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Algebraic Semantics and Model Completeness for Intuitionistic Public Announcement Logic Minghui Ma, Alessandra Palmigiano, Mehrnoosh Sadrzadeh TACL 2011, Marseille 28 July 2011 Minghui Ma, Alessandra Palmigiano, Mehrnoosh Sadrzadeh Algebraic

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Algebraic Semantics and Model Completeness for Intuitionistic Public Announcement Logic

Minghui Ma, Alessandra Palmigiano, Mehrnoosh Sadrzadeh

TACL 2011, Marseille

28 July 2011

Minghui Ma, Alessandra Palmigiano, Mehrnoosh Sadrzadeh Algebraic Semantics and Model Completeness for Intuitionistic Public

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PAL

Minghui Ma, Alessandra Palmigiano, Mehrnoosh Sadrzadeh Algebraic Semantics and Model Completeness for Intuitionistic Public

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PAL

The simplest dynamic epistemic logic.

Minghui Ma, Alessandra Palmigiano, Mehrnoosh Sadrzadeh Algebraic Semantics and Model Completeness for Intuitionistic Public

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PAL

The simplest dynamic epistemic logic. Language

Minghui Ma, Alessandra Palmigiano, Mehrnoosh Sadrzadeh Algebraic Semantics and Model Completeness for Intuitionistic Public

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PAL

The simplest dynamic epistemic logic. Language

ϕ ::= p ∈ AtProp | ¬ϕ | ϕ ∨ ψ | ϕ | αϕ.

Minghui Ma, Alessandra Palmigiano, Mehrnoosh Sadrzadeh Algebraic Semantics and Model Completeness for Intuitionistic Public

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PAL

The simplest dynamic epistemic logic. Language

ϕ ::= p ∈ AtProp | ¬ϕ | ϕ ∨ ψ | ϕ | αϕ.

Axioms

Minghui Ma, Alessandra Palmigiano, Mehrnoosh Sadrzadeh Algebraic Semantics and Model Completeness for Intuitionistic Public

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PAL

The simplest dynamic epistemic logic. Language

ϕ ::= p ∈ AtProp | ¬ϕ | ϕ ∨ ψ | ϕ | αϕ.

Axioms

αp ↔ (α ∧ p)

α¬ϕ ↔ (α ∧ ¬αϕ)

α(ϕ ∨ ψ) ↔ (αϕ ∨ αψ)

αϕ ↔ (α ∧ (α ∧ αϕ)).

Minghui Ma, Alessandra Palmigiano, Mehrnoosh Sadrzadeh Algebraic Semantics and Model Completeness for Intuitionistic Public

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PAL

The simplest dynamic epistemic logic. Language

ϕ ::= p ∈ AtProp | ¬ϕ | ϕ ∨ ψ | ϕ | αϕ.

Axioms

αp ↔ (α ∧ p)

α¬ϕ ↔ (α ∧ ¬αϕ)

α(ϕ ∨ ψ) ↔ (αϕ ∨ αψ)

αϕ ↔ (α ∧ (α ∧ αϕ)).

Not amenable to a standard algebraic treatment.

Minghui Ma, Alessandra Palmigiano, Mehrnoosh Sadrzadeh Algebraic Semantics and Model Completeness for Intuitionistic Public

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Semantics of PAL

Minghui Ma, Alessandra Palmigiano, Mehrnoosh Sadrzadeh Algebraic Semantics and Model Completeness for Intuitionistic Public

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Semantics of PAL

PAL-models are S5 Kripke models: M = (W, R, V)

Minghui Ma, Alessandra Palmigiano, Mehrnoosh Sadrzadeh Algebraic Semantics and Model Completeness for Intuitionistic Public

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Semantics of PAL

PAL-models are S5 Kripke models: M = (W, R, V) M, w αϕ iff

Minghui Ma, Alessandra Palmigiano, Mehrnoosh Sadrzadeh Algebraic Semantics and Model Completeness for Intuitionistic Public

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Semantics of PAL

PAL-models are S5 Kripke models: M = (W, R, V) M, w αϕ iff M, w α

Minghui Ma, Alessandra Palmigiano, Mehrnoosh Sadrzadeh Algebraic Semantics and Model Completeness for Intuitionistic Public

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Semantics of PAL

PAL-models are S5 Kripke models: M = (W, R, V) M, w αϕ iff M, w α and Mα, w ϕ,

Minghui Ma, Alessandra Palmigiano, Mehrnoosh Sadrzadeh Algebraic Semantics and Model Completeness for Intuitionistic Public

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Semantics of PAL

PAL-models are S5 Kripke models: M = (W, R, V) M, w αϕ iff M, w α and Mα, w ϕ, Relativized model Mα = (Wα, Rα, Vα):

Minghui Ma, Alessandra Palmigiano, Mehrnoosh Sadrzadeh Algebraic Semantics and Model Completeness for Intuitionistic Public

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Semantics of PAL

PAL-models are S5 Kripke models: M = (W, R, V) M, w αϕ iff M, w α and Mα, w ϕ, Relativized model Mα = (Wα, Rα, Vα): Wα = [

[α] ]M,

Minghui Ma, Alessandra Palmigiano, Mehrnoosh Sadrzadeh Algebraic Semantics and Model Completeness for Intuitionistic Public

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Semantics of PAL

PAL-models are S5 Kripke models: M = (W, R, V) M, w αϕ iff M, w α and Mα, w ϕ, Relativized model Mα = (Wα, Rα, Vα): Wα = [

[α] ]M,

Rα = R ∩ (Wα × Wα),

Minghui Ma, Alessandra Palmigiano, Mehrnoosh Sadrzadeh Algebraic Semantics and Model Completeness for Intuitionistic Public

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Semantics of PAL

PAL-models are S5 Kripke models: M = (W, R, V) M, w αϕ iff M, w α and Mα, w ϕ, Relativized model Mα = (Wα, Rα, Vα): Wα = [

[α] ]M,

Rα = R ∩ (Wα × Wα), Vα(p) = V(p) ∩ Wα.

Minghui Ma, Alessandra Palmigiano, Mehrnoosh Sadrzadeh Algebraic Semantics and Model Completeness for Intuitionistic Public

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Methodology based on duality theory:

Minghui Ma, Alessandra Palmigiano, Mehrnoosh Sadrzadeh Algebraic Semantics and Model Completeness for Intuitionistic Public

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Methodology based on duality theory:

Dualize epistemic update on Kripke models to epistemic update on algebras.

Minghui Ma, Alessandra Palmigiano, Mehrnoosh Sadrzadeh Algebraic Semantics and Model Completeness for Intuitionistic Public

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Methodology based on duality theory:

Dualize epistemic update on Kripke models to epistemic update on algebras. Generalize epistemic update on algebras to much wider classes of algebras.

Minghui Ma, Alessandra Palmigiano, Mehrnoosh Sadrzadeh Algebraic Semantics and Model Completeness for Intuitionistic Public

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Methodology based on duality theory:

Dualize epistemic update on Kripke models to epistemic update on algebras. Generalize epistemic update on algebras to much wider classes of algebras. Dualize back to relational models for non classically based logics.

Minghui Ma, Alessandra Palmigiano, Mehrnoosh Sadrzadeh Algebraic Semantics and Model Completeness for Intuitionistic Public

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Algebraic models

Minghui Ma, Alessandra Palmigiano, Mehrnoosh Sadrzadeh Algebraic Semantics and Model Completeness for Intuitionistic Public

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Algebraic models

An algebraic model is a tuple M = (A, V) s.t. A is a monadic Heyting algebra and V : AtProp → A.

Minghui Ma, Alessandra Palmigiano, Mehrnoosh Sadrzadeh Algebraic Semantics and Model Completeness for Intuitionistic Public

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Algebraic models

An algebraic model is a tuple M = (A, V) s.t. A is a monadic Heyting algebra and V : AtProp → A. For every A and every a ∈ A, define the equivalence relation ≡a:

Minghui Ma, Alessandra Palmigiano, Mehrnoosh Sadrzadeh Algebraic Semantics and Model Completeness for Intuitionistic Public

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Algebraic models

An algebraic model is a tuple M = (A, V) s.t. A is a monadic Heyting algebra and V : AtProp → A. For every A and every a ∈ A, define the equivalence relation ≡a: for every b, c ∈ A, b ≡a c iff b ∧ a = c ∧ a.

Minghui Ma, Alessandra Palmigiano, Mehrnoosh Sadrzadeh Algebraic Semantics and Model Completeness for Intuitionistic Public

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Algebraic models

An algebraic model is a tuple M = (A, V) s.t. A is a monadic Heyting algebra and V : AtProp → A. For every A and every a ∈ A, define the equivalence relation ≡a: for every b, c ∈ A, b ≡a c iff b ∧ a = c ∧ a. Let [b]a be the equivalence class of b ∈ A. Let

Aa := A/≡a Aa is ordered: [b] ≤ [c] iff b′ ≤A c′ for some b′ ∈ [b] and some

c′ ∈ [c].

Minghui Ma, Alessandra Palmigiano, Mehrnoosh Sadrzadeh Algebraic Semantics and Model Completeness for Intuitionistic Public

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Algebraic models

An algebraic model is a tuple M = (A, V) s.t. A is a monadic Heyting algebra and V : AtProp → A. For every A and every a ∈ A, define the equivalence relation ≡a: for every b, c ∈ A, b ≡a c iff b ∧ a = c ∧ a. Let [b]a be the equivalence class of b ∈ A. Let

Aa := A/≡a Aa is ordered: [b] ≤ [c] iff b′ ≤A c′ for some b′ ∈ [b] and some

c′ ∈ [c]. Let πa : A → Aa be the canonical projection.

Minghui Ma, Alessandra Palmigiano, Mehrnoosh Sadrzadeh Algebraic Semantics and Model Completeness for Intuitionistic Public

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Properties of the (pseudo)-congruence

Minghui Ma, Alessandra Palmigiano, Mehrnoosh Sadrzadeh Algebraic Semantics and Model Completeness for Intuitionistic Public

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Properties of the (pseudo)-congruence

For every A and every a ∈ A,

Minghui Ma, Alessandra Palmigiano, Mehrnoosh Sadrzadeh Algebraic Semantics and Model Completeness for Intuitionistic Public

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Properties of the (pseudo)-congruence

For every A and every a ∈ A,

≡a is a congruence if A is a BA / HA / BDL / Fr.

Minghui Ma, Alessandra Palmigiano, Mehrnoosh Sadrzadeh Algebraic Semantics and Model Completeness for Intuitionistic Public

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Properties of the (pseudo)-congruence

For every A and every a ∈ A,

≡a is a congruence if A is a BA / HA / BDL / Fr. ≡a is not a congruence w.r.t. modal operators.

Minghui Ma, Alessandra Palmigiano, Mehrnoosh Sadrzadeh Algebraic Semantics and Model Completeness for Intuitionistic Public

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Properties of the (pseudo)-congruence

For every A and every a ∈ A,

≡a is a congruence if A is a BA / HA / BDL / Fr. ≡a is not a congruence w.r.t. modal operators.

For every b ∈ A there exists a unique c ∈ A s.t. c ∈ [b]a and c ≤ a.

Minghui Ma, Alessandra Palmigiano, Mehrnoosh Sadrzadeh Algebraic Semantics and Model Completeness for Intuitionistic Public

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Properties of the (pseudo)-congruence

For every A and every a ∈ A,

≡a is a congruence if A is a BA / HA / BDL / Fr. ≡a is not a congruence w.r.t. modal operators.

For every b ∈ A there exists a unique c ∈ A s.t. c ∈ [b]a and c ≤ a. Crucial remark Each ≡a-equivalence class has a canonical representant. Hence, the map i′ : Aa → A given by [b] → b ∧ a is injective.

Minghui Ma, Alessandra Palmigiano, Mehrnoosh Sadrzadeh Algebraic Semantics and Model Completeness for Intuitionistic Public

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Modalities of the pseudo-quotient

Minghui Ma, Alessandra Palmigiano, Mehrnoosh Sadrzadeh Algebraic Semantics and Model Completeness for Intuitionistic Public

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Modalities of the pseudo-quotient

Let (A, , ) be a HAO. Define for every b ∈ A,

Minghui Ma, Alessandra Palmigiano, Mehrnoosh Sadrzadeh Algebraic Semantics and Model Completeness for Intuitionistic Public

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Modalities of the pseudo-quotient

Let (A, , ) be a HAO. Define for every b ∈ A,

a[b] := [(b ∧ a) ∧ a] = [(b ∧ a)]. a[b] := [a → (a → b)] = [(a → b)].

Minghui Ma, Alessandra Palmigiano, Mehrnoosh Sadrzadeh Algebraic Semantics and Model Completeness for Intuitionistic Public

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Modalities of the pseudo-quotient

Let (A, , ) be a HAO. Define for every b ∈ A,

a[b] := [(b ∧ a) ∧ a] = [(b ∧ a)]. a[b] := [a → (a → b)] = [(a → b)].

For every HAO (A, , ) and every a ∈ A,

Minghui Ma, Alessandra Palmigiano, Mehrnoosh Sadrzadeh Algebraic Semantics and Model Completeness for Intuitionistic Public

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Modalities of the pseudo-quotient

Let (A, , ) be a HAO. Define for every b ∈ A,

a[b] := [(b ∧ a) ∧ a] = [(b ∧ a)]. a[b] := [a → (a → b)] = [(a → b)].

For every HAO (A, , ) and every a ∈ A,

a, a are normal modal operators.

Minghui Ma, Alessandra Palmigiano, Mehrnoosh Sadrzadeh Algebraic Semantics and Model Completeness for Intuitionistic Public

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Modalities of the pseudo-quotient

Let (A, , ) be a HAO. Define for every b ∈ A,

a[b] := [(b ∧ a) ∧ a] = [(b ∧ a)]. a[b] := [a → (a → b)] = [(a → b)].

For every HAO (A, , ) and every a ∈ A,

a, a are normal modal operators.

If (A, , ) is an MHA, then (Aa, a, a) is an MHA.

Minghui Ma, Alessandra Palmigiano, Mehrnoosh Sadrzadeh Algebraic Semantics and Model Completeness for Intuitionistic Public

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Modalities of the pseudo-quotient

Let (A, , ) be a HAO. Define for every b ∈ A,

a[b] := [(b ∧ a) ∧ a] = [(b ∧ a)]. a[b] := [a → (a → b)] = [(a → b)].

For every HAO (A, , ) and every a ∈ A,

a, a are normal modal operators.

If (A, , ) is an MHA, then (Aa, a, a) is an MHA. If A = F + for some Kripke frame F , then Aa BAO F a+.

Minghui Ma, Alessandra Palmigiano, Mehrnoosh Sadrzadeh Algebraic Semantics and Model Completeness for Intuitionistic Public

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Interpreting dynamic modalities in algebraic models

Minghui Ma, Alessandra Palmigiano, Mehrnoosh Sadrzadeh Algebraic Semantics and Model Completeness for Intuitionistic Public

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Interpreting dynamic modalities in algebraic models

Let i : Mα ֒→ M.

Minghui Ma, Alessandra Palmigiano, Mehrnoosh Sadrzadeh Algebraic Semantics and Model Completeness for Intuitionistic Public

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Interpreting dynamic modalities in algebraic models

Let i : Mα ֒→ M. The satisfaction condition M, w αϕ iff M, w α and Mα, w ϕ : can be equivalently written as follows:

Minghui Ma, Alessandra Palmigiano, Mehrnoosh Sadrzadeh Algebraic Semantics and Model Completeness for Intuitionistic Public

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Interpreting dynamic modalities in algebraic models

Let i : Mα ֒→ M. The satisfaction condition M, w αϕ iff M, w α and Mα, w ϕ : can be equivalently written as follows: w ∈ [

[αϕ] ]M

iff

∃w′ ∈ Wα s.t. i(w′) = w ∈ [ [α] ]M and w′ ∈ [ [ϕ] ]Mα.

Minghui Ma, Alessandra Palmigiano, Mehrnoosh Sadrzadeh Algebraic Semantics and Model Completeness for Intuitionistic Public

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Interpreting dynamic modalities in algebraic models

Let i : Mα ֒→ M. The satisfaction condition M, w αϕ iff M, w α and Mα, w ϕ : can be equivalently written as follows: w ∈ [

[αϕ] ]M

iff

∃w′ ∈ Wα s.t. i(w′) = w ∈ [ [α] ]M and w′ ∈ [ [ϕ] ]Mα.

Because i : Mα ֒→ M is injective, then w′ ∈ [

[ϕ] ]Mα

iff w = i(w′) ∈ i[[

[ϕ] ]Mα].

Minghui Ma, Alessandra Palmigiano, Mehrnoosh Sadrzadeh Algebraic Semantics and Model Completeness for Intuitionistic Public

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Interpreting dynamic modalities in algebraic models

Let i : Mα ֒→ M. The satisfaction condition M, w αϕ iff M, w α and Mα, w ϕ : can be equivalently written as follows: w ∈ [

[αϕ] ]M

iff

∃w′ ∈ Wα s.t. i(w′) = w ∈ [ [α] ]M and w′ ∈ [ [ϕ] ]Mα.

Because i : Mα ֒→ M is injective, then w′ ∈ [

[ϕ] ]Mα

iff w = i(w′) ∈ i[[

[ϕ] ]Mα].

Hence: w ∈ [

[αϕ] ]M

iff w ∈ [

[α] ]M ∩ i[[ [ϕ] ]Mα],

Minghui Ma, Alessandra Palmigiano, Mehrnoosh Sadrzadeh Algebraic Semantics and Model Completeness for Intuitionistic Public

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Interpreting dynamic modalities in algebraic models

Let i : Mα ֒→ M. The satisfaction condition M, w αϕ iff M, w α and Mα, w ϕ : can be equivalently written as follows: w ∈ [

[αϕ] ]M

iff

∃w′ ∈ Wα s.t. i(w′) = w ∈ [ [α] ]M and w′ ∈ [ [ϕ] ]Mα.

Because i : Mα ֒→ M is injective, then w′ ∈ [

[ϕ] ]Mα

iff w = i(w′) ∈ i[[

[ϕ] ]Mα].

Hence: w ∈ [

[αϕ] ]M

iff w ∈ [

[α] ]M ∩ i[[ [ϕ] ]Mα],

from which we get

[ [αϕ] ]M = [ [α] ]M ∩ i[[ [ϕ] ]Mα] = [ [α] ]M ∩ i′([ [ϕ] ]Mα).

(1)

Minghui Ma, Alessandra Palmigiano, Mehrnoosh Sadrzadeh Algebraic Semantics and Model Completeness for Intuitionistic Public

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Interpreting dynamic modalities in algebraic models

Minghui Ma, Alessandra Palmigiano, Mehrnoosh Sadrzadeh Algebraic Semantics and Model Completeness for Intuitionistic Public

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Interpreting dynamic modalities in algebraic models

For every algebraic model M = (A, V), the extension map

[ [·] ]M : Fm → A is defined recursively as follows:

Minghui Ma, Alessandra Palmigiano, Mehrnoosh Sadrzadeh Algebraic Semantics and Model Completeness for Intuitionistic Public

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Interpreting dynamic modalities in algebraic models

For every algebraic model M = (A, V), the extension map

[ [·] ]M : Fm → A is defined recursively as follows: [ [p] ]M =

V(p)

[ [⊥] ]M = ⊥A [ [⊤] ]M = ⊤A [ [ϕ ∨ ψ] ]M = [ [ϕ] ]M ∨A [ [ψ] ]M [ [ϕ ∧ ψ] ]M = [ [ϕ] ]M ∧A [ [ψ] ]M [ [ϕ → ψ] ]M = [ [ϕ] ]M →A [ [ψ] ]M [ [ϕ] ]M = A[ [ϕ] ]M [ [ϕ] ]M = A[ [ϕ] ]M [ [αϕ] ]M = [ [α] ]M ∧A i′([ [ϕ] ]Mα) [ [[α]ϕ] ]M = [ [α] ]M →A i′([ [ϕ] ]Mα)

Minghui Ma, Alessandra Palmigiano, Mehrnoosh Sadrzadeh Algebraic Semantics and Model Completeness for Intuitionistic Public

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Interpreting dynamic modalities in algebraic models

For every algebraic model M = (A, V), the extension map

[ [·] ]M : Fm → A is defined recursively as follows: [ [p] ]M =

V(p)

[ [⊥] ]M = ⊥A [ [⊤] ]M = ⊤A [ [ϕ ∨ ψ] ]M = [ [ϕ] ]M ∨A [ [ψ] ]M [ [ϕ ∧ ψ] ]M = [ [ϕ] ]M ∧A [ [ψ] ]M [ [ϕ → ψ] ]M = [ [ϕ] ]M →A [ [ψ] ]M [ [ϕ] ]M = A[ [ϕ] ]M [ [ϕ] ]M = A[ [ϕ] ]M [ [αϕ] ]M = [ [α] ]M ∧A i′([ [ϕ] ]Mα) [ [[α]ϕ] ]M = [ [α] ]M →A i′([ [ϕ] ]Mα)

Mα := (Aα, Vα) s.t. Aα = A[

[α] ]M and Vα : AtProp → Aα is π ◦ V, i.e.

[ [p] ]Mα = Vα(p) = π(V(p)) = π([ [p] ]M) for every p.

Minghui Ma, Alessandra Palmigiano, Mehrnoosh Sadrzadeh Algebraic Semantics and Model Completeness for Intuitionistic Public

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Intuitionistic PAL

Minghui Ma, Alessandra Palmigiano, Mehrnoosh Sadrzadeh Algebraic Semantics and Model Completeness for Intuitionistic Public

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Intuitionistic PAL

ϕ ::= p ∈ AtProp | ⊥ | ⊤ | ϕ∨ψ | ϕ∧ψ | ϕ → ψ | ϕ | ϕ | αϕ | [α]ϕ.

Minghui Ma, Alessandra Palmigiano, Mehrnoosh Sadrzadeh Algebraic Semantics and Model Completeness for Intuitionistic Public

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Intuitionistic PAL

ϕ ::= p ∈ AtProp | ⊥ | ⊤ | ϕ∨ψ | ϕ∧ψ | ϕ → ψ | ϕ | ϕ | αϕ | [α]ϕ.

Interaction with logical constants Preservation of facts

α⊥ = ⊥ αp = α ∧ p [α]⊤ = ⊤ [α]p = α → p

Interaction with disjunction Interaction with conjunction

α(ϕ ∨ ψ) = αϕ ∨ αψ α(ϕ ∧ ψ) = αϕ ∧ αψ [α](ϕ ∨ ψ) = α → (αϕ ∨ αψ) [α](ϕ ∧ ψ) = [α]ϕ ∧ [α]ψ

Interaction with implication

α(ϕ → ψ) = α ∧ (αϕ → αψ) [α](ϕ → ψ) = αϕ → αψ

Interaction with Interaction with

αϕ = α ∧ αϕ αϕ = α ∧ [α]ϕ [α]ϕ = α → αϕ [α]ϕ = α → [α]ϕ

Minghui Ma, Alessandra Palmigiano, Mehrnoosh Sadrzadeh Algebraic Semantics and Model Completeness for Intuitionistic Public

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Results

Minghui Ma, Alessandra Palmigiano, Mehrnoosh Sadrzadeh Algebraic Semantics and Model Completeness for Intuitionistic Public

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Results

IPAL is sound w.r.t. algebraic models (A, V).

Minghui Ma, Alessandra Palmigiano, Mehrnoosh Sadrzadeh Algebraic Semantics and Model Completeness for Intuitionistic Public

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Results

IPAL is sound w.r.t. algebraic models (A, V). IPAL is complete w.r.t. relational models:

Minghui Ma, Alessandra Palmigiano, Mehrnoosh Sadrzadeh Algebraic Semantics and Model Completeness for Intuitionistic Public

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Results

IPAL is sound w.r.t. algebraic models (A, V). IPAL is complete w.r.t. relational models: (W, ≤, R, V)

Minghui Ma, Alessandra Palmigiano, Mehrnoosh Sadrzadeh Algebraic Semantics and Model Completeness for Intuitionistic Public

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Results

IPAL is sound w.r.t. algebraic models (A, V). IPAL is complete w.r.t. relational models: (W, ≤, R, V)

W is a nonempty set;

Minghui Ma, Alessandra Palmigiano, Mehrnoosh Sadrzadeh Algebraic Semantics and Model Completeness for Intuitionistic Public

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Results

IPAL is sound w.r.t. algebraic models (A, V). IPAL is complete w.r.t. relational models: (W, ≤, R, V)

W is a nonempty set; ≤ is a partial order on W;

Minghui Ma, Alessandra Palmigiano, Mehrnoosh Sadrzadeh Algebraic Semantics and Model Completeness for Intuitionistic Public

SLIDE 61

Results

IPAL is sound w.r.t. algebraic models (A, V). IPAL is complete w.r.t. relational models: (W, ≤, R, V)

W is a nonempty set; ≤ is a partial order on W; R is an (equivalence) relation on W s.t. (R ◦≥) ⊆ (≥◦R) (≤◦R) ⊆ (R ◦≤) R = (≥◦R)∩(R ◦≤);

Minghui Ma, Alessandra Palmigiano, Mehrnoosh Sadrzadeh Algebraic Semantics and Model Completeness for Intuitionistic Public

SLIDE 62

Results

IPAL is sound w.r.t. algebraic models (A, V). IPAL is complete w.r.t. relational models: (W, ≤, R, V)

W is a nonempty set; ≤ is a partial order on W; R is an (equivalence) relation on W s.t. (R ◦≥) ⊆ (≥◦R) (≤◦R) ⊆ (R ◦≤) R = (≥◦R)∩(R ◦≤); V(p) is a down-set (or an up-set) of (W, ≤).

Minghui Ma, Alessandra Palmigiano, Mehrnoosh Sadrzadeh Algebraic Semantics and Model Completeness for Intuitionistic Public

SLIDE 63

Results

IPAL is sound w.r.t. algebraic models (A, V). IPAL is complete w.r.t. relational models: (W, ≤, R, V)

W is a nonempty set; ≤ is a partial order on W; R is an (equivalence) relation on W s.t. (R ◦≥) ⊆ (≥◦R) (≤◦R) ⊆ (R ◦≤) R = (≥◦R)∩(R ◦≤); V(p) is a down-set (or an up-set) of (W, ≤).

Epistemic updates defined exactly in the same way as in the Boolean case.

Minghui Ma, Alessandra Palmigiano, Mehrnoosh Sadrzadeh Algebraic Semantics and Model Completeness for Intuitionistic Public

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Results

IPAL is sound w.r.t. algebraic models (A, V). IPAL is complete w.r.t. relational models: (W, ≤, R, V)

W is a nonempty set; ≤ is a partial order on W; R is an (equivalence) relation on W s.t. (R ◦≥) ⊆ (≥◦R) (≤◦R) ⊆ (R ◦≤) R = (≥◦R)∩(R ◦≤); V(p) is a down-set (or an up-set) of (W, ≤).

Epistemic updates defined exactly in the same way as in the Boolean case. Work in progress: Intuitionistic account of Muddy Children Puzzle.

Minghui Ma, Alessandra Palmigiano, Mehrnoosh Sadrzadeh Algebraic Semantics and Model Completeness for Intuitionistic Public