[PPT] - Free algebras via a functor Sam van Gool on partial algebras Free PowerPoint Presentation

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Free alg’s via functor on partial alg’s Dion Coumans and Sam van Gool Free algebra step-by-step Free image-total functor Application to KB

Free algebras via a functor

n partial algebras

Dion Coumans and Sam van Gool Topology, Algebra and Categories in Logic (TACL) 26 – 30 July 2011 Marseilles, France

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Free alg’s via functor on partial alg’s Dion Coumans and Sam van Gool Free algebra step-by-step Free image-total functor Application to KB

Logic via algebra

Algebraic logic L, signature σ, variety VL of σ-algebras

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Free alg’s via functor on partial alg’s Dion Coumans and Sam van Gool Free algebra step-by-step Free image-total functor Application to KB

Logic via algebra

Algebraic logic L, signature σ, variety VL of σ-algebras
Studying the logic L Studying finitely generated free

VL-algebras

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Free alg’s via functor on partial alg’s Dion Coumans and Sam van Gool Free algebra step-by-step Free image-total functor Application to KB

Logic via algebra

Algebraic logic L, signature σ, variety VL of σ-algebras
Studying the logic L Studying finitely generated free

VL-algebras Language Fσ(x1, . . . , xm)

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Free alg’s via functor on partial alg’s Dion Coumans and Sam van Gool Free algebra step-by-step Free image-total functor Application to KB

Logic via algebra

Algebraic logic L, signature σ, variety VL of σ-algebras
Studying the logic L Studying finitely generated free

VL-algebras Language Fσ(x1, . . . , xm)

[·]⊣⊢L

Logic FVL(x1, . . . , xm)

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Free alg’s via functor on partial alg’s Dion Coumans and Sam van Gool Free algebra step-by-step Free image-total functor Application to KB

Free algebra as colimit of a chain

In many cases, a variety (VL)− of reducts is

well-understood and locally finite, e.g.:

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Free alg’s via functor on partial alg’s Dion Coumans and Sam van Gool Free algebra step-by-step Free image-total functor Application to KB

Free algebra as colimit of a chain

In many cases, a variety (VL)− of reducts is

well-understood and locally finite, e.g.:

Modal algebras = Boolean algebras + ,

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Free alg’s via functor on partial alg’s Dion Coumans and Sam van Gool Free algebra step-by-step Free image-total functor Application to KB

Free algebra as colimit of a chain

In many cases, a variety (VL)− of reducts is

well-understood and locally finite, e.g.:

Modal algebras = Boolean algebras + ,
Heyting algebras = Distributive lattices + →,

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Free alg’s via functor on partial alg’s Dion Coumans and Sam van Gool Free algebra step-by-step Free image-total functor Application to KB

Free algebra as colimit of a chain

In many cases, a variety (VL)− of reducts is

well-understood and locally finite, e.g.:

Modal algebras = Boolean algebras + ,
Heyting algebras = Distributive lattices + →,
. . .

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Free alg’s via functor on partial alg’s Dion Coumans and Sam van Gool Free algebra step-by-step Free image-total functor Application to KB

Free algebra as colimit of a chain

In many cases, a variety (VL)− of reducts is

well-understood and locally finite, e.g.:

Modal algebras = Boolean algebras + ,
Heyting algebras = Distributive lattices + →,
. . .
Regard FVL(x1, . . . , xn) as colimit of a chain of finite

algebras in the reduced signature, and add the additional

peration(s) step-by-step:

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Free alg’s via functor on partial alg’s Dion Coumans and Sam van Gool Free algebra step-by-step Free image-total functor Application to KB

Language

· · ·

T1 T0

[·]⊣⊢L

Logic

· · ·

B1 B0

Tn: formulas in variables x1, . . . , xm of rank ≤ n in
peration f
Bn: L-equivalence classes of formulas in Tn

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Free alg’s via functor on partial alg’s Dion Coumans and Sam van Gool Free algebra step-by-step Free image-total functor Application to KB

Free algebra as colimit of a chain

Language Tn

· · ·

T1 T0

[·]⊣⊢L

Logic Bn

· · ·

B1 B0

Tn: formulas in variables x1, . . . , xm of rank ≤ n in
peration f
Bn: L-equivalence classes of formulas in Tn

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Free alg’s via functor on partial alg’s Dion Coumans and Sam van Gool Free algebra step-by-step Free image-total functor Application to KB

Free algebra as colimit of a chain

Language

· · ·

Tn

· · ·

T1 T0

[·]⊣⊢L

Logic

· · ·

Bn

· · ·

B1 B0

Tn: formulas in variables x1, . . . , xm of rank ≤ n in
peration f
Bn: L-equivalence classes of formulas in Tn

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Free alg’s via functor on partial alg’s Dion Coumans and Sam van Gool Free algebra step-by-step Free image-total functor Application to KB

Free algebra as colimit of a chain

Language

· · ·

Tn

· · ·

T1 T0

[·]⊣⊢L

Logic f

· · ·

Bn

· · ·

B1 B0

Tn: formulas in variables x1, . . . , xm of rank ≤ n in
peration f
Bn: L-equivalence classes of formulas in Tn

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Free alg’s via functor on partial alg’s Dion Coumans and Sam van Gool Free algebra step-by-step Free image-total functor Application to KB

Free algebra as colimit of a chain

Language

· · ·

Tn

· · ·

T1 T0

[·]⊣⊢L

Logic f

· · ·

Bn

· · ·

B1 B0 f

Tn: formulas in variables x1, . . . , xm of rank ≤ n in
peration f
Bn: L-equivalence classes of formulas in Tn

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Free alg’s via functor on partial alg’s Dion Coumans and Sam van Gool Free algebra step-by-step Free image-total functor Application to KB

Free algebra as colimit of a chain

Language

· · ·

Tn

· · ·

T1 T0

[·]⊣⊢L

Logic f f f

· · ·

Bn

· · ·

B1 B0 f f f

Tn: formulas in variables x1, . . . , xm of rank ≤ n in
peration f
Bn: L-equivalence classes of formulas in Tn

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Free alg’s via functor on partial alg’s Dion Coumans and Sam van Gool Free algebra step-by-step Free image-total functor Application to KB

Free algebra as colimit of a chain

Language

· · ·

Tn

· · ·

T1 T0

[·]⊣⊢L

Logic f f f

· · ·

Bn

· · ·

B1 B0 f f f

Tn: formulas in variables x1, . . . , xm of rank ≤ n in
peration f
Bn: L-equivalence classes of formulas in Tn

FVL(x1, . . . , xm) = colimn≥0 Bn

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Free alg’s via functor on partial alg’s Dion Coumans and Sam van Gool Free algebra step-by-step Free image-total functor Application to KB

Research Question

· · ·

Bn

· · ·

B1 B0 Can Bn+1 be obtained from Bn by a uniform method?

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Free alg’s via functor on partial alg’s Dion Coumans and Sam van Gool Free algebra step-by-step Free image-total functor Application to KB

Research Question

· · ·

Bn

· · ·

B1 B0 Can Bn+1 be obtained from Bn by a uniform method?

Yes, if the variety is defined by pure rank 1 equations

[N. Bezhanishvili, Kurz]

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Free alg’s via functor on partial alg’s Dion Coumans and Sam van Gool Free algebra step-by-step Free image-total functor Application to KB

Research Question

· · ·

Bn

· · ·

B1 B0 Can Bn+1 be obtained from Bn by a uniform method?

Yes, if the variety is defined by pure rank 1 equations

[N. Bezhanishvili, Kurz]

Yes, in some particular cases outside this class: S4 modal

algebras [Ghilardi], Heyting algebras [Ghilardi, N. Bezhanishvili & Gehrke].

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Free alg’s via functor on partial alg’s Dion Coumans and Sam van Gool Free algebra step-by-step Free image-total functor Application to KB

Research Question

· · ·

Bn

· · ·

B1 B0 Can Bn+1 be obtained from Bn by a uniform method?

Yes, if the variety is defined by pure rank 1 equations

[N. Bezhanishvili, Kurz]

Yes, in some particular cases outside this class: S4 modal

algebras [Ghilardi], Heyting algebras [Ghilardi, N. Bezhanishvili & Gehrke].

Not always, since logics can be undecidable.

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SLIDE 27

Free alg’s via functor on partial alg’s Dion Coumans and Sam van Gool Free algebra step-by-step Free image-total functor Application to KB

Research Question

· · ·

Bn

· · ·

B1 B0 Can Bn+1 be obtained from Bn by a uniform method?

Yes, if the variety is defined by pure rank 1 equations

[N. Bezhanishvili, Kurz]

Yes, in some particular cases outside this class: S4 modal

algebras [Ghilardi], Heyting algebras [Ghilardi, N. Bezhanishvili & Gehrke].

Not always, since logics can be undecidable.
We give general sufficient conditions under which this is

possible (known cases follow as particular instances).

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Free alg’s via functor on partial alg’s Dion Coumans and Sam van Gool Free algebra step-by-step Free image-total functor Application to KB

Partial algebras

In the chain, Bn+1 is a partial algebra, where the domain
f the operation f is Bn.

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Free alg’s via functor on partial alg’s Dion Coumans and Sam van Gool Free algebra step-by-step Free image-total functor Application to KB

Partial algebras

In the chain, Bn+1 is a partial algebra, where the domain
f the operation f is Bn.
The variety V is contained in a category pV of partial

algebras for the variety V.

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Free alg’s via functor on partial alg’s Dion Coumans and Sam van Gool Free algebra step-by-step Free image-total functor Application to KB

Partial algebras

In the chain, Bn+1 is a partial algebra, where the domain
f the operation f is Bn.
The variety V is contained in a category pV of partial

algebras for the variety V.

A homomorphism h : A → B of partial algebras is a

function which preserves all total operations, and preserves the partial operation f whenever defined.

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Free alg’s via functor on partial alg’s Dion Coumans and Sam van Gool Free algebra step-by-step Free image-total functor Application to KB

Partial algebras

In the chain, Bn+1 is a partial algebra, where the domain
f the operation f is Bn.
The variety V is contained in a category pV of partial

algebras for the variety V.

A homomorphism h : A → B of partial algebras is a

function which preserves all total operations, and preserves the partial operation f whenever defined.

A homomorphism h : A → B is image-total if the image of

h is contained in the domain of fB.

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Free alg’s via functor on partial alg’s Dion Coumans and Sam van Gool Free algebra step-by-step Free image-total functor Application to KB

Free image-total functor

Definition

A functor F : pV → pV is free image-total if there is a component-wise image-total natural transformation η : 1pV → F such that, for all image-total h : A → B, there exists a unique

¯

h : FA → B making the following diagram commute: A

ηA ✲ FA

B

¯

h ❄ h ✲

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Free alg’s via functor on partial alg’s Dion Coumans and Sam van Gool Free algebra step-by-step Free image-total functor Application to KB

Free image-total functor

Main theorem

Theorem

Let η : 1 → F be a free image-total functor and A0 ∈ pV. Let Aω be the partial algebra-colimit of the image-total chain

{ηFn(A0) : Fn(A0) → Fn+1(A0)}n≥0.

If Aω is in V, then Aω is the free total V-algebra over A0.

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SLIDE 34

Free alg’s via functor on partial alg’s Dion Coumans and Sam van Gool Free algebra step-by-step Free image-total functor Application to KB

Free image-total functor

Main theorem

Theorem

Let η : 1 → F be a free image-total functor and A0 ∈ pV. Let Aω be the partial algebra-colimit of the image-total chain

{ηFn(A0) : Fn(A0) → Fn+1(A0)}n≥0.

If Aω is in V, then Aω is the free total V-algebra over A0.

Proof.

Category-theoretic arguments.

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SLIDE 35

Free alg’s via functor on partial alg’s Dion Coumans and Sam van Gool Free algebra step-by-step Free image-total functor Application to KB

Free image-total functor

Main theorem

Theorem

Let η : 1 → F be a free image-total functor and A0 ∈ pV. Let Aω be the partial algebra-colimit of the image-total chain

{ηFn(A0) : Fn(A0) → Fn+1(A0)}n≥0.

If Aω is in V, then Aω is the free total V-algebra over A0.

Proof.

Category-theoretic arguments.

Now, to apply this theorem:

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SLIDE 36

Free alg’s via functor on partial alg’s Dion Coumans and Sam van Gool Free algebra step-by-step Free image-total functor Application to KB

Free image-total functor

Main theorem

Theorem

Let η : 1 → F be a free image-total functor and A0 ∈ pV. Let Aω be the partial algebra-colimit of the image-total chain

{ηFn(A0) : Fn(A0) → Fn+1(A0)}n≥0.

If Aω is in V, then Aω is the free total V-algebra over A0.

Proof.

Category-theoretic arguments.

Now, to apply this theorem:
We construct a free image-total functor for any set of

quasi-equations,

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SLIDE 37

Free alg’s via functor on partial alg’s Dion Coumans and Sam van Gool Free algebra step-by-step Free image-total functor Application to KB

Free image-total functor

Main theorem

Theorem

Let η : 1 → F be a free image-total functor and A0 ∈ pV. Let Aω be the partial algebra-colimit of the image-total chain

{ηFn(A0) : Fn(A0) → Fn+1(A0)}n≥0.

If Aω is in V, then Aω is the free total V-algebra over A0.

Proof.

Category-theoretic arguments.

Now, to apply this theorem:
We construct a free image-total functor for any set of

quasi-equations,

We give sufficient conditions under which Aω ∈ V.

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SLIDE 38

Free alg’s via functor on partial alg’s Dion Coumans and Sam van Gool Free algebra step-by-step Free image-total functor Application to KB

Free image-total functor

Construction

Let E be a set of quasi-equations (of rank at most 1)

axiomatizing the variety V.

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SLIDE 39

Free alg’s via functor on partial alg’s Dion Coumans and Sam van Gool Free algebra step-by-step Free image-total functor Application to KB

Free image-total functor

Construction

Let E be a set of quasi-equations (of rank at most 1)

axiomatizing the variety V.

For A ∈ pV, define

FE(A) := [A + FV−(fA)]/θA

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SLIDE 40

Free alg’s via functor on partial alg’s Dion Coumans and Sam van Gool Free algebra step-by-step Free image-total functor Application to KB

Free image-total functor

Construction

Let E be a set of quasi-equations (of rank at most 1)

axiomatizing the variety V.

For A ∈ pV, define

FE(A) := [A + FV−(fA)]/θA

V−: reduct of V to the signature of total operations,

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SLIDE 41

Free alg’s via functor on partial alg’s Dion Coumans and Sam van Gool Free algebra step-by-step Free image-total functor Application to KB

Free image-total functor

Construction

Let E be a set of quasi-equations (of rank at most 1)

axiomatizing the variety V.

For A ∈ pV, define

FE(A) := [A + FV−(fA)]/θA

V−: reduct of V to the signature of total operations,
fA: formal elements {fa : a ∈ A}, yielding partial operation

a → fa for a ∈ A,

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SLIDE 42

Free alg’s via functor on partial alg’s Dion Coumans and Sam van Gool Free algebra step-by-step Free image-total functor Application to KB

Free image-total functor

Construction

Let E be a set of quasi-equations (of rank at most 1)

axiomatizing the variety V.

For A ∈ pV, define

FE(A) := [A + FV−(fA)]/θA

V−: reduct of V to the signature of total operations,
fA: formal elements {fa : a ∈ A}, yielding partial operation

a → fa for a ∈ A,

θA: smallest pV-congruence on A + FV−(fA) containing

fAa, fa, for all a ∈ dom(fA).

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SLIDE 43

Free alg’s via functor on partial alg’s Dion Coumans and Sam van Gool Free algebra step-by-step Free image-total functor Application to KB

Free image-total functor

Construction

Let E be a set of quasi-equations (of rank at most 1)

axiomatizing the variety V.

For A ∈ pV, define

FE(A) := [A + FV−(fA)]/θA

V−: reduct of V to the signature of total operations,
fA: formal elements {fa : a ∈ A}, yielding partial operation

a → fa for a ∈ A,

θA: smallest pV-congruence on A + FV−(fA) containing

fAa, fa, for all a ∈ dom(fA).

ηA is the composite

A ֌ A + FV−(fA) ։ FE(A).

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Free alg’s via functor on partial alg’s Dion Coumans and Sam van Gool Free algebra step-by-step Free image-total functor Application to KB

Free image-total functor

Lemma

FE is a free image-total functor with universal arrow η. Furthermore, if A0 ∈ pV is such that each component

ηFn

E(A0) : Fn

E(A0) → Fn+1 E

(A0) is an embedding, then Aω ∈ V.

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SLIDE 45

Free alg’s via functor on partial alg’s Dion Coumans and Sam van Gool Free algebra step-by-step Free image-total functor Application to KB

Free image-total functor

Lemma

FE is a free image-total functor with universal arrow η. Furthermore, if A0 ∈ pV is such that each component

ηFn

E(A0) : Fn

E(A0) → Fn+1 E

(A0) is an embedding, then Aω ∈ V. Proof.

Uses universal algebra for partial algebras.

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Free alg’s via functor on partial alg’s Dion Coumans and Sam van Gool Free algebra step-by-step Free image-total functor Application to KB

Free image-total functor

Lemma

FE is a free image-total functor with universal arrow η. Furthermore, if A0 ∈ pV is such that each component

ηFn

E(A0) : Fn

E(A0) → Fn+1 E

(A0) is an embedding, then Aω ∈ V. Proof.

Uses universal algebra for partial algebras.

Corollary

If A0 ∈ pV is such that each component

ηFn

E(A0) : Fn

E(A0) → Fn+1 E

(A0) is an embedding, then Aω is the

free total V-algebra over A0.

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SLIDE 47

Free alg’s via functor on partial alg’s Dion Coumans and Sam van Gool Free algebra step-by-step Free image-total functor Application to KB

The variety KB

Signature = ⊥, ⊤, ∨, ∧, ¬,

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Free alg’s via functor on partial alg’s Dion Coumans and Sam van Gool Free algebra step-by-step Free image-total functor Application to KB

The variety KB

Signature = ⊥, ⊤, ∨, ∧, ¬,
Axioms = Boolean algebras +

⊥ = ⊥ (x ∨ y) = x ∨ y

x ≤ ¬y → y ≤ ¬x.

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SLIDE 49

Free alg’s via functor on partial alg’s Dion Coumans and Sam van Gool Free algebra step-by-step Free image-total functor Application to KB

The variety KB

Signature = ⊥, ⊤, ∨, ∧, ¬,
Axioms = Boolean algebras +

⊥ = ⊥ (x ∨ y) = x ∨ y

x ≤ ¬y → y ≤ ¬x.

(Finite) Duality theory:

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SLIDE 50

Free alg’s via functor on partial alg’s Dion Coumans and Sam van Gool Free algebra step-by-step Free image-total functor Application to KB

The variety KB

Signature = ⊥, ⊤, ∨, ∧, ¬,
Axioms = Boolean algebras +

⊥ = ⊥ (x ∨ y) = x ∨ y

x ≤ ¬y → y ≤ ¬x.

(Finite) Duality theory:
KB algebras ↔ Sets with a symmetric relation

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SLIDE 51

Free alg’s via functor on partial alg’s Dion Coumans and Sam van Gool Free algebra step-by-step Free image-total functor Application to KB

The variety KB

Signature = ⊥, ⊤, ∨, ∧, ¬,
Axioms = Boolean algebras +

⊥ = ⊥ (x ∨ y) = x ∨ y

x ≤ ¬y → y ≤ ¬x.

(Finite) Duality theory:
KB algebras ↔ Sets with a symmetric relation
Partial KB algebras ↔ Sets with an equivalence relation ∼

and a quasi-symmetric relation R satisfying R ◦ ∼ ⊆ R

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SLIDE 52

Free alg’s via functor on partial alg’s Dion Coumans and Sam van Gool Free algebra step-by-step Free image-total functor Application to KB

The functor FKB

By definition, for a partial KB algebra A,

FKB(A) = [A + FBA(A)]/θA.

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SLIDE 53

Free alg’s via functor on partial alg’s Dion Coumans and Sam van Gool Free algebra step-by-step Free image-total functor Application to KB

The functor FKB

By definition, for a partial KB algebra A,

FKB(A) = [A + FBA(A)]/θA. A Duality X

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SLIDE 54

Free alg’s via functor on partial alg’s Dion Coumans and Sam van Gool Free algebra step-by-step Free image-total functor Application to KB

The functor FKB

By definition, for a partial KB algebra A,

FKB(A) = [A + FBA(A)]/θA. A Duality X FBA(A)

+ × P(X)

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SLIDE 55

Free alg’s via functor on partial alg’s Dion Coumans and Sam van Gool Free algebra step-by-step Free image-total functor Application to KB

The functor FKB

By definition, for a partial KB algebra A,

FKB(A) = [A + FBA(A)]/θA.

GKB(X)

A Duality X FBA(A)

+ × P(X)

A FBA(A)

+ /θA

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SLIDE 56

Free alg’s via functor on partial alg’s Dion Coumans and Sam van Gool Free algebra step-by-step Free image-total functor Application to KB

The functor FKB

By definition, for a partial KB algebra A,

FKB(A) = [A + FBA(A)]/θA.

GKB(X)

A Duality X FBA(A)

+ × P(X)

A FBA(A)

+ /θA

Using correspondence theory, one can explicitly calculate

a first-order definition of the points in GKB(X, R, ∼)

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SLIDE 57

Free alg’s via functor on partial alg’s Dion Coumans and Sam van Gool Free algebra step-by-step Free image-total functor Application to KB

The functor FKB

By definition, for a partial KB algebra A,

FKB(A) = [A + FBA(A)]/θA.

GKB(X)

A Duality X FBA(A)

+ × P(X)

A FBA(A)

+ /θA

Using correspondence theory, one can explicitly calculate

a first-order definition of the points in GKB(X, R, ∼)

These points are normal forms for KB.

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SLIDE 58

Free alg’s via functor on partial alg’s Dion Coumans and Sam van Gool Free algebra step-by-step Free image-total functor Application to KB

The chain for KB

First steps

X0 p

¬p

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SLIDE 59

Free alg’s via functor on partial alg’s Dion Coumans and Sam van Gool Free algebra step-by-step Free image-total functor Application to KB

The chain for KB

First steps

X0 p

¬p

GKB(X0)

p ∧ p ∧ ¬p p ∧ p ∧ ¬¬p p ∧ ¬p ∧ ¬p p ∧ ¬p ∧ ¬¬p ¬p ∧ p ∧ ¬p ¬p ∧ p ∧ ¬¬p ¬p ∧ ¬p ∧ ¬p ¬p ∧ ¬p ∧ ¬¬p

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SLIDE 60

Free alg’s via functor on partial alg’s Dion Coumans and Sam van Gool Free algebra step-by-step Free image-total functor Application to KB

The chain for KB

(part of) G2

KB(X0) 14 / 16

SLIDE 61

Free alg’s via functor on partial alg’s Dion Coumans and Sam van Gool Free algebra step-by-step Free image-total functor Application to KB

The chain for S4

First steps

X0

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Free alg’s via functor on partial alg’s Dion Coumans and Sam van Gool Free algebra step-by-step Free image-total functor Application to KB

The chain for S4

First steps

X0 GS4(X0)

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SLIDE 63

Free alg’s via functor on partial alg’s Dion Coumans and Sam van Gool Free algebra step-by-step Free image-total functor Application to KB

The chain for S4

First steps

X0 GS4(X0) G2

S4(X0)

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SLIDE 64

Free alg’s via functor on partial alg’s Dion Coumans and Sam van Gool Free algebra step-by-step Free image-total functor Application to KB

The chain for S4

First steps

X0 GS4(X0) G2

S4(X0)

(...)

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SLIDE 65

Free alg’s via functor on partial alg’s Dion Coumans and Sam van Gool Free algebra step-by-step Free image-total functor Application to KB

Free algebras via a functor

n partial algebras

Dion Coumans and Sam van Gool Topology, Algebra and Categories in Logic (TACL) 26 – 30 July 2011 Marseilles, France

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