[PPT] - Discrete duality for downset lattices and their Positive Modal PowerPoint Presentation

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Duality and Residuation Sam van Gool Discrete Duality Positive Modal Logic Residuation Research Question

Discrete duality for downset lattices and their residuated operations

Sam van Gool 30 October 2010 The categorical flow of information University of Oxford

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Duality and Residuation Sam van Gool Discrete Duality Positive Modal Logic Residuation Research Question

Motivation

Labelled transition systems:

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Duality and Residuation Sam van Gool Discrete Duality Positive Modal Logic Residuation Research Question

Motivation

Labelled transition systems:

States (S, ≤),

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Duality and Residuation Sam van Gool Discrete Duality Positive Modal Logic Residuation Research Question

Motivation

Labelled transition systems:

States (S, ≤), Actions A∗,

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Duality and Residuation Sam van Gool Discrete Duality Positive Modal Logic Residuation Research Question

Motivation

Labelled transition systems:

States (S, ≤), Actions A∗, Transitions R ⊆ S × A∗ × S.

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Duality and Residuation Sam van Gool Discrete Duality Positive Modal Logic Residuation Research Question

Motivation

Labelled transition systems:

States (S, ≤), Actions A∗, Transitions R ⊆ S × A∗ × S.

Logical description:

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Duality and Residuation Sam van Gool Discrete Duality Positive Modal Logic Residuation Research Question

Motivation

Labelled transition systems:

States (S, ≤), Actions A∗, Transitions R ⊆ S × A∗ × S.

Logical description:

1 Without negation,

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Duality and Residuation Sam van Gool Discrete Duality Positive Modal Logic Residuation Research Question

Motivation

Labelled transition systems:

States (S, ≤), Actions A∗, Transitions R ⊆ S × A∗ × S.

Logical description:

1 Without negation, 2 With converse,

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Duality and Residuation Sam van Gool Discrete Duality Positive Modal Logic Residuation Research Question

Motivation

Labelled transition systems:

States (S, ≤), Actions A∗, Transitions R ⊆ S × A∗ × S.

Logical description:

1 Without negation, 2 With converse, 3 With • as primitive operation.

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Duality and Residuation Sam van Gool Discrete Duality Positive Modal Logic Residuation Research Question

Presentation outline

1

Discrete Duality

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Duality and Residuation Sam van Gool Discrete Duality Positive Modal Logic Residuation Research Question

Presentation outline

1

Discrete Duality

2

Positive Modal Logic

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Duality and Residuation Sam van Gool Discrete Duality Positive Modal Logic Residuation Research Question

Presentation outline

1

Discrete Duality

2

Positive Modal Logic

3

Residuation

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Duality and Residuation Sam van Gool Discrete Duality Positive Modal Logic Residuation Research Question

Presentation outline

1

Discrete Duality

2

Positive Modal Logic

3

Residuation

4

Research Question

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Duality and Residuation Sam van Gool Discrete Duality Positive Modal Logic Residuation Research Question

Discrete Duality

Kripke semantics for modal logic

Discrete duality for complete atomic Boolean algebras

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Duality and Residuation Sam van Gool Discrete Duality Positive Modal Logic Residuation Research Question

Discrete Duality

Kripke semantics for modal logic

Discrete duality for complete atomic Boolean algebras

CABA ⇆ Set

B → At(B) P(X) →

X

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Duality and Residuation Sam van Gool Discrete Duality Positive Modal Logic Residuation Research Question

Discrete Duality

Kripke semantics for modal logic

Discrete duality for complete atomic Boolean algebras

CABA ⇆ Set

B → At(B) P(X) →

X

Complete operator on B ↔ relation R on At(B):

xRy ⇐⇒ x ≤ y

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Duality and Residuation Sam van Gool Discrete Duality Positive Modal Logic Residuation Research Question

Discrete Duality

Kripke semantics for modal logic

Discrete duality for complete atomic Boolean algebras

CABA ⇆ Set

B → At(B) P(X) →

X

Complete operator on B ↔ relation R on At(B):

xRy ⇐⇒ x ≤ y

Canonical extension: every BA with operator embeds in

a CABA with complete operator

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Duality and Residuation Sam van Gool Discrete Duality Positive Modal Logic Residuation Research Question

Discrete Duality

Kripke semantics for modal logic

Discrete duality for complete atomic Boolean algebras

CABA ⇆ Set

B → At(B) P(X) →

X

Complete operator on B ↔ relation R on At(B):

xRy ⇐⇒ x ≤ y

Canonical extension: every BA with operator embeds in

a CABA with complete operator

“Canonical Extension + Discrete Duality = Frame Completeness”

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Duality and Residuation Sam van Gool Discrete Duality Positive Modal Logic Residuation Research Question

Discrete Duality

Distributive lattices

Discrete duality for doubly algebraic distributive lattices

(Birkhoff 1940)

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Duality and Residuation Sam van Gool Discrete Duality Positive Modal Logic Residuation Research Question

Discrete Duality

Distributive lattices

Discrete duality for doubly algebraic distributive lattices

(Birkhoff 1940) DADLat ⇆ Poset

D → (J(D), ≤D) D(P) → (P, ≤)

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Duality and Residuation Sam van Gool Discrete Duality Positive Modal Logic Residuation Research Question

Discrete Duality

Distributive lattices

Discrete duality for doubly algebraic distributive lattices

(Birkhoff 1940) DADLat ⇆ Poset

D → (J(D), ≤D) D(P) → (P, ≤)

Complete operator on D ↔ relation R on J(D) with

≤ ◦ R ◦ ≤ = R (order-compatible)

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Duality and Residuation Sam van Gool Discrete Duality Positive Modal Logic Residuation Research Question

Discrete Duality

Distributive lattices

Discrete duality for doubly algebraic distributive lattices

(Birkhoff 1940) DADLat ⇆ Poset

D → (J(D), ≤D) D(P) → (P, ≤)

Complete operator on D ↔ relation R on J(D) with

≤ ◦ R ◦ ≤ = R (order-compatible)

Canonical extension: every DLat with operator embeds

in a DADLat with complete operator

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Duality and Residuation Sam van Gool Discrete Duality Positive Modal Logic Residuation Research Question

Positive Modal Logic

Condensed Chronology

Dunn (1995): modal logic in the absence of negation

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Duality and Residuation Sam van Gool Discrete Duality Positive Modal Logic Residuation Research Question

Positive Modal Logic

Condensed Chronology

Dunn (1995): modal logic in the absence of negation
Celani, Jansana (1997): simple Kripke-style semantics

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Duality and Residuation Sam van Gool Discrete Duality Positive Modal Logic Residuation Research Question

Positive Modal Logic

Condensed Chronology

Dunn (1995): modal logic in the absence of negation
Celani, Jansana (1997): simple Kripke-style semantics
Gehrke, Nagahashi, Venema (2004): general framework

for distributive-lattice-based modal logic

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Duality and Residuation Sam van Gool Discrete Duality Positive Modal Logic Residuation Research Question

Positive Modal Logic

Algebraic definition

A positive modal algebra (PMA) is a distributive lattice D

equipped with two unary operations and , such that

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Duality and Residuation Sam van Gool Discrete Duality Positive Modal Logic Residuation Research Question

Positive Modal Logic

Algebraic definition

A positive modal algebra (PMA) is a distributive lattice D

equipped with two unary operations and , such that

⊤ = ⊤, (a ∧ b) = a ∧ b,

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Duality and Residuation Sam van Gool Discrete Duality Positive Modal Logic Residuation Research Question

Positive Modal Logic

Algebraic definition

A positive modal algebra (PMA) is a distributive lattice D

equipped with two unary operations and , such that

⊤ = ⊤, ⊥ = ⊥, (a ∧ b) = a ∧ b, (a ∨ b) = a ∨ b,

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Duality and Residuation Sam van Gool Discrete Duality Positive Modal Logic Residuation Research Question

Positive Modal Logic

Algebraic definition

A positive modal algebra (PMA) is a distributive lattice D

equipped with two unary operations and , such that

⊤ = ⊤, ⊥ = ⊥, (a ∧ b) = a ∧ b, (a ∨ b) = a ∨ b, (a ∨ b) ≤ a ∨ b,

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Duality and Residuation Sam van Gool Discrete Duality Positive Modal Logic Residuation Research Question

Positive Modal Logic

Algebraic definition

A positive modal algebra (PMA) is a distributive lattice D

equipped with two unary operations and , such that

⊤ = ⊤, ⊥ = ⊥, (a ∧ b) = a ∧ b, (a ∨ b) = a ∨ b, (a ∨ b) ≤ a ∨ b, a ∧ b ≤ (a ∧ b).

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Duality and Residuation Sam van Gool Discrete Duality Positive Modal Logic Residuation Research Question

Positive Modal Logic

Algebraic definition

A positive modal algebra (PMA) is a distributive lattice D

equipped with two unary operations and , such that

⊤ = ⊤, ⊥ = ⊥, (a ∧ b) = a ∧ b, (a ∨ b) = a ∨ b, (a ∨ b) ≤ a ∨ b, a ∧ b ≤ (a ∧ b).

Two interaction axioms replacing = ¬¬

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Duality and Residuation Sam van Gool Discrete Duality Positive Modal Logic Residuation Research Question

Positive Modal Logic

Algebraic definition

A positive modal algebra (PMA) is a distributive lattice D

equipped with two unary operations and , such that

⊤ = ⊤, ⊥ = ⊥, (a ∧ b) = a ∧ b, (a ∨ b) = a ∨ b, (a ∨ b) ≤ a ∨ b, a ∧ b ≤ (a ∧ b).

Two interaction axioms replacing = ¬¬
Positive modal logic K+ : logic of positive modal

algebras.

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Duality and Residuation Sam van Gool Discrete Duality Positive Modal Logic Residuation Research Question

Positive Modal Logic

Kripke semantics for K+

Call a PMA (D, , ) perfect if D is doubly algebraic,

preserves all joins, and preserves all meets.

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Duality and Residuation Sam van Gool Discrete Duality Positive Modal Logic Residuation Research Question

Positive Modal Logic

Kripke semantics for K+

Call a PMA (D, , ) perfect if D is doubly algebraic,

preserves all joins, and preserves all meets.

Canonical extension magic: every PMA embeds in a

perfect PMA.

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Duality and Residuation Sam van Gool Discrete Duality Positive Modal Logic Residuation Research Question

Positive Modal Logic

Kripke semantics for K+

Call a PMA (D, , ) perfect if D is doubly algebraic,

preserves all joins, and preserves all meets.

Canonical extension magic: every PMA embeds in a

perfect PMA.

Consequence: automatic Kripke-style semantics for K+

via discrete duality

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Duality and Residuation Sam van Gool Discrete Duality Positive Modal Logic Residuation Research Question

Positive Modal Logic

Kripke semantics for K+

Call a PMA (D, , ) perfect if D is doubly algebraic,

preserves all joins, and preserves all meets.

Canonical extension magic: every PMA embeds in a

perfect PMA.

Consequence: automatic Kripke-style semantics for K+

via discrete duality A Kripke frame for K+ is a poset (P, ≤) equipped with two binary relations R and R, satisfying

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Duality and Residuation Sam van Gool Discrete Duality Positive Modal Logic Residuation Research Question

Positive Modal Logic

Kripke semantics for K+

Call a PMA (D, , ) perfect if D is doubly algebraic,

preserves all joins, and preserves all meets.

Canonical extension magic: every PMA embeds in a

perfect PMA.

Consequence: automatic Kripke-style semantics for K+

via discrete duality A Kripke frame for K+ is a poset (P, ≤) equipped with two binary relations R and R, satisfying

≤ ◦ R ◦ ≤ = R,

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Duality and Residuation Sam van Gool Discrete Duality Positive Modal Logic Residuation Research Question

Positive Modal Logic

Kripke semantics for K+

Call a PMA (D, , ) perfect if D is doubly algebraic,

preserves all joins, and preserves all meets.

Canonical extension magic: every PMA embeds in a

perfect PMA.

Consequence: automatic Kripke-style semantics for K+

via discrete duality A Kripke frame for K+ is a poset (P, ≤) equipped with two binary relations R and R, satisfying

≤ ◦ R ◦ ≤ = R, ≥ ◦ R ◦ ≥ = R,

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Duality and Residuation Sam van Gool Discrete Duality Positive Modal Logic Residuation Research Question

Positive Modal Logic

Kripke semantics for K+

Call a PMA (D, , ) perfect if D is doubly algebraic,

preserves all joins, and preserves all meets.

Canonical extension magic: every PMA embeds in a

perfect PMA.

Consequence: automatic Kripke-style semantics for K+

via discrete duality A Kripke frame for K+ is a poset (P, ≤) equipped with two binary relations R and R, satisfying

≤ ◦ R ◦ ≤ = R, ≥ ◦ R ◦ ≥ = R,

R ⊆ (R ∩ R) ◦ ≥,

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Duality and Residuation Sam van Gool Discrete Duality Positive Modal Logic Residuation Research Question

Positive Modal Logic

Kripke semantics for K+

Call a PMA (D, , ) perfect if D is doubly algebraic,

preserves all joins, and preserves all meets.

Canonical extension magic: every PMA embeds in a

perfect PMA.

Consequence: automatic Kripke-style semantics for K+

via discrete duality A Kripke frame for K+ is a poset (P, ≤) equipped with two binary relations R and R, satisfying

≤ ◦ R ◦ ≤ = R, ≥ ◦ R ◦ ≥ = R,

R ⊆ (R ∩ R) ◦ ≥, R ⊆ (R ∩ R) ◦ ≤.

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Duality and Residuation Sam van Gool Discrete Duality Positive Modal Logic Residuation Research Question

Residuation

Unary operation

If : D → D preserves all joins, let its upper adjoint.

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Duality and Residuation Sam van Gool Discrete Duality Positive Modal Logic Residuation Research Question

Residuation

Unary operation

If : D → D preserves all joins, let its upper adjoint.
Write (P, ≤, R) for the corresponding Kripke frame.

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Duality and Residuation Sam van Gool Discrete Duality Positive Modal Logic Residuation Research Question

Residuation

Unary operation

If : D → D preserves all joins, let its upper adjoint.
Write (P, ≤, R) for the corresponding Kripke frame.
Let ϕ ∈ D.

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Duality and Residuation Sam van Gool Discrete Duality Positive Modal Logic Residuation Research Question

Residuation

Unary operation

If : D → D preserves all joins, let its upper adjoint.
Write (P, ≤, R) for the corresponding Kripke frame.
Let ϕ ∈ D.

x |= ϕ ⇐⇒ x ≤ ϕ,

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Duality and Residuation Sam van Gool Discrete Duality Positive Modal Logic Residuation Research Question

Residuation

Unary operation

If : D → D preserves all joins, let its upper adjoint.
Write (P, ≤, R) for the corresponding Kripke frame.
Let ϕ ∈ D.

x |= ϕ ⇐⇒ x ≤ ϕ,

⇐⇒ ∃y : x R y and y |= ϕ.

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Duality and Residuation Sam van Gool Discrete Duality Positive Modal Logic Residuation Research Question

Residuation

Unary operation

If : D → D preserves all joins, let its upper adjoint.
Write (P, ≤, R) for the corresponding Kripke frame.
Let ϕ ∈ D.

x |= ϕ ⇐⇒ x ≤ ϕ,

⇐⇒ ∃y : x R y and y |= ϕ.

x |= ϕ ⇐⇒ x ≤ ϕ,

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Duality and Residuation Sam van Gool Discrete Duality Positive Modal Logic Residuation Research Question

Residuation

Unary operation

If : D → D preserves all joins, let its upper adjoint.
Write (P, ≤, R) for the corresponding Kripke frame.
Let ϕ ∈ D.

x |= ϕ ⇐⇒ x ≤ ϕ,

⇐⇒ ∃y : x R y and y |= ϕ.

x |= ϕ ⇐⇒ x ≤ ϕ,

⇐⇒ x ≤ ϕ,

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Duality and Residuation Sam van Gool Discrete Duality Positive Modal Logic Residuation Research Question

Residuation

Unary operation

If : D → D preserves all joins, let its upper adjoint.
Write (P, ≤, R) for the corresponding Kripke frame.
Let ϕ ∈ D.

x |= ϕ ⇐⇒ x ≤ ϕ,

⇐⇒ ∃y : x R y and y |= ϕ.

x |= ϕ ⇐⇒ x ≤ ϕ,

⇐⇒ x ≤ ϕ, ⇐⇒ ∀y : y R x implies y |= ϕ.

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Duality and Residuation Sam van Gool Discrete Duality Positive Modal Logic Residuation Research Question

Residuation

Unary operation

If : D → D preserves all joins, let its upper adjoint.
Write (P, ≤, R) for the corresponding Kripke frame.
Let ϕ ∈ D.

x |= ϕ ⇐⇒ x ≤ ϕ,

⇐⇒ ∃y : x R y and y |= ϕ.

x |= ϕ ⇐⇒ x ≤ ϕ,

⇐⇒ x ≤ ϕ, ⇐⇒ ∀y : y R x implies y |= ϕ.

Conclusion: R = R−1

.

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Duality and Residuation Sam van Gool Discrete Duality Positive Modal Logic Residuation Research Question

Residuation

Unary operation

If : D → D preserves all joins, let its upper adjoint.
Write (P, ≤, R) for the corresponding Kripke frame.
Let ϕ ∈ D.

x |= ϕ ⇐⇒ x ≤ ϕ,

⇐⇒ ∃y : x R y and y |= ϕ.

x |= ϕ ⇐⇒ x ≤ ϕ,

⇐⇒ x ≤ ϕ, ⇐⇒ ∀y : y R x implies y |= ϕ.

Conclusion: R = R−1

.

Note that ≤ ◦ R ◦ ≤ = R iff ≥ ◦ R ◦ ≥ = R.

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Duality and Residuation Sam van Gool Discrete Duality Positive Modal Logic Residuation Research Question

Residuation

Unary vs. Binary operations

Operations on D Relations on P

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Duality and Residuation Sam van Gool Discrete Duality Positive Modal Logic Residuation Research Question

Residuation

Unary vs. Binary operations

Operations on D Relations on P Unary Binary R with ≤ ◦ R ◦ ≤ = R Upper adjoint R = R−1

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Duality and Residuation Sam van Gool Discrete Duality Positive Modal Logic Residuation Research Question

Residuation

Unary vs. Binary operations

Operations on D Relations on P Unary Binary R with ≤ ◦ R ◦ ≤ = R Upper adjoint R = R−1

Binary •

Ternary R• with ≤ ◦ R• ◦ (≤ × ≤) = R• Upper adjoints /, \ ...

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Duality and Residuation Sam van Gool Discrete Duality Positive Modal Logic Residuation Research Question

Residuation

Unary vs. Binary operations

Operations on D Relations on P Unary Binary R with ≤ ◦ R ◦ ≤ = R Upper adjoint R = R−1

Binary •

Ternary R• with ≤ ◦ R• ◦ (≤ × ≤) = R• Upper adjoints /, \ ... n-ary f n + 1-ary Rf with ≤ ◦ Rf ◦ (≤)n = Rf Upper adjoints f#

i

...

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Duality and Residuation Sam van Gool Discrete Duality Positive Modal Logic Residuation Research Question

Residuation

Binary operation

• : D0 × D1 → D2 preserving all joins in each coordinate.

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Duality and Residuation Sam van Gool Discrete Duality Positive Modal Logic Residuation Research Question

Residuation

Binary operation

• : D0 × D1 → D2 preserving all joins in each coordinate.
Fixing a coordinate, • has upper adjoints \ and /:

a0 • a1 ≤ a2 ⇐⇒ a0 ≤ a2\a1

⇐⇒ a1 ≤ a0/a2

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Duality and Residuation Sam van Gool Discrete Duality Positive Modal Logic Residuation Research Question

Residuation

Binary operation

• : D0 × D1 → D2 preserving all joins in each coordinate.
Fixing a coordinate, • has upper adjoints \ and /:

a0 • a1 ≤ a2 ⇐⇒ a0 ≤ a2\a1

⇐⇒ a1 ≤ a0/a2

With a calculation similar to unary case:

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Duality and Residuation Sam van Gool Discrete Duality Positive Modal Logic Residuation Research Question

Residuation

Binary operation

• : D0 × D1 → D2 preserving all joins in each coordinate.
Fixing a coordinate, • has upper adjoints \ and /:

a0 • a1 ≤ a2 ⇐⇒ a0 ≤ a2\a1

⇐⇒ a1 ≤ a0/a2

With a calculation similar to unary case:

x0 |= ϕ2\ϕ1 ⇐⇒

∀x1∀x2(x1 |= ϕ1 and R•(x0, x1, x2) implies x2 |= ϕ2)

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Duality and Residuation Sam van Gool Discrete Duality Positive Modal Logic Residuation Research Question

Residuation

Binary operation

• : D0 × D1 → D2 preserving all joins in each coordinate.
Fixing a coordinate, • has upper adjoints \ and /:

a0 • a1 ≤ a2 ⇐⇒ a0 ≤ a2\a1

⇐⇒ a1 ≤ a0/a2

With a calculation similar to unary case:

x0 |= ϕ2\ϕ1 ⇐⇒

∀x1∀x2(x1 |= ϕ1 and R•(x0, x1, x2) implies x2 |= ϕ2)

Conclusion: R\ and R/ are permutations of Rf

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Duality and Residuation Sam van Gool Discrete Duality Positive Modal Logic Residuation Research Question

Residuation

Binary operation

• : D0 × D1 → D2 preserving all joins in each coordinate.
Fixing a coordinate, • has upper adjoints \ and /:

a0 • a1 ≤ a2 ⇐⇒ a0 ≤ a2\a1

⇐⇒ a1 ≤ a0/a2

With a calculation similar to unary case:

x0 |= ϕ2\ϕ1 ⇐⇒

∀x1∀x2(x1 |= ϕ1 and R•(x0, x1, x2) implies x2 |= ϕ2)

Conclusion: R\ and R/ are permutations of Rf
R• is order-compatible iff R\ is order-compatible iff R/ is
rder-compatible

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Duality and Residuation Sam van Gool Discrete Duality Positive Modal Logic Residuation Research Question

Research Question

Positive modal logic:

2 negation-dual operations (, ) from 1 binary relation R

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Duality and Residuation Sam van Gool Discrete Duality Positive Modal Logic Residuation Research Question

Research Question

Positive modal logic:

2 negation-dual operations (, ) from 1 binary relation R

Residuation:

n residuated operations from 1 n-ary relation R

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Duality and Residuation Sam van Gool Discrete Duality Positive Modal Logic Residuation Research Question

Research Question

Positive modal logic:

2 negation-dual operations (, ) from 1 binary relation R

Residuation:

n residuated operations from 1 n-ary relation R

Question:

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Duality and Residuation Sam van Gool Discrete Duality Positive Modal Logic Residuation Research Question

Research Question

Positive modal logic:

2 negation-dual operations (, ) from 1 binary relation R

Residuation:

n residuated operations from 1 n-ary relation R

Question:
Axiomatisation of n-ary positive modal logic with

residuated operations?

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SLIDE 65

Duality and Residuation Sam van Gool Discrete Duality Positive Modal Logic Residuation Research Question

Research Question

Positive modal logic:

2 negation-dual operations (, ) from 1 binary relation R

Residuation:

n residuated operations from 1 n-ary relation R

Question:
Axiomatisation of n-ary positive modal logic with

residuated operations?

Adding epistemic modalities?

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Duality and Residuation Sam van Gool Discrete Duality Positive Modal Logic Residuation Research Question

Discrete duality for downset lattices and their residuated operations

Sam van Gool 30 October 2010 The categorical flow of information University of Oxford

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