Positive modal separation logics Fredrik Dahlqvist University - PowerPoint PPT Presentation

Positive modal separation logics Fredrik Dahlqvist University College London Resource Reasoning Meeting 13 January 2016

Positive modal logic First studied by Jon Michael Dunn in 1995

Positive modal logic First studied by Jon Michael Dunn in 1995 a ::= p | a ∧ a | a ∨ a | ♦ a | � a , p ∈ V

Positive modal logic First studied by Jon Michael Dunn in 1995 a ::= p | a ∧ a | a ∨ a | ♦ a | � a , p ∈ V Axioms: distribution laws ( K ) and Interaction Axioms : ♦ a ∧ � b ⊢ ♦ ( a ∧ b ) , � ( a ∨ b ) ⊢ ♦ a ∨ � b

Positive modal logic First studied by Jon Michael Dunn in 1995 a ::= p | a ∧ a | a ∨ a | ♦ a | � a , p ∈ V Axioms: distribution laws ( K ) and Interaction Axioms : ♦ a ∧ � b ⊢ ♦ ( a ∧ b ) , � ( a ∨ b ) ⊢ ♦ a ∨ � b Strong completeness w.r.t. Kripke frames

Positive modal logic First studied by Jon Michael Dunn in 1995 a ::= p | a ∧ a | a ∨ a | ♦ a | � a , p ∈ V Axioms: distribution laws ( K ) and Interaction Axioms : ♦ a ∧ � b ⊢ ♦ ( a ∧ b ) , � ( a ∨ b ) ⊢ ♦ a ∨ � b Strong completeness w.r.t. Kripke frames Problem: incompletness in the presence of axioms, e.g. add ♦♦ a ⊢ ♦ a to the logic and � a ⊢ �� a is valid but not derivable.

Positive modal logic 1996: Celani and Jansana offer a solution by altering the semantics.

Positive modal logic 1996: Celani and Jansana offer a solution by altering the semantics. New semantics in terms of ordered Kripke frames with compatibility requirements between R and � . Valuations in upsets .

Positive modal logic 1996: Celani and Jansana offer a solution by altering the semantics. New semantics in terms of ordered Kripke frames with compatibility requirements between R and � . Valuations in upsets . Strong completeness preserved.

Positive modal logic 1996: Celani and Jansana offer a solution by altering the semantics. New semantics in terms of ordered Kripke frames with compatibility requirements between R and � . Valuations in upsets . Strong completeness preserved. This solves the problem: ♦♦ p | = ♦ p iff ( R ; � ) is transitive � p | = �� p iff ( R ; � ) is transitive

Positive modal logic 2005: Gehrke, Nagahshi and Venema define a related ordered Kripke semantics with two relations: R ♦ , R � and compatibility relations with � .

Positive modal logic 2005: Gehrke, Nagahshi and Venema define a related ordered Kripke semantics with two relations: R ♦ , R � and compatibility relations with � . 2015: Semantics based on coalgebraic ideas: ordered Kripke frames, valuations in upsets, convex relations R ♦ , R � . w | = ♦ p if ∃ wR ♦ x , x | = p = � p if ∀ wR � x , x | w | = p

Positive modal logic 2005: Gehrke, Nagahshi and Venema define a related ordered Kripke semantics with two relations: R ♦ , R � and compatibility relations with � . 2015: Semantics based on coalgebraic ideas: ordered Kripke frames, valuations in upsets, convex relations R ♦ , R � . w | = ♦ p if ∃ wR ♦ x , x | = p = � p if ∀ wR � x , x | w | = p All these semantics are related. Coalgebraic semantics: start with R ♦ , R � and use Interaction axioms to prove one R is enough.

Positive modal logic Working with positive modal logic is a bit different. R ♦ x x | = p R ♦ , R � w | = ♦ p ∧ � q w z z | = p ∧ q y y | = q R �

Positive modal logic Working with positive modal logic is a bit different. R ♦ x x | = p R ♦ , R � w | = ♦ p ∧ � q w z z | = p ∧ q y y | = q R � Strong completeness. Positive ML is strongly complete w.r.t. to Kripke frames with two convex relations R ♦ , R � and upset valuation validating Interaction axioms. Moreover: w | = R ♦ × R � a iff w | = ( R ♦ ∩ R � ) × ( R ♦ ∩ R � ) a

‘Separation logic’ as positive ML a ::= I | p | a ∗ a | a − ∗ a | a ∗ − a , p ∈ V

‘Separation logic’ as positive ML a ::= I | p | a ∗ a | a − ∗ a | a ∗ − a , p ∈ V Models: posets with convex binary relations and downset of ‘special points’: w | = I if w ∈ I w | = p ∗ q if ∃ wR ∗ ( x , y ) , x | = p and y | = q w | = p − ∗ q if ∀ wR − ∗ ( x , y ) , x | = p implies y | = q = p ∗ − q if ∀ wR ∗ − ( x , y ) , y | w | = p implies x | = q

‘Separation logic’ as positive ML a ::= I | p | a ∗ a | a − ∗ a | a ∗ − a , p ∈ V Models: posets with convex binary relations and downset of ‘special points’: w | = I if w ∈ I w | = p ∗ q if ∃ wR ∗ ( x , y ) , x | = p and y | = q w | = p − ∗ q if ∀ wR − ∗ ( x , y ) , x | = p implies y | = q = p ∗ − q if ∀ wR ∗ − ( x , y ) , y | w | = p implies x | = q Axioms: distribution laws of ∗ , − ∗ , ∗ − (think K ) plus a ∗ I ⊣⊢ a , I ∗ a ⊣⊢ a 4 ( c ∗ − b ) ∗ a ⊢ c ∗ −( a ∗ b ) 1 I ⊢ a − ∗ a , I ⊢ a ∗ − a 5 ( a ∗ − b ) ∗ b ⊢ a 2 a ∗ ( b − ∗ c ) ⊢ ( a ∗ b − ∗ ) c b ∗ ( b − ∗ a ) ⊢ a 3 6

‘Separation logic’ as positive ML Strong completeness of ‘separation logic’ Positive ‘separation logic’ is strongly complete w.r.t. Kripke frames with convex ternary relations R ∗ , R − ∗ , R ∗ − validating its axioms. This means that it is complete w.r.t. to Kripke frames with a single convex ternary relation R w | = p ∗ q iff ∃ wR ( x , y ) , x | = p and y | = q ∗ q iff ∀ xR ( w , y ) , x | w | = p − = p implies y | = q w | = p ∗ − q iff ∀ yR ( x , w ) , y | = p implies x | = q

‘Separation logic’ as positive ML Strong completeness of ‘separation logic’ Positive ‘separation logic’ is strongly complete w.r.t. Kripke frames with convex ternary relations R ∗ , R − ∗ , R ∗ − validating its axioms. This means that it is complete w.r.t. to Kripke frames with a single convex ternary relation R w | = p ∗ q iff ∃ wR ( x , y ) , x | = p and y | = q ∗ q iff ∀ xR ( w , y ) , x | w | = p − = p implies y | = q w | = p ∗ − q iff ∀ yR ( x , w ) , y | = p implies x | = q Much more general result: residuation is preserved under canonical extension on boolean algebras, distributive lattices, semi-lattices and even posets!

‘Separation logic’ as positive ML Some posets with convex ternary relation and ‘identities’:

‘Separation logic’ as positive ML Some posets with convex ternary relation and ‘identities’: Take W = { f : N + ⇀ f N } with f � g whenever f = g ↾ dom f , I = { Id U | U ∈ P f ( N + ) } and f R ( g , h ) iff dom g ∩ dom h = ∅ , g � f , h � f

‘Separation logic’ as positive ML Some posets with convex ternary relation and ‘identities’: Take W = { f : N + ⇀ f N } with f � g whenever f = g ↾ dom f , I = { Id U | U ∈ P f ( N + ) } and f R ( g , h ) iff dom g ∩ dom h = ∅ , g � f , h � f For ( P , ◦ , I ) a partial monoid, take W = P with a � b if ∃ c , a ◦ c = b , I = I and aR ( b , c ) iff b ∗ c � a

‘Separation logic’ as positive ML Some posets with convex ternary relation and ‘identities’: Take W = { f : N + ⇀ f N } with f � g whenever f = g ↾ dom f , I = { Id U | U ∈ P f ( N + ) } and f R ( g , h ) iff dom g ∩ dom h = ∅ , g � f , h � f For ( P , ◦ , I ) a partial monoid, take W = P with a � b if ∃ c , a ◦ c = b , I = I and aR ( b , c ) iff b ∗ c � a For ( P , � , ◦ , I ) an ordered partial monoid: same as above with native order.

‘Separation logic’ as positive ML Some posets with convex ternary relation and ‘identities’: Take W = { f : N + ⇀ f N } with f � g whenever f = g ↾ dom f , I = { Id U | U ∈ P f ( N + ) } and f R ( g , h ) iff dom g ∩ dom h = ∅ , g � f , h � f For ( P , ◦ , I ) a partial monoid, take W = P with a � b if ∃ c , a ◦ c = b , I = I and aR ( b , c ) iff b ∗ c � a For ( P , � , ◦ , I ) an ordered partial monoid: same as above with native order. For any set X , take W = { S ⊆ X × X } with � given by ⊆ , I = { Id U | U ⊆ X } and SR ( T 1 , T 2 ) whenever T 1 ; T 2 ⊆ S .

Modularity Given a modal signature Σ , let L Σ be the associated positive modal language.

Modularity Given a modal signature Σ , let L Σ be the associated positive modal language. Given two signatures Σ 1 , Σ 2 , define L Σ 1 ⊕ L Σ 2 = L Σ 1 � Σ 2 the fusion of L Σ 1 and L Σ 2

Modularity Given a modal signature Σ , let L Σ be the associated positive modal language. Given two signatures Σ 1 , Σ 2 , define L Σ 1 ⊕ L Σ 2 = L Σ 1 � Σ 2 the fusion of L Σ 1 and L Σ 2 The coalgebraic method used to prove strong completeness is modular:

Modularity Given a modal signature Σ , let L Σ be the associated positive modal language. Given two signatures Σ 1 , Σ 2 , define L Σ 1 ⊕ L Σ 2 = L Σ 1 � Σ 2 the fusion of L Σ 1 and L Σ 2 The coalgebraic method used to prove strong completeness is modular: Strong completeness is modular Let Σ 1 , Σ 2 be two signatures and Ax 1 , Ax 2 be sets of canonical axioms in L Σ 1 and L Σ 2 which include distribution laws, then L Σ 1 ⊕ L Σ 2 / { Ax 1 ∪ Ax 2 } is strongly complete w.r.t. to Kripke frames with convex n -ary relations R σ , σ ∈ Σ 1 ∪ Σ 1 validating the axioms in Ax 1 ∪ Ax 2 .

Positive Modal Separation Logics Idea: describe evolving resources by combining positive modal logics with positive separation logic and keep strong completeness.