Coalgebraic Logics: A Computer Science Perspective Dirk Pattinson, Imperial College London
Part I: Coalgebraic Logics: Motivation and Some Results
A Computer Science View Coalgebraic Logics: Describe computational phenomena with modal logics • State Transition Systems → Hennessy-Milner Logic • Probabilistic Effects → Probabilistic Modal Logic → Coalition Logic • Games • Ontologies . . . → Description Logic . . . Logical Aspects Computer Science Aspects • completeness • Genericity: development of uniform • complexity proofs/algorithms/tools? • Modularity: synthesis of complex • cut elimination • interpolation . . . systems from simple building blocks August 9, 2007 1
A Cook’s Tour Through Modal Semantics ~p p C → P ( C ) × P ( A ) Kripke Models. p ~p 2 p C → B ( C ) × P ( A ) Multigraphs. 4 p B ( X ) = { f : X → N | supp( f ) finite } ~p 0.2 p C → D ( C ) × P ( A ) Probabilistic Systems. 0.8 p D ( X ) = { µ : X → [0 , 1] | � x ∈ X µ ( x ) = 1 } August 9, 2007 2
Unifying Feature: Coalgebraic Semantics All examples are instances of Coalgebras ( C, γ : C → TC ) where T : Set → Set is an endofunctor, the signature functor . (Dually, T -algebras are pairs ( A, α : TA → A ) ) Intuition. • coalgebras are generalised transition systems • morphisms of coalgebras are generalised p -morphisms Computer Science Concerns • Genericity: Prove things once and for all, parametric in T • Modularity: Construct complex functors from simple ingredients August 9, 2007 3
Coalgebraic Semantics of Modal Logics Given: T : Set → Set Question: What’s the “right” logic for T -coalgebras? • should generalise well-known cases, e.g. K, probabilistic/graded modal logic, coalition logic • theory should be parametric in T ❀ uniform theorems that apply to a large class of logics Semantically: What’s a modal operator, or: what is � ✷ φ � ? August 9, 2007 4
Moss’ Coalgebraic Logic I Kripke Frames: C → P ( C ) T -coalgebras: C → T ( C ) Concrete Syntax Concrete Syntax φ, ψ ∈ L Φ ∈ TL φ, ψ ∈ L Φ ∈ P ( L ) ⊥ ∈ L φ → ψ ∈ L ∇ Φ ∈ L ⊥ ∈ L φ → ψ ∈ L ∇ Φ ∈ L Modal Semantics Modal Semantics c | = ∇ Φ ⇐ ⇒ ( γ ( c ) , Φ) ∈ T ( | =) c | = ∇ Φ ⇐ ⇒ ( γ ( c ) , Φ) ∈ P ( | =) Abstract Syntax: Abstract Syntax: = F ( L ) = 1 + L 2 + T ( L ) = F ( L ) = 1 + L 2 + P ( L ) L ∼ L ∼ Algebraic Semantics Algebraic Semantics F ( L ) F ( P ( C )) F ( L ) F ( P ( C )) γ ˆ γ ˆ i i P ( C ) P ( C ) L L � · � � · � ∇ Φ = ✷ � Φ ∧ ✸ Φ Need: F -algebra structure F ( P ( C )) → P ( C ) August 9, 2007 5
Moss’ Coalgebraic Logic II Algebraic Semantics of Coalgebraic Logic: 1 + ( P C ) 2 + T ( P C ) 1 + L 2 + TL [ ⊥ , → , ˆ γ ] i P ( C ) L � · � γ − 1 δ where ˆ γ : T ( P C ) − → P ( TC ) − → P ( C ) � �� � distributive law M Representation Theorem: � n A n × X n ։ TX , e.g. X − → TX gives algebraic semantics of Unary Modalities: γ − 1 M δ P ( C ) − → T ( P C ) − → P ( TC ) − → P ( C ) � �� � unary modality August 9, 2007 6
Coalgebraic Semantics of Modal Logics Structures for T coalgebras determine the semantics of modal operators: they assign a nbhd frame translation or, equivalently, a predicate lifting � M � : TC → PP ( C ) � M � : P ( C ) → P ( TC ) to every modal operator M of the language, parametric in C . Together with a T -coalgebra ( C, γ ) this gives a boolean algebra with operator neighbourhood frame � M � PP ( C ) γ � M � P ( TC ) 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 ∈ � Mφ � iff � φ � ∈ � M � ◦ γ ( � φ � ) c ∈ � Mφ � ⇐ ⇒ γ ( c ) ∈ � M � ( � φ � ) August 9, 2007 7
Examples Neighbourhood Frames, i.e. coalgebras C → PP ( C ) � ✷ � = id : PP ( C ) → PP ( C ) � �� � (identical nbhd frame translation) T C Kripke Frames, ie. coalgebras C → P ( C ) via boolean algebras with operators viewed as neighbourhood frames T C T C � �� � � �� � � ✷ � : P ( C ) → P P ( C ) � ✷ � : P ( C ) → PP ( C ) �→ { c ′ : c ′ ⊇ c } �→ { c ′ : c ′ ⊆ c } c c Probabilistic Transition Systems , i.e. coalgebras C → D C T C � �� � � L p � : P ( C ) → P D ( C ) (algebraic perspective) c �→ { µ : C → [0 , 1] : µ ( c ) ≥ p } August 9, 2007 8
Genericity I: Expressivity Easy, but important: Coalgebraic Logics are bisimulation invariant. Hennessy-Milner Property: Bisimulation coincides with logical equivalence over image finite transition systems. • what is image finite for T -coalgebras? • additional condition(s) on the logic (e.g. exclude empty set of operators) Theorem (P , 2001) If T is ω -accessible and the modal structure is separating , i.e. for predicate liftings TC ∋ t �→ { � M � ( c ) : c ⊆ C, M modal op } is injective, then the induced logic has the Hennessy-Milner property. Theorem (Schroeder, 2005) Admitting polyadic modalities, the structure that comprises all predicate liftings is separating. August 9, 2007 9
Genericity II: Completeness Deduction for Coalgebraic Logics: propositional logic plus a set R of one-step rules φ/ψ : φ propositional, ψ clause over Ma, a ∈ V Intuition. Rules axiomatise those nbhd frames that come from coalgebras One Step Derivability of χ (propositional over { Mx : x ⊆ X } ) over a set X • TX | = χ defined inductively by � Mx � = � M � ( x ) • R X ⊢ χ iff { ψσ : X | = φσ, φ/ψ ∈ R} ⊢ PL χ R is one-step sound (complete) if TX | = χ whenever (only if) R X ⊢ χ Theorem (P , 2003, Schroeder 2006) Soundness and weak completeness are implied by their one-step counterparts. Theorem (Schroeder 2006) The set of axioms that is one-step sound is one-step complete. August 9, 2007 10
Genericity III: Complexity Shallow Model Construction for T -coalgebras: inductively strip off modalities ∀ φ/ψ ∈ R.ψσ → χ = ⇒ ¬ φσ satisfiable Countermodel of φσ ’s ⇑ ⇓ ¬ χ satisfiable Countermodel of χ Crucial Requirement is Resolution Closure of R : derivable consequences are derivable using a single rule. Theorem. (Schroeder/P , 2006) If R is resolution closed and rule matching is in NP , then satisfiability is in PSPACE. Example. K, KD, Coalition Logic, GML, PML, Majority Logic are in PSPACE. August 9, 2007 11
Construction of Resolution Closed Sets Example: K axiomatised by rules a a ∧ b → c ✷ a ✷ a ∧ ✷ b → ✷ c Rule Resolution: a ∧ b → c c ∧ d → e ✷ a ∧ ✷ b → ✷ c ✷ c ∧ ✷ d → ✷ e Resolving the conclusions at c ( a ∧ b → c ) ∧ ( c ∧ d → e ) ✷ a ∧ ✷ b ∧ ✷ d → ✷ e Eliminating c from the premise: a ∧ b ∧ d → e ✷ a ∧ ✷ b ∧ ✷ d → ✷ e (This converges to a cut-free sequent-calculus . . . ) August 9, 2007 12
Modularity Example. Combining Probabilities and Non-Determinism • • a b 0 . 6 0 . 4 a ◦ ◦ ◦ ◦ • 0 . 2 0 . 8 0 . 5 a 0 . 8 1 0 . 5 b 0 . 2 • • • • • • ◦ • ◦ Simple Segala Systems Alternating Systems Coalgebraic Interpretation C → P ( A × D ( C )) C → P ( A × C ) + D ( C ) Semantics of Combination. Functor Composition – ingredients represent features. Logic Combinations. Mimic Functor Composition August 9, 2007 13
Logics for Combined Systems Simple Segala Systems: C → P ( A × D ( C )) L n ∋ φ ::= ⊤ | φ 1 ∧ φ 2 | ¬ φ | ✷ a ψ (nondeterministic formulas; ψ ∈ L u , a ∈ A ) L u ∋ ψ ::= ⊤ | ψ 1 ∧ ψ 2 | ¬ ψ | L p φ (probabilistic formulas; φ ∈ L n , p ∈ [0 , 1] ∩ Q ) . Alternating Systems: C → P ( A × C ) + D ( C ) L o ∋ ρ ::= ⊤ | ρ 1 ∧ ρ 2 | ¬ ρ | φ + ψ (alternating formulas; φ ∈ L u , ψ ∈ L n ) L u ∋ φ ::= ⊤ | φ 1 ∧ φ 2 | ¬ φ | L p ρ (probabilistic formulas; ρ ∈ L o , p ∈ [0 , 1] ∩ Q ) L n ∋ ψ ::= ⊤ | ψ 1 ∧ ψ 2 | ¬ ψ 2 | ✷ a ρ (nondeterministic formulas; ρ ∈ L o , a ∈ A ) Semantics by Example: given γ : C → P ( A × C ) + D ( C ) � φ + ψ � = γ − 1 ( � φ � + � ψ � ) ⊆ C • ( L o ) • ( L u ) � L p ρ � = � L p � ( � ρ � ) ⊆ D C • ( L n ) � ✷ a ρ � = � ✷ a � ( � ρ � ) ⊆ P ( A × C ) August 9, 2007 14
Modularity I: Expressivity Features: Basic Building Blocks comprising • an endofunctor F : Set n → Set • typed modal operators M : i 1 , . . . , i k • predicate liftings � M � : P ( X 1 ) × · · · × P ( X k ) → P F ( X 1 , . . . , X k ) Example 1: Uncertainty Example 2: Binary Choice • � : Set 2 → Set • D : Set → Set • L p : 1 ( p ∈ [0 , 1] ∩ Q ) • + : 1 , 2 • � L p � as before • � + � : ( x, y ) �→ x + y Theorem (Cirstea, 2000) The logic associated with any combination of features that are ω -accessible and separating has the Hennessy-Milner property. August 9, 2007 15
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