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 ∃ r 1 , r 2 ∈ R such that r = r 1 • r 2 and r 1 � A and r 2 � 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 ∃ r 1 , r 2 ∈ R such that r = r 1 • r 2 and r 1 � A and r 2 � B A ∗ B A B ∗ B iff ∀ r ′ ∈ R , if r ′ � A then r • r ′ � B ◮ r � A − B A − ∗ B A

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

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 ∗ ψ iff for all r ′ ∈ Nom such that r • r ′ ↓ and r ′ � K φ , we ◮ r � K φ − have r • r ′ � K ψ − ψ iff for all r ′ ∈ Nom such that r ′ • r ↓ and r ′ � K φ , we ◮ r � K φ ∗ 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 . S x : φ S @ x ( φ ) � BI labelled tableaux HRL unlabelled tableaux

A tableau method for HRL Additive Rules T @ x ( φ ∧ ψ ) F @ x ( φ ∧ ψ ) � T ∧� � F ∧� T @ x ( φ ) , T @ x ( ψ ) F @ x ( φ ) | F @ x ( ψ ) T @ x ( φ ∨ ψ ) F @ x ( φ ∨ ψ ) � T ∨� � F ∨� T @ x ( φ ) | T @ x ( ψ ) F @ x ( φ ) , F @ x ( ψ ) T @ x ( φ → ψ ) F @ x ( φ → ψ ) � T →� � F →� F @ x ( φ ) | T @ x ( ψ ) T @ x ( φ ) , F @ x ( ψ ) T @ x ( ¬ φ ) F @ x ( ¬ φ ) � T ¬� � F ¬� F @ x ( φ ) T @ x ( φ ) x is a nominal.

A tableau method for HRL Multiplicative Rules T @ x ( φ ∗ ψ ) F @ x ( φ ∗ ψ ) , T @ x ( y ∗ z ) � T ∗� � F ∗� T @ c i ( φ ) , T @ c j ( ψ ) , T @ x ( c i ∗ c j ) F @ y ( φ ) | F @ z ( ψ ) T @ x ( φ − ∗ ψ ) , T @ z ( x ∗ y ) F @ x ( φ − ∗ ψ ) � T − ∗� � F − ∗� F @ y ( φ ) | T @ z ( ψ ) T @ c i ( φ ) , F @ c j ( ψ ) , T @ c j ( x ∗ c i ) T @ x ( φ ∗ − ψ ) , T @ z ( y ∗ x ) F @ x ( φ ∗ − ψ ) � T ∗ −� � F ∗ −� F @ y ( φ ) | T @ z ( ψ ) T @ c i ( φ ) , F @ c j ( ψ ) , T @ c j ( c i ∗ x ) x , y , z are nominals and c i , c j are new nominals.

A tableau method for HRL Label Rules S @ x (@ y ( φ )) � i r � � @ � T @ x ( x ) S @ y ( φ ) T @ x ( y ) S @ x ( φ ) , T @ x ( y ) � i s � � i t � T @ y ( x ) S @ y ( φ ) S @ x ( φ [ y ]) , T @ y ( z ∗ t ) S @ x ( φ [ y ∗ z ]) � i + � � i −� S @ x ( φ [ y ∗ z / c i ]) , T @ c i ( y ∗ z ) S @ x ( φ [ y / z ∗ t ]) S @ x ( φ [ y ]) , T @ y ( z ) � i p � S @ x ( φ [ y / z ]) x , y , z , t are nominals and c i is a new nominal.

A tableau method for HRL Closure A tableau for a formula φ is a tableau for { F @ c 1 ( φ ) } where c 1 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 @ c 1 (@ i ( A ) ∧ ( i ∗ B ) → A ∗ B )

A tableau method for HRL Tableau example F @ x ( φ → ψ ) F @ c 1 (@ i ( A ) ∧ ( i ∗ B ) → A ∗ B ) � F →� T @ x ( φ ) , F @ x ( ψ )

A tableau method for HRL Tableau example F @ x ( φ → ψ ) F @ c 1 (@ i ( A ) ∧ ( i ∗ B ) → A ∗ B ) � F →� T @ x ( φ ) , F @ x ( ψ ) T @ c 1 (@ i ( A ) ∧ ( i ∗ B )) F @ c 1 ( A ∗ B )

A tableau method for HRL Tableau example T @ x ( φ ∧ ψ ) F @ c 1 (@ i ( A ) ∧ ( i ∗ B ) → A ∗ B ) � T ∧� T @ x ( φ ) , T @ x ( ψ ) T @ c 1 (@ i ( A ) ∧ ( i ∗ B )) F @ c 1 ( A ∗ B )