Definable Model Classes in Polynomial Coalgebraic Logic Rob - PowerPoint PPT Presentation

Definable Model Classes in Polynomial Coalgebraic Logic Rob Goldblatt Victoria University of Wellington Workshop on Coalgebraic Logic, Oxford, August 2007 Coalgebraic Logic, Oxford ’07 1 / 37

Theme: Lift ideas and results from propositional modal logic to polynomial coalgebraic logic. Issue: how to handle infinite sets of “observables” ? Coalgebraic Logic, Oxford ’07 2 / 37

Theme: Lift ideas and results from propositional modal logic to polynomial coalgebraic logic. Issue: how to handle infinite sets of “observables” ? Coalgebraic Logic, Oxford ’07 2 / 37

References Observational ultraproducts of polynomial coalgebras. Annals of Pure and Applied Logic , 123:235–290, 2003. Enlargements of polynomial coalgebras. In Rod Downey et al., editors, Proceedings of the 7th and 8th Asian Logic Conferences , pages 152–192. World Scientific, 2003. Duality for some categories of coalgebras. Algebra Universalis , 46(3):389–416, 2001. with David Friggens A modal proof theory for final polynomial coalgebras. Theoretical Computer Science , 360:1–22, 2006. see also www.mcs.vuw.ac.nz/˜rob Coalgebraic Logic, Oxford ’07 3 / 37

T -Coalgebras T : Set → Set is a functor on the category Set of sets and functions Definition A T -coalgebra ( A, α ) is given by a function of the form α � TA A A is the state set α is the transition structure. Coalgebraic Logic, Oxford ’07 4 / 37

� � � Morphism of T -Coalgebras f � ( B, β ) ( A, α ) given by a function f for which f A B α β Tf � TB TA β ◦ f = Tf ◦ α Coalgebraic Logic, Oxford ’07 5 / 37

T : Set → Set Polynomial functors constructed from the identity functor Id : A �→ A and/or ¯ constant functors D : A �→ D , by forming products T 1 × T 2 : A �→ T 1 A × T 2 A , coproducts (disjoint unions) T 1 + T 2 : A �→ T 1 A + T 2 A , T D : A �→ ( TA ) D exponential functors with constant exponent D . Definition A α − → TA have polynomial T . Polynomial coalgebras Coalgebraic Logic, Oxford ’07 6 / 37

T : Set → Set Polynomial functors constructed from the identity functor Id : A �→ A and/or ¯ constant functors D : A �→ D , by forming products T 1 × T 2 : A �→ T 1 A × T 2 A , coproducts (disjoint unions) T 1 + T 2 : A �→ T 1 A + T 2 A , T D : A �→ ( TA ) D exponential functors with constant exponent D . Definition A α − → TA have polynomial T . Polynomial coalgebras Coalgebraic Logic, Oxford ’07 6 / 37

Syntax for polynomial T Notation: M : S means M is a term of type S , with S a component functor of T . Definition of Terms for T Variables: v : S any S c : ¯ if c ∈ D (observable elements) Constants: D , State Parameter: s : Id (one only) tr ( M ) : T , if M : Id Transition: ...over Coalgebraic Logic, Oxford ’07 7 / 37

Syntax for polynomial T Notation: M : S means M is a term of type S , with S a component functor of T . Definition of Terms for T Variables: v : S any S c : ¯ if c ∈ D (observable elements) Constants: D , State Parameter: s : Id (one only) tr ( M ) : T , if M : Id Transition: ...over Coalgebraic Logic, Oxford ’07 7 / 37

... continued Products: � M 1 , M 2 � : S 1 × S 2 , if M j : S j π j M : S j , if M : S 1 × S 2 if v : ¯ λvM : S D , Exponentials: D and M : S if M : S D and N : ¯ M ( N ) : S , D Coproducts: ι j M : S 1 + S 2 , if M : S j [ case N of v 1 in M 1 or v 2 in M 2 ] : S if N : S 1 + S 2 , v j : S j , M j : S Coalgebraic Logic, Oxford ’07 8 / 37

Semantics of Terms In a T -coalgebra ( A, α ) , the denotation/ interpretation of a term M : S with free variables v 1 : S 1 ,. . . , v n : S n , is a function [ [ M ] ] α : A × S 1 A × · · · × S n A → SA. Definition ground term: has no free variables [ [ M ] ] α : A → SA. Example A α [ [ tr ( s ) ] ] α is − → TA . Coalgebraic Logic, Oxford ’07 9 / 37

Semantics of Terms In a T -coalgebra ( A, α ) , the denotation/ interpretation of a term M : S with free variables v 1 : S 1 ,. . . , v n : S n , is a function [ [ M ] ] α : A × S 1 A × · · · × S n A → SA. Definition ground term: has no free variables [ [ M ] ] α : A → SA. Example A α [ [ tr ( s ) ] ] α is − → TA . Coalgebraic Logic, Oxford ’07 9 / 37

Semantics of Terms In a T -coalgebra ( A, α ) , the denotation/ interpretation of a term M : S with free variables v 1 : S 1 ,. . . , v n : S n , is a function [ [ M ] ] α : A × S 1 A × · · · × S n A → SA. Definition ground term: has no free variables [ [ M ] ] α : A → SA. Example A α [ [ tr ( s ) ] ] α is − → TA . Coalgebraic Logic, Oxford ’07 9 / 37

Ground observable (GO) term: a ground term of “observable” type ¯ D , some D . Ground equation: M 1 ≈ M 2 with M 1 , M 2 ground terms of same type. Truth-sets of ground equations: � M 1 ≈ M 2 � α is the set { x ∈ A : [ [ M 1 ] ] α ( x ) = [ [ M 2 ] ] α ( x ) } of all states in coalgebra ( A, α ) at which the equation M 1 ≈ M 2 is true. Coalgebraic Logic, Oxford ’07 10 / 37

Ground observable (GO) term: a ground term of “observable” type ¯ D , some D . Ground equation: M 1 ≈ M 2 with M 1 , M 2 ground terms of same type. Truth-sets of ground equations: � M 1 ≈ M 2 � α is the set { x ∈ A : [ [ M 1 ] ] α ( x ) = [ [ M 2 ] ] α ( x ) } of all states in coalgebra ( A, α ) at which the equation M 1 ≈ M 2 is true. Coalgebraic Logic, Oxford ’07 10 / 37

Ground observable (GO) term: a ground term of “observable” type ¯ D , some D . Ground equation: M 1 ≈ M 2 with M 1 , M 2 ground terms of same type. Truth-sets of ground equations: � M 1 ≈ M 2 � α is the set { x ∈ A : [ [ M 1 ] ] α ( x ) = [ [ M 2 ] ] α ( x ) } of all states in coalgebra ( A, α ) at which the equation M 1 ≈ M 2 is true. Coalgebraic Logic, Oxford ’07 10 / 37

Ground formula: built from ground equations by logical connectives ¬ , ∧ . �¬ ϕ � α = A − � ϕ � α � ϕ 1 ∧ ϕ 2 � α = � ϕ 1 � α ∩ � ϕ 2 � α . Truth/satisfaction relation: x ∈ � ϕ � α . α, x | = ϕ means � ϕ � α = A . α | = ϕ means Ground observable (GO) formula: built from equations between GO terms. Coalgebraic Logic, Oxford ’07 11 / 37

Ground formula: built from ground equations by logical connectives ¬ , ∧ . �¬ ϕ � α = A − � ϕ � α � ϕ 1 ∧ ϕ 2 � α = � ϕ 1 � α ∩ � ϕ 2 � α . Truth/satisfaction relation: x ∈ � ϕ � α . α, x | = ϕ means � ϕ � α = A . α | = ϕ means Ground observable (GO) formula: built from equations between GO terms. Coalgebraic Logic, Oxford ’07 11 / 37

Ground formula: built from ground equations by logical connectives ¬ , ∧ . �¬ ϕ � α = A − � ϕ � α � ϕ 1 ∧ ϕ 2 � α = � ϕ 1 � α ∩ � ϕ 2 � α . Truth/satisfaction relation: x ∈ � ϕ � α . α, x | = ϕ means � ϕ � α = A . α | = ϕ means Ground observable (GO) formula: built from equations between GO terms. Coalgebraic Logic, Oxford ’07 11 / 37

GO formulas equations polynomial coalgebras = abstract algebras The GO terms and formulas provide a natural language for specifying properties of polynomial coalgebras. characterizing morphisms in terms of term-value preservation. characterizing the bisimilarity relation of observational indistinguishability of states by satisfaction of the same formulas (Hennessy-Milner property). Coalgebraic Logic, Oxford ’07 12 / 37

Modally Definable Classes of Kripke Frames Theorem Let K be a class of Kripke frames that is closed under ultrapowers. Then K is modally definable iff it is closed under subframes, p-morphic images and disjoint unions; and its complement is closed under ultrafilter extensions (i.e. it reflects ultrafilter extensions). Note: can weaken K is closed under ultrapowers to K is closed under ultrafilter extensions, because . . . Coalgebraic Logic, Oxford ’07 13 / 37

Modally Definable Classes of Kripke Frames Theorem Let K be a class of Kripke frames that is closed under ultrapowers. Then K is modally definable iff it is closed under subframes, p-morphic images and disjoint unions; and its complement is closed under ultrafilter extensions (i.e. it reflects ultrafilter extensions). Note: can weaken K is closed under ultrapowers to K is closed under ultrafilter extensions, because . . . Coalgebraic Logic, Oxford ’07 13 / 37

� �� � � the ultrafilter extension ue F of frame F is a p-morphic image of a suitably saturated ultrapower of F : F I /U � � � � � � � Φ � � � � � � � � ue F F Φ : f U �→ { X ⊆ F : f ∈ U X } Coalgebraic Logic, Oxford ’07 14 / 37

Venema’s analogue for Kripke models Theorem A class of Kripke models is modally definable iff it is closed under images of bisimulation relations and disjoint unions, and is invariant under ultrafilter extensions. Note: here can replace invariance under ultrafilter extensions by invariance under ultrapowers. Coalgebraic Logic, Oxford ’07 15 / 37

Venema’s analogue for Kripke models Theorem A class of Kripke models is modally definable iff it is closed under images of bisimulation relations and disjoint unions, and is invariant under ultrafilter extensions. Note: here can replace invariance under ultrafilter extensions by invariance under ultrapowers. Coalgebraic Logic, Oxford ’07 15 / 37