Introduction The quest for Axioms The dark side of the moon Dynamic Coalgebraic Modalities Raul Andres Leal 1 & Helle Hvid Hansen 2 1 ILLC Universiteit van Amsterdam 2 Eindhoven University of Technology, Centrum Wiskunde & Informatica, Amsterdam CMCS 10, 2010 Cyprus
Introduction The quest for Axioms The dark side of the moon Outline Introduction 1 The quest for Axioms 2 The dark side of the moon 3
Introduction The quest for Axioms The dark side of the moon Outline Introduction 1 The quest for Axioms 2 The dark side of the moon 3
Introduction The quest for Axioms The dark side of the moon The long term aim Plane: KLM 1951, requires landing authorisation at Schiphol Airport. Tower: Schiphol tower, KLM 1951 descend to 1000mt. Expect heavy rain and strong tail wind during landing. Plane: KLM 1951, Schiphol what should be landing procedure under these weather conditions? Tower: Schiphol Tower, KLM 1951 after lowering the landing gear keep tail rudder still this will keep the aircraft stable.
Introduction The quest for Axioms The dark side of the moon The long term aim State based system Plane: KLM 1951, requires landing authorisation at Schiphol Airport. Tower: Schiphol tower, KLM 1951 descend to 1000mt. Expect heavy rain and strong tail wind during landing. Plane: KLM 1951, Schiphol what should be landing procedure under these weather conditions? Tower: Schiphol Tower, KLM 1951 after lowering the landing gear keep tail rudder still this will keep the aircraft stable.
Introduction The quest for Axioms The dark side of the moon The long term aim State based system State of the system Plane: KLM 1951, requires landing authorisation at Schiphol Airport. Tower: Schiphol tower, KLM 1951 descend to 1000mt. Expect heavy rain and strong tail wind during landing. Plane: KLM 1951, Schiphol what should be landing procedure under these weather conditions? Tower: Schiphol Tower, KLM 1951 after lowering the landing gear keep tail rudder still this will keep the aircraft stable.
Introduction The quest for Axioms The dark side of the moon The long term aim State based system State of the system Plane: KLM 1951, requires landing authorisation at Schiphol Airport. Tower: Schiphol tower, KLM 1951 descend to 1000mt. Expect heavy rain and strong tail wind during landing. Plane: KLM 1951, Schiphol what should be landing procedure under these weather conditions? Tower: Schiphol Tower, KLM 1951 after lowering the landing gear keep tail rudder still this will keep the aircraft stable. Programs
Introduction The quest for Axioms The dark side of the moon The long term aim State based system State of the system Plane: KLM 1951, requires landing authorisation at Schiphol Airport. Tower: Schiphol tower, KLM 1951 descend to 1000mt. Expect heavy rain and strong tail wind during landing. Plane: KLM 1951, Schiphol what should be landing procedure under these weather conditions? Tower: Schiphol Tower, KLM 1951 AFTER lowering the landing gear keep tail rudder still this will keep the aircraft stable. Programs
Introduction The quest for Axioms The dark side of the moon The long term aim State based system State of the system Plane: KLM 1951, requires landing authorisation at Schiphol Airport. Tower: Schiphol tower, KLM 1951 descend to 1000mt. Expect heavy rain and strong tail wind during Modal Statement landing. Plane: KLM 1951, Schiphol what should be landing procedure under these weather conditions? Tower: Schiphol Tower, KLM 1951 AFTER lowering the landing gear keep tail rudder still this will keep the aircraft stable. Programs
Introduction The quest for Axioms The dark side of the moon From Planes to Kripke Frames Labelled transition systems. These are coalgebras → ( P S ) L S − Monotone neighborhoods This yields to PDL, we reason about programs. Game/Coalition Frames These are coalgebras → ( M S ) L S − This yields to Game Logic, we reason about strategic ability in 2-player games
Introduction The quest for Axioms The dark side of the moon From Planes to Kripke Frames Labelled transition systems. These are coalgebras → ( P S ) L S − This yields to PDL, we reason about programs. Game/Coalition Frames These are coalgebras → ( M S ) L S − This yields to Game Logic, we reason about strategic ability in 2-player games
Introduction The quest for Axioms The dark side of the moon From Planes to Kripke Frames (free) algebra of regular expressions Labelled transition systems. These are coalgebras → ( P S ) L S − This yields to PDL, we reason about programs. Game/Coalition Frames These are coalgebras → ( M S ) L S − This yields to Game Logic, we reason about strategic ability in 2-player games
Introduction The quest for Axioms The dark side of the moon From Planes to Kripke Frames (free) algebra of regular expressions Labelled transition systems. These are coalgebras → ( P S ) L S − This yields to PDL, we reason about programs. Game/Coalition Frames These are coalgebras → ( M S ) L S − This yields to Game Logic, we reason about strategic ability in 2-player games (free) algebra of game expressions
Introduction The quest for Axioms The dark side of the moon Double perspective Algebraic Perspective → ( GS ) S σ : L − Structure + Dynamics Coalgebraic Perspective → ( GS ) L � σ : S − Behavior + Modalities
Introduction The quest for Axioms The dark side of the moon Double perspective Algebraic Perspective → ( GS ) S σ : L − Structure + Dynamics Coalgebraic Perspective → ( GS ) L � σ : S − Behavior + Modalities
Introduction The quest for Axioms The dark side of the moon Dynamic Modalities Intuition s � λ α ϕ means “in state s , after α , ϕ holds”. PDL s � � α ϕ means “in state s , after transition α , ϕ holds”. Game Logic s � ♦ α ϕ means “in state s , player 1 has a strategy in game α to bring about ϕ ”.
Introduction The quest for Axioms The dark side of the moon Dynamic Modalities Intuition s � λ α ϕ means “in state s , after α , ϕ holds”. PDL s � � α ϕ means “in state s , after transition α , ϕ holds”. Game Logic s � ♦ α ϕ means “in state s , player 1 has a strategy in game α to bring about ϕ ”.
Introduction The quest for Axioms The dark side of the moon Dynamic Modalities Intuition s � λ α ϕ means “in state s , after α , ϕ holds”. PDL s � � α ϕ means “in state s , after transition α , ϕ holds”. Game Logic s � ♦ α ϕ means “in state s , player 1 has a strategy in game α to bring about ϕ ”.
Introduction The quest for Axioms The dark side of the moon Dynamic Modalities Intuition s � λ α ϕ means “in state s , after α , ϕ holds”. Labelling Given a predicate lifting λ : Q − → Q G and α ∈ L , the α labelling of λ is a predicate lifting λ α : Q − → Q G L given by λ α ( U ) = { δ ∈ G ( S ) L | δ ( α ) ∈ λ ( U ) }
Introduction The quest for Axioms The dark side of the moon Outline Introduction 1 The quest for Axioms 2 The dark side of the moon 3
Introduction The quest for Axioms The dark side of the moon Describing composition of actions PDL Take � = λ , why does λ α ; β ϕ ⇐ ⇒ λ α λ β ϕ hold? Predicate transformers → ( GS ) L consider Given σ : S − λ α → Q ( GS ) L σ − 1 ([ α ] σ ) S Q S − − − → Q S the equivalence above follows from [ α ; β ] σ = [ α ] σ ◦ [ β ] σ
Introduction The quest for Axioms The dark side of the moon Describing composition of actions PDL Take � = λ , why does [ α ; β ] ϕ ⇐ ⇒ [ α ][ β ] ϕ hold? Predicate transformers → ( GS ) L consider Given σ : S − λ α → Q ( GS ) L σ − 1 ([ α ] σ ) S Q S − − − → Q S the equivalence above follows from [ α ; β ] σ = [ α ] σ ◦ [ β ] σ
Introduction The quest for Axioms The dark side of the moon Describing composition of actions PDL Take � = λ , why does λ α ; β ϕ ⇐ ⇒ λ α λ β ϕ hold? Predicate transformers → ( GS ) L consider Given σ : S − λ α → Q ( GS ) L σ − 1 ([ α ] σ ) S Q S − − − → Q S the equivalence above follows from [ α ; β ] σ = [ α ] σ ◦ [ β ] σ
Introduction The quest for Axioms The dark side of the moon Describing composition of actions PDL Take � = λ , why does λ α ; β ϕ ⇐ ⇒ λ α λ β ϕ hold? Predicate transformers → ( GS ) L consider Given σ : S − λ α → Q ( GS ) L σ − 1 ([ α ] σ ) S Q S − − − → Q S the equivalence above follows from [ α ; β ] σ = [ α ] σ ◦ [ β ] σ
Introduction The quest for Axioms The dark side of the moon Monads for composition Theorem Let λ be a predicate lifting. If sequential composition is interpreted as Kleisli composition, then λ α ; β ϕ ⇐ ⇒ λ α λ β ϕ holds if one of the following conditions hold . . . the transpose � λ : G − → QQ is a monad morphism. The algebra Y ( λ ) : G 2 − → 2 is a G-algebra (monads).
Introduction The quest for Axioms The dark side of the moon Monads for composition Theorem Let λ be a predicate lifting. If sequential composition is interpreted as Kleisli composition, then [ α ; β ] σ = [ α ] σ ◦ [ β ] σ holds if one of the following conditions hold . . . the transpose � λ : G − → QQ is a monad morphism. The algebra Y ( λ ) : G 2 − → 2 is a G-algebra (monads).
Recommend
More recommend
