Venema Co-Oxford 2007 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 • : � ∅ (= ⊥ ) if P � = P ′ ( P • Φ) ∧ ( P ′ • Φ ′ ) ≡ ⊳ Φ ′ P • { ϕ ∧ ϕ ′ | ( ϕ, ϕ ′ ) ∈ Z } if P = P ′ � Z ∈ Φ ⊲ A modal distributive law 10
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 11
Venema Co-Oxford 2007 Game semantics for ML Position Player Legitimate moves ( ϕ 1 ∨ ϕ 2 , s ) ∃ { ( ϕ 1 , s ) , ( ϕ 2 , s ) } ( ϕ 1 ∧ ϕ 2 , s ) ∀ { ( ϕ 1 , s ) , ( ϕ 2 , 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
Venema Co-Oxford 2007 Game semantics for ML ∇ Position Player Legitimate moves ( ϕ 1 ∨ ϕ 2 , s ) ∃ { ( ϕ 1 , s ) , ( ϕ 2 , s ) } ( ϕ 1 ∧ ϕ 2 , s ) ∀ { ( ϕ 1 , s ) , ( ϕ 2 , 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
Venema Co-Oxford 2007 Strategic normal forms ◮ ‘static’ distributive law: ϕ ∧ ( ψ 1 ∨ ψ 2 ) ≡ ( ϕ ∧ ψ 1 ) ∨ ( ϕ ∧ ψ 2 ) ∀∃ ∃∀ A game-theoretical perspective 14
Venema Co-Oxford 2007 Strategic normal forms ◮ ‘static’ distributive law: ϕ ∧ ( ψ 1 ∨ ψ 2 ) ≡ ( ϕ ∧ ψ 1 ) ∨ ( ϕ ∧ ψ 2 ) ∀∃ ∃∀ ◮ modal distributive law: ∇{ ϕ ∧ ϕ ′ | ( ϕ, ϕ ′ ) ∈ Z } � ∇ Φ ∧ ∇ Φ ′ ≡ ⊳ Φ ′ Z ∈ Φ ⊲ ∀∃∀ ∃∃∀∀ 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
Venema Co-Oxford 2007 Bisimulation Quantifiers 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. 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
Venema Co-Oxford 2007 Bisimulation Quantifiers & Uniform interpolation Proposition Let ϕ , ψ be modal formulas, p not occurring in ψ . Then = ˜ • ϕ | ∃ p.ϕ = ψ iff ˜ • ϕ | ∃ 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 = ψ iff ˜ • ϕ | ∃ p.ϕ | = ψ Corollary (‘Uniform Interpolation’) Let ϕ , χ be formulas with ϕ | = ψ. Assume Var ( ϕ ) \ Var ( ψ ) = { p 1 , . . . , p n } . Then = ˜ ϕ | ∃ p 1 · · · p n .ϕ | = ψ. Uniform interpolation 17
Venema Co-Oxford 2007 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
Venema Co-Oxford 2007 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
Venema Co-Oxford 2007 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
Venema Co-Oxford 2007 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
Venema Co-Oxford 2007 A Fundamental Result ◮ Key result in Rabin’s decidability proof for S n S: • not the Complementation Lemma, but . . . Coalgebra Automata 21
Venema Co-Oxford 2007 A Fundamental Result ◮ Key result in Rabin’s decidability proof for S n S: • not the Complementation Lemma, but . . . • the simulation of alternating tree automata by nondeterministic ones Coalgebra Automata 21
Venema Co-Oxford 2007 A Fundamental Result ◮ Key result in Rabin’s decidability proof for S n S: • 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
Venema Co-Oxford 2007 A Fundamental Result ◮ Key result in Rabin’s decidability proof for S n S: • 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
Venema Co-Oxford 2007 A Fundamental Result ◮ Key result in Rabin’s decidability proof for S n S: • 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
Venema Co-Oxford 2007 A Fundamental Result ◮ Key result in Rabin’s decidability proof for S n S: • 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
Venema Co-Oxford 2007 A Fundamental Result ◮ Key result in Rabin’s decidability proof for S n S: • 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
Venema Co-Oxford 2007 Automata & Fixpoint Logics Theorem (Arnold & Niwi´ nski) Automata 22
Venema Co-Oxford 2007 Automata & Fixpoint Logics Theorem (Arnold & Niwi´ nski) Elimination of conjunction is preserved under adding fixpoint operators! Automata 22
Venema Co-Oxford 2007 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) µ ML and µ CML − (based on � , • ) are effectively equi-expressive. Automata 22
Venema Co-Oxford 2007 Axiomatizing Fixpoint Logics (joint work with Luigi Santocanale) ◮ A connective ♯ ( p 1 , . . . , p n ) is a flat fixpoint connective if its semantics is given by the least fixpoint of a modal formula γ ( x, p 1 , . . . , p n ) : ♯ ( p 1 , . . . , p n ) ≡ µx.γ ( x, p 1 , . . . , p n ) ◮ Examples: �∗� p ≡ µx.p ∨ ✸ x , pUq ≡ µx.q ∨ ( p ∧ ✸ x ) . Automata 23
Venema Co-Oxford 2007 Axiomatizing Fixpoint Logics (joint work with Luigi Santocanale) ◮ A connective ♯ ( p 1 , . . . , p n ) is a flat fixpoint connective if its semantics is given by the least fixpoint of a modal formula γ ( x, p 1 , . . . , p n ) : ♯ ( p 1 , . . . , p n ) ≡ µx.γ ( x, p 1 , . . . , p n ) ◮ Examples: �∗� p ≡ µx.p ∨ ✸ x , pUq ≡ µx.q ∨ ( p ∧ ✸ x ) . ◮ Given set Γ of modal formulas, ML Γ is extension of ML with { ♯ γ | γ ∈ Γ } . ◮ Example: CTL. Automata 23
Venema Co-Oxford 2007 Axiomatizing Fixpoint Logics (joint work with Luigi Santocanale) ◮ A connective ♯ ( p 1 , . . . , p n ) is a flat fixpoint connective if its semantics is given by the least fixpoint of a modal formula γ ( x, p 1 , . . . , p n ) : ♯ ( p 1 , . . . , p n ) ≡ µx.γ ( x, p 1 , . . . , p n ) ◮ Examples: �∗� p ≡ µx.p ∨ ✸ x , pUq ≡ µx.q ∨ ( p ∧ ✸ x ) . ◮ Given set Γ of modal formulas, ML Γ is extension of ML with { ♯ γ | γ ∈ Γ } . ◮ Example: CTL. Theorem Sound and complete axiom systems for ML Γ , uniform and effective in Γ . Automata 23
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 24
Venema Co-Oxford 2007 Axiomatizing ∇ (joint work with Alessandra Palmigiano) Axiomatizing ∇ 25
Venema Co-Oxford 2007 Axiomatizing ∇ (joint work with Alessandra Palmigiano) ◮ (Equi-expressiveness with ML trivially provides axiomatization) Axiomatizing ∇ 25
Venema Co-Oxford 2007 Axiomatizing ∇ (joint work with Alessandra Palmigiano) ◮ (Equi-expressiveness with ML trivially provides axiomatization) ◮ Aim: Axiomatize ∇ ‘in its own terms’ Axiomatizing ∇ 25
Venema Co-Oxford 2007 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
Venema Co-Oxford 2007 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
Venema Co-Oxford 2007 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 . Axiomatizing ∇ 26
Venema Co-Oxford 2007 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 , ∇ 2 a. ∇ α ∧ ∇ β ≤ � {∇{ a ∧ b | ( a, b ) ∈ Z } | Z ∈ α ⊲ ⊳ β } , ∇ 2 b. ⊤ ≤ ∇ ∅ ∨ ∇{⊤} , ∇ 3 a. If ⊥ ∈ α , then ∇ α ≤ ⊥ , ∇ 3 b. ∇ 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 } . Axiomatizing ∇ 27
Venema Co-Oxford 2007 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 ∇ . Axiomatizing ∇ 28
Venema Co-Oxford 2007 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 ∇ . Axiomatizing ∇ 28
Venema Co-Oxford 2007 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 ∇ . Axiomatizing ∇ 28
Venema Co-Oxford 2007 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 ) � � � � � � ∇{ � β | β ∈ B } | B ∈ FRDB ( A ) ∇ α | α ∈ A ≤ ( ∇ 2 ) � ∇{ � α | α ∈ A } ≤ {∇ β | ∈ is full on β and A } . ( ∇ 3 ) Axiomatizing ∇ 29
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 30
Venema Co-Oxford 2007 Almost all of this has been generalized to the level of coalgebras (for weak pullback-preserving set functors) Coalgebra 31
Venema Co-Oxford 2007 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
Venema Co-Oxford 2007 Kripke Structures as Coalgebras Coalgebra 32
Venema Co-Oxford 2007 Kripke Structures as Coalgebras ◮ Represent R ⊆ S × S as map σ R : S → ℘ ( S ) : σ R ( s ) := { t ∈ S | Rst } . ◮ Kripke frame � S, R � ∼ coalgebra � S, σ R � Coalgebra 32
Venema Co-Oxford 2007 Kripke Structures as Coalgebras ◮ Represent R ⊆ S × S as map σ R : S → ℘ ( S ) : σ R ( s ) := { t ∈ S | Rst } . ◮ Kripke frame � S, R � ∼ coalgebra � S, σ R � ◮ Kripke model = Kripke frame + assignment (valuation) ◮ A valuation is a map V : X → ℘ ( S ) Coalgebra 32
Venema Co-Oxford 2007 Kripke Structures as Coalgebras ◮ Represent R ⊆ S × S as map σ R : S → ℘ ( S ) : σ R ( s ) := { t ∈ S | Rst } . ◮ Kripke frame � S, R � ∼ coalgebra � S, σ R � ◮ Kripke model = Kripke frame + assignment (valuation) ◮ A valuation is a map V : X → ℘ ( S ) ◮ represent this as a map σ V : S → ℘ ( X ) : σ V ( s ) := { p ∈ X | s ∈ V ( p ) } . Coalgebra 32
Venema Co-Oxford 2007 Kripke Structures as Coalgebras ◮ Represent R ⊆ S × S as map σ R : S → ℘ ( S ) : σ R ( s ) := { t ∈ S | Rst } . ◮ Kripke frame � S, R � ∼ coalgebra � S, σ R � ◮ Kripke model = Kripke frame + assignment (valuation) ◮ A valuation is a map V : X → ℘ ( S ) ◮ represent this as a map σ V : S → ℘ ( X ) : σ V ( s ) := { p ∈ X | s ∈ V ( p ) } . ◮ Combine σ V and σ R into map σ V,R : S → ℘ ( X ) × ℘ ( S ) : ◮ Kripke model � S, R, V � ∼ coalgebra � S, σ V,R � Coalgebra 32
Venema Co-Oxford 2007 Coalgebra ◮ Coalgebra is a general mathematical theory for evolving state-based systems Coalgebra 33
Venema Co-Oxford 2007 Coalgebra ◮ Coalgebra is a general mathematical theory for evolving state-based systems ◮ It provides a natural framework for notions like • behavior Coalgebra 33
Venema Co-Oxford 2007 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
Venema Co-Oxford 2007 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
Venema Co-Oxford 2007 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 → F S � , where F is the type of the coalgebra. Coalgebra 33
Venema Co-Oxford 2007 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 → F S � , where F is the type of the coalgebra. ◮ Sufficiently general to model notions like: input, output, non-determinism, interaction, probability, . . . Coalgebra 33
Venema Co-Oxford 2007 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 → F S � , 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 K X , with K X S = ℘ ( X ) × ℘ ( S ) Type of Kripke frames is K, with K S = ℘ ( S ) Coalgebra 33
Venema Co-Oxford 2007 Examples ◮ C -streams: F S = C × S ◮ finite words: F S = C × ( S ⊎ {↓} ) ◮ finite trees: F S = C × (( S × S ) ⊎ {↓} ) ◮ deterministic automata: F S = { 0 , 1 } × S C ◮ labeled transition systems: F S = ( ℘S ) A ◮ (non-wellfounded) sets: F S = ℘S ◮ topologies: F S = ℘℘ ( S ) Coalgebra 34
Venema Co-Oxford 2007 Coalgebra and Modal Logic Coalgebra 35
Venema Co-Oxford 2007 Coalgebra and Modal Logic ◮ Coalgebras are a natural generalization of Kripke structures Coalgebra 35
Venema Co-Oxford 2007 Coalgebra and Modal Logic ◮ Coalgebras are a natural generalization of Kripke structures ◮ Modal Logic ∗ = Equational Logic Coalgebra Algebra Coalgebra 35
Venema Co-Oxford 2007 Coalgebra and Modal Logic ◮ Coalgebras are a natural generalization of Kripke structures ◮ Modal Logic ∗ = Equational Logic Coalgebra Algebra * with fixpoint operators Coalgebra 35
Venema Co-Oxford 2007 Relation Lifting ◮ K S := ℘ ( S ) ◮ Kripke frame is pair � S, σ : S → K S � ◮ Lift Z ⊆ S × S ′ to K ( Z ) ⊆ K S × K S ′ : 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 ) . Coalgebra 36
Venema Co-Oxford 2007 Relation Lifting ◮ K S := ℘ ( S ) ◮ Kripke frame is pair � S, σ : S → K S � ◮ Lift Z ⊆ S × S ′ to K ( Z ) ⊆ K S × K S ′ : 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 . Coalgebra 36
Venema Co-Oxford 2007 Relation Lifting ◮ K S := ℘ ( S ) ◮ Kripke frame is pair � S, σ : S → K S � ◮ Lift Z ⊆ S × S ′ to K ( Z ) ⊆ K S × K S ′ : 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 ( � ) . Coalgebra 36
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