A Dynamic Distributive Law Yde Venema Universiteit van Amsterdam - - PowerPoint PPT Presentation

A Dynamic Distributive Law

Yde Venema Universiteit van Amsterdam staff.science.uva.nl/∼yde August 10, 2007 Coalgebraic Logic Workshop Oxford

(Largely based on joint work with Marta Bilkova, Clemens Kupke, Alexander Kurz, Alessandra Palmigiano, Luigi Santocanale)

Venema Co-Oxford 2007

Overview

◮ Introduction: reorganizing modal logic ◮ A modal distributive law ◮ A game-theoretical perspective ◮ Uniform interpolation ◮ Automata ◮ Axiomatizing ∇ ◮ A coalgebraic generalization ◮ Concluding remarks

Overview 1

Venema Co-Oxford 2007

Overview

◮ Introduction: reorganizing modal logic ◮ A modal distributive law ◮ A game-theoretical perspective ◮ Uniform interpolation ◮ Automata ◮ Axiomatizing ∇ ◮ A coalgebraic generalization ◮ Concluding remarks

Overview 2

Venema Co-Oxford 2007

The Cover Modality ∇

◮ Define the language ML of standard modal logic by

ϕ ::= p | ¬p | ⊥ | ⊤ | ϕ ∨ ϕ | ϕ ∧ ϕ | ✸ϕ | ✷ϕ

Introduction 3

Venema Co-Oxford 2007

The Cover Modality ∇

◮ Define the language ML of standard modal logic by

ϕ ::= p | ¬p | ⊥ | ⊤ | ϕ ∨ ϕ | ϕ ∧ ϕ | ✸ϕ | ✷ϕ

◮ Given set Φ of formulas, define

∇Φ := ✷

• Φ ∧
• ✸Φ

(here ✸Φ := {✸ϕ | ϕ ∈ Φ})

Introduction 3

Venema Co-Oxford 2007

The Cover Modality ∇

◮ Define the language ML of standard modal logic by

ϕ ::= p | ¬p | ⊥ | ⊤ | ϕ ∨ ϕ | ϕ ∧ ϕ | ✸ϕ | ✷ϕ

◮ Given set Φ of formulas, define

∇Φ := ✷

• Φ ∧
• ✸Φ

(here ✸Φ := {✸ϕ | ϕ ∈ Φ})

◮ History: (predicate logic), Fine, Moss, Abramsky, Walukiewicz

Introduction 3

Venema Co-Oxford 2007

The Cover Modality ∇

◮ Define the language ML of standard modal logic by

ϕ ::= p | ¬p | ⊥ | ⊤ | ϕ ∨ ϕ | ϕ ∧ ϕ | ✸ϕ | ✷ϕ

◮ Given set Φ of formulas, define

∇Φ := ✷

• Φ ∧
• ✸Φ

(here ✸Φ := {✸ϕ | ϕ ∈ Φ})

◮ History: (predicate logic), Fine, Moss, Abramsky, Walukiewicz ◮ Define the language ML∇ by

ϕ ::= p | ¬p | ⊥ | ⊤ | ϕ ∨ ϕ | ϕ ∧ ϕ | ∇Φ

Introduction 3

Venema Co-Oxford 2007

Semantics

Fix a Kripke model S = S, R, V . S, s ∇Φ iff for all t ∈ R[s] there is a ϕ ∈ Φ with S, t ϕ and for all ϕ ∈ Φ there is a t ∈ R[s] with S, t ϕ

Introduction 4

Venema Co-Oxford 2007

Semantics

Fix a Kripke model S = S, R, V . S, s ∇Φ iff for all t ∈ R[s] there is a ϕ ∈ Φ with S, t ϕ and for all ϕ ∈ Φ there is a t ∈ R[s] with S, t ϕ Call a relation Z full on two sets A and B if ∀a ∈ A∃b. ∈ BZab and ∀b ∈ B∃a. ∈ AZab.

Introduction 4

Venema Co-Oxford 2007

Semantics

Fix a Kripke model S = S, R, V . S, s ∇Φ iff for all t ∈ R[s] there is a ϕ ∈ Φ with S, t ϕ and for all ϕ ∈ Φ there is a t ∈ R[s] with S, t ϕ Call a relation Z full on two sets A and B if ∀a ∈ A∃b. ∈ BZab and ∀b ∈ B∃a. ∈ AZab. Theorem (Moss) S, s ∇Φ iff the satisfaction relation is full on R[s] and Φ

Introduction 4

Venema Co-Oxford 2007

Semantics

Fix a Kripke model S = S, R, V . S, s ∇Φ iff for all t ∈ R[s] there is a ϕ ∈ Φ with S, t ϕ and for all ϕ ∈ Φ there is a t ∈ R[s] with S, t ϕ Call a relation Z full on two sets A and B if ∀a ∈ A∃b. ∈ BZab and ∀b ∈ B∃a. ∈ AZab. Theorem (Moss) S, s ∇Φ iff the satisfaction relation is full on R[s] and Φ iff there is a Z ⊆ which is full on R[s] and Φ

Introduction 4

Venema Co-Oxford 2007

Reorganizing Modal Logic

Conversely, express ✷ and ✸ in terms of ∇ ✸ϕ ≡ ∇{ϕ, ⊤} ✷ϕ ≡ ∇∅ ∨ ∇{ϕ}.

Introduction 5

Venema Co-Oxford 2007

Reorganizing Modal Logic

Conversely, express ✷ and ✸ in terms of ∇ ✸ϕ ≡ ∇{ϕ, ⊤} ✷ϕ ≡ ∇∅ ∨ ∇{ϕ}. Theorem The languages ML and ML∇ are effectively equi-expressive.

Introduction 5

Venema Co-Oxford 2007

Overview

◮ Introduction: reorganizing modal logic ◮ A modal distributive law ◮ A game-theoretical perspective ◮ Uniform interpolation ◮ Automata ◮ Axiomatizing ∇ ◮ A coalgebraic generalization ◮ Concluding remarks

Overview 6

Venema Co-Oxford 2007

A modal distributive law

A modal distributive law 7

Venema Co-Oxford 2007

A modal distributive law

Theorem For any sets Φ, Φ′ of formulas, ∇Φ ∧ ∇Φ′ ≡

• Z∈Φ⊲

⊳Φ′

∇{ϕ ∧ ϕ′ | (ϕ, ϕ′) ∈ Z}, where Φ ⊲ ⊳ Φ′ is the set of full relations between Φ and Φ′.

A modal distributive law 7

Venema Co-Oxford 2007

A modal distributive law

Theorem For any sets Φ, Φ′ of formulas, ∇Φ ∧ ∇Φ′ ≡

• Z∈Φ⊲

⊳Φ′

∇{ϕ ∧ ϕ′ | (ϕ, ϕ′) ∈ Z}, where Φ ⊲ ⊳ Φ′ is the set of full relations between Φ and Φ′. Proof of ‘⇒’: Suppose S, s ∇Φ ∧ ∇Φ′.

A modal distributive law 7

Venema Co-Oxford 2007

A modal distributive law

Theorem For any sets Φ, Φ′ of formulas, ∇Φ ∧ ∇Φ′ ≡

• Z∈Φ⊲

⊳Φ′

∇{ϕ ∧ ϕ′ | (ϕ, ϕ′) ∈ Z}, where Φ ⊲ ⊳ Φ′ is the set of full relations between Φ and Φ′. Proof of ‘⇒’: Suppose S, s ∇Φ ∧ ∇Φ′. Define Z ⊆ Φ × Φ′ as Z := {(ϕ, ϕ′) | S, t ϕ ∧ ϕ′ for some t ∈ R[s]}.

A modal distributive law 7

Venema Co-Oxford 2007

A modal distributive law

Theorem For any sets Φ, Φ′ of formulas, ∇Φ ∧ ∇Φ′ ≡

• Z∈Φ⊲

⊳Φ′

∇{ϕ ∧ ϕ′ | (ϕ, ϕ′) ∈ Z}, where Φ ⊲ ⊳ Φ′ is the set of full relations between Φ and Φ′. Proof of ‘⇒’: Suppose S, s ∇Φ ∧ ∇Φ′. Define Z ⊆ Φ × Φ′ as Z := {(ϕ, ϕ′) | S, t ϕ ∧ ϕ′ for some t ∈ R[s]}. Claim 1: Z is full on Φ and Φ′.

A modal distributive law 7

Venema Co-Oxford 2007

A modal distributive law

Theorem For any sets Φ, Φ′ of formulas, ∇Φ ∧ ∇Φ′ ≡

• Z∈Φ⊲

⊳Φ′

∇{ϕ ∧ ϕ′ | (ϕ, ϕ′) ∈ Z}, where Φ ⊲ ⊳ Φ′ is the set of full relations between Φ and Φ′. Proof of ‘⇒’: Suppose S, s ∇Φ ∧ ∇Φ′. Define Z ⊆ Φ × Φ′ as Z := {(ϕ, ϕ′) | S, t ϕ ∧ ϕ′ for some t ∈ R[s]}. Claim 1: Z is full on Φ and Φ′. Claim 2: S, s ∇{ϕ ∧ ϕ′ | (ϕ, ϕ′) ∈ Z}.

A modal distributive law 7

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Reorganizing Modal Logic: proposition part

◮ Fix (finite) set X of proposition letters

A modal distributive law 8

Venema Co-Oxford 2007

Reorganizing Modal Logic: proposition part

◮ Fix (finite) set X of proposition letters ◮ Define local description connective ⊙: given set P ⊆ X, put

⊙P :=

• p∈P

p ∧

• p∈X\P

¬p

A modal distributive law 8

Venema Co-Oxford 2007

Reorganizing Modal Logic: proposition part

◮ Fix (finite) set X of proposition letters ◮ Define local description connective ⊙: given set P ⊆ X, put

⊙P :=

• p∈P

p ∧

• p∈X\P

¬p

◮ Conversely, for every q ∈ X, have

q ≡

• q∈P

⊙P

A modal distributive law 8

Venema Co-Oxford 2007

Reorganizing Modal Logic: proposition part

◮ Fix (finite) set X of proposition letters ◮ Define local description connective ⊙: given set P ⊆ X, put

⊙P :=

• p∈P

p ∧

• p∈X\P

¬p

◮ Conversely, for every q ∈ X, have

q ≡

• q∈P

⊙P Proposition ML is effectively equi-expressive with the language given by ϕ ::= ⊙P |

• Φ |
• Φ | ∇Φ

A modal distributive law 8

Venema Co-Oxford 2007

Coalgebraic Modal Logic

Define distributed conjunction •: P • Φ := ⊙P ∧ ∇Φ

A modal distributive law 9

Venema Co-Oxford 2007

Coalgebraic Modal Logic

Define distributed conjunction •: P • Φ := ⊙P ∧ ∇Φ Define the language CML by ϕ ::=

• Φ |
• Φ | P • Φ

A modal distributive law 9

Venema Co-Oxford 2007

Coalgebraic Modal Logic

Define distributed conjunction •: P • Φ := ⊙P ∧ ∇Φ Define the language CML by ϕ ::=

• Φ |
• Φ | P • Φ

Conversely, express ⊙P ≡ P • ∅ ∨ P • {⊤} ∇Φ ≡

• P ⊆X

P • Φ

A modal distributive law 9

Venema Co-Oxford 2007

Coalgebraic Modal Logic

Define distributed conjunction •: P • Φ := ⊙P ∧ ∇Φ Define the language CML by ϕ ::=

• Φ |
• Φ | P • Φ

Conversely, express ⊙P ≡ P • ∅ ∨ P • {⊤} ∇Φ ≡

• P ⊆X

P • Φ Proposition The languages ML and CML are effectively equi-expressive.

A modal distributive law 9

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Modal Distributive Normal Forms

A modal distributive law 10

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Modal Distributive Normal Forms

◮ Define the language CML− by

ϕ ::=

• Φ | P • Φ

Theorem The languages ML and CML− are effectively equi-expressive.

A modal distributive law 10

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Modal Distributive Normal Forms

◮ Define the language CML− by

ϕ ::=

• Φ | P • Φ

Theorem The languages ML and CML− are effectively equi-expressive. Proof via modal distributive law for •: (P •Φ)∧(P ′•Φ′) ≡    ∅ (= ⊥) if P = P ′

• Z∈Φ⊲

⊳Φ′ P • {ϕ ∧ ϕ′ | (ϕ, ϕ′) ∈ Z}

if P = P ′

A modal distributive law 10

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Overview

◮ Introduction: reorganizing modal logic ◮ A modal distributive law ◮ A game-theoretical perspective ◮ Uniform interpolation ◮ Automata ◮ Axiomatizing ∇ ◮ A coalgebraic generalization ◮ Concluding remarks

Overview 11

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Game semantics for ML

Position Player Legitimate moves (ϕ1 ∨ ϕ2, s) ∃ {(ϕ1, s), (ϕ2, s)} (ϕ1 ∧ ϕ2, s) ∀ {(ϕ1, s), (ϕ2, s)} (✸ϕ, s) ∃ {(ϕ, t) | t ∈ R[s]} (✷ϕ, s) ∀ {(ϕ, 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) ∃ ∅

A game-theoretical perspective 12

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Game semantics for ML∇

Position Player Legitimate moves (ϕ1 ∨ ϕ2, s) ∃ {(ϕ1, s), (ϕ2, s)} (ϕ1 ∧ ϕ2, s) ∀ {(ϕ1, s), (ϕ2, s)} (∇Φ, s) ∃ {Z ⊆ S × Fmas | Z ∈ Φ ⊲ ⊳ R[s]} Z⊆ S × Fmas ∀ {(s, ϕ) | (s, ϕ) ∈ Z} (⊥, s) ∃ ∅ (⊤, s) ∀ ∅ (p, s), s ∈ V (p) ∀ ∅ (p, s), s ∈ V (p) ∃ ∅ (¬p, s), s ∈ V (p) ∀ ∅ (¬p, s), s ∈ V (p) ∃ ∅

A game-theoretical perspective 13

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Strategic normal forms

◮ ‘static’ distributive law:

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

A game-theoretical perspective 14

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Strategic normal forms

◮ ‘static’ distributive law:

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

◮ modal distributive law:

∇Φ ∧ ∇Φ′ ≡

• Z∈Φ⊲

⊳Φ′

∇{ϕ ∧ ϕ′ | (ϕ, ϕ′) ∈ Z} ∀∃∀ ∃∃∀∀

A game-theoretical perspective 14

Venema Co-Oxford 2007

Overview

◮ Introduction: reorganizing modal logic ◮ A modal distributive law ◮ A game-theoretical perspective ◮ Uniform interpolation ◮ Automata ◮ Axiomatizing ∇ ◮ A coalgebraic generalization ◮ Concluding remarks

Overview 15

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Bisimulation Quantifiers

Uniform interpolation 16

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Bisimulation Quantifiers

◮ Fix set X of proposition letters ◮ Syntax: if ϕ is a formula, then so is ˜

∃p.ϕ

◮ Semantics:

S, s ∃p.ϕ iff S′, s′ ϕ for some S′, s′ ↔p S, s, where ↔p denotes bisimilarity wrt X \ {p}-formulas.

Uniform interpolation 16

Venema Co-Oxford 2007

Bisimulation Quantifiers

◮ Fix set X of proposition letters ◮ Syntax: if ϕ is a formula, then so is ˜

∃p.ϕ

◮ Semantics:

S, s ∃p.ϕ iff S′, s′ ϕ for some S′, s′ ↔p S, s, where ↔p denotes bisimilarity wrt X \ {p}-formulas.

◮ Example: ˜

∃p(✸p ∧ ✸¬p) ≡ ✸⊤.

Uniform interpolation 16

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Bisimulation Quantifiers & Uniform interpolation

Proposition Let ϕ, ψ be modal formulas, p not occurring in ψ. Then

• ϕ |

= ˜ ∃p.ϕ

• ϕ |

= ψ iff ˜ ∃p.ϕ | = ψ

Uniform interpolation 17

Venema Co-Oxford 2007

Bisimulation Quantifiers & Uniform interpolation

Proposition Let ϕ, ψ be modal formulas, p not occurring in ψ. Then

• ϕ |

= ˜ ∃p.ϕ

• ϕ |

= ψ iff ˜ ∃p.ϕ | = ψ Corollary (‘Uniform Interpolation’) Let ϕ, χ be formulas with ϕ | = ψ. Assume Var(ϕ) \ Var(ψ) = {p1, . . . , pn}. Then ϕ | = ˜ ∃p1 · · · pn.ϕ | = ψ.

Uniform interpolation 17

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Uniform interpolation of ML

Theorem Modal logic has uniform interpolation. Proof sketch

Uniform interpolation 18

Venema Co-Oxford 2007

Uniform interpolation of ML

Theorem Modal logic has uniform interpolation. Proof sketch ˜ ∃ is definable in CML−, and hence in ML:

Uniform interpolation 18

Venema Co-Oxford 2007

Uniform interpolation of ML

Theorem Modal logic has uniform interpolation. Proof sketch ˜ ∃ is definable in CML−, and hence in ML:

• ˜

∃p(ϕ ∨ ψ) ≡ ˜ ∃p.ϕ ∨ ˜ ∃p.ψ

Uniform interpolation 18

Venema Co-Oxford 2007

Uniform interpolation of ML

Theorem Modal logic has uniform interpolation. Proof sketch ˜ ∃ is definable in CML−, and hence in ML:

• ˜

∃p(ϕ ∨ ψ) ≡ ˜ ∃p.ϕ ∨ ˜ ∃p.ψ

• ˜

∃p.∇Φ ≡ ∇˜ ∃p.Φ

Uniform interpolation 18

Venema Co-Oxford 2007

Uniform interpolation of ML

Theorem Modal logic has uniform interpolation. Proof sketch ˜ ∃ is definable in CML−, and hence in ML:

• ˜

∃p(ϕ ∨ ψ) ≡ ˜ ∃p.ϕ ∨ ˜ ∃p.ψ

• ˜

∃p.∇Φ ≡ ∇˜ ∃p.Φ

• ˜

∃p.⊙P ≡ ⊙(P \ {p}) ∨ ⊙(P ∪ {p})

Uniform interpolation 18

Venema Co-Oxford 2007

Uniform interpolation of ML

Theorem Modal logic has uniform interpolation. Proof sketch ˜ ∃ is definable in CML−, and hence in ML:

• ˜

∃p(ϕ ∨ ψ) ≡ ˜ ∃p.ϕ ∨ ˜ ∃p.ψ

• ˜

∃p.∇Φ ≡ ∇˜ ∃p.Φ

• ˜

∃p.⊙P ≡ ⊙(P \ {p}) ∨ ⊙(P ∪ {p})

• ˜

∃p.(P • Φ) ≡ P • ˜ ∃p.Φ ∨ (P ∪ {p}) • ˜ ∃p.Φ

Uniform interpolation 18

Venema Co-Oxford 2007

Overview

◮ Introduction: reorganizing modal logic ◮ A modal distributive law ◮ A game-theoretical perspective ◮ Uniform interpolation ◮ Automata ◮ Axiomatizing ∇ ◮ A coalgebraic generalization ◮ Concluding remarks

Overview 19

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Automata Theory

◮ automata: finite devices classifying potentially infinite objects ◮ strong connections with (fixpoint/second order) logic

Slogan: formulas are automata

◮ rich history: B¨

uchi, Rabin, Walukiewicz, . . .

◮ applications in model checking

Coalgebra Automata 20

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Automata Theory

◮ automata: finite devices classifying potentially infinite objects ◮ strong connections with (fixpoint/second order) logic

Slogan: formulas are automata

◮ rich history: B¨

uchi, Rabin, Walukiewicz, . . .

◮ applications in model checking

Automata can be classified according to

Coalgebra Automata 20

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Automata Theory

◮ automata: finite devices classifying potentially infinite objects ◮ strong connections with (fixpoint/second order) logic

Slogan: formulas are automata

◮ rich history: B¨

uchi, Rabin, Walukiewicz, . . .

◮ applications in model checking

Automata can be classified according to

◮ objects on which they operate (words/trees/graphs, . . . ) ◮ transition structure: deterministic/nondeterministic/alternating ◮ acceptance condition: B¨

uchi/Muller/parity/. . .

Coalgebra Automata 20

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A Fundamental Result

◮ Key result in Rabin’s decidability proof for SnS:

• not the Complementation Lemma, but . . .

Coalgebra Automata 21

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A Fundamental Result

◮ Key result in Rabin’s decidability proof for SnS:

• not the Complementation Lemma, but . . .
• the simulation of alternating tree automata by nondeterministic ones

Coalgebra Automata 21

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A Fundamental Result

◮ Key result in Rabin’s decidability proof for SnS:

• not the Complementation Lemma, but . . .
• the simulation of alternating tree automata by nondeterministic ones

◮ Logically, this corresponds to the elimination of conjunctions

Coalgebra Automata 21

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A Fundamental Result

◮ Key result in Rabin’s decidability proof for SnS:

• not the Complementation Lemma, but . . .
• the simulation of alternating tree automata by nondeterministic ones

◮ Logically, this corresponds to the elimination of conjunctions

For the modal µ-calculus,

Coalgebra Automata 21

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A Fundamental Result

◮ Key result in Rabin’s decidability proof for SnS:

• not the Complementation Lemma, but . . .
• the simulation of alternating tree automata by nondeterministic ones

◮ Logically, this corresponds to the elimination of conjunctions

For the modal µ-calculus,

◮ Janin & Walukiewicz introduced modal µ-automata . . .

Coalgebra Automata 21

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A Fundamental Result

◮ Key result in Rabin’s decidability proof for SnS:

• not the Complementation Lemma, but . . .
• the simulation of alternating tree automata by nondeterministic ones

◮ Logically, this corresponds to the elimination of conjunctions

For the modal µ-calculus,

◮ Janin & Walukiewicz introduced modal µ-automata . . . ◮ . . . and proved a corresponding simulation result . . .

Coalgebra Automata 21

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A Fundamental Result

◮ Key result in Rabin’s decidability proof for SnS:

• not the Complementation Lemma, but . . .
• the simulation of alternating tree automata by nondeterministic ones

◮ Logically, this corresponds to the elimination of conjunctions

For the modal µ-calculus,

◮ Janin & Walukiewicz introduced modal µ-automata . . . ◮ . . . and proved a corresponding simulation result . . . ◮ . . . which lies as the heart of all results on the modal µ-calculus.

Coalgebra Automata 21

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Automata & Fixpoint Logics

Theorem (Arnold & Niwi´ nski)

Automata 22

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Automata & Fixpoint Logics

Theorem (Arnold & Niwi´ nski) Elimination of conjunction is preserved under adding fixpoint operators!

Automata 22

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Automata & Fixpoint Logics

Theorem (Arnold & Niwi´ nski) Elimination of conjunction is preserved under adding fixpoint operators! Hence, by the modal distributive law, conjunctions can be eliminated from the modal µ-calculus. Corollary (Janin & Walukiewicz) µML and µCML− (based on , •) are effectively equi-expressive.

Automata 22

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Axiomatizing Fixpoint Logics

(joint work with Luigi Santocanale)

◮ A connective ♯(p1, . . . , pn) is a flat fixpoint connective if its semantics is

given by the least fixpoint of a modal formula γ(x, p1, . . . , pn): ♯(p1, . . . , pn) ≡ µx.γ(x, p1, . . . , pn)

◮ Examples: ∗p ≡ µx.p ∨ ✸x, pUq ≡ µx.q ∨ (p ∧ ✸x).

Automata 23

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Axiomatizing Fixpoint Logics

(joint work with Luigi Santocanale)

◮ A connective ♯(p1, . . . , pn) is a flat fixpoint connective if its semantics is

given by the least fixpoint of a modal formula γ(x, p1, . . . , pn): ♯(p1, . . . , pn) ≡ µx.γ(x, p1, . . . , pn)

◮ Examples: ∗p ≡ µx.p ∨ ✸x, pUq ≡ µx.q ∨ (p ∧ ✸x). ◮ Given set Γ of modal formulas, MLΓ is extension of ML with {♯γ | γ ∈ Γ}. ◮ Example: CTL.

Automata 23

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Axiomatizing Fixpoint Logics

(joint work with Luigi Santocanale)

◮ A connective ♯(p1, . . . , pn) is a flat fixpoint connective if its semantics is

given by the least fixpoint of a modal formula γ(x, p1, . . . , pn): ♯(p1, . . . , pn) ≡ µx.γ(x, p1, . . . , pn)

◮ Examples: ∗p ≡ µx.p ∨ ✸x, pUq ≡ µx.q ∨ (p ∧ ✸x). ◮ Given set Γ of modal formulas, MLΓ is extension of ML with {♯γ | γ ∈ Γ}. ◮ Example: CTL.

Theorem Sound and complete axiom systems for MLΓ, uniform and effective in Γ.

Automata 23

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Overview

◮ Introduction: reorganizing modal logic ◮ A modal distributive law ◮ A game-theoretical perspective ◮ Uniform interpolation ◮ Automata ◮ Axiomatizing ∇ ◮ A coalgebraic generalization ◮ Concluding remarks

Overview 24

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Axiomatizing ∇

(joint work with Alessandra Palmigiano)

Axiomatizing ∇ 25

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Axiomatizing ∇

(joint work with Alessandra Palmigiano)

◮ (Equi-expressiveness with ML trivially provides axiomatization)

Axiomatizing ∇ 25

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Axiomatizing ∇

(joint work with Alessandra Palmigiano)

◮ (Equi-expressiveness with ML trivially provides axiomatization) ◮ Aim: Axiomatize ∇ ‘in its own terms’

Axiomatizing ∇ 25

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Axiomatizing ∇

(joint work with Alessandra Palmigiano)

◮ (Equi-expressiveness with ML trivially provides axiomatization) ◮ Aim: Axiomatize ∇ ‘in its own terms’ ◮ Observation: axiomatization of ∇ is independent to that of negation ◮ Change setting to positive modal logic: (= ¬-free residu of classical ML)

Axiomatizing ∇ 25

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Axiomatizing ∇

(joint work with Alessandra Palmigiano)

◮ (Equi-expressiveness with ML trivially provides axiomatization) ◮ Aim: Axiomatize ∇ ‘in its own terms’ ◮ Observation: axiomatization of ∇ is independent to that of negation ◮ Change setting to positive modal logic: (= ¬-free residu of classical ML) ◮ Our approach is algebraic.

Axiomatizing ∇ 25

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Algebraic approach

◮ Positive modal algebra: structure A = A, ∧, ∨, ⊤, ⊥, ✸, ✷ with

• A := A, ∧, ∨, ⊤, ⊥ a distributive lattice, and
• ✷, ✸ unary operations on A satisfying:

✸(a ∨ b) = ✸a ∨ ✸b ✸⊥ = ⊥ ✷(a ∧ b) = ✷a ∧ ✷b ✷⊤ = ⊤ ✷a ∧ ✸b ≤ ✸(a ∧ b) ✷(a ∨ b) ≤ ✷a ∨ ✸b

◮ Modal algebra: A = A, ∧, ∨, ⊤, ⊥, ¬, ✸, ✷ with

• A, ∧, ∨, ⊤, ⊥, ¬ a Boolean algebra
• ✷ and ✸ satisfy, in addition to the axioms above:

¬✸a = ✷¬a.

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Axioms for ∇

Positive modal ∇-algebra: A = A, ∧, ∨, ⊤, ⊥, ∇ with

◮ A, ∧, ∨, ⊤, ⊥ a distributive lattice, and ∇ satisfying ◮ ∇1. If ≤ is full on α and β, then ∇α ≤ ∇β,

∇2a. ∇α ∧ ∇β ≤ {∇{a ∧ b | (a, b) ∈ Z} | Z ∈ α ⊲ ⊳ β}, ∇2b. ⊤ ≤ ∇∅ ∨ ∇{⊤}, ∇3a. If ⊥ ∈ α, then ∇α ≤ ⊥, ∇3b. ∇α ∪ {a ∨ b} ≤ ∇(α ∪ {a}) ∨ ∇(α ∪ {b}) ∨ ∇(α ∪ {a, b}). Modal ∇-algebra: A = A, ∧, ∨, ⊤, ⊥, ¬, ∇ with

◮ A, ∧, ∨, ⊤, ⊥, ¬ a Boolean algebra, and ∇ satisfying ∇1 – ∇3 and: ◮ ∇4. ¬∇α = ∇{ ¬α, ⊤} ∨ ∇∅ ∨ {∇{¬a} | a ∈ α}.

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Results

◮ Given a PMA A = A, ∧, ∨, ⊤, ⊥, ✸, ✷, define ∇α := ✷ α ∧ ✸α,

and put A∇ := A, ∧, ∨, ⊤, ⊥, ∇.

◮ Conversely, given a PMA∇ B, ∧, ∨, ⊤, ⊥, ∇), define ✸a := ∇{a, ⊤}

and ✷a := ∇∅ ∨ ∇{a}, and put B✸ := B, ∧, ∨, ⊤, ⊥, ✸, ✷.

◮ Extend to maps: f ∇ := f and f ✸ := f whenever applicable.

Theorem The functors (·)∇ and (·)✸

• give a categorical isomorphism between the categories PMA and PMA∇,
• and similarly for the categories MA and MA∇.

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Results

◮ Given a PMA A = A, ∧, ∨, ⊤, ⊥, ✸, ✷, define ∇α := ✷ α ∧ ✸α,

and put A∇ := A, ∧, ∨, ⊤, ⊥, ∇.

◮ Conversely, given a PMA∇ B, ∧, ∨, ⊤, ⊥, ∇), define ✸a := ∇{a, ⊤}

and ✷a := ∇∅ ∨ ∇{a}, and put B✸ := B, ∧, ∨, ⊤, ⊥, ✸, ✷.

◮ Extend to maps: f ∇ := f and f ✸ := f whenever applicable.

Theorem The functors (·)∇ and (·)✸

• give a categorical isomorphism between the categories PMA and PMA∇,
• and similarly for the categories MA and MA∇.

Corollary ∇1 – ∇4 form a complete axiomatization of ∇.

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Results

◮ Given a PMA A = A, ∧, ∨, ⊤, ⊥, ✸, ✷, define ∇α := ✷ α ∧ ✸α,

and put A∇ := A, ∧, ∨, ⊤, ⊥, ∇.

◮ Conversely, given a PMA∇ B, ∧, ∨, ⊤, ⊥, ∇), define ✸a := ∇{a, ⊤}

and ✷a := ∇∅ ∨ ∇{a}, and put B✸ := B, ∧, ∨, ⊤, ⊥, ✸, ✷.

◮ Extend to maps: f ∇ := f and f ✸ := f whenever applicable.

Theorem The functors (·)∇ and (·)✸

• give a categorical isomorphism between the categories PMA and PMA∇,
• and similarly for the categories MA and MA∇.

Corollary ∇1 – ∇4 form a complete axiomatization of ∇. Corollary Description of topological Vietoris construction in terms of ∇.

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Carioca Axioms for ∇

(joint work with Marta Bilkova & Alessandra Palmigiano)

A set B ∈ ℘℘(S) is a full redistribution of a set A ∈ ℘℘(S) if

• B = A
• β ∩ α = ∅ for all β ∈ B and all α ∈ A

The set of redistributions of A is denoted as FRDB(A). ∇-Axioms: If ≤ is full on α and β, then ∇α ≤ ∇β. (∇1) ∇α | α ∈ A

• ≤

∇{β | β ∈ B} | B ∈ FRDB(A)

• (∇2)

∇{α | α ∈ A} ≤

• {∇β | ∈ is full on β and A}.

(∇3)

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Overview

◮ Introduction: reorganizing modal logic ◮ A modal distributive law ◮ A game-theoretical perspective ◮ Uniform interpolation ◮ Automata ◮ Axiomatizing ∇ ◮ A coalgebraic generalization ◮ Concluding remarks

Overview 30

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Almost all of this has been generalized to the level of coalgebras (for weak pullback-preserving set functors)

Coalgebra 31

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Almost all of this has been generalized to the level of coalgebras (for weak pullback-preserving set functors)

(partly joint work with Clemens Kupke & Alexander Kurz)

Coalgebra 31

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Kripke Structures as Coalgebras

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Kripke Structures as Coalgebras

◮ Represent R ⊆ S × S as map σR : S → ℘(S):

σR(s) := {t ∈ S | Rst}.

◮ Kripke frame S, R ∼ coalgebra S, σR

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Kripke Structures as Coalgebras

◮ Represent R ⊆ S × S as map σR : S → ℘(S):

σR(s) := {t ∈ S | Rst}.

◮ Kripke frame S, R ∼ coalgebra S, σR ◮ Kripke model = Kripke frame + assignment (valuation) ◮ A valuation is a map V : X → ℘(S)

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Kripke Structures as Coalgebras

◮ Represent R ⊆ S × S as map σR : S → ℘(S):

σR(s) := {t ∈ S | Rst}.

◮ Kripke frame S, R ∼ coalgebra S, σR ◮ Kripke model = Kripke frame + assignment (valuation) ◮ A valuation is a map V : X → ℘(S) ◮ represent this as a map σV : S → ℘(X):

σV (s) := {p ∈ X | s ∈ V (p)}.

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Kripke Structures as Coalgebras

◮ Represent R ⊆ S × S as map σR : S → ℘(S):

σR(s) := {t ∈ S | Rst}.

◮ Kripke frame S, R ∼ coalgebra S, σR ◮ Kripke model = Kripke frame + assignment (valuation) ◮ A valuation is a map V : X → ℘(S) ◮ represent this as a map σV : S → ℘(X):

σV (s) := {p ∈ X | s ∈ V (p)}.

◮ Combine σV and σR into map σV,R : S → ℘(X) × ℘(S): ◮ Kripke model S, R, V ∼ coalgebra S, σV,R

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Coalgebra

◮ Coalgebra is

a general mathematical theory for evolving state-based systems

Coalgebra 33

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Coalgebra

◮ Coalgebra is

a general mathematical theory for evolving state-based systems

◮ It provides a natural framework for notions like

• behavior

Coalgebra 33

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Coalgebra

◮ Coalgebra is

a general mathematical theory for evolving state-based systems

◮ It provides a natural framework for notions like

• behavior
• bisimulation/behavioral equivalence

Coalgebra 33

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Coalgebra

◮ Coalgebra is

a general mathematical theory for evolving state-based systems

◮ It provides a natural framework for notions like

• behavior
• bisimulation/behavioral equivalence
• invariants

Coalgebra 33

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Coalgebra

◮ Coalgebra is

a general mathematical theory for evolving state-based systems

◮ It provides a natural framework for notions like

• behavior
• bisimulation/behavioral equivalence
• invariants

◮ A coalgebra is a structure S = S, σ : S → FS,

where F is the type of the coalgebra.

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Coalgebra

◮ Coalgebra is

a general mathematical theory for evolving state-based systems

◮ It provides a natural framework for notions like

• behavior
• bisimulation/behavioral equivalence
• invariants

◮ A coalgebra is a structure S = S, σ : S → FS,

where F is the type of the coalgebra.

◮ Sufficiently general to model notions like:

input, output, non-determinism, interaction, probability, . . .

Coalgebra 33

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Coalgebra

◮ Coalgebra is

a general mathematical theory for evolving state-based systems

◮ It provides a natural framework for notions like

• behavior
• bisimulation/behavioral equivalence
• invariants

◮ A coalgebra is a structure S = S, σ : S → FS,

where F is the type of the coalgebra.

◮ Sufficiently general to model notions like:

input, output, non-determinism, interaction, probability, . . .

◮ Type of Kripke models is KX, with KXS = ℘(X) × ℘(S)

Type of Kripke frames is K, with KS = ℘(S)

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Examples

◮ C-streams: FS = C × S ◮ finite words: FS = C × (S ⊎ {↓}) ◮ finite trees: FS = C × ((S × S) ⊎ {↓}) ◮ deterministic automata: FS = {0, 1} × SC ◮ labeled transition systems: FS = (℘S)A ◮ (non-wellfounded) sets: FS = ℘S ◮ topologies: FS = ℘℘(S)

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Coalgebra and Modal Logic

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Coalgebra and Modal Logic

◮ Coalgebras are a natural generalization of Kripke structures

Coalgebra 35

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Coalgebra and Modal Logic

◮ Coalgebras are a natural generalization of Kripke structures ◮

Modal Logic∗ Coalgebra = Equational Logic Algebra

Coalgebra 35

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Coalgebra and Modal Logic

◮ Coalgebras are a natural generalization of Kripke structures ◮

Modal Logic∗ Coalgebra = Equational Logic Algebra

* with fixpoint operators

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Relation Lifting

◮ KS := ℘(S) ◮ Kripke frame is pair S, σ : S → KS ◮ Lift Z ⊆ S × S′ to K(Z) ⊆ KS × KS′:

K(Z) := {(T, T ′) | ∀t ∈ T∃t′ ∈ T ′.Ztt′ and ∀t′ ∈ T ′∃t ∈ T.Ztt′}

◮ Z is full on T and T ′ iff (T, T ′) ∈ K(Z).

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Relation Lifting

◮ KS := ℘(S) ◮ Kripke frame is pair S, σ : S → KS ◮ Lift Z ⊆ S × S′ to K(Z) ⊆ KS × KS′:

K(Z) := {(T, T ′) | ∀t ∈ T∃t′ ∈ T ′.Ztt′ and ∀t′ ∈ T ′∃t ∈ T.Ztt′}

◮ Z is full on T and T ′ iff (T, T ′) ∈ K(Z).

Proposition

◮ Z is a bisimulation iff (σ(s), σ′(s′)) ∈ K(Z) for all (s, s′) ∈ Z.

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Relation Lifting

◮ KS := ℘(S) ◮ Kripke frame is pair S, σ : S → KS ◮ Lift Z ⊆ S × S′ to K(Z) ⊆ KS × KS′:

K(Z) := {(T, T ′) | ∀t ∈ T∃t′ ∈ T ′.Ztt′ and ∀t′ ∈ T ′∃t ∈ T.Ztt′}

◮ Z is full on T and T ′ iff (T, T ′) ∈ K(Z).

Proposition

◮ Z is a bisimulation iff (σ(s), σ′(s′)) ∈ K(Z) for all (s, s′) ∈ Z. ◮ S, s ∇Φ iff (σ(s), Φ) ∈ K().

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Moss’ Coalgebraic Logic

◮ Moss: generalize this to (almost) arbitrary functor

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Moss’ Coalgebraic Logic

◮ Moss: generalize this to (almost) arbitrary functor ◮ Define the language CMLF by

ϕ ::= ⊥ | ⊤ | ϕ ∨ ϕ | ϕ ∧ ϕ | ∇Fα where α ∈ F(Fma)

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Moss’ Coalgebraic Logic

◮ Moss: generalize this to (almost) arbitrary functor ◮ Define the language CMLF by

ϕ ::= ⊥ | ⊤ | ϕ ∨ ϕ | ϕ ∧ ϕ | ∇Fα where α ∈ F(Fma)

◮ Semantics: S, s ∇Fα iff (σ(s), α) ∈ F().

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Moss’ Coalgebraic Logic

◮ Moss: generalize this to (almost) arbitrary functor ◮ Define the language CMLF by

ϕ ::= ⊥ | ⊤ | ϕ ∨ ϕ | ϕ ∧ ϕ | ∇Fα where α ∈ F(Fma)

◮ Semantics: S, s ∇Fα iff (σ(s), α) ∈ F(). ◮ The ‘nabla for Kripke models’ is: •!

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The coalgebraic distributive law

◮ Consider : ℘(Fma) → Fma, then F : F℘(Fma) → FFma

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The coalgebraic distributive law

◮ Consider : ℘(Fma) → Fma, then F : F℘(Fma) → FFma ◮ Ξ ∈ F℘S is a redistribution of A ∈ ℘FS if α(F∈S)Ξ, for all α ∈ A.

• {∇Fα | α ∈ A} ≡
• {∇F(F
• )(Ξ) | Ξ a redistribution of A}

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Axioms for ∇F

(joint work with Clemens Kupke & Alexander Kurz)

Axiomatizing ∇ 39

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Axioms for ∇F

(joint work with Clemens Kupke & Alexander Kurz)

◮ Consider , : ℘(Fma) → Fma, then F : F℘(Fma) → FFma ◮ Ξ ∈ F℘S is a redistribution of A ∈ ℘FS if α(F∈S)Ξ, for all α ∈ A.

Axioms: From αF(≤)β derive ∇α ≤ ∇β. (∇1)

• {∇Fα | α ∈ A} =
• {∇F(F
• )(Ξ) | Ξ a redistribution of A}

(∇2) ∇{α | α ∈ A} ≤

• {∇β | βF(∈)A}.

(∇3)

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Axioms for ∇F

(joint work with Clemens Kupke & Alexander Kurz)

◮ Consider , : ℘(Fma) → Fma, then F : F℘(Fma) → FFma ◮ Ξ ∈ F℘S is a redistribution of A ∈ ℘FS if α(F∈S)Ξ, for all α ∈ A.

Axioms: From αF(≤)β derive ∇α ≤ ∇β. (∇1)

• {∇Fα | α ∈ A} =
• {∇F(F
• )(Ξ) | Ξ a redistribution of A}

(∇2) ∇{α | α ∈ A} ≤

• {∇β | βF(∈)A}.

(∇3) Completeness is on its way

Axiomatizing ∇ 39

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Overview

◮ Introduction: reorganizing modal logic ◮ A modal distributive law ◮ A game-theoretical perspective ◮ Uniform interpolation ◮ Automata ◮ Axiomatizing ∇ ◮ A coalgebraic generalization ◮ Concluding remarks

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Concluding remarks

The modal distributive law is a fundamental principle,

Concluding remarks 41

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Concluding remarks

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

Concluding remarks 41

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Concluding remarks

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

◮ logic

Concluding remarks 41

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Concluding remarks

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

◮ logic ◮ game theory

Concluding remarks 41

Venema Co-Oxford 2007

Concluding remarks

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

◮ logic ◮ game theory ◮ automata theory

Concluding remarks 41

Venema Co-Oxford 2007

Concluding remarks

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

◮ logic ◮ game theory ◮ automata theory ◮ coalgebra

Concluding remarks 41

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Concluding remarks

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

◮ logic ◮ game theory ◮ automata theory ◮ coalgebra ◮ . . .

Concluding remarks 41

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Further research

Concluding remarks 42

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Further research

◮ proof theory

Concluding remarks 42

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Further research

◮ proof theory ◮ completeness for fixpoint logics

Concluding remarks 42

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Further research

◮ proof theory ◮ completeness for fixpoint logics ◮ algebraic aspects of ∇

Concluding remarks 42

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Further research

◮ proof theory ◮ completeness for fixpoint logics ◮ algebraic aspects of ∇ ◮ logics for coalgebra

Concluding remarks 42

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Further research

◮ proof theory ◮ completeness for fixpoint logics ◮ algebraic aspects of ∇ ◮ logics for coalgebra ◮ role of negation

Concluding remarks 42

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Further research

◮ proof theory ◮ completeness for fixpoint logics ◮ algebraic aspects of ∇ ◮ logics for coalgebra ◮ role of negation ◮ constructive content

Concluding remarks 42