Q) ∧ P) ⊢
  • Q

By (i).

The Formal Dual of Birkhoff’s Completeness Theorem – p.17/26

slide-78
SLIDE 78

The modal operator

(i) If P ⊢ Q then

  • P ⊢
  • Q;

(ii)

  • P ⊢ P;

(iii)

  • P ⊢
  • P;

(iv)

  • (P → Q) ⊢
  • P →
  • Q;

Proof.

  • ((P → Q) ∧ P) ⊢
  • Q
  • (P → Q) ∧
  • P ⊢
  • Q

Because

  • preserves meets.

The Formal Dual of Birkhoff’s Completeness Theorem – p.17/26

slide-79
SLIDE 79

The modal operator

(i) If P ⊢ Q then

  • P ⊢
  • Q;

(ii)

  • P ⊢ P;

(iii)

  • P ⊢
  • P;

(iv)

  • (P → Q) ⊢
  • P →
  • Q;

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

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SLIDE 80

Invariant coequations

Let f :A, α

B, β and P A be given. We let ∃fP

denote the image of the composite P

A B.

P

  • ∃fP
  • A

B

The Formal Dual of Birkhoff’s Completeness Theorem – p.18/26

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SLIDE 81

Invariant coequations

Let P ⊆ UHC. We say that P is endomorphism-invariant just in case, for every “repainting” p:UHC

C,

equivalently, every homomorphism p:HC

HC, we have

∃

pP ≤ P.

The Formal Dual of Birkhoff’s Completeness Theorem – p.18/26

slide-82
SLIDE 82

Invariant coequations

Let P ⊆ UHC. We say that P is endomorphism-invariant just in case, for every “repainting” p:UHC

C,

equivalently, every homomorphism p:HC

HC, we have

∃c∈UHC( p(c) = x ∧ P(c)) ⊢ P(x).

The Formal Dual of Birkhoff’s Completeness Theorem – p.18/26

slide-83
SLIDE 83

Invariant coequations

Let P ⊆ UHC. We say that P is endomorphism-invariant just in case, for every “repainting” p:UHC

C,

equivalently, every homomorphism p:HC

HC, we have

∃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

slide-84
SLIDE 84

Definition of

Let P ⊆ UHC. Define IP = {Q ≤ UHC | ∀p:HC

HC(∃pQ ≤ P)}.

The Formal Dual of Birkhoff’s Completeness Theorem – p.19/26

slide-85
SLIDE 85

Definition of

Let P ⊆ UHC. Define IP = {Q ≤ UHC | ∀p:HC

HC(∃pQ ≤ P)}.

That is, IP 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

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SLIDE 86

Definition of

Let P ⊆ UHC. Define IP = {Q ≤ UHC | ∀p:HC

HC(∃pQ ≤ P)}.

That is, IP 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 ∈ IP, then Q ⊢ P.

The Formal Dual of Birkhoff’s Completeness Theorem – p.19/26

slide-87
SLIDE 87

Definition of

Let P ⊆ UHC. Define IP = {Q ≤ UHC | ∀p:HC

HC(∃pQ ≤ P)}.

We define a functor

✁

:Sub(UHC)

Sub(UHC) by

✁

P =

  • IP.

Then

✁

P is the greatest invariant subobject of UHC con- tained in P.

The Formal Dual of Birkhoff’s Completeness Theorem – p.19/26

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SLIDE 88

Definition of

We define a functor

✁

:Sub(UHC)

Sub(UHC) by

✁

P =

  • IP.

That is,

✁

P satisfies the following:

  • For all p:HC

HC, ∃p

✁

P ⊢

✁

P.

The Formal Dual of Birkhoff’s Completeness Theorem – p.19/26

slide-89
SLIDE 89

Definition of

We define a functor

✁

:Sub(UHC)

Sub(UHC) by

✁

P =

  • IP.

That is,

✁

P satisfies the following:

  • For all p:HC

HC, ∃p

✁

P ⊢

✁

P.

  • If Q ⊢ P and for all p:HC

HC, ∃pQ ⊢ Q, then

Q ⊢

✁

P.

The Formal Dual of Birkhoff’s Completeness Theorem – p.19/26

slide-90
SLIDE 90

Example (cont.)

The coequation P.

The Formal Dual of Birkhoff’s Completeness Theorem – p.20/26

slide-91
SLIDE 91

Example (cont.)

P is not invariant.

The Formal Dual of Birkhoff’s Completeness Theorem – p.20/26

slide-92
SLIDE 92

Example (cont.)

The coequation

✁

P.

The Formal Dual of Birkhoff’s Completeness Theorem – p.20/26

slide-93
SLIDE 93

is S4

One can show that

✁

is an S4 operator. (i) If P ⊢ Q then

✁

P ⊢

✁

Q; (ii)

✁

P ⊢ P; (iii)

✁

P ⊢

✁ ✁

P; (iv)

✁

(P → Q) ⊢

✁

P →

✁

Q;

The Formal Dual of Birkhoff’s Completeness Theorem – p.21/26

slide-94
SLIDE 94

is S4

One can show that

✁

is an S4 operator. (i) If P ⊢ Q then

✁

P ⊢

✁

Q; (ii)

✁

P ⊢ P; (iii)

✁

P ⊢

✁ ✁

P; (iv)

✁

(P → Q) ⊢

✁

P →

✁

Q; (i) - (iii) follow from the fact that

✁

is a comonad, as before.

The Formal Dual of Birkhoff’s Completeness Theorem – p.21/26

slide-95
SLIDE 95

is S4

One can show that

✁

is an S4 operator. (i) If P ⊢ Q then

✁

P ⊢

✁

Q; (ii)

✁

P ⊢ P; (iii)

✁

P ⊢

✁ ✁

P; (iv)

✁

(P → Q) ⊢

✁

P →

✁

Q; (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

slide-96
SLIDE 96

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
  • perator
  • VIII. The
✁
  • perator
  • IX. The invariance theorem
  • X. Commutativity of
  • ,
✁

The Formal Dual of Birkhoff’s Completeness Theorem – p.22/26

slide-97
SLIDE 97

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
  • perator
  • VIII. The
✁
  • perator
  • IX. The invariance theorem
  • X. Commutativity of
  • ,
✁

The Formal Dual of Birkhoff’s Completeness Theorem – p.22/26

slide-98
SLIDE 98

The invariance theorem

  • Lemma. A, α |

= P iff A, α | =

  • P.

The Formal Dual of Birkhoff’s Completeness Theorem – p.23/26

slide-99
SLIDE 99

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

slide-100
SLIDE 100

The invariance theorem

  • Lemma. A, α |

= P iff A, α | =

  • ✁

P.

  • Lemma. [
✁

P] | = P.

The Formal Dual of Birkhoff’s Completeness Theorem – p.23/26

slide-101
SLIDE 101

The invariance theorem

  • Lemma. A, α |

= P iff A, α | =

  • ✁

P.

  • Lemma. [
✁

P] | = P. Lemma.

  • ✁

P ≤

✁
  • P , i.e., if P is invariant,then

so is

  • P .

The Formal Dual of Birkhoff’s Completeness Theorem – p.23/26

slide-102
SLIDE 102

The invariance theorem

  • Lemma. A, α |

= P iff A, α | =

  • ✁

P.

  • Lemma. [
✁

P] | = P. Lemma.

  • ✁

P ≤

✁
  • P.
  • Theorem. Gen Mod P =
  • ✁

P.

The Formal Dual of Birkhoff’s Completeness Theorem – p.23/26

slide-103
SLIDE 103

The invariance theorem

  • Lemma. A, α |

= P iff A, α | =

  • ✁

P.

  • Lemma. [
✁

P] | = P. Lemma.

  • ✁

P ≤

✁
  • P.
  • Theorem. Gen Mod P =
  • ✁

P.

  • Proof. From the above, we see that Mod P |

=

  • ✁

P.

The Formal Dual of Birkhoff’s Completeness Theorem – p.23/26

slide-104
SLIDE 104

The invariance theorem

  • Lemma. A, α |

= P iff A, α | =

  • ✁

P.

  • Lemma. [
✁

P] | = P. Lemma.

  • ✁

P ≤

✁
  • P.
  • Theorem. Gen Mod P =
  • ✁

P.

  • Proof. From the above, we see that Mod P |

=

  • ✁

P. Suppose that Mod P | = Q. Then [

✁

P] | = Q.

The Formal Dual of Birkhoff’s Completeness Theorem – p.23/26

slide-105
SLIDE 105

The invariance theorem

  • Lemma. A, α |

= P iff A, α | =

  • ✁

P.

  • Lemma. [
✁

P] | = P. Lemma.

  • ✁

P ≤

✁
  • P.
  • Theorem. Gen Mod P =
  • ✁

P.

  • Proof. From the above, we see that Mod P |

=

  • ✁

P. Suppose that Mod P | = Q. Then [

✁

P] | = Q. Hence: U[

✁

P]

  • UHC

Q

  • The Formal Dual of Birkhoff’s Completeness Theorem – p.23/26
slide-106
SLIDE 106

The invariance theorem

  • Lemma. A, α |

= P iff A, α | =

  • ✁

P.

  • Lemma. [
✁

P] | = P. Lemma.

  • ✁

P ≤

✁
  • P.
  • Theorem. Gen Mod P =
  • ✁

P.

  • Proof. From the above, we see that Mod P |

=

  • ✁

P. Suppose that Mod P | = Q. Then [

✁

P] | = Q. Hence:

  • ✁

P

  • UHC

Q

  • That is,
  • ✁

P ⊢ Q.

The Formal Dual of Birkhoff’s Completeness Theorem – p.23/26

slide-107
SLIDE 107

Commutativity of ,

As we saw (without proof), Lemma.

  • ✁

P ≤

✁
  • P.

That is, the greatest subcoalgebra of an endomorphism in- variant predicate is itself invariant.

The Formal Dual of Birkhoff’s Completeness Theorem – p.24/26

slide-108
SLIDE 108

Commutativity of ,

As we saw (without proof), Lemma.

  • ✁

P ≤

✁
  • P.

Question: When is that an equality?

The Formal Dual of Birkhoff’s Completeness Theorem – p.24/26

slide-109
SLIDE 109

Commutativity of ,

As we saw (without proof), Lemma.

  • ✁

P ≤

✁
  • P.
  • Theorem. If Γ preserves non-empty intersections,

then

  • ✁

P =

✁
  • P.

The Formal Dual of Birkhoff’s Completeness Theorem – p.24/26

slide-110
SLIDE 110

Commutativity of ,

As we saw (without proof), Lemma.

  • ✁

P ≤

✁
  • P.
  • Theorem. If Γ preserves non-empty intersections,

then

  • ✁

P =

✁
  • P.

In this case, subcoalgebras are closed under arbitrary inter- sections.

The Formal Dual of Birkhoff’s Completeness Theorem – p.24/26

slide-111
SLIDE 111

A counterexample

Consider the functor F :Set

Set taking a set X to the

filters on X. F does not preserve non-empty intersections.

The Formal Dual of Birkhoff’s Completeness Theorem – p.25/26

slide-112
SLIDE 112

A counterexample

Consider the functor F :Set

Set taking a set X to the

filters on X. A topological space X may be considered as a F-coalgebra, via the structure map X

FX taking an

element x ∈ X to the neighborhood filter containing x.

The Formal Dual of Birkhoff’s Completeness Theorem – p.25/26

slide-113
SLIDE 113

A counterexample

Consider the functor F :Set

Set taking a set X to the

filters on X. A topological space X may be considered as a F-coalgebra, via the structure map X

FX taking an

element x ∈ X to the neighborhood filter containing x. We will show an example of a space X together with a “coequation” P ⊆ X such that

  • ✁

P =

✁
  • P, i.e.,
  • ✁

P ≤

✁
  • P.

The Formal Dual of Birkhoff’s Completeness Theorem – p.25/26

slide-114
SLIDE 114

A counterexample

We will show an example of a space X together with a “coequation” P ⊆ X such that

  • ✁

P =

✁
  • P, i.e.,
  • ✁

P ≤

✁
  • P.

Consider the real interval (0, 1], topologized with open sets

  • f the form (x, 1] for x ∈ X.

The Formal Dual of Birkhoff’s Completeness Theorem – p.25/26

slide-115
SLIDE 115

A counterexample

We will show an example of a space X together with a “coequation” P ⊆ X such that

  • ✁

P =

✁
  • P, i.e.,
  • ✁

P ≤

✁
  • P.

Consider the real interval (0, 1], topologized with open sets

  • f the form (x, 1] for x ∈ X. It is not difficult to show that

the only non-trivial endo-invariant subset of (0, 1] is {1}.

The Formal Dual of Birkhoff’s Completeness Theorem – p.25/26

slide-116
SLIDE 116

A counterexample

We will show an example of a space X together with a “coequation” P ⊆ X such that

  • ✁

P =

✁
  • P, i.e.,
  • ✁

P ≤

✁
  • P.

Consider the real interval (0, 1], topologized with open sets

  • f the form (x, 1] for x ∈ X. It is not difficult to show that

the only non-trivial endo-invariant subset of (0, 1] is {1}. Let P = (1

2, 1].

The Formal Dual of Birkhoff’s Completeness Theorem – p.25/26

slide-117
SLIDE 117

A counterexample

We will show an example of a space X together with a “coequation” P ⊆ X such that

  • ✁

P =

✁
  • P, i.e.,
  • ✁

P ≤

✁
  • P.

Let P = (1

2, 1]. Then

  • P = P and so
✁
  • P = {1}.

The Formal Dual of Birkhoff’s Completeness Theorem – p.25/26

slide-118
SLIDE 118

A counterexample

We will show an example of a space X together with a “coequation” P ⊆ X such that

  • ✁

P =

✁
  • P, i.e.,
  • ✁

P ≤

✁
  • P.

Let P = (1

2, 1]. Then

  • P = P and so
✁
  • P = {1}. On

the other hand,

✁

P = {1}, and so

  • ✁

P = ∅.

The Formal Dual of Birkhoff’s Completeness Theorem – p.25/26

slide-119
SLIDE 119

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
  • perator
  • VIII. The
✁
  • perator
  • IX. The invariance theorem
  • X. Commutativity of
  • ,
✁

The Formal Dual of Birkhoff’s Completeness Theorem – p.26/26

slide-120
SLIDE 120

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
  • perator
  • VIII. The
✁
  • perator
  • IX. The invariance theorem
  • X. Commutativity of
  • ,
✁

The Formal Dual of Birkhoff’s Completeness Theorem – p.26/26