Coalgebras and Modal Logics: an Overview Dirk Pattinson, Imperial College London CMCS 2010, Paphos, Cyprus
Part I: Examples or: Why should I care? May 26, 2010 1
A Cook’s Tour Through Modal Semantics ~p p C → P ( C ) Kripke Frames. p ~p 2 p C → B ( C ) Multigraph Frames. 4 p B ( X ) = { f : X → N | supp( f ) finite } ~p 0.2 p C → D ( C ) Probabilistic Frames. 0.8 p D ( X ) = { µ : X → [0 , 1] | � x ∈ X µ ( x ) = 1 } May 26, 2010 2
More Examples Neighbourhood Frames. C → PP ( C ) = N ( C ) mapping each world c ∈ C to a set of neighbourhoods Game Frames over a set N of agents � C → { (( S n ) n ∈ N , f ) | f : S n → C } = G ( C ) n associating to each state c ∈ C a strategic game with strategy sets S n and outcome function f Conditional Frames. C → { f : P ( C ) → P ( C ) | f a function } = C ( C ) where every state yields a selection function that assigns properties to conditions May 26, 2010 3
Coalgebras and Modalites: A Non-Definition Coalgebras are about successors . T -coalgebras are pairs ( C, γ ) where γ : C → TC maps states to successors. Write Coalg ( T ) for the collection of T -coalgebras. states = elements c ∈ C properties of states = subsets A ⊆ C successors = elements γ ( c ) ∈ TC properties of successors = subsets ♥ A ⊆ TC Modal Operators are about properties of successors, so � φ 1 � , . . . , � φ n � ⊆ C � ♥ ( φ 1 , . . . , φ n ) � ⊆ TC with the intended interpretation c | = ♥ ( φ 1 , . . . , φ n ) iff γ ( c ) ∈ � ♥ φ 1 , . . . , φ n � . May 26, 2010 4
Part II: Approaches to Syntax and Semantics or: What’s a modal operator? May 26, 2010 5
Moss’ Coalgebraic Logic: The Synthetic Approach Idea. ♥ reflects the action of T on sets: ‘import’ semantics into syntax Concrete Syntax Abstract Syntax: Φ ⊆ f L φ ∈ L Φ ∈ T ω L L ∼ = F ( L ) = P f ( L ) + L + T ω ( L ) � Φ ∈ L ¬ φ ∈ L ∇ Φ ∈ L Modal Semantics Algebraic Semantics F ( P ( C )) F ( L ) c | = ∇ Φ ⇐ ⇒ ( γ ( c ) , Φ) ∈ T ( | =) ˆ γ i P ( C ) L � · � relative to T -coalgebra ( C, γ : C → TC ) where T ω is the finitary part of T May 26, 2010 6
Synthetic Semantics Explained Relation Lifting: from states to successors R TR π 1 π 2 T π 1 T π 2 �→ X Y TX TY Formal Definition. (Assume T preserves weak pullbacks to make things work) TR = { ( Tπ 1 ( w ) , Tπ 2 ( w )) | w ∈ TR } ⊆ TX × TY Modal Semantics. Assume that | = is already given for ‘ingredients’ of α ∈ TL c | = ∇ α ⇐ ⇒ ( γ ( c ) , α ) ∈ T ( | =) for c ∈ C and ( C, γ : C → TC ) ∈ Coalg ( T ) . Thm. [Moss, 1999] L has the Hennessy-Milner Property. May 26, 2010 7
Example: Coalgebraic Logic of Multigraphs Modal Operators for B X = { f : X → N | supp( f ) finite } α : L → N and supp( α ) finite ∇ α ∈ L Satisfaction. c | = ∇ α ⇐ ⇒ ( γ ( c ) , α ) ∈ T ( | =) ⇐ ⇒ the ‘magic square’ x 1 x 2 · · · x k � • m j = γ ( c )( x j ) is multiplicity of x j φ 1 w 1 . . . . • w i = α ( φ i ) is weight of φ i . . φ n w n • x/φ -entry is 0 if x �| = φ Σ m 1 m 2 . . . m n can be filled according to the rules on the right. May 26, 2010 8
Synthetic Semantics, Algebraically Syntax as initial algebra. L ∼ = P f ( L ) + LT ( L ) Semantics as algebra morphism P f ( L ) + L + TL P f ( P ( C )) + P ( C ) + T P ( C ) 1+1+ ρ C P f ( L ) + P ( C ) + P ( TC ) i [ T , ( · ) c ,γ − 1 ] P ( C ) L � · � where ρ C : T P ( C ) → P ( TC ) is ’ lifted membership ’, i.e. ρ C (Φ) = { t ∈ TC | ( t, Φ) ∈ T ( ∈ ) } where ǫ C ⊆ C × P ( C ) is membership (for T = B a ’magic square’ problem) May 26, 2010 9
Logics via Liftings: The Organic Approach Idea. take ♥ what we want it to mean: grow your own modalities T -Structures then define the semantics of modalities: they assign a nbhd frame translation or, equivalently, a predicate lifting � ♥ � : P ( C ) n → P ( TC ) � ♥ � : TC → P ( P ( C ) n ) to every modal operator ♥ of the language, parametric in C . Together with a T -coalgebra ( C, γ ) this gives (in the unary case) a boolean algebra with operator neighbourhood frame � ♥ � PP ( C ) � ♥ � P ( TC ) γ γ − 1 C TC P ( C ) P ( C ) Induced Coalgebraic Semantics � φ � ⊆ C of a modal formula equivalent algebraic viewpoint from a modal perspective c ∈ � ♥ φ � iff � φ � ∈ � ♥ � ◦ γ ( � φ � ) c ∈ � ♥ φ � ⇐ ⇒ γ ( c ) ∈ � ♥ � ( � φ � ) May 26, 2010 10
Example: The Logic of Multigraphs Modal Operators for B X = { µ : X → N | supp( µ ) finite } Our Choice. ♥ ( φ, ψ ) , intended meaning ‘at least 5 times as much φ ’s than ψ ’s’ Associated Lifting. � ♥ � X ( A, B ) = { µ ∈ B X | µ ( A ) ≥ 5 · µ ( B ) } where µ ( A ) = � x ∈ A µ ( x ) Satisfaction. c | = ♥ ( φ, ψ ) ⇐ ⇒ µ ( � φ � ) ≥ 5 · µ ( � ψ � ) where µ = γ ( c ) is the local weighting as seen from point c . (i.e. one can pick and choose the primitives but has to define their meaning) May 26, 2010 11
Part III: Reasoning in Coalgebraic Logics or: What’s a good proof system? May 26, 2010 12
Synthetic Approach: One Proof Calculus for All Recall. Semantics as algebra morphism P f ( L ) + L + TL P f P ( C ) + P ( C ) + T P ( C ) 1+1+ ρ C P f P ( C ) + P ( C ) + P T ( C ) i [ T , ( · ) c ,γ − 1 ] P ( C ) L � · � where ρ C : T P ( C ) → P ( TC ) is ρ C (Φ) = { t ∈ TC | ( t, Φ) ∈ T ( ∈ ) } Slim Redistributions. ’import’ the action of ρ into the proof system. Φ ∈ T P ( X ) redistribution of A ∈ P ( TX ) ⇐ ⇒ A ⊆ ρ X (Φ) Call Φ slim if Φ ∈ P ω T ω ( A ) (i.e. Φ only re-arranges material from A ) Notation. SRD ( A ) = { Φ ∈ T P ( A ) | Φ slim redistribution of A } May 26, 2010 13
Redistributions of Multisets Redistributions of B X = { f : X → N | supp( f ) finite } Φ : P ( X ) → f N ∈ BP X redistribution of A ∈ P ( X → f N ) = P ( B X ) ⇐ ⇒ A only contains f : X → f N that allow to fill the ’magic square’ · · · x 1 x 2 x k � • x/S -entry is 0 if x �∈ S S 1 w 1 . . . . • m j is f -multiplicity of x j . . S n w n • w i is Φ -weight of S i Σ m 1 m 2 . . . m n Φ is slim if each nozero S i only contains nonzero x j s relative to some element of A May 26, 2010 14
The Synthetic Proof System Synthetic Proofs. • judegements are inequalities a ≤ b for a, b ∈ L • propositional logic and cut: from a ≤ b and b ≤ c infer a ≤ c Modal Proof Rules. ⊤ ≤ � φ { a ∧ ∇ α ′ ≤ ⊥ | α ′ ∈ T ω ( φ ) \ { α }} α ≤ β ( ∇ 1) ( ∇ 4) ∇ α ≤ ∇ β a ≤ ∇ α {∇ ( T � )(Φ) ≤ a | Φ ∈ SRD ( A ) } {∇ α ≤ a | ( α, Φ) ∈ T ( ∈ ) } ( ∇ 2) ( ∇ 3) � {∇ α | α ∈ A } ≤ a ∇ ( T � )Φ ≤ a where a ∈ L, α, β ∈ T ω L, A ∈ P ω T ω ( L ) and Φ ∈ T ω P ω ( L ) . Thm. [Kupke, Kurz, Venema 2009] The synthetic system is sound and complete over T -coalgebras. May 26, 2010 15
Organic: Proof Systems for Homegrown Modalities Recall. Language L given by operators ♥ , semantics by � ♥ � : P ( X ) → P ( TX ) Proof Systems in terms of sequents: Γ ⊆ L with � Γ � = � { � A � | A ∈ Γ } One-step Rules ( specific for each choice of ♥ s) � Γ 1 � ∩ · · · ∩ � Γ n � ⊆ X Γ 1 . . . Γ n property of states ∼ ∼ � Γ 0 � ⊆ TX Γ 0 property of successors where • Γ 1 , . . . , Γ n ⊆ V ∪ ¬ V are propositional over a set V of variables • Γ 0 ⊆ {♥ ( p 1 , . . . , p n ) | ♥ n -ary } ∪ {¬♥ ( p 1 , . . . , p n ) | ♥ n -ary } Crucial: need Coherence Conditions between proof rules and semantics May 26, 2010 16
Organic Modalities: Coherence Conditions Consider a set X and a valuation τ : V → P ( X ) . Coherence: matching between rules and semantics at one-step level Propositional Sequents Γ ⊆ V ∪ ¬ V Γ τ -valid ⇐ ⇒ � Γ � τ = X where � p � τ = τ ( p ) Modalised Sequents Γ ⊆ {±♥ ( p 1 , . . . , p n ) | ♥ n -ary } Γ τ -valid ⇐ ⇒ � Γ � τ = TX where � ♥ ( p 1 , . . . , p n ) � τ = � ♥ � ( τ ( p 1 ) , . . . , τ ( p n )) where ± indicates possible negation. Coherence relates τ -validity of premises with τ -validity of conclusions May 26, 2010 17
Organic Modalities: Coherence Conditions One-Step Soundness of a set R of one-step rules: for all τ : V → P ( X ) Γ 1 , . . . , Γ n τ -valid = ⇒ Γ 0 τ -valid for all Γ 1 . . . Γ n / Γ 0 ∈ R One-Step Completeness of a set R of one-step rules: for all τ : V → P ( X ) ⇒ ∃ Γ 1 . . . Γ n ∈ R (Γ i σ τ -valid and Γ 0 σ ⊆ Γ) Γ τ -valid = Γ 0 for some renaming σ : V → V , for all Γ ⊆ f {±♥ ( p 1 , . . . , p n ) | ♥ n -ary } . Thm. [P , 2003, Schröder 2007] One-step soundness and one-step completeness imply soundness and (cut-free) completeness, respectively, when augmented with propositional reasoning. May 26, 2010 18
Recommend
More recommend
