Moss coalgebraic logic for preordered coalgebras Marta B lkov a - - PowerPoint PPT Presentation

Introduction Base Relation lifting to Rel(Pre) ∧∨ distributive law Cover modalities in Pre Expressivity

Moss’ coalgebraic logic for preordered coalgebras

Marta B´ ılkov´ a joint work with Jiˇ r´ ı Velebil

MB, JV ALCOP, 18 April 2013 1/32

Introduction Base Relation lifting to Rel(Pre) ∧∨ distributive law Cover modalities in Pre Expressivity

Coalgebras in Set

A T-coalgebra is a c : X − → TX

MB, JV ALCOP, 18 April 2013 2/32

Introduction Base Relation lifting to Rel(Pre) ∧∨ distributive law Cover modalities in Pre Expressivity

Coalgebras in Set

A T-coalgebra is a c : X − → TX Examples Deterministic and nondeterministic automata Q − → QA × 2 and Q − → (PQ)A × 2 Kripke frames for modal logics: c : X − → PX ... many others

MB, JV ALCOP, 18 April 2013 2/32

Introduction Base Relation lifting to Rel(Pre) ∧∨ distributive law Cover modalities in Pre Expressivity

Moss’ logic in Set

Cover modality An expressive language for T-coalgebras — extend language of BA with a single modality ∇ : TL − → L

MB, JV ALCOP, 18 April 2013 3/32

Introduction Base Relation lifting to Rel(Pre) ∧∨ distributive law Cover modalities in Pre Expressivity

Moss’ logic in Set

Cover modality An expressive language for T-coalgebras — extend language of BA with a single modality ∇ : TL − → L Semantics For T : Set − → Set and a coalgebra c : X − → TX x ∇α iff c(x)T α Where T : Rel − → Rel is a lifting of T

MB, JV ALCOP, 18 April 2013 3/32

Introduction Base Relation lifting to Rel(Pre) ∧∨ distributive law Cover modalities in Pre Expressivity

Moss’ logic in Set

Results for T finitary and preserving weak pulbacks expressivity for bisimulation axiomatization by modal distributive laws completeness nice proof theory

MB, JV ALCOP, 18 April 2013 3/32

Introduction Base Relation lifting to Rel(Pre) ∧∨ distributive law Cover modalities in Pre Expressivity

Base for finitary functors

base of α in TX is the smallest finite subset Y of X, with α in TY . baseX(α) =

• {Z ⊆fin X | α ∈ TZ}

If T weakly preserves preimages and intersectionsa, base is a natural transformation baseX : TX − → PωX

• aP. Gumm, From T-coalgebras to filter structures and transition systems,

Calco 05

MB, JV ALCOP, 18 April 2013 4/32

Introduction Base Relation lifting to Rel(Pre) ∧∨ distributive law Cover modalities in Pre Expressivity

Base for finitary functors

base of α in TX is the smallest finite subset Y of X, with α in TY . baseX(α) =

• {Z ⊆fin X | α ∈ TZ}

If T weakly preserves preimages and intersectionsa, base is a natural transformation baseX : TX − → PωX

• aP. Gumm, From T-coalgebras to filter structures and transition systems,

Calco 05

MB, JV ALCOP, 18 April 2013 4/32

Introduction Base Relation lifting to Rel(Pre) ∧∨ distributive law Cover modalities in Pre Expressivity

The category Pre

Pre has preordered sets as objects, and monotone maps as arrows. Preorders and 2-functors form a 2-category Pre The 2-cell f − → g given by the point-wise order. Factorisation system Pre possesses a factorisaton system (E , M) (in the sense of category theory enriched in posets) consisting of E = monotone surjections and M = monotone maps reflecting order.

MB, JV ALCOP, 18 April 2013 5/32

Introduction Base Relation lifting to Rel(Pre) ∧∨ distributive law Cover modalities in Pre Expressivity

The category Pre

Pre has preordered sets as objects, and monotone maps as arrows. Preorders and 2-functors form a 2-category Pre The 2-cell f − → g given by the point-wise order. Factorisation system Pre possesses a factorisaton system (E , M) (in the sense of category theory enriched in posets) consisting of E = monotone surjections and M = monotone maps reflecting order.

MB, JV ALCOP, 18 April 2013 5/32

Introduction Base Relation lifting to Rel(Pre) ∧∨ distributive law Cover modalities in Pre Expressivity

The category Pre

Pre has preordered sets as objects, and monotone maps as arrows. Preorders and 2-functors form a 2-category Pre The 2-cell f − → g given by the point-wise order. Factorisation system Pre possesses a factorisaton system (E , M) (in the sense of category theory enriched in posets) consisting of E = monotone surjections and M = monotone maps reflecting order.

MB, JV ALCOP, 18 April 2013 5/32

Introduction Base Relation lifting to Rel(Pre) ∧∨ distributive law Cover modalities in Pre Expressivity

Functors in Pre

T ∂ is the dual of T, defined T ∂A = (TAop)op LX = [X op, 2], thus L∂X = [X, 2]op = UX Kripke Polynomial Functors T ::= constX | Id | T ∂ | T + T | T × T | LT

MB, JV ALCOP, 18 April 2013 6/32

Introduction Base Relation lifting to Rel(Pre) ∧∨ distributive law Cover modalities in Pre Expressivity

Coalgebras in Pre

A T-coalgebra is a monotone map c : X − → TX

MB, JV ALCOP, 18 April 2013 7/32

Introduction Base Relation lifting to Rel(Pre) ∧∨ distributive law Cover modalities in Pre Expressivity

Coalgebras in Pre

A T-coalgebra is a monotone map c : X − → TX Examples Deterministic and nondeterministic ordered automataa Frames for positive modal logics Frames for distributive substructural logics

• aD. Perrin, J.E. Pin, Infinite words, 2004.

MB, JV ALCOP, 18 April 2013 7/32

Introduction Base Relation lifting to Rel(Pre) ∧∨ distributive law Cover modalities in Pre Expressivity

Coalgebras in Pre

A T-coalgebra is a monotone map c : X − → TX Deterministic ordered automata are coalgebras for the functor IdA × 2, i.e. c : Q − → QA × 2

MB, JV ALCOP, 18 April 2013 7/32

Introduction Base Relation lifting to Rel(Pre) ∧∨ distributive law Cover modalities in Pre Expressivity

Coalgebras in Pre

A T-coalgebra is a monotone map c : X − → TX Nondeterministic ordered automata are coalgebras c : Q − → (PQ)A × 2

MB, JV ALCOP, 18 April 2013 7/32

Introduction Base Relation lifting to Rel(Pre) ∧∨ distributive law Cover modalities in Pre Expressivity

Coalgebras in Pre

A T-coalgebra is a monotone map c : X − → TX Frames for positive modal logics with adjoint modalities R : X

✕ X such that R(x, y) ∧ x′ ≤ x ∧ y ≤ y′ −

→ R(x′y′) are coalgebras for the functors U and L, i.e. c : X − → UX and d : X − → LX defined by c(x) = {y | R(x, y)} and d(x) = {y | R(y, x)}

MB, JV ALCOP, 18 April 2013 7/32

Introduction Base Relation lifting to Rel(Pre) ∧∨ distributive law Cover modalities in Pre Expressivity

Coalgebras in Pre

A T-coalgebra is a monotone map c : X − → TX Frames for distributive substructural logics R : X

✕ X × X

(the fusion fragment) are coalgebras for the functor L(Id × Id), i.e. c : X − → L(X × X)

MB, JV ALCOP, 18 April 2013 7/32

Introduction Base Relation lifting to Rel(Pre) ∧∨ distributive law Cover modalities in Pre Expressivity

Moss’ logic in Pre

Cover modality An expressive language for T-coalgebras — extend language of DL with ∇ : T ∂L − → L

MB, JV ALCOP, 18 April 2013 8/32

Introduction Base Relation lifting to Rel(Pre) ∧∨ distributive law Cover modalities in Pre Expressivity

Moss’ logic in Pre

Cover modality An expressive language for T-coalgebras — extend language of DL with ∇ : T ∂L − → L Semantics For T : Pre − → Pre and a coalgebra c : X − → TX x ∇α iff c(x)T ∂ α Where T ∂ : Rel(Pre) − → Rel(Pre) is a lifting of T ∂a

aBilkova, Kurz, Petri¸

san, Velebil CALCO 2010

MB, JV ALCOP, 18 April 2013 8/32

Introduction Base Relation lifting to Rel(Pre) ∧∨ distributive law Cover modalities in Pre Expressivity

Moss’ logic in Pre

Results for T finitary and preserving exact squares expressivity for simulation axiomatization by distributive laws – this talk completeness – not yet nice proof theory – not yet

MB, JV ALCOP, 18 April 2013 8/32

Introduction Base Relation lifting to Rel(Pre) ∧∨ distributive law Cover modalities in Pre Expressivity

Subobjects

Functor Sub Sub(X) is the essentially small preorder of all M-subobjects of X preordered by the existence of a factorisation Z

h

• f
• Z ′
• f ′
• X

Subfin(X) is the full subpreorder of Sub(X) spanned by f : Z

X

having Z finite

MB, JV ALCOP, 18 April 2013 9/32

Introduction Base Relation lifting to Rel(Pre) ∧∨ distributive law Cover modalities in Pre Expressivity

Subobjects

Sub on maps Z

• f
• h ∗ (Z)
• h∗(f )
• X

h

Y

Sub(h) on f : Z

X is given by the factorisation of hf .

MB, JV ALCOP, 18 April 2013 9/32

Introduction Base Relation lifting to Rel(Pre) ∧∨ distributive law Cover modalities in Pre Expressivity

Base

T preserving monotone maps in M induces a functor T ∗

X : Subfin(X) −

→ Sub(TX) T : Pre − → Pre admits a base , if a free object α : Z

X (called

the base of α) w.r.t. T ∗

X exists on every α : Z

TX

For each α with Z finite there is α such that for each f ′

• Z
• α
• Z ′
• f ′
• X

iff Z

• α
• TZ ′
• Tf ′
• TX

MB, JV ALCOP, 18 April 2013 10/32

Introduction Base Relation lifting to Rel(Pre) ∧∨ distributive law Cover modalities in Pre Expressivity

Examples - L The base of α : 1

LX

is base(α) : 1

X

the image of base(α) is the finite preorder of generators of the lowerset given by the image of α, the 1 is the shape of the generators.

MB, JV ALCOP, 18 April 2013 11/32

Introduction Base Relation lifting to Rel(Pre) ∧∨ distributive law Cover modalities in Pre Expressivity

Examples - L The base of α : 2

LX

which picks u ⊆ v is base(α) : 2

X

the image of base(α) is the finite preorder of generators of u, generators of v, and the preorder between them

MB, JV ALCOP, 18 April 2013 11/32

Introduction Base Relation lifting to Rel(Pre) ∧∨ distributive law Cover modalities in Pre Expressivity

Properties - ”α is in T-of-base-of-α”

• Z
• base(α)
• Z
• base(α)
• X

iff Z

• α
• T

Z

• Tbase(α)
• TX

MB, JV ALCOP, 18 April 2013 11/32

Introduction Base Relation lifting to Rel(Pre) ∧∨ distributive law Cover modalities in Pre Expressivity

Properties baseX : Subfin(TX) − → Subfin(X) relation lifting restricts to (images of) bases

MB, JV ALCOP, 18 April 2013 11/32

Introduction Base Relation lifting to Rel(Pre) ∧∨ distributive law Cover modalities in Pre Expressivity

Monotone relations as spans A monotone relation from A to B is R : A

✕ B

understood as a monotone map r : Bop × A − → 2

MB, JV ALCOP, 18 April 2013 12/32

Introduction Base Relation lifting to Rel(Pre) ∧∨ distributive law Cover modalities in Pre Expressivity

Monotone relations as spans A monotone relation from A to B is R : A

✕ B

understood as a monotone map r : Bop × A − → 2 In Pre tabulated by a span (a discrete fibration) R

d0

• d1
• B

A

MB, JV ALCOP, 18 April 2013 12/32

Introduction Base Relation lifting to Rel(Pre) ∧∨ distributive law Cover modalities in Pre Expressivity

Monotone relations as spans A monotone relation from A to B is R : A

✕ B

understood as a monotone map r : Bop × A − → 2 In Pre tabulated by a span (a discrete fibration) R

d0

• d1
• B

A where [d0, d1] is an order embedding

MB, JV ALCOP, 18 April 2013 12/32

Introduction Base Relation lifting to Rel(Pre) ∧∨ distributive law Cover modalities in Pre Expressivity

Monotone relations as spans A monotone relation from A to B is R : A

✕ B

understood as a monotone map r : Bop × A − → 2 In Pre tabulated by a span (a discrete fibration) R

d0

• d1
• B

A R(a, b) ∧ a′ ≤ a ∧ b ≤ b′ − → R(a′, b′)

MB, JV ALCOP, 18 April 2013 12/32

Introduction Base Relation lifting to Rel(Pre) ∧∨ distributive law Cover modalities in Pre Expressivity

Opposite (converse) Rop : Bop

✕ Aop

understood as r′ : A × Bop − → 2

MB, JV ALCOP, 18 April 2013 13/32

Introduction Base Relation lifting to Rel(Pre) ∧∨ distributive law Cover modalities in Pre Expressivity

Opposite (converse) Rop : Bop

✕ Aop

understood as r′ : A × Bop − → 2 Rop

dop

1

• dop

Aop Bop where Rop(a, b) = R(b, a)

MB, JV ALCOP, 18 April 2013 13/32

Introduction Base Relation lifting to Rel(Pre) ∧∨ distributive law Cover modalities in Pre Expressivity

Composition The composition A

✕

R

B ✕

S

C

is computed by R · S(c, a) =

• b

(R(b, a) ∧ S(c, b))

MB, JV ALCOP, 18 April 2013 14/32

Introduction Base Relation lifting to Rel(Pre) ∧∨ distributive law Cover modalities in Pre Expressivity

Graph and opgraph For a monotone map f : A − → B we define two ”graph” relations A

✕

f⋄

B

B

✕

f ⋄

A

by the formulas f⋄(b, a) iff b ≤B fa and f ⋄(a, b) iff fa ≤B b.

MB, JV ALCOP, 18 April 2013 15/32

Introduction Base Relation lifting to Rel(Pre) ∧∨ distributive law Cover modalities in Pre Expressivity

Membership relations ∈A: LA

✕ A

and (∋A) = (∈Aop)op ∋A: A

✕ UA

MB, JV ALCOP, 18 April 2013 16/32

Introduction Base Relation lifting to Rel(Pre) ∧∨ distributive law Cover modalities in Pre Expressivity

Relation lifting

Theorem For a 2-functor T : Pre − → Pre the following are equivalent:

1 There is a 2-functor T : Rel(Pre) −

→ Rel(Pre) such that Rel(Pre)

T

Rel(Pre)

Pre

T

• (−)⋄
• Pre

(−)⋄

• (1)

2 The functor T preserves exact squares. 3 There is a distributive law T · L −

→ L · T

MB, JV ALCOP, 18 April 2013 17/32

Introduction Base Relation lifting to Rel(Pre) ∧∨ distributive law Cover modalities in Pre Expressivity

Distributive laws δ : T · L − → L · T given by δX(Φ) = {α | αT(∈)Φ} δ∂ : T ∂ · U − → U · T ∂ given by δ∂

X(Ψ) = {α | ΨT(∋)α}

MB, JV ALCOP, 18 April 2013 18/32

Introduction Base Relation lifting to Rel(Pre) ∧∨ distributive law Cover modalities in Pre Expressivity

Lifting Represent R as a span R

d0

• d1
• B

A and put T(R) to be the composite TA

✕

(Td1)⋄ TR

✕

(Td0)⋄ TB

MB, JV ALCOP, 18 April 2013 19/32

Introduction Base Relation lifting to Rel(Pre) ∧∨ distributive law Cover modalities in Pre Expressivity

Examples of lifting R : A

✕ B

The lowerset functor: L(R)(β, α) = ∀b(b ∈ β − → ∃a(a ∈ α ∧ R(b, a))) The upperset functor: U(R)(β, α) = ∀a(α ∋ a − → ∃b(β ∋ b ∧ R(b, a)))

MB, JV ALCOP, 18 April 2013 20/32

Introduction Base Relation lifting to Rel(Pre) ∧∨ distributive law Cover modalities in Pre Expressivity

Lemma Relation lifting satisfies the following properties: (i) Tf⋄ = (Tf )⋄, Tf ⋄ = (Tf )⋄ (ii) T ∂R = (TRop)op (iii) If η : T − → T ′ is natural, then η⋄ : T

✕ T ′ is lax

natural.

MB, JV ALCOP, 18 April 2013 21/32

Introduction Base Relation lifting to Rel(Pre) ∧∨ distributive law Cover modalities in Pre Expressivity

Language Fix a preorder At of propositional letters and the following

• perators with their corresponding (finitary) arities:

: UL − → L : LL − → L ∇ : T ∂L − → L The language is obtained as the free algebra of the (finitary) functor U + L + T ∂ over the preorder At.

MB, JV ALCOP, 18 April 2013 22/32

Introduction Base Relation lifting to Rel(Pre) ∧∨ distributive law Cover modalities in Pre Expressivity

Semantics of the propositional part Given c : X − → TX and a valuation relation val : At

✕ X op

we obtain the semantics : L

✕ X op inductively as follows:

MB, JV ALCOP, 18 April 2013 23/32

Introduction Base Relation lifting to Rel(Pre) ∧∨ distributive law Cover modalities in Pre Expressivity

Semantics of the propositional part Given c : X − → TX and a valuation relation val : At

✕ X op

we obtain the semantics : L

✕ X op inductively as follows:

MB, JV ALCOP, 18 April 2013 23/32

Introduction Base Relation lifting to Rel(Pre) ∧∨ distributive law Cover modalities in Pre Expressivity

Semantics of the propositional part Given c : X − → TX and a valuation relation val : At

✕ X op

we obtain the semantics : L

✕ X op inductively as follows:

x

• ϕ

= ∀a(ϕ ∋ a − → x a) x

• ϕ

= ∃a(a ∈ ϕ ∧ x a)

MB, JV ALCOP, 18 April 2013 23/32

Introduction Base Relation lifting to Rel(Pre) ∧∨ distributive law Cover modalities in Pre Expressivity

Semantics of the propositional part Given c : X − → TX and a valuation relation val : At

✕ X op

we obtain the semantics : L

✕ X op inductively as follows:

x

• ϕ

= ↓x U ϕ x

• ψ

= ↑x L ψ

MB, JV ALCOP, 18 April 2013 23/32

Introduction Base Relation lifting to Rel(Pre) ∧∨ distributive law Cover modalities in Pre Expressivity

Semantics of the propositional part Given c : X − → TX and a valuation relation val : At

✕ X op

we obtain the semantics : L

✕ X op inductively as follows:

x

• ϕ

= ↓x U ϕ x

• ψ

= ↑x L ψ a ≤ b iff ∀x(x a − → x b)

MB, JV ALCOP, 18 April 2013 23/32

Introduction Base Relation lifting to Rel(Pre) ∧∨ distributive law Cover modalities in Pre Expressivity

• distributive law

Φ in ULL:

• Φ∋ψ
• a∈ψ

a ≡

• ϕU∈Φ
• ϕ∋b

b

• (U
• )Φ

≡

• (L
• )δΦ

δ : UL − → LU

MB, JV ALCOP, 18 April 2013 24/32

Introduction Base Relation lifting to Rel(Pre) ∧∨ distributive law Cover modalities in Pre Expressivity

Semantics of the modal part x

• ϕ

= ↓x U ϕ x

• ψ

= ↑x L ψ x ∇T ∂α = c(x) T ∂ α

MB, JV ALCOP, 18 April 2013 25/32

Introduction Base Relation lifting to Rel(Pre) ∧∨ distributive law Cover modalities in Pre Expressivity

Examples

Positive modal logic with adjoint modalities ✷ and c : X − → UX and d : X − → LX x ∇Lα iff ∀y(c(x) ∋ y − → ∃a(a ∈ α ∧ y a)) x ∇Uβ iff ∀b(β ∋ b − → ∃y(y ∈ d(x) ∧ y b)) and therefore ∇Lα ≡ ✷

• α and ∇Uβ ≡
• β.

MB, JV ALCOP, 18 April 2013 26/32

Introduction Base Relation lifting to Rel(Pre) ∧∨ distributive law Cover modalities in Pre Expressivity

Examples

Fusion fragment of distributive substructural logics c : X − → L(X × X) x ∇α iff ∀(a, a′)(α ∋ (a, a′) − → ∃(y, y′)((y, y′) ∈ c(x) ∧ y a ∧ y′ a′)). Therefore ∇α ≡

• α∋(a,a′)

(a ⊗ a′).

MB, JV ALCOP, 18 April 2013 26/32

Introduction Base Relation lifting to Rel(Pre) ∧∨ distributive law Cover modalities in Pre Expressivity

Distributive law δ∂ : T ∂ · U − → U · T ∂

MB, JV ALCOP, 18 April 2013 27/32

Introduction Base Relation lifting to Rel(Pre) ∧∨ distributive law Cover modalities in Pre Expressivity

Distributive law δ∂ : T ∂ · U − → U · T ∂ Redistributions in Pre For A in UT ∂L, a lowerset srd(A) of slim redistributions of A consists of all Φ: (i) Φ is in T ∂Ubase(A) (ii) for each α with A ∋ α it holds that Φ T ∂(∋) α

MB, JV ALCOP, 18 April 2013 27/32

Introduction Base Relation lifting to Rel(Pre) ∧∨ distributive law Cover modalities in Pre Expressivity

Example: Fix c : X − → TX and consider x

A∋α

∇α.

MB, JV ALCOP, 18 April 2013 28/32

Introduction Base Relation lifting to Rel(Pre) ∧∨ distributive law Cover modalities in Pre Expressivity

Example: Fix c : X − → TX and consider x

A∋α

∇α. : L

✕ X op

MB, JV ALCOP, 18 April 2013 28/32

Introduction Base Relation lifting to Rel(Pre) ∧∨ distributive law Cover modalities in Pre Expressivity

Example: Fix c : X − → TX and consider x

A∋α

∇α. th : X op − → UL

MB, JV ALCOP, 18 April 2013 28/32

Introduction Base Relation lifting to Rel(Pre) ∧∨ distributive law Cover modalities in Pre Expressivity

Example: Fix c : X − → TX and consider x

A∋α

∇α. thA : X op − → Ubase(A)

MB, JV ALCOP, 18 April 2013 28/32

Introduction Base Relation lifting to Rel(Pre) ∧∨ distributive law Cover modalities in Pre Expressivity

Example: Fix c : X − → TX and consider x

A∋α

∇α. T ∂thA : (TX)op − → T ∂Ubase(A)

MB, JV ALCOP, 18 April 2013 28/32

Introduction Base Relation lifting to Rel(Pre) ∧∨ distributive law Cover modalities in Pre Expressivity

Example: Fix c : X − → TX and consider x

A∋α

∇α. Φx = T ∂thA(c(x))

MB, JV ALCOP, 18 April 2013 28/32

Introduction Base Relation lifting to Rel(Pre) ∧∨ distributive law Cover modalities in Pre Expressivity

Example: Fix c : X − → TX and consider x

A∋α

∇α. Φx = T ∂thA(c(x)) We claim that Φx is a slim redistribution of A

MB, JV ALCOP, 18 April 2013 28/32

Introduction Base Relation lifting to Rel(Pre) ∧∨ distributive law Cover modalities in Pre Expressivity

Example: Fix c : X − → TX and consider x

A∋α

∇α. Φx = T ∂thA(c(x)) Which is: for each α with (A ∋ α) we have (Φx T ∂(∋) α)

MB, JV ALCOP, 18 April 2013 28/32

Introduction Base Relation lifting to Rel(Pre) ∧∨ distributive law Cover modalities in Pre Expressivity

Example: Fix c : X − → TX and consider x

A∋α

∇α. Φx = T ∂thA(c(x)) Which is: for each α with (A ∋ α) we have (Φx T ∂(∋) α) base(A)

✟

• ❯

∋

• X op

Ubase(A)

✥

th⋄

A

• →

T ∂base(A)

✠

T ∂()

• ❯

T ∂(∋)

• (TX)op

T ∂Ubase(A)

✤

(T ∂thA)⋄

• MB, JV

ALCOP, 18 April 2013 28/32

Introduction Base Relation lifting to Rel(Pre) ∧∨ distributive law Cover modalities in Pre Expressivity

Modal Distributive laws

• αT ∂(∈)Ψ

∇α ≡ ∇(T ∂ )Ψ

• A∋α

∇α ≡

• Φ∈srd(A)

∇(T ∂ )Φ

MB, JV ALCOP, 18 April 2013 29/32

Introduction Base Relation lifting to Rel(Pre) ∧∨ distributive law Cover modalities in Pre Expressivity

Modal Distributive laws

• (L∇)(δT ∂Ψ)

≡ ∇(T ∂ )Ψ

• (U∇)A

≡

• L(∇T ∂

)srd(A)

MB, JV ALCOP, 18 April 2013 30/32

Introduction Base Relation lifting to Rel(Pre) ∧∨ distributive law Cover modalities in Pre Expressivity

Simulation Let c : X − → TX and d : Y − → TY be given, together with valuations c and d. We call a relation R : Y

✕ X a

T-simulation of c by d iff R(x, y) implies TR(c(x), d(y)) for each atom p in At, x c p implies y d p (c, x, c) T-simulates (d, y, d), notation (c, x, c) (d, y, d), iff there exists a T-simulation R with R(x, y). (c, x) is modally stronger than (d, y), notation (c, x) ✁ (d, y), iff x c a implies y d a, for each formula a

MB, JV ALCOP, 18 April 2013 31/32

Introduction Base Relation lifting to Rel(Pre) ∧∨ distributive law Cover modalities in Pre Expressivity

Simulation Let c : X − → TX and d : Y − → TY be given, together with valuations c and d. We call a relation R : Y

✕ X a

T-simulation of c by d iff R(x, y) implies TR(c(x), d(y)) for each atom p in At, x c p implies y d p (c, x, c) T-simulates (d, y, d), notation (c, x, c) (d, y, d), iff there exists a T-simulation R with R(x, y). (c, x) is modally stronger than (d, y), notation (c, x) ✁ (d, y), iff x c a implies y d a, for each formula a

MB, JV ALCOP, 18 April 2013 31/32

Introduction Base Relation lifting to Rel(Pre) ∧∨ distributive law Cover modalities in Pre Expressivity

Simulation Let c : X − → TX and d : Y − → TY be given, together with valuations c and d. We call a relation R : Y

✕ X a

T-simulation of c by d iff R(x, y) implies TR(c(x), d(y)) for each atom p in At, x c p implies y d p (c, x, c) T-simulates (d, y, d), notation (c, x, c) (d, y, d), iff there exists a T-simulation R with R(x, y). (c, x) is modally stronger than (d, y), notation (c, x) ✁ (d, y), iff x c a implies y d a, for each formula a

MB, JV ALCOP, 18 April 2013 31/32

Introduction Base Relation lifting to Rel(Pre) ∧∨ distributive law Cover modalities in Pre Expressivity

Expressivity for T finitary Lemma (i) (c, x, c) (d, y, d) implies (c, x, c) ✁ (d, y, d). (ii) (c, x, c) ✁ (d, y, d) implies (c, x, c) (d, y, d). The relation ✁ : Y

✕ X is a T-simulation.

MB, JV ALCOP, 18 April 2013 32/32