Internalizing labels in BI logics Meeting TICAMORE Marseille Pierre - - PowerPoint PPT Presentation

Internalizing labels in BI logics

Meeting TICAMORE Marseille Pierre Kimmel November 14, 2017

Introduction

BI logics

BI

O’Hearn & Pym, 1999

Resource sharing and separation : ∗, −

∗

Intuitionistic logic

∧,∨,→

Introduction

BI logics

BI | BBI

O’Hearn & Pym, 1999

Resource sharing and separation : ∗, −

∗

Intuitionistic logic | Classical logic

∧,∨,→

Introduction

BBI semantics

◮ R set of resources, • composition, e neutral element

֒ → r φ ֒ → resources : knowledge, space, general context...

Introduction

BBI semantics

◮ R set of resources, • composition, e neutral element

֒ → r φ ֒ → resources : knowledge, space, general context...

◮ r A∗ B iff ∃r1,r2 ∈ R such that r = r1 • r2 and r1 A and r2 B

A B A∗ B

Introduction

BBI semantics

◮ R set of resources, • composition, e neutral element

֒ → r φ ֒ → resources : knowledge, space, general context...

◮ r A∗ B iff ∃r1,r2 ∈ R such that r = r1 • r2 and r1 A and r2 B

A B A∗ B

◮ r A−

∗ B iff ∀r′ ∈ R, if r′ A then r • r′ B

A A−

∗ B

B

Introduction

BBI example ((((E ∧ F)∗ G)∧ D)∗ C)∧ A∧ B

A,B D E,F G C

Introduction

Hybrid logic

Modal logic (especially temporal)

◮ φ : For all states that follow, φ is valid ◮ ♦φ : There exists a state that follow where φ is valid

Introduction

Hybrid logic

Modal logic (especially temporal)

◮ φ : For all states that follow, φ is valid ◮ ♦φ : There exists a state that follow where φ is valid

֒ → Quantifiers over states, no way to capture a precise state.

Introduction

Hybrid logic

Modal logic (especially temporal)

◮ φ : For all states that follow, φ is valid ◮ ♦φ : There exists a state that follow where φ is valid

֒ → Quantifiers over states, no way to capture a precise state.

Prior, 1967 / Blackburn, 2006

⇒ Hybrid logic : addition of state labels in the syntax

◮ @s(φ) : φ is valid at state s

Introduction

Motivations

Why not do the same with BBI ?

Introduction

Motivations

Why not do the same with BBI ?

֒ → Hybrid Resource Logic : BBI + location operators from Hybrid

Logic

Introduction

Motivations

Why not do the same with BBI ?

֒ → Hybrid Resource Logic : BBI + location operators from Hybrid

Logic

⇒ Extends expressiveness (similarly to Hybrid Logics)

Introduction

Motivations

Why not do the same with BBI ?

֒ → Hybrid Resource Logic : BBI + location operators from Hybrid

Logic

⇒ Extends expressiveness (similarly to Hybrid Logics) ⇒ Allows axiomatisation of BBI properties

Introduction

Contributions

◮ A new logic to reason on sharing and separating resources ◮ Syntax including location operators with resource labels ◮ Weaker semantics than BBI, added properties through axioms ◮ Axioms allow to recapture BBI expressiveness and some variants ◮ Extended expressiveness through location operator ◮ Tableau method without labels (soundness/completeness)

HRL logic

HRL logic

Syntax

Set of propositional symbols : Prop Set of resource symbols or nominals : Nom HRL language is defined by the following grammar : X ::=p ∈ Prop

|⊤ |⊥ |¬X |X ∧ X |X ∨ X |X → X |I |X ∗ X |X − ∗ X |X ∗ − X |i ∈ Nom |@i(X)

Note : differentiation between −

∗ and ∗ − is necessary because composition won’t

always be commutative.

HRL logic

Semantics Definition (Weak resource structure)

A weak resource structure associated to Nom is a triple

R = (•,e,∼) such that:

◮ e ∈ Nom ; ◮ • : Nom × Nom ⇀ Nom ; ◮ ∼ is an equivalence relation on Nom compatible with •.

Definition (Interpretation)

An interpretation of Prop for R is a function · : Prop → P(Nom) which is monotone on Prop, which means for all p ∈ Prop, for all r,r′ ∈ Nom, if r ∼ r′ and r ∈ p then r′ ∈ p.

HRL logic

Semantics Definition (Model)

A model of HRL is a triple K = (R ,·,K ) where R = (•,e,∼) is a weak resource structure on Nom, · is an interpretation of Prop for R and K ⊆ L × Nom is defined by :

◮ r K p iff r ∈ p ◮ r K φ∧ψ iff r K φ and r K ψ ◮ r K φ∗ψ iff there exist r′,r′′ ∈ Nom such that r′ • r′′ ↓ and

r′ • r′′ ∼ r and r′ K φ and r′′ K ψ

◮ r K φ−

∗ψ iff for all r′ ∈ Nom such that r • r′ ↓ and r′ K φ, we

have r • r′ K ψ

◮ r K φ∗

−ψ iff for all r′ ∈ Nom such that r′ • r ↓ and r′ K φ, we

have r′ • r K ψ

◮ r K i iff r ∼ i ◮ r K @i(φ) iff i K φ

HRL logic

HBBI logic Definition (HBBI logic)

HBBI logic is the fragment of HRL where the following axioms are valid for any i,j,k ∈ Nom :

(BI)n ≡ @i(i ∗I) (BI)c ≡ j ∗ k → k ∗ j (BI)a ≡ j ∗(k ∗ l) → (j ∗ k)∗ l Theorem (Semantic equivalence between HBBI and BBI)

Let φ be a BI formula. If any model of BBI is built on Nom, then BBI φ iff HBBI φ.

Note : in HBBI, A−

∗ B ≡ A∗ − B

A tableau method for HRL

A tableau method for HRL

Formulae and SS Definition (Labelled formulae, Set of statements)

A labelled formula is a pair (S,Φ) with S ∈ {T,F} and Φ a HRL-formula of the form Φ = @x(φ) where x ∈ Nom et φ ∈ L. We note S @x(φ) a labelled formula (S,@x(φ)). A Set of Statements or SS, noted F is a set of labelled formulae. The alphabet of F , noted A(F ) is the set of nominals appearing in F .

A tableau method for HRL

Formulae and SS Definition (Labelled formulae, Set of statements)

A labelled formula is a pair (S,Φ) with S ∈ {T,F} and Φ a HRL-formula of the form Φ = @x(φ) where x ∈ Nom et φ ∈ L. We note S @x(φ) a labelled formula (S,@x(φ)). A Set of Statements or SS, noted F is a set of labelled formulae. The alphabet of F , noted A(F ) is the set of nominals appearing in F .

Sx : φ

• S @x(φ)

BI labelled tableaux HRL unlabelled tableaux

A tableau method for HRL

Additive Rules T @x(φ∧ψ)

T∧

T @x(φ), T @x(ψ) F @x(φ∧ψ)

F∧

F @x(φ) | F @x(ψ) T @x(φ∨ψ)

T∨

T @x(φ) | T @x(ψ) F @x(φ∨ψ)

F∨

F @x(φ), F @x(ψ) T @x(φ → ψ)

T →

F @x(φ) | T @x(ψ) F @x(φ → ψ)

F →

T @x(φ), F @x(ψ) T @x(¬φ)

T¬

F @x(φ) F @x(¬φ)

F¬

T @x(φ)

x is a nominal.

A tableau method for HRL

Multiplicative Rules

T @x(φ∗ψ)

T∗

T @ci(φ), T @cj(ψ), T @x(ci ∗ cj) F @x(φ∗ψ),T @x(y ∗ z)

F∗

F @y(φ) | F @z(ψ) T @x(φ− ∗ψ),T @z(x ∗ y)

T− ∗

F @y(φ) | T @z(ψ) F @x(φ− ∗ψ)

F− ∗

T @ci(φ), F @cj(ψ), T @cj(x ∗ ci) T @x(φ∗ −ψ),T @z(y ∗ x)

T∗ −

F @y(φ) | T @z(ψ) F @x(φ∗ −ψ)

F∗ −

T @ci(φ), F @cj(ψ), T @cj(ci ∗ x)

x,y,z are nominals and ci,cj are new nominals.

A tableau method for HRL

Label Rules S @x(@y(φ))

@

S @y(φ)

ir

T @x(x) T @x(y)

is

T @y(x) S @x(φ),T @x(y)

it

S @y(φ) S @x(φ[y ∗ z])

i+

S @x(φ[y ∗ z/ci]), T @ci(y ∗ z) S @x(φ[y]),T @y(z ∗ t)

i−

S @x(φ[y/z ∗ t]) S @x(φ[y]),T @y(z)

ip

S @x(φ[y/z])

x,y,z,t are nominals and ci is a new nominal.

A tableau method for HRL

Closure

A tableau for a formula φ is a tableau for {F @c1(φ)} where c1 is a nominal not appearing in φ.

Definition (Closure)

A SS F is closed if one of the following is verified (for φ ∈ L and x ∈ Nom) :

• 1. T @x(φ) ∈ F and F @x(φ) ∈ F
• 2. T @x(⊥) ∈ F
• 3. F @x(⊤) ∈ F

A SS is opened if it’s not closed A tableau is closed if all its branches (its SS) are closed. A tableau-proof for a formula φ is a closed tableau for φ.

A tableau method for HRL

Properties of the method Theorem (Soundness)

If there exists a proof for a HRL-formula φ, then it is valid.

Proof.

Through realisability of branches.

Theorem (Completeness)

Let φ be a HRL-formula. If φ is valid, then there is a proof of φ.

Proof.

Through construction of a Hintikka branch and extraction of counter-model from this saturated branch.

A tableau method for HRL

Tableau example

F @c1(@i(A)∧(i ∗ B) → A∗ B)

A tableau method for HRL

Tableau example

F @c1(@i(A)∧(i ∗ B) → A∗ B)

F @x(φ → ψ)

F →

T @x(φ), F @x(ψ)

A tableau method for HRL

Tableau example

F @c1(@i(A)∧(i ∗ B) → A∗ B) T @c1(@i(A)∧(i ∗ B)) F @c1(A∗ B)

F @x(φ → ψ)

F →

T @x(φ), F @x(ψ)

A tableau method for HRL

Tableau example

F @c1(@i(A)∧(i ∗ B) → A∗ B) T @c1(@i(A)∧(i ∗ B)) F @c1(A∗ B)

T @x(φ∧ψ)

T∧

T @x(φ), T @x(ψ)

A tableau method for HRL

Tableau example

F @c1(@i(A)∧(i ∗ B) → A∗ B) T @c1(@i(A)∧(i ∗ B)) F @c1(A∗ B) T @c1(@i(A)) T @c1(i ∗ B)

T @x(φ∧ψ)

T∧

T @x(φ), T @x(ψ)

A tableau method for HRL

Tableau example

F @c1(@i(A)∧(i ∗ B) → A∗ B) T @c1(@i(A)∧(i ∗ B)) F @c1(A∗ B) T @c1(@i(A)) T @c1(i ∗ B)

S @x(@y(φ))

@

S @y(φ)

A tableau method for HRL

Tableau example

F @c1(@i(A)∧(i ∗ B) → A∗ B) T @c1(@i(A)∧(i ∗ B)) F @c1(A∗ B) T @c1(@i(A)) T @c1(i ∗ B) T @i(A)

S @x(@y(φ))

@

S @y(φ)

A tableau method for HRL

Tableau example

F @c1(@i(A)∧(i ∗ B) → A∗ B) T @c1(@i(A)∧(i ∗ B)) F @c1(A∗ B) T @c1(@i(A)) T @c1(i ∗ B) T @i(A)

T @x(φ∗ψ)

T∗

T @ci(φ), T @cj(ψ), T @x(ci ∗ cj)

A tableau method for HRL

Tableau example

F @c1(@i(A)∧(i ∗ B) → A∗ B) T @c1(@i(A)∧(i ∗ B)) F @c1(A∗ B) T @c1(@i(A)) T @c1(i ∗ B) T @i(A) T @c2(i) T @c3(B) T @c1(c2 ∗ c3)

T @x(φ∗ψ)

T∗

T @ci(φ), T @cj(ψ), T @x(ci ∗ cj)

A tableau method for HRL

Tableau example

F @c1(@i(A)∧(i ∗ B) → A∗ B) T @c1(@i(A)∧(i ∗ B)) F @c1(A∗ B) T @c1(@i(A)) T @c1(i ∗ B) T @i(A) T @c2(i) T @c3(B) T @c1(c2 ∗ c3)

F @x(φ∗ψ),T @x(y ∗ z)

F∗

F @y(φ) | F @z(ψ)

A tableau method for HRL

Tableau example

F @c1(@i(A)∧(i ∗ B) → A∗ B) T @c1(@i(A)∧(i ∗ B)) F @c1(A∗ B) T @c1(@i(A)) T @c1(i ∗ B) T @i(A) T @c2(i) T @c3(B) T @c1(c2 ∗ c3) F @c2(A) F @c3(B)

F @x(φ∗ψ),T @x(y ∗ z)

F∗

F @y(φ) | F @z(ψ)

A tableau method for HRL

Tableau example

F @c1(@i(A)∧(i ∗ B) → A∗ B) T @c1(@i(A)∧(i ∗ B)) F @c1(A∗ B) T @c1(@i(A)) T @c1(i ∗ B) T @i(A) T @c2(i) T @c3(B) T @c1(c2 ∗ c3) F @c2(A) F @c3(B) ×

A tableau method for HRL

Tableau example

F @c1(@i(A)∧(i ∗ B) → A∗ B) T @c1(@i(A)∧(i ∗ B)) F @c1(A∗ B) T @c1(@i(A)) T @c1(i ∗ B) T @i(A) T @c2(i) T @c3(B) T @c1(c2 ∗ c3) F @c2(A) F @c3(B) ×

A tableau method for HRL

Tableau example

F @c1(@i(A)∧(i ∗ B) → A∗ B) T @c1(@i(A)∧(i ∗ B)) F @c1(A∗ B) T @c1(@i(A)) T @c1(i ∗ B) T @i(A) T @c2(i) T @c3(B) T @c1(c2 ∗ c3) F @c2(A) F @c3(B) ×

S @x(φ),T @x(y)

it

S @y(φ)

A tableau method for HRL

Tableau example

F @c1(@i(A)∧(i ∗ B) → A∗ B) T @c1(@i(A)∧(i ∗ B)) F @c1(A∗ B) T @c1(@i(A)) T @c1(i ∗ B) T @i(A) T @c2(i) T @c3(B) T @c1(c2 ∗ c3) F @c2(A) F @c3(B) ×

T @x(y)

is

T @y(x)

A tableau method for HRL

Tableau example

F @c1(@i(A)∧(i ∗ B) → A∗ B) T @c1(@i(A)∧(i ∗ B)) F @c1(A∗ B) T @c1(@i(A)) T @c1(i ∗ B) T @i(A) T @c2(i) T @c3(B) T @c1(c2 ∗ c3) F @c2(A) F @c3(B) T @i(c2) ×

T @x(y)

is

T @y(x)

A tableau method for HRL

Tableau example

F @c1(@i(A)∧(i ∗ B) → A∗ B) T @c1(@i(A)∧(i ∗ B)) F @c1(A∗ B) T @c1(@i(A)) T @c1(i ∗ B) T @i(A) T @c2(i) T @c3(B) T @c1(c2 ∗ c3) F @c2(A) F @c3(B) T @i(c2) ×

S @x(φ),T @x(y)

it

S @y(φ)

A tableau method for HRL

Tableau example

F @c1(@i(A)∧(i ∗ B) → A∗ B) T @c1(@i(A)∧(i ∗ B)) F @c1(A∗ B) T @c1(@i(A)) T @c1(i ∗ B) T @i(A) T @c2(i) T @c3(B) T @c1(c2 ∗ c3) F @c2(A) F @c3(B) T @i(c2) × T @c2(A)

S @x(φ),T @x(y)

it

S @y(φ)

A tableau method for HRL

Tableau example

F @c1(@i(A)∧(i ∗ B) → A∗ B) T @c1(@i(A)∧(i ∗ B)) F @c1(A∗ B) T @c1(@i(A)) T @c1(i ∗ B) T @i(A) T @c2(i) T @c3(B) T @c1(c2 ∗ c3) F @c2(A) F @c3(B) T @i(c2) × T @c2(A) ×

A tableau method for HRL

HBBI tableaux

HBBI axioms

(BI)n ≡ @i(i ∗I) (BI)c ≡ j ∗ k → k ∗ j (BI)a ≡ j ∗(k ∗ l) → (j ∗ k)∗ l

A tableau method for HRL

HBBI tableaux

HBBI axioms HBBI tableaux rules

(BI)n ≡ @i(i ∗I)

BIn

T @x(x ∗I) (BI)c ≡ j ∗ k → k ∗ j

• T @x(y ∗ z)

BIc

T @x(z ∗ y) (BI)a ≡ j ∗(k ∗ l) → (j ∗ k)∗ l T @x(y ∗(z ∗ t))

BIa

T @x((y ∗ z)∗ t)

◮ Soundness is conserved ◮ Completeness have to be studied

Expressiveness

Expressiveness

Why use HRL ?

Expressiveness

Why use HRL ? ⇒ Extended expressiveness

Expressiveness

Why use HRL ? ⇒ Extended expressiveness ⇒ Modular logic : addition of axioms

Expressiveness

An example of modelling with BBI

Expressiveness

An example of modelling with BBI

◮ Set of elementary resources coding for coins :

Res = {e,e1,e2,c50,c20,c10}

Expressiveness

An example of modelling with BBI

◮ Set of elementary resources coding for coins :

Res = {e,e1,e2,c50,c20,c10}

◮ Set of resources R, closure of Res by •

Expressiveness

An example of modelling with BBI

◮ Set of elementary resources coding for coins :

Res = {e,e1,e2,c50,c20,c10}

◮ Set of resources R, closure of Res by • ◮ Set of propositions coding for objects to buy :

Prop = {Obj(0.30),Obj(1.70),Obj(2)}

Expressiveness

An example of modelling with BBI

◮ Set of elementary resources coding for coins :

Res = {e,e1,e2,c50,c20,c10}

◮ Set of resources R, closure of Res by • ◮ Set of propositions coding for objects to buy :

Prop = {Obj(0.30),Obj(1.70),Obj(2)}

◮ Equivalence relation : two resources are equivalent if they

represent the same sum of money. E.G. : e2 ∼ e1 • c50 • c20 • c20 • c10

Expressiveness

An example of modelling with BBI

r φ if the sum represented by r allows to perform exactly the

• perations described by φ.

Expressiveness

An example of modelling with BBI

r φ if the sum represented by r allows to perform exactly the

• perations described by φ.

Examples :

Expressiveness

An example of modelling with BBI

r φ if the sum represented by r allows to perform exactly the

• perations described by φ.

Examples :

• 1. c20 • c10 Obj(0.30)

Expressiveness

An example of modelling with BBI

r φ if the sum represented by r allows to perform exactly the

• perations described by φ.

Examples :

• 1. c20 • c10 Obj(0.30)
• 2. c20 • c10 Obj(1.70) −

∗ Obj(2)

Expressiveness

An example of modelling with BBI

r φ if the sum represented by r allows to perform exactly the

• perations described by φ.

Examples :

• 1. c20 • c10 Obj(0.30)
• 2. c20 • c10 Obj(1.70) −

∗ Obj(2)

• 3. e2 Obj(2) ∧(Obj(0.30) ∗ Obj(1.70))

Expressiveness

An example of modelling with BBI

r φ if the sum represented by r allows to perform exactly the

• perations described by φ.

Examples :

• 1. c20 • c10 Obj(0.30)
• 2. c20 • c10 Obj(1.70) −

∗ Obj(2)

• 3. e2 Obj(2) ∧(Obj(0.30) ∗ Obj(1.70))
• 4. e1 Obj(0.30) ∗⊤

Expressiveness

More expressiveness with HBBI

We develop the same example with HBBI (so that everything we’ve stated is still valid). The set of nominal is the set of elementary resources (Nom = Res).

Expressiveness

More expressiveness with HBBI

We develop the same example with HBBI (so that everything we’ve stated is still valid). The set of nominal is the set of elementary resources (Nom = Res).

◮ Using nominals to state properties of resources and propositions.

Expressiveness

More expressiveness with HBBI

We develop the same example with HBBI (so that everything we’ve stated is still valid). The set of nominal is the set of elementary resources (Nom = Res).

◮ Using nominals to state properties of resources and propositions.

e2 (e1 ∗ e1)∧ Obj(2)

Expressiveness

More expressiveness with HBBI

We develop the same example with HBBI (so that everything we’ve stated is still valid). The set of nominal is the set of elementary resources (Nom = Res).

◮ Using nominals to state properties of resources and propositions.

e2 (e1 ∗ e1)∧ Obj(2)

◮ Using extra nominals as variables (capturing money return).

Expressiveness

More expressiveness with HBBI

We develop the same example with HBBI (so that everything we’ve stated is still valid). The set of nominal is the set of elementary resources (Nom = Res).

◮ Using nominals to state properties of resources and propositions.

e2 (e1 ∗ e1)∧ Obj(2)

◮ Using extra nominals as variables (capturing money return).

e2 (Obj(0.30) ∗ x)∧@x(Obj(1.70))

Expressiveness

More expressiveness with HBBI

We develop the same example with HBBI (so that everything we’ve stated is still valid). The set of nominal is the set of elementary resources (Nom = Res).

◮ Using nominals to state properties of resources and propositions.

e2 (e1 ∗ e1)∧ Obj(2)

◮ Using extra nominals as variables (capturing money return).

e2 (Obj(0.30) ∗ x)∧@x(Obj(1.70)) Even further : e2 • e1 (Obj(0.30) ∗ x)∧@x(Obj(1.70) ∗ y)

Expressiveness

BBI extensions and restrictions

HRL + (BI)n + (BI)c + (BI)a = HBBI

Expressiveness

BBI extensions and restrictions

HRL + (BI)n + (BI)c + (BI)a = HBBI HRL + (BI)n + (BI)a = non commutative HBBI

Expressiveness

BBI extensions and restrictions

HRL + (BI)n + (BI)c + (BI)a = HBBI HRL + (BI)n + (BI)a = non commutative HBBI HBBI + (Inv) = HBBI with invertible resources

(Inv) ≡ (i ∗⊤)∧I

Expressiveness

BBI extensions and restrictions

HRL + (BI)n + (BI)c + (BI)a = HBBI HRL + (BI)n + (BI)a = non commutative HBBI HBBI + (Inv) = HBBI with invertible resources

(Inv) ≡ (i ∗⊤)∧I ↔

Inv

T @x((y ∗⊤)∧I)

Expressiveness

BBI extensions and restrictions

HRL + (BI)n + (BI)c + (BI)a = HBBI HRL + (BI)n + (BI)a = non commutative HBBI HBBI + (Inv) = HBBI with invertible resources

(Inv) ≡ (i ∗⊤)∧I ↔

Inv

T @x((y ∗⊤)∧I)

Note : in (i ∗ x)∧I, x is the invert of i.

Conclusion and perspectives

Contribution :

◮ A new logic, HRL, weaker than BBI but with internalized labels ◮ A tableau method for HRL (sound and complete) ◮ An extension of HRL, HBBI, that matches exactly BBI

Perspectives :

◮ Extended expressiveness of BI logics ◮ Easy extensions and restrictions to new logics, with tableau

method