A modal distributive law Yde Venema ILLC, UvA, Amsterdam - PowerPoint PPT Presentation

The cover modality ∇ Syntax If α is a finite set of formulas then ∇ α is a formula. Semantics Fix a Kripke model S = � S , R , V � . S , s � ∇ α iff for all t ∈ R [ s ] there is an a ∈ α with S , t � a and for all a ∈ α there is a t ∈ R [ s ] with S , t � a . Informally: α and R [ s ] cover one another.

The cover modality ∇ Syntax If α is a finite set of formulas then ∇ α is a formula. Semantics Fix a Kripke model S = � S , R , V � . S , s � ∇ α iff for all t ∈ R [ s ] there is an a ∈ α with S , t � a and for all a ∈ α there is a t ∈ R [ s ] with S , t � a . Informally: α and R [ s ] cover one another. History ◮ model theory: Hintikka, Scott, . . . ◮ modal logic: Fine’s normal forms ◮ ∇ as primitive: Barwise & Moss/Janin & Walukiewicz

Reconstructing modal logic

Reconstructing modal logic Observe � � ∇ α ≡ � α ∧ ♦ α (where ♦ α := { ♦ a | a ∈ α } ).

Reconstructing modal logic Observe � � ∇ α ≡ � α ∧ ♦ α (where ♦ α := { ♦ a | a ∈ α } ). Conversely: ♦ a ≡ ∇{ a , ⊤} � a ≡ ∇ ∅ ∨ ∇{ a }

Reconstructing modal logic Observe � � ∇ α ≡ � α ∧ ♦ α (where ♦ α := { ♦ a | a ∈ α } ). Conversely: ♦ a ≡ ∇{ a , ⊤} � a ≡ ∇ ∅ ∨ ∇{ a } Define the language L of modal logic (in negation normal form) by a ::= p | ¬ p | ⊥ | ⊤ | a ∨ a | a ∧ a | ♦ a | � a

Reconstructing modal logic Observe � � ∇ α ≡ � α ∧ ♦ α (where ♦ α := { ♦ a | a ∈ α } ). Conversely: ♦ a ≡ ∇{ a , ⊤} � a ≡ ∇ ∅ ∨ ∇{ a } Define the language L of modal logic (in negation normal form) by a ::= p | ¬ p | ⊥ | ⊤ | a ∨ a | a ∧ a | ♦ a | � a Define the language L ∇ by a ::= p | ¬ p | ⊥ | ⊤ | a ∨ a | a ∧ a | ∇ α

Reconstructing modal logic Observe � � ∇ α ≡ � α ∧ ♦ α (where ♦ α := { ♦ a | a ∈ α } ). Conversely: ♦ a ≡ ∇{ a , ⊤} � a ≡ ∇ ∅ ∨ ∇{ a } Define the language L of modal logic (in negation normal form) by a ::= p | ¬ p | ⊥ | ⊤ | a ∨ a | a ∧ a | ♦ a | � a Define the language L ∇ by a ::= p | ¬ p | ⊥ | ⊤ | a ∨ a | a ∧ a | ∇ α Proposition The languages L and L ∇ are effectively equi-expressive.

Coalgebraic Generalization

Coalgebraic Generalization Fundamental Observation (Moss, 1999) Apply relation lifting to the binary relation � ⊆ S × L ∇ : S , s � ∇ α iff ( R [ s ] , α ) ∈ P( � )

Coalgebraic Generalization Fundamental Observation (Moss, 1999) Apply relation lifting to the binary relation � ⊆ S × L ∇ : S , s � ∇ α iff ( R [ s ] , α ) ∈ P( � ) This paves the way for coalgebraic generalizations of modal logic!

Overview ◮ Introduction ◮ Modal logic via ∇ ◮ A modal distributive law ◮ Games ◮ Logic ◮ Algebra ◮ Automata ◮ Topology ◮ Final remarks

A modal distributive law Definition Given sets α, α ′ , a relation Z ⊆ α × α ′ is full on α, α ′ , if ( α, α ′ ) ∈ P Z . Notation: Z ∈ α ⊲ ⊳ α ′ .

A modal distributive law Definition Given sets α, α ′ , a relation Z ⊆ α × α ′ is full on α, α ′ , if ( α, α ′ ) ∈ P Z . Notation: Z ∈ α ⊲ ⊳ α ′ . Theorem For any sets α, α ′ of formulas, ∇ α ∧ ∇ α ′ ≡ ∇{ a ∧ a ′ | ( a , a ′ ) ∈ Z } , � (MDL) Z ∈ α⊲ ⊳α ′

A modal distributive law Definition Given sets α, α ′ , a relation Z ⊆ α × α ′ is full on α, α ′ , if ( α, α ′ ) ∈ P Z . Notation: Z ∈ α ⊲ ⊳ α ′ . Theorem For any sets α, α ′ of formulas, ∇ α ∧ ∇ α ′ ≡ ∇{ a ∧ a ′ | ( a , a ′ ) ∈ Z } , � (MDL) Z ∈ α⊲ ⊳α ′ Proof of ‘ ⇒ ’: Suppose S , s � ∇ α ∧ ∇ α ′ .

A modal distributive law Definition Given sets α, α ′ , a relation Z ⊆ α × α ′ is full on α, α ′ , if ( α, α ′ ) ∈ P Z . Notation: Z ∈ α ⊲ ⊳ α ′ . Theorem For any sets α, α ′ of formulas, ∇ α ∧ ∇ α ′ ≡ ∇{ a ∧ a ′ | ( a , a ′ ) ∈ Z } , � (MDL) Z ∈ α⊲ ⊳α ′ Proof of ‘ ⇒ ’: Suppose S , s � ∇ α ∧ ∇ α ′ . Define Z s ⊆ α × α ′ as Z s := { ( a , a ′ ) | S , t � a ∧ a ′ for some t ∈ R [ s ] } .

A modal distributive law Definition Given sets α, α ′ , a relation Z ⊆ α × α ′ is full on α, α ′ , if ( α, α ′ ) ∈ P Z . Notation: Z ∈ α ⊲ ⊳ α ′ . Theorem For any sets α, α ′ of formulas, ∇ α ∧ ∇ α ′ ≡ ∇{ a ∧ a ′ | ( a , a ′ ) ∈ Z } , � (MDL) Z ∈ α⊲ ⊳α ′ Proof of ‘ ⇒ ’: Suppose S , s � ∇ α ∧ ∇ α ′ . Define Z s ⊆ α × α ′ as Z s := { ( a , a ′ ) | S , t � a ∧ a ′ for some t ∈ R [ s ] } . Then S , s � ∇{ a ∧ a ′ | ( a , a ′ ) ∈ Z s } ,

A modal distributive law Definition Given sets α, α ′ , a relation Z ⊆ α × α ′ is full on α, α ′ , if ( α, α ′ ) ∈ P Z . Notation: Z ∈ α ⊲ ⊳ α ′ . Theorem For any sets α, α ′ of formulas, ∇ α ∧ ∇ α ′ ≡ ∇{ a ∧ a ′ | ( a , a ′ ) ∈ Z } , � (MDL) Z ∈ α⊲ ⊳α ′ Proof of ‘ ⇒ ’: Suppose S , s � ∇ α ∧ ∇ α ′ . Define Z s ⊆ α × α ′ as Z s := { ( a , a ′ ) | S , t � a ∧ a ′ for some t ∈ R [ s ] } . Then S , s � ∇{ a ∧ a ′ | ( a , a ′ ) ∈ Z s } , and ( α, α ′ ) ∈ P Z .

A modal distributive law Definition Given sets α, α ′ , a relation Z ⊆ α × α ′ is full on α, α ′ , if ( α, α ′ ) ∈ P Z . Notation: Z ∈ α ⊲ ⊳ α ′ . Theorem For any sets α, α ′ of formulas, ∇ α ∧ ∇ α ′ ≡ ∇{ a ∧ a ′ | ( a , a ′ ) ∈ Z } , � (MDL) Z ∈ α⊲ ⊳α ′ Proof of ‘ ⇒ ’: Suppose S , s � ∇ α ∧ ∇ α ′ . Define Z s ⊆ α × α ′ as Z s := { ( a , a ′ ) | S , t � a ∧ a ′ for some t ∈ R [ s ] } . Then S , s � ∇{ a ∧ a ′ | ( a , a ′ ) ∈ Z s } , and ( α, α ′ ) ∈ P Z . Note This theorem enables us to (almost) eliminate conjunctions!

Elimination of conjunctions Assume that X is finite. For Π ⊆ X, abbreviate � � ⊙ Π := p ∧ ¬ p . p ∈ Π p ∈ X \ Π

Elimination of conjunctions Assume that X is finite. For Π ⊆ X, abbreviate � � ⊙ Π := p ∧ ¬ p . p ∈ Π p ∈ X \ Π Define the language L − ∇ by a ::= ⊥ | ⊤ | a ∨ a | ⊙ Π ∧ ∇ α Only special conjunctions are allowed in L − ∇ !

Elimination of conjunctions Assume that X is finite. For Π ⊆ X, abbreviate � � ⊙ Π := p ∧ ¬ p . p ∈ Π p ∈ X \ Π Define the language L − ∇ by a ::= ⊥ | ⊤ | a ∨ a | ⊙ Π ∧ ∇ α Only special conjunctions are allowed in L − ∇ ! Key Theorem The languages L and L − ∇ are effectively equi-expressive.

Overview ◮ Introduction ◮ Modal logic via ∇ ◮ A modal distributive law ◮ Games ◮ Logic ◮ Algebra ◮ Automata ◮ Topology ◮ Final remarks

Game semantics for L Position Player Legitimate moves ( a 1 ∨ a 2 , s ) ∃ { ( a 1 , s ) , ( a 2 , s ) } ( a 1 ∧ a 2 , s ) ∀ { ( a 1 , s ) , ( a 2 , s ) } ( ♦ a , s ) ∃ { ( a , t ) | t ∈ R [ s ] } ( � a , s ) ∀ { ( a , t ) | t ∈ R [ s ] } ( ⊥ , s ) ∃ ∅ ( ⊤ , s ) ∀ ∅ ( p , s ) , s ∈ V ( p ) ∀ ∅ ( p , s ) , s �∈ V ( p ) ∃ ∅ ( ¬ p , s ) , s �∈ V ( p ) ∀ ∅ ( ¬ p , s ) , s ∈ V ( p ) ∃ ∅

Game semantics for L ∇ Position Player Legitimate moves ( a 1 ∨ a 2 , s ) ∃ { ( a 1 , s ) , ( a 2 , s ) } ( a 1 ∧ a 2 , s ) ∀ { ( a 1 , s ) , ( a 2 , s ) } ( ∇ α, s ) ∃ { Z ⊆ S × L ∇ | Z ∈ α ⊲ ⊳ R [ s ] } Z ⊆ L ∇ × S ∀ { ( a , s ) | ( a , s ) ∈ Z } ( ⊥ , s ) ∃ ∅ ( ⊤ , s ) ∀ ∅ ( p , s ) , s ∈ V ( p ) ∀ ∅ ( p , s ) , s �∈ V ( p ) ∃ ∅ ( ¬ p , s ) , s �∈ V ( p ) ∀ ∅ ( ¬ p , s ) , s ∈ V ( p ) ∃ ∅

Game semantics for L ∇ Position Player Legitimate moves ( a 1 ∨ a 2 , s ) ∃ { ( a 1 , s ) , ( a 2 , s ) } ( a 1 ∧ a 2 , s ) ∀ { ( a 1 , s ) , ( a 2 , s ) } ( ∇ α, s ) ∃ { Z ⊆ S × L ∇ | Z ∈ α ⊲ ⊳ R [ s ] } Z ⊆ L ∇ × S ∀ { ( a , s ) | ( a , s ) ∈ Z } ( ⊥ , s ) ∃ ∅ ( ⊤ , s ) ∀ ∅ ( p , s ) , s ∈ V ( p ) ∀ ∅ ( p , s ) , s �∈ V ( p ) ∃ ∅ ( ¬ p , s ) , s �∈ V ( p ) ∀ ∅ ( ¬ p , s ) , s ∈ V ( p ) ∃ ∅ Note the asymmetry!

Strategic normal forms ◮ propositional distributive law: a ∧ ( ψ 1 ∨ ψ 2 ) ≡ ( a ∧ ψ 1 ) ∨ ( a ∧ ψ 2 ) ∀∃ ∃∀

Strategic normal forms ◮ propositional distributive law: a ∧ ( ψ 1 ∨ ψ 2 ) ≡ ( a ∧ ψ 1 ) ∨ ( a ∧ ψ 2 ) ∀∃ ∃∀ ◮ modal distributive law: ∇{ a ∧ a ′ | ( a , a ′ ) ∈ Z } ∇ α ∧ ∇ α ′ � ≡ Z ∈ α⊲ ⊳α ′ ∀∃∀ ∃∃∀∀

Scattered strategies Compare the formulas � a ∧ ♦ b and ∇{ a , b } .

Scattered strategies Compare the formulas � a ∧ ♦ b and ∇{ a , b } . Call a strategy for ∃ scattered if at a position ( ∇ α, s ) she always picks a relation Z = Gr ( z ) for a function z : R [ s ] → L ∇ .

Scattered strategies Compare the formulas � a ∧ ♦ b and ∇{ a , b } . Call a strategy for ∃ scattered if at a position ( ∇ α, s ) she always picks a relation Z = Gr ( z ) for a function z : R [ s ] → L ∇ . Key Observation WLOG we may assume ∃ uses scattered strategies in L − ∇ . (This is modulo bisimilarity.)

Scattered strategies Compare the formulas � a ∧ ♦ b and ∇{ a , b } . Call a strategy for ∃ scattered if at a position ( ∇ α, s ) she always picks a relation Z = Gr ( z ) for a function z : R [ s ] → L ∇ . Key Observation WLOG we may assume ∃ uses scattered strategies in L − ∇ . (This is modulo bisimilarity.) This reduces the power of ∀ to that of a pathfinder.

Overview ◮ Introduction ◮ Modal logic via ∇ ◮ A modal distributive law ◮ Games ◮ Logic ◮ Algebra ◮ Automata ◮ Topology ◮ Final remarks

Axiomatization Use algebraic format, work with inequalities a ≤ b . Write a ⊑ b for ⊢ a ≤ b

Axiomatization Use algebraic format, work with inequalities a ≤ b . Write a ⊑ b for ⊢ a ≤ b ◮ Derivation rule (for α, α ′ ∈ L ∇ ) From ( α, α ′ ) ∈ P( ⊑ ) derive ∇ α ≤ ∇ α ′ ( ∇ 1)

Axiomatization Use algebraic format, work with inequalities a ≤ b . Write a ⊑ b for ⊢ a ≤ b ◮ Derivation rule (for α, α ′ ∈ L ∇ ) From ( α, α ′ ) ∈ P( ⊑ ) derive ∇ α ≤ ∇ α ′ ( ∇ 1) View disjunction as � : P ω L ∇ → L ∇ , then P ω � : P ω P ω L ∇ → P ω L ∇ , � ) { α 1 , . . . , α n } = { � α 1 , . . . , � α n } . with (P ω

Axiomatization Use algebraic format, work with inequalities a ≤ b . Write a ⊑ b for ⊢ a ≤ b ◮ Derivation rule (for α, α ′ ∈ L ∇ ) From ( α, α ′ ) ∈ P( ⊑ ) derive ∇ α ≤ ∇ α ′ ( ∇ 1) View disjunction as � : P ω L ∇ → L ∇ , then P ω � : P ω P ω L ∇ → P ω L ∇ , � ) { α 1 , . . . , α n } = { � α 1 , . . . , � α n } . with (P ω With ∈ ⊆ L ∇ × P ω L ∇ , obtain P ∈ ⊆ P ω L ∇ × P ω P ω

Axiomatization Use algebraic format, work with inequalities a ≤ b . Write a ⊑ b for ⊢ a ≤ b ◮ Derivation rule (for α, α ′ ∈ L ∇ ) From ( α, α ′ ) ∈ P( ⊑ ) derive ∇ α ≤ ∇ α ′ ( ∇ 1) View disjunction as � : P ω L ∇ → L ∇ , then P ω � : P ω P ω L ∇ → P ω L ∇ , � ) { α 1 , . . . , α n } = { � α 1 , . . . , � α n } . with (P ω With ∈ ⊆ L ∇ × P ω L ∇ , obtain P ∈ ⊆ P ω L ∇ × P ω P ω ◮ Axiom (for Φ ∈ P ω P ω L ∇ ): � � � ∇ (P � )(Φ) ≤ ∇ β | β P ∈ Φ ( ∇ 3)

Carioca Axioms for ∇ Definition Let C be the axiom system obtained by adding to classical propositional logic the following three axioms/rules: From ( α, α ′ ) ∈ P( ⊑ ) derive ∇ α ≤ ∇ α ′ ( ∇ 1)

Carioca Axioms for ∇ Definition Let C be the axiom system obtained by adding to classical propositional logic the following three axioms/rules: From ( α, α ′ ) ∈ P( ⊑ ) derive ∇ α ≤ ∇ α ′ ( ∇ 1) For Γ ∈ P ω P ω L ∇ : � � � � {∇ γ | γ ∈ Γ } ≤ ∇ (P � )Φ | Φ ∈ SRD (Γ) ( ∇ 2)

Carioca Axioms for ∇ Definition Let C be the axiom system obtained by adding to classical propositional logic the following three axioms/rules: From ( α, α ′ ) ∈ P( ⊑ ) derive ∇ α ≤ ∇ α ′ ( ∇ 1) For Γ ∈ P ω P ω L ∇ : � � � � {∇ γ | γ ∈ Γ } ≤ ∇ (P � )Φ | Φ ∈ SRD (Γ) ( ∇ 2) For Φ ∈ P ω P ω L ∇ : � � � ∇ (P � )(Φ) ≤ ∇ β | β P ∈ Φ ( ∇ 3)

Completeness Theorem (B´ ılkov´ a, Palmigiano & V.) C is sound and complete wrt the Kripke semantics.

Completeness Theorem (B´ ılkov´ a, Palmigiano & V.) C is sound and complete wrt the Kripke semantics. Kurz, Kupke & V.: This generalizes to arbitrary weak pullback preserving functors.

Completeness Theorem (B´ ılkov´ a, Palmigiano & V.) C is sound and complete wrt the Kripke semantics. Kurz, Kupke & V.: This generalizes to arbitrary weak pullback preserving functors. B´ ılkov´ a, Palmigiano & V. developed cut-free Gentzen proof systems for ∇ , both in the modal and the general coalgebraic setting.

Modal model theory Disjunctive normal forms have applications in the model theory of modal (fixpoint) logics.

Modal model theory Disjunctive normal forms have applications in the model theory of modal (fixpoint) logics. Eg Fontaine and V. obtained decidability and syntactic characterization results for various semantic properties of modal ( µ -calculus) formulas.

Overview ◮ Introduction ◮ Modal logic via ∇ ◮ A modal distributive law ◮ Games ◮ Logic ◮ Algebra ◮ Automata ◮ Topology ◮ Final remarks

Modal Algebras Definition A modal algebra is an algebra � B , ⊥ , ¬ , ∨ , ♦ � with • � B , ⊥ , ¬ , ∨� a Boolean algebra and • ♦ : B → B a map that preserves finite joins.

Modal Algebras Definition A modal algebra is an algebra � B , ⊥ , ¬ , ∨ , ♦ � with • � B , ⊥ , ¬ , ∨� a Boolean algebra and • ♦ : B → B a map that preserves finite joins. Let MA be the variety of modal algebras.

Modal Algebras Definition A modal algebra is an algebra � B , ⊥ , ¬ , ∨ , ♦ � with • � B , ⊥ , ¬ , ∨� a Boolean algebra and • ♦ : B → B a map that preserves finite joins. Let MA be the variety of modal algebras. Fact MA algebraizes modal logic.

Modal Algebras Definition A modal algebra is an algebra � B , ⊥ , ¬ , ∨ , ♦ � with • � B , ⊥ , ¬ , ∨� a Boolean algebra and • ♦ : B → B a map that preserves finite joins. Let MA be the variety of modal algebras. Fact MA algebraizes modal logic. Free modal algebras F (X) have many residuation properties.

Modal Algebras Definition A modal algebra is an algebra � B , ⊥ , ¬ , ∨ , ♦ � with • � B , ⊥ , ¬ , ∨� a Boolean algebra and • ♦ : B → B a map that preserves finite joins. Let MA be the variety of modal algebras. Fact MA algebraizes modal logic. Free modal algebras F (X) have many residuation properties. Definition A map f : A → B is residuated by/left adjoint to g : B → A if fa ≤ b a ≤ gb

Uniform Interpolation Theorem (Ghilardi/Visser) e ❥ F (X) F (X ∪ { p } )

Uniform Interpolation Theorem (Ghilardi/Visser) e ❥ F (X) F (X ∪ { p } ) ⊥ ❨ ∃ p

Uniform Interpolation Theorem (Ghilardi/Visser) e ❥ F (X) F (X ∪ { p } ) ⊥ ❨ ∃ p I.e. there is a map ∃ p : L (X ∪ { p } ) → L (X) such that a | = b ∃ p . a | = b for all a ∈ L (X), b ∈ L (X ∪ { p } ).

Uniform Interpolation Theorem (Ghilardi/Visser) e ❥ F (X) F (X ∪ { p } ) ⊥ ❨ ∃ p I.e. there is a map ∃ p : L (X ∪ { p } ) → L (X) such that a | = b ∃ p . a | = b for all a ∈ L (X), b ∈ L (X ∪ { p } ). Corollary Uniform Interpolation Let a , b be formulas with a | = b .

Uniform Interpolation Theorem (Ghilardi/Visser) e ❥ F (X) F (X ∪ { p } ) ⊥ ❨ ∃ p I.e. there is a map ∃ p : L (X ∪ { p } ) → L (X) such that a | = b ∃ p . a | = b for all a ∈ L (X), b ∈ L (X ∪ { p } ). Corollary Uniform Interpolation Let a , b be formulas with a | = b . Assume Var( a ) \ Var( b ) = { p 1 , . . . , p n } .

Uniform Interpolation Theorem (Ghilardi/Visser) e ❥ F (X) F (X ∪ { p } ) ⊥ ❨ ∃ p I.e. there is a map ∃ p : L (X ∪ { p } ) → L (X) such that a | = b ∃ p . a | = b for all a ∈ L (X), b ∈ L (X ∪ { p } ). Corollary Uniform Interpolation Let a , b be formulas with a | = b . Assume Var( a ) \ Var( b ) = { p 1 , . . . , p n } . Then a | = ∃ p 1 · · · ∃ p n . a | = b .

Uniform Interpolation via ∇ There is a map ∃ p : L (X ∪ { p } ) → L (X) such that for all a ∈ L (X), b ∈ L (X ∪ { p } ) a | = b ∃ p . a | = b

Uniform Interpolation via ∇ There is a map ∃ p : L (X ∪ { p } ) → L (X) such that for all a ∈ L (X), b ∈ L (X ∪ { p } ) a | = b ∃ p . a | = b Proof In L − ∇ we can define ∃ p inductively:

Uniform Interpolation via ∇ There is a map ∃ p : L (X ∪ { p } ) → L (X) such that for all a ∈ L (X), b ∈ L (X ∪ { p } ) a | = b ∃ p . a | = b Proof In L − ∇ we can define ∃ p inductively: ◮ ∃ p ( a ∨ b ) ≡ ∃ p . a ∨ ∃ p . b

Uniform Interpolation via ∇ There is a map ∃ p : L (X ∪ { p } ) → L (X) such that for all a ∈ L (X), b ∈ L (X ∪ { p } ) a | = b ∃ p . a | = b Proof In L − ∇ we can define ∃ p inductively: ◮ ∃ p ( a ∨ b ) ≡ ∃ p . a ∨ ∃ p . b ◮ ∃ p . ∇ α ≡ ∇∃ p .α

Uniform Interpolation via ∇ There is a map ∃ p : L (X ∪ { p } ) → L (X) such that for all a ∈ L (X), b ∈ L (X ∪ { p } ) a | = b ∃ p . a | = b Proof In L − ∇ we can define ∃ p inductively: ◮ ∃ p ( a ∨ b ) ≡ ∃ p . a ∨ ∃ p . b ◮ ∃ p . ∇ α ≡ ∇∃ p .α This generalizes to the modal µ -calculus (d’Agostino & Hollenberg)

Uniform Interpolation via ∇ There is a map ∃ p : L (X ∪ { p } ) → L (X) such that for all a ∈ L (X), b ∈ L (X ∪ { p } ) a | = b ∃ p . a | = b Proof In L − ∇ we can define ∃ p inductively: ◮ ∃ p ( a ∨ b ) ≡ ∃ p . a ∨ ∃ p . b ◮ ∃ p . ∇ α ≡ ∇∃ p .α This generalizes to the modal µ -calculus (d’Agostino & Hollenberg) and to the coalgebraic setting (Kissig, Kupke & V.)

Other residuation properties Theorem (Ghilardi/Santocanale) The diamond operator of a free modal algebra is residuated.