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

A modal distributive law

Yde Venema ILLC, UvA, Amsterdam http://staff.science.uva.nl/~yde Logic Colloquium 2011

Collaborators

This talk is based on joint work with many collaborators, including: Marta B´ ılkov´ a, Balder ten Cate, Willem Conradie, Gaelle Fontaine, Christian Kissig, Clemens Kupke, Alexander Kurz, Maarten Marx, Alessandra Palmigiano, Luigi Santocanale, Jiˇ r´ ı Velebil, Steve Vickers, Jacob Vosmaer

Point of Talk

Modal Distributive Law:

• {∇γ | γ ∈ Γ} ≡
• {∇(P)Φ | Φ ∈ SRD(Γ)}

(∇2)

Point of Talk

Modal Distributive Law:

• {∇γ | γ ∈ Γ} ≡
• {∇(P)Φ | Φ ∈ SRD(Γ)}

(∇2) The Modal Distributive Law is a key principle of Universal Coalgebra

Distributivity

Propositional Distributive Law: a ∧ (b ∨ c) ≡ (a ∧ b) ∨ (a ∧ c),

Distributivity

Propositional Distributive Law: a ∧ (b ∨ c) ≡ (a ∧ b) ∨ (a ∧ c),

• r
• {α | α ∈ A} =
• {Im(γ) | γ ∈ Choice(A)}

Distributivity

Propositional Distributive Law: a ∧ (b ∨ c) ≡ (a ∧ b) ∨ (a ∧ c),

• r
• {α | α ∈ A} =
• {Im(γ) | γ ∈ Choice(A)}

Modal Distributive Law: ∇α ∧ ∇α′ ≡

• Z∈α⊲

⊳α′

∇{a ∧ a′ | (a, a′) ∈ Z}, (MDL)

Distributivity

Propositional Distributive Law: a ∧ (b ∨ c) ≡ (a ∧ b) ∨ (a ∧ c),

• r
• {α | α ∈ A} =
• {Im(γ) | γ ∈ Choice(A)}

Modal Distributive Law: ∇α ∧ ∇α′ ≡

• Z∈α⊲

⊳α′

∇{a ∧ a′ | (a, a′) ∈ Z}, (MDL)

• r
• {∇γ | γ ∈ Γ} ≡
• {∇(P)Φ | Φ ∈ SRD(Γ)}

Coalgebra

◮ Coalgebra (Aczel, Rutten, . . . ) is a general mathematical theory for evolving systems

Coalgebra

◮ Coalgebra (Aczel, Rutten, . . . ) is a general mathematical theory for evolving systems ◮ It provides a natural framework for notions like behavior, invariants,

• bservational equivalence, . . .

Coalgebra

◮ Coalgebra (Aczel, Rutten, . . . ) is a general mathematical theory for evolving systems ◮ It provides a natural framework for notions like behavior, invariants,

• bservational equivalence, . . .

◮ A coalgebra is a structure S = S, σ : S → TS, where T : Set → Set is the type of the coalgebra.

Coalgebra

◮ Coalgebra (Aczel, Rutten, . . . ) is a general mathematical theory for evolving systems ◮ It provides a natural framework for notions like behavior, invariants,

• bservational equivalence, . . .

◮ A coalgebra is a structure S = S, σ : S → TS, where T : Set → Set is the type of the coalgebra. (More generally, given T : C → C, study T-coalgebras over C)

Coalgebra

◮ Coalgebra (Aczel, Rutten, . . . ) is a general mathematical theory for evolving systems ◮ It provides a natural framework for notions like behavior, invariants,

• bservational equivalence, . . .

◮ A coalgebra is a structure S = S, σ : S → TS, where T : Set → Set is the type of the coalgebra. (More generally, given T : C → C, study T-coalgebras over C) ◮ Sufficiently general to model notions like: input, output, non-determinism, interaction, probability, . . .

Coalgebra

◮ Coalgebra (Aczel, Rutten, . . . ) is a general mathematical theory for evolving systems ◮ It provides a natural framework for notions like behavior, invariants,

• bservational equivalence, . . .

◮ A coalgebra is a structure S = S, σ : S → TS, where T : Set → Set is the type of the coalgebra. (More generally, given T : C → C, study T-coalgebras over C) ◮ Sufficiently general to model notions like: input, output, non-determinism, interaction, probability, . . . ◮ Examples include streams, trees, deterministic automata, Kripke structures, labelled transition systems, Markov chains, . . .

Coalgebra

◮ Coalgebra (Aczel, Rutten, . . . ) is a general mathematical theory for evolving systems ◮ It provides a natural framework for notions like behavior, invariants,

• bservational equivalence, . . .

◮ A coalgebra is a structure S = S, σ : S → TS, where T : Set → Set is the type of the coalgebra. (More generally, given T : C → C, study T-coalgebras over C) ◮ Sufficiently general to model notions like: input, output, non-determinism, interaction, probability, . . . ◮ Examples include streams, trees, deterministic automata, Kripke structures, labelled transition systems, Markov chains, . . . ◮ Universal Coalgebra: develop theory of coalgebras uniformly in T

Coalgebraic Logic

To specify and reason about behavior, need coalgebraic logic.

Coalgebraic Logic

To specify and reason about behavior, need coalgebraic logic. Algebra Equational Logic = Coalgebra

Coalgebraic Logic

To specify and reason about behavior, need coalgebraic logic. Algebra Equational Logic = Coalgebra Modal Logicµ

Coalgebraic Logic

To specify and reason about behavior, need coalgebraic logic. Algebra Equational Logic = Coalgebra Modal Logicµ

µ: with fixpoint operators

Relation Lifting

Let P denote the covariant powerset functor on Set: for f : S → S′ we have Pf : PS → PS′ by Pf (X) := {fx | x ∈ X}.

Relation Lifting

Let P denote the covariant powerset functor on Set: for f : S → S′ we have Pf : PS → PS′ by Pf (X) := {fx | x ∈ X}. Definition Given Z ⊆ S × S′, define its (Egli-Milner) lifting PZ ⊆ PS × PS′ by

Relation Lifting

Let P denote the covariant powerset functor on Set: for f : S → S′ we have Pf : PS → PS′ by Pf (X) := {fx | x ∈ X}. Definition Given Z ⊆ S × S′, define its (Egli-Milner) lifting PZ ⊆ PS × PS′ by PZ := {(α, α′) |∀a ∈ α.∃a′ ∈ α′.Zaa′ & ∀a′ ∈ α′.∃a ∈ α.Zaa′}

Relation Lifting

Let P denote the covariant powerset functor on Set: for f : S → S′ we have Pf : PS → PS′ by Pf (X) := {fx | x ∈ X}. Definition Given Z ⊆ S × S′, define its (Egli-Milner) lifting PZ ⊆ PS × PS′ by PZ := {(α, α′) |∀a ∈ α.∃a′ ∈ α′.Zaa′ & ∀a′ ∈ α′.∃a ∈ α.Zaa′} Proposition ◮ P(∆S) = ∆PS ◮ P(Z˘) = (PZ)˘ ◮ P(Z0; Z1) = (PZ0); (PZ1) ◮ P(Grf ) = Gr(Pf ) ◮ . . .

Relation Lifting

Let P denote the covariant powerset functor on Set: for f : S → S′ we have Pf : PS → PS′ by Pf (X) := {fx | x ∈ X}. Definition Given Z ⊆ S × S′, define its (Egli-Milner) lifting PZ ⊆ PS × PS′ by PZ := {(α, α′) |∀a ∈ α.∃a′ ∈ α′.Zaa′ & ∀a′ ∈ α′.∃a ∈ α.Zaa′} Proposition ◮ P(∆S) = ∆PS ◮ P(Z˘) = (PZ)˘ ◮ P(Z0; Z1) = (PZ0); (PZ1) ◮ P(Grf ) = Gr(Pf ) ◮ . . . Most of this can be extended to an arbitrary set functor T : Set → Set.

Overview

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

Overview

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

Modal Logic

Syntax Fix a (finite) set X of proposition letters. The set of modal formulas over X is given by a ::= p ∈ X | ⊥ | ¬a | a ∨ a | ♦a

Modal Logic

Syntax Fix a (finite) set X of proposition letters. The set of modal formulas over X is given by a ::= p ∈ X | ⊥ | ¬a | a ∨ a | ♦a Abbreviate a := ¬♦¬a.

Modal Logic

Syntax Fix a (finite) set X of proposition letters. The set of modal formulas over X is given by a ::= p ∈ X | ⊥ | ¬a | a ∨ a | ♦a Abbreviate a := ¬♦¬a. Semantics A Kripke frame is a pair S, R with R ⊆ S × S.

Modal Logic

Syntax Fix a (finite) set X of proposition letters. The set of modal formulas over X is given by a ::= p ∈ X | ⊥ | ¬a | a ∨ a | ♦a Abbreviate a := ¬♦¬a. Semantics A Kripke frame is a pair S, R with R ⊆ S × S. Such R induces a map R[·] : S → P(S) given by R[s] := {t ∈ S | Rst}

Modal Logic

Syntax Fix a (finite) set X of proposition letters. The set of modal formulas over X is given by a ::= p ∈ X | ⊥ | ¬a | a ∨ a | ♦a Abbreviate a := ¬♦¬a. Semantics A Kripke frame is a pair S, R with R ⊆ S × S. Such R induces a map R[·] : S → P(S) given by R[s] := {t ∈ S | Rst} Hence Kripke frames are coalgebras for the powerset functor P.

Modal truth/satisfaction

A valuation on a Kripke frame S is an assignment V : X → P(S).

Modal truth/satisfaction

A valuation on a Kripke frame S is an assignment V : X → P(S). A Kripke model is a triple S = S, R, V .

Modal truth/satisfaction

A valuation on a Kripke frame S is an assignment V : X → P(S). A Kripke model is a triple S = S, R, V . Given a Kripke model S, inductively define a truth/satisfaction relation between states and formulas: S, s p if s ∈ V (p) S, s ⊥ S, s ¬a if S, s a S, s a ∨ b if S, s a or S, s b S, s ♦a if S, t a for some t ∈ R[s] Hence S, s a if S, t a for all t ∈ R[s]

The cover modality ∇

Syntax If α is a finite set of formulas then ∇α is a formula.

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′) ∈ PZ. 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′) ∈ PZ. Notation: Z ∈ α ⊲ ⊳ α′. Theorem For any sets α, α′ of formulas, ∇α ∧ ∇α′ ≡

• Z∈α⊲

⊳α′

∇{a ∧ a′ | (a, a′) ∈ Z}, (MDL)

A modal distributive law

Definition Given sets α, α′, a relation Z ⊆ α × α′ is full on α, α′, if (α, α′) ∈ PZ. Notation: Z ∈ α ⊲ ⊳ α′. Theorem For any sets α, α′ of formulas, ∇α ∧ ∇α′ ≡

• Z∈α⊲

⊳α′

∇{a ∧ a′ | (a, a′) ∈ Z}, (MDL) 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′) ∈ PZ. Notation: Z ∈ α ⊲ ⊳ α′. Theorem For any sets α, α′ of formulas, ∇α ∧ ∇α′ ≡

• Z∈α⊲

⊳α′

∇{a ∧ a′ | (a, a′) ∈ Z}, (MDL) Proof of ‘⇒’: Suppose S, s ∇α ∧ ∇α′. Define Zs ⊆ α × α′ as Zs := {(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′) ∈ PZ. Notation: Z ∈ α ⊲ ⊳ α′. Theorem For any sets α, α′ of formulas, ∇α ∧ ∇α′ ≡

• Z∈α⊲

⊳α′

∇{a ∧ a′ | (a, a′) ∈ Z}, (MDL) Proof of ‘⇒’: Suppose S, s ∇α ∧ ∇α′. Define Zs ⊆ α × α′ as Zs := {(a, a′) | S, t a ∧ a′ for some t ∈ R[s]}. Then S, s ∇{a ∧ a′ | (a, a′) ∈ Zs},

A modal distributive law

Definition Given sets α, α′, a relation Z ⊆ α × α′ is full on α, α′, if (α, α′) ∈ PZ. Notation: Z ∈ α ⊲ ⊳ α′. Theorem For any sets α, α′ of formulas, ∇α ∧ ∇α′ ≡

• Z∈α⊲

⊳α′

∇{a ∧ a′ | (a, a′) ∈ Z}, (MDL) Proof of ‘⇒’: Suppose S, s ∇α ∧ ∇α′. Define Zs ⊆ α × α′ as Zs := {(a, a′) | S, t a ∧ a′ for some t ∈ R[s]}. Then S, s ∇{a ∧ a′ | (a, a′) ∈ Zs}, and (α, α′) ∈ PZ.

A modal distributive law

Definition Given sets α, α′, a relation Z ⊆ α × α′ is full on α, α′, if (α, α′) ∈ PZ. Notation: Z ∈ α ⊲ ⊳ α′. Theorem For any sets α, α′ of formulas, ∇α ∧ ∇α′ ≡

• Z∈α⊲

⊳α′

∇{a ∧ a′ | (a, a′) ∈ Z}, (MDL) Proof of ‘⇒’: Suppose S, s ∇α ∧ ∇α′. Define Zs ⊆ α × α′ as Zs := {(a, a′) | S, t a ∧ a′ for some t ∈ R[s]}. Then S, s ∇{a ∧ a′ | (a, a′) ∈ Zs}, and (α, α′) ∈ PZ. Note This theorem enables us to (almost) eliminate conjunctions!

Elimination of conjunctions

Assume that X is finite. For Π ⊆ X, abbreviate ⊙Π :=

• p∈Π

p ∧

• p∈X\Π

¬p.

Elimination of conjunctions

Assume that X is finite. For Π ⊆ X, abbreviate ⊙Π :=

• p∈Π

p ∧

• p∈X\Π

¬p. 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∈Π

p ∧

• p∈X\Π

¬p. 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 (a1 ∨ a2, s) ∃ {(a1, s), (a2, s)} (a1 ∧ a2, s) ∀ {(a1, s), (a2, 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 (a1 ∨ a2, s) ∃ {(a1, s), (a2, s)} (a1 ∧ a2, s) ∀ {(a1, s), (a2, 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 (a1 ∨ a2, s) ∃ {(a1, s), (a2, s)} (a1 ∧ a2, s) ∀ {(a1, s), (a2, 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 ∧ (ψ1 ∨ ψ2) ≡ (a ∧ ψ1) ∨ (a ∧ ψ2) ∀∃ ∃∀

Strategic normal forms

◮ propositional distributive law:

a ∧ (ψ1 ∨ ψ2) ≡ (a ∧ ψ1) ∨ (a ∧ ψ2) ∀∃ ∃∀

◮ modal distributive law:

∇α ∧ ∇α′ ≡

• Z∈α⊲

⊳α′

∇{a ∧ a′ | (a, a′) ∈ Z} ∀∃∀ ∃∃∀∀

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 : PωL∇ → L∇, then Pω : PωPωL∇ → PωL∇, with (Pω ){α1, . . . , αn} = { α1, . . . , αn}.

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 : PωL∇ → L∇, then Pω : PωPωL∇ → PωL∇, with (Pω ){α1, . . . , αn} = { α1, . . . , αn}. With ∈ ⊆ L∇ × PωL∇, obtain P∈ ⊆ PωL∇ × PωPω

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 : PωL∇ → L∇, then Pω : PωPωL∇ → PωL∇, with (Pω ){α1, . . . , αn} = { α1, . . . , αn}. With ∈ ⊆ L∇ × PωL∇, obtain P∈ ⊆ PωL∇ × PωPω ◮ Axiom (for Φ ∈ PωPωL∇): ∇(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 ∈ PωPωL∇:

• {∇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 ∈ PωPωL∇:

• {∇γ | γ ∈ Γ} ≤

∇(P)Φ | Φ ∈ SRD(Γ)

• (∇2)

For Φ ∈ PωPωL∇: ∇(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) F(X) F(X ∪ {p}) ❥ e

Uniform Interpolation

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

Uniform Interpolation

Theorem (Ghilardi/Visser) F(X) F(X ∪ {p}) ❥ e ⊥ ❨ ∃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) F(X) F(X ∪ {p}) ❥ e ⊥ ❨ ∃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) F(X) F(X ∪ {p}) ❥ e ⊥ ❨ ∃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) = {p1, . . . , pn}.

Uniform Interpolation

Theorem (Ghilardi/Visser) F(X) F(X ∪ {p}) ❥ e ⊥ ❨ ∃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) = {p1, . . . , pn}. Then a | = ∃p1 · · · ∃pn.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.

Other residuation properties

Theorem (Ghilardi/Santocanale) The diamond operator of a free modal algebra is residuated. Theorem (Santocanale & V.) The nabla operator of a free modal algebra is ω-residuated.

Other residuation properties

Theorem (Ghilardi/Santocanale) The diamond operator of a free modal algebra is residuated. Theorem (Santocanale & V.) The nabla operator of a free modal algebra is ω-residuated. Hence, free modal algebras are generated by ω-residuated primitive

• perations.

Other residuation properties

Theorem (Ghilardi/Santocanale) The diamond operator of a free modal algebra is residuated. Theorem (Santocanale & V.) The nabla operator of a free modal algebra is ω-residuated. Hence, free modal algebras are generated by ω-residuated primitive

• perations.

Theorem (B´ ılkova, Velebil & Venema) This generalizes to the coalgebraic setting.

Overview

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

Rabin’s Theorem

Theorem (Rabin) S2S is decidable. Proof sketch Inductively define a construction ϕ → Aϕ transforming S2S-formulas into tree automata,

Rabin’s Theorem

Theorem (Rabin) S2S is decidable. Proof sketch Inductively define a construction ϕ → Aϕ transforming S2S-formulas into tree automata, and prove that ϕ is satisfiable iff L(Aϕ) = ∅.

Rabin’s Theorem

Theorem (Rabin) S2S is decidable. Proof sketch Inductively define a construction ϕ → Aϕ transforming S2S-formulas into tree automata, and prove that ϕ is satisfiable iff L(Aϕ) = ∅. Emptiness for tree automata is (relatively easily) decidable.

Rabin’s Theorem

Theorem (Rabin) S2S is decidable. Proof sketch Inductively define a construction ϕ → Aϕ transforming S2S-formulas into tree automata, and prove that ϕ is satisfiable iff L(Aϕ) = ∅. Emptiness for tree automata is (relatively easily) decidable. Hard part of proof:

Rabin’s Theorem

Theorem (Rabin) S2S is decidable. Proof sketch Inductively define a construction ϕ → Aϕ transforming S2S-formulas into tree automata, and prove that ϕ is satisfiable iff L(Aϕ) = ∅. Emptiness for tree automata is (relatively easily) decidable. Hard part of proof: Complementation Lemma providing a construction A → Ac such that Ac accepts exactly the trees that A rejects.

Tree Automata

Fix a finite alphabet/color set C and consider the class TreeC of C-labelled binary trees. Write 2 = {0, 1}.

Tree Automata

Fix a finite alphabet/color set C and consider the class TreeC of C-labelled binary trees. Write 2 = {0, 1}. A (nondeterministic) tree automaton is a quadruple A = A, aI ∈ A, ∆ : A × C → P(A × A), Acc with A finite.

Tree Automata

Fix a finite alphabet/color set C and consider the class TreeC of C-labelled binary trees. Write 2 = {0, 1}. A (nondeterministic) tree automaton is a quadruple A = A, aI ∈ A, ∆ : A × C → P(A × A), Acc with A finite. A run of A on a tree τ : 2∗ → C is a tree ρ : 2∗ → A such that

◮ (ρ(s0), ρ(s1) ∈ ∆(ρ(s), τ(s)) for all s ∈ 2∗.

Tree Automata

Fix a finite alphabet/color set C and consider the class TreeC of C-labelled binary trees. Write 2 = {0, 1}. A (nondeterministic) tree automaton is a quadruple A = A, aI ∈ A, ∆ : A × C → P(A × A), Acc with A finite. A run of A on a tree τ : 2∗ → C is a tree ρ : 2∗ → A such that

◮ (ρ(s0), ρ(s1) ∈ ∆(ρ(s), τ(s)) for all s ∈ 2∗.

Such a run is accepting if all infinite branches meet the acceptance condition Acc ⊆ Aω.

Tree Automata

Fix a finite alphabet/color set C and consider the class TreeC of C-labelled binary trees. Write 2 = {0, 1}. A (nondeterministic) tree automaton is a quadruple A = A, aI ∈ A, ∆ : A × C → P(A × A), Acc with A finite. A run of A on a tree τ : 2∗ → C is a tree ρ : 2∗ → A such that

◮ (ρ(s0), ρ(s1) ∈ ∆(ρ(s), τ(s)) for all s ∈ 2∗.

Such a run is accepting if all infinite branches meet the acceptance condition Acc ⊆ Aω. Acceptance of tree automata can also be defined in terms of an (infinite!) 2-player game in which ∀ is only the pathfinder.

Alternating Tree Automata

Alternating Tree Automata

Muller & Schupp defined alternating tree automata: where acceptance is defined in terms of a more symmetric 2-player game (∃/∨, ∀/∧).

Alternating Tree Automata

Muller & Schupp defined alternating tree automata: where acceptance is defined in terms of a more symmetric 2-player game (∃/∨, ∀/∧). Alternating automata admit an easy Complementation Lemma,

Alternating Tree Automata

Muller & Schupp defined alternating tree automata: where acceptance is defined in terms of a more symmetric 2-player game (∃/∨, ∀/∧). Alternating automata admit an easy Complementation Lemma, but now decidability of emptiness is hard . . .

Alternating Tree Automata

Muller & Schupp defined alternating tree automata: where acceptance is defined in terms of a more symmetric 2-player game (∃/∨, ∀/∧). Alternating automata admit an easy Complementation Lemma, but now decidability of emptiness is hard . . . Simulation Theorem (Muller & Schupp) There is a construction transforming an alternating automaton into an equivalent nondeterministic one.

Alternating Tree Automata

Muller & Schupp defined alternating tree automata: where acceptance is defined in terms of a more symmetric 2-player game (∃/∨, ∀/∧). Alternating automata admit an easy Complementation Lemma, but now decidability of emptiness is hard . . . Simulation Theorem (Muller & Schupp) There is a construction transforming an alternating automaton into an equivalent nondeterministic one. Proof idea Power construction + results on ω-automata + . . .

Alternating Tree Automata

Muller & Schupp defined alternating tree automata: where acceptance is defined in terms of a more symmetric 2-player game (∃/∨, ∀/∧). Alternating automata admit an easy Complementation Lemma, but now decidability of emptiness is hard . . . Simulation Theorem (Muller & Schupp) There is a construction transforming an alternating automaton into an equivalent nondeterministic one. Proof idea Power construction + results on ω-automata + . . . . . . Modal Distributive Law (Binary Tree version)!

Alternating Tree Automata

Muller & Schupp defined alternating tree automata: where acceptance is defined in terms of a more symmetric 2-player game (∃/∨, ∀/∧). Alternating automata admit an easy Complementation Lemma, but now decidability of emptiness is hard . . . Simulation Theorem (Muller & Schupp) There is a construction transforming an alternating automaton into an equivalent nondeterministic one. Proof idea Power construction + results on ω-automata + . . . . . . Modal Distributive Law (Binary Tree version)! Similar ideas underly the introduction of ∇ by Janin & Walukiewicz in the modal µ-calculus. Their simulation theorem is the basis of many results on Lµ.

Alternating Tree Automata

Muller & Schupp defined alternating tree automata: where acceptance is defined in terms of a more symmetric 2-player game (∃/∨, ∀/∧). Alternating automata admit an easy Complementation Lemma, but now decidability of emptiness is hard . . . Simulation Theorem (Muller & Schupp) There is a construction transforming an alternating automaton into an equivalent nondeterministic one. Proof idea Power construction + results on ω-automata + . . . . . . Modal Distributive Law (Binary Tree version)! Similar ideas underly the introduction of ∇ by Janin & Walukiewicz in the modal µ-calculus. Their simulation theorem is the basis of many results on Lµ. Kupke and V. lifted the Simulation Theorem to Universal Coalgebra.

Overview

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

Stone Duality - Extended

Theorem (Stone) BA ≡op Stone.

Stone Duality - Extended

Theorem (Stone) BA ≡op Stone. What about modal algebras?

Stone Duality - Extended

Theorem (Stone) BA ≡op Stone. What about modal algebras? From work by (Esakia and) Abramsky we

• btain duality for modal logic as an Algebra|Coalgebra duality:

BA Stone ❥ S ❨ ˘ P

Stone Duality - Extended

Theorem (Stone) BA ≡op Stone. What about modal algebras? From work by (Esakia and) Abramsky we

• btain duality for modal logic as an Algebra|Coalgebra duality:

BA Stone ❥ S ❨ ˘ P ❘ M ✠ V

Stone Duality - Extended

Theorem (Stone) BA ≡op Stone. What about modal algebras? From work by (Esakia and) Abramsky we

• btain duality for modal logic as an Algebra|Coalgebra duality:

BA Stone ❥ S ❨ ˘ P ❘ M ✠ V Modal Logic dualizes/axiomatizes the Vietoris functor

Stone Duality - Extended

Theorem (Stone) BA ≡op Stone. What about modal algebras? From work by (Esakia and) Abramsky we

• btain duality for modal logic as an Algebra|Coalgebra duality:

BA Stone ❥ S ❨ ˘ P ❘ M ✠ V Modal Logic dualizes/axiomatizes the Vietoris functor C induces a functor M on the category BA of Boolean algebras: MB := BA{∇β | β ∈ PB} | ∇1 − 3

Stone Duality - Extended

Theorem (Stone) BA ≡op Stone. What about modal algebras? From work by (Esakia and) Abramsky we

• btain duality for modal logic as an Algebra|Coalgebra duality:

BA Stone ❥ S ❨ ˘ P ❘ M ✠ V Modal Logic dualizes/axiomatizes the Vietoris functor C induces a functor M on the category BA of Boolean algebras: MB := BA{∇β | β ∈ PB} | ∇1 − 3 An M-algebra is a pair B, β : MB → B.

Stone Duality - Extended

Theorem (Stone) BA ≡op Stone. What about modal algebras? From work by (Esakia and) Abramsky we

• btain duality for modal logic as an Algebra|Coalgebra duality:

BA Stone ❥ S ❨ ˘ P ❘ M ✠ V Modal Logic dualizes/axiomatizes the Vietoris functor C induces a functor M on the category BA of Boolean algebras: MB := BA{∇β | β ∈ PB} | ∇1 − 3 An M-algebra is a pair B, β : MB → B. Theorem (Kupke, Kurz & V) MA ∼ = ALgBA(M).

The Vietoris construction

The Vietoris construction

◮ Let X = X, τ be a topological space. ◮ K(X) denotes the collection of compact sets, and for a ∈ τ, define ∋a := {F ∈ K(X) | F ∩ a = ∅} [∋]a := {F ∈ K(X) | F ⊆ a}

The Vietoris construction

◮ Let X = X, τ be a topological space. ◮ K(X) denotes the collection of compact sets, and for a ∈ τ, define ∋a := {F ∈ K(X) | F ∩ a = ∅} [∋]a := {F ∈ K(X) | F ⊆ a} ◮ Generate the Vietoris topology υτ on K(X) from {∋a, [∋] | a ∈ τ}.

The Vietoris construction

◮ Let X = X, τ be a topological space. ◮ K(X) denotes the collection of compact sets, and for a ∈ τ, define ∋a := {F ∈ K(X) | F ∩ a = ∅} [∋]a := {F ∈ K(X) | F ⊆ a} ◮ Generate the Vietoris topology υτ on K(X) from {∋a, [∋] | a ∈ τ}. ◮ V(X) := K(X), υτ is the Vietoris space of X.

The Vietoris construction

◮ Let X = X, τ be a topological space. ◮ K(X) denotes the collection of compact sets, and for a ∈ τ, define ∋a := {F ∈ K(X) | F ∩ a = ∅} [∋]a := {F ∈ K(X) | F ⊆ a} ◮ Generate the Vietoris topology υτ on K(X) from {∋a, [∋] | a ∈ τ}. ◮ V(X) := K(X), υτ is the Vietoris space of X. Fact The Vietoris construction preserves various properties, including:

• compactness
• compact Hausdorfness
• Stone-ness

Variation: Pointfree Topology

Frames/Locales provide pointfree versions of topologies.

Variation: Pointfree Topology

Frames/Locales provide pointfree versions of topologies. KRFr KHaus ❥ S ❨ ˘ P

Variation: Pointfree Topology

Frames/Locales provide pointfree versions of topologies. KRFr KHaus ❥ S ❨ ˘ P ✠ V

Variation: Pointfree Topology

Frames/Locales provide pointfree versions of topologies. KRFr KHaus ❥ S ❨ ˘ P ✠ V ❘ J

Variation: Pointfree Topology

Frames/Locales provide pointfree versions of topologies. KRFr KHaus ❥ S ❨ ˘ P ✠ V ❘ J Geometric modal logic dualizes/axiomatizes the Vietoris functor (Johnstone)

Vietoris pointfree (Johnstone)

Given a frame L, define L := {a | a ∈ L} and L♦ := {♦a | a ∈ L}. VL := FrL ⊎ L♦ | ( A) =

a∈A a

(A ∈ PωL) ♦( A) =

a∈A ♦a

(A ∈ PωL) a ∧ ♦b ≤ ♦(a ∧ b) (a ∨ b) ≤ a ∨ ♦b ( A) =

a∈A a

(A ∈ PL directed) ♦( A) =

a∈A ♦a

(A ∈ PL directed)

Vietoris via ∇

Vietoris via ∇

Fix a standard set functor T that preserves weak pullbacks.

Vietoris via ∇

Fix a standard set functor T that preserves weak pullbacks. Define the T-powerlocale of a frame L as JTL := FrTωL | (∇1), (∇2), (∇3),

Vietoris via ∇

Fix a standard set functor T that preserves weak pullbacks. Define the T-powerlocale of a frame L as JTL := FrTωL | (∇1), (∇2), (∇3), where the relations are as follows:

Vietoris via ∇

Fix a standard set functor T that preserves weak pullbacks. Define the T-powerlocale of a frame L as JTL := FrTωL | (∇1), (∇2), (∇3), where the relations are as follows: (∇1) ∇α ≤ ∇β (α T≤ β)

Vietoris via ∇

Fix a standard set functor T that preserves weak pullbacks. Define the T-powerlocale of a frame L as JTL := FrTωL | (∇1), (∇2), (∇3), where the relations are as follows: (∇1) ∇α ≤ ∇β (α T≤ β) (∇2)

• γ∈Γ∇γ ≤
• {∇(T )Ψ | Ψ ∈ SRD(Γ)}

(Γ ∈ PωTωL)

Vietoris via ∇

Fix a standard set functor T that preserves weak pullbacks. Define the T-powerlocale of a frame L as JTL := FrTωL | (∇1), (∇2), (∇3), where the relations are as follows: (∇1) ∇α ≤ ∇β (α T≤ β) (∇2)

• γ∈Γ∇γ ≤
• {∇(T )Ψ | Ψ ∈ SRD(Γ)}

(Γ ∈ PωTωL) (∇3) ∇(T)Φ ≤

• {∇β | β T∈ Φ}

(Φ ∈ TωPL)

Some results

Some results

Theorem (V., Vickers & Vosmaer) ◮ JT provides a functor on the category Fr of frames.

Some results

Theorem (V., Vickers & Vosmaer) ◮ JT provides a functor on the category Fr of frames. ◮ JT generalizes Johnstone’s J: J = JP.

Some results

Theorem (V., Vickers & Vosmaer) ◮ JT provides a functor on the category Fr of frames. ◮ JT generalizes Johnstone’s J: J = JP. ◮ JT preserves regularity, zero-dimensionality, and Stone-ness.

Some results

Theorem (V., Vickers & Vosmaer) ◮ JT provides a functor on the category Fr of frames. ◮ JT generalizes Johnstone’s J: J = JP. ◮ JT preserves regularity, zero-dimensionality, and Stone-ness. Conjecture If T preserves finite sets, then JT preserves compactness.

Some results

Theorem (V., Vickers & Vosmaer) ◮ JT provides a functor on the category Fr of frames. ◮ JT generalizes Johnstone’s J: J = JP. ◮ JT preserves regularity, zero-dimensionality, and Stone-ness. Conjecture If T preserves finite sets, then JT preserves compactness.

(In fact, our ∇-presentation is very compatible with that of Vietoris!)

Overview

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

Summary

Summary

The modal distributive law is a fundamental principle,

Summary

The modal distributive law is a fundamental principle, with many applications/manifestations/generalizations:

◮ logic ◮ algebra ◮ automata theory ◮ topology ◮ . . .

Summary

The modal distributive law is a fundamental principle, with many applications/manifestations/generalizations:

◮ logic ◮ algebra ◮ automata theory ◮ topology ◮ . . .

This generalizes to wide coalgebraic setting.

Ongoing & Further research

Ongoing & Further research

◮ algebraic aspects of ∇ ◮ universal automata theory ◮ point-free topology ◮ completeness for fixpoint logics ◮ extending the coalgebraic scope ◮ . . .

Thank you!