Simulations in Coalgebra Bart Jacobs and Jesse Hughes { bart,jesseh - PowerPoint PPT Presentation

Simulations in Coalgebra Bart Jacobs and Jesse Hughes { bart,jesseh } @cs.kun.nl. University of Nijmegen Simulations in Coalgebra – p.1/16

Outline I. Simulations, bisimulations, two-way simulations Simulations in Coalgebra – p.2/16

Outline I. Simulations, bisimulations, two-way simulations II. Orders on functors Simulations in Coalgebra – p.2/16

Outline I. Simulations, bisimulations, two-way simulations II. Orders on functors III. Lax relation lifting Simulations in Coalgebra – p.2/16

Outline I. Simulations, bisimulations, two-way simulations II. Orders on functors III. Lax relation lifting IV. Two-way simulations Simulations in Coalgebra – p.2/16

Outline I. Simulations, bisimulations, two-way simulations II. Orders on functors III. Lax relation lifting IV. Two-way simulations V. DPCO structure on final coalgebras Simulations in Coalgebra – p.2/16

Outline I. Simulations, bisimulations, two-way simulations II. Orders on functors III. Lax relation lifting IV. Two-way simulations V. DPCO structure on final coalgebras VI. Summary Simulations in Coalgebra – p.2/16

Simulations, etc. Let R be a relation on coalgebras � A, α � and � B, β � . Simulations in Coalgebra – p.3/16

Simulations, etc. Let R be a relation on coalgebras � A, α � and � B, β � . � a ′ , there is R is a simulation iff, whenever aRb and a � b ′ where a ′ Rb ′ . b Simulations in Coalgebra – p.3/16

Simulations, etc. Let R be a relation on coalgebras � A, α � and � B, β � . � a ′ , there is R is a simulation iff, whenever aRb and a � b ′ where a ′ Rb ′ . b ⇔ ∃ R . aRb and R is a a b Similarity simulation. Simulations in Coalgebra – p.3/16

Simulations, etc. Let R be a relation on coalgebras � A, α � and � B, β � . � a ′ , there is R is a simulation iff, whenever aRb and a � b ′ where a ′ Rb ′ . b ⇔ ∃ R . aRb and R is a a b Similarity simulation. ∃ R . aRb and R , R op are a ↔ b ⇔ Bisimilarity simulations Simulations in Coalgebra – p.3/16

Simulations, etc. Let R be a relation on coalgebras � A, α � and � B, β � . � a ′ , there is R is a simulation iff, whenever aRb and a � b ′ where a ′ Rb ′ . b ⇔ ∃ R . aRb and R is a a b Similarity simulation. ∃ R . aRb and R , R op are a ↔ b ⇔ Bisimilarity simulations a ∼ b ⇔ a b and b a Two-way similarity Simulations in Coalgebra – p.3/16

Sequences Consider FX = 1 + × X . Final F -coalgebra: (possibly finite) sequences over . Simulations in Coalgebra – p.4/16

Sequences Consider FX = 1 + × X . Final F -coalgebra: (possibly finite) sequences over . “Standard” similarity 1 τ ⇔ σ is a prefix of τ. σ � 1 � 2 � 3 � 5 1 � 1 � 2 � 3 � 5 � 8 � . . . 1 Simulations in Coalgebra – p.4/16

Sequences Consider FX = 1 + × X . Final F -coalgebra: (possibly finite) sequences over . Another similarity 2 τ ⇔ len ( σ ) = len ( τ ) and for each n < len ( σ ) , σ σ ( n ) ≤ τ ( n ) . � 1 � 1 � 2 � 3 � 5 � . . . 0 � 1 � 2 � 3 � 5 � 8 � . . . 1 Simulations in Coalgebra – p.4/16

Sequences Consider FX = 1 + × X . Final F -coalgebra: (possibly finite) sequences over . Another similarity 2 τ ⇔ len ( σ ) = len ( τ ) and for each n < len ( σ ) , σ σ ( n ) ≤ τ ( n ) . � 1 � 1 � 2 � 3 � 5 � . . . 0 ≤ ≤ ≤ ≤ ≤ ≤ � 1 � 2 � 3 � 5 � 8 � . . . 1 Simulations in Coalgebra – p.4/16

Sequences Consider FX = 1 + × X . Final F -coalgebra: (possibly finite) sequences over . Similarity via composition σ ( 2 ◦ 1 ) τ ⇔ len ( σ ) ≤ len ( τ ) and for all n ≤ len ( σ ) , σ ( n ) ≤ τ ( n ) . � 1 � 1 � 2 � 3 � 5 0 � 1 � 2 � 3 � 5 � 8 � . . . 1 Simulations in Coalgebra – p.4/16

Sequences Consider FX = 1 + × X . Final F -coalgebra: (possibly finite) sequences over . What structure suffices to describe these examples of similarity? Simulations in Coalgebra – p.4/16

� � � Our starting point: Orders on functors An order on a functor F : Set � Set is a functor ⊑ : Set � PreOrd such that this diagram commutes. PreOrd � � ⊑ � � � � � � � � � � � Set Set F Simulations in Coalgebra – p.5/16

� � � Our starting point: Orders on functors An order on a functor F : Set � Set is a functor ⊑ : Set � PreOrd such that this diagram commutes. PreOrd � � ⊑ � � � � � � � � � � � Set Set F This means: • For each set X , we have a preorder ⊑ X on FX ; Simulations in Coalgebra – p.5/16

� � � Our starting point: Orders on functors An order on a functor F : Set � Set is a functor ⊑ : Set � PreOrd such that this diagram commutes. PreOrd � � ⊑ � � � � � � � � � � � Set Set F This means: • For each set X , we have a preorder ⊑ X on FX ; • For each map f : X � Y , the map Ff : FX � FY is monotone. Simulations in Coalgebra – p.5/16

� � � Our starting point: Orders on functors An order on a functor F : Set � Set is a functor ⊑ : Set � PreOrd such that this diagram commutes. PreOrd � � ⊑ � � � � � � � � � � � Set Set F This means: • For each set X , we have a preorder ⊑ X on FX ; • For each map f : X � Y , the map Ff : FX � FY is monotone. An order on F yields a notion of F -similarity. Simulations in Coalgebra – p.5/16

� � � Excursion: bisimulations A functor F : Set � Set has a (canonical) associated relation lifting: Rel ( F ) Rel Rel � Set × Set Set × Set F × F Simulations in Coalgebra – p.6/16

� � � � � � � Excursion: bisimulations A functor F : Set � Set has a (canonical) associated relation lifting: Rel ( F ) Rel Rel � Set × Set Set × Set F × F This can be defined via image factorization: � � Rel ( F )( R ) FR � FA × FB F ( A × B ) � Fπ 1 , Fπ 2 � Simulations in Coalgebra – p.6/16

� � � � � Excursion: bisimulations A bisimulation over F -coalgebras � A, α � and � B, β � is a Rel ( F ) -coalgebra: Rel ( F )( R ) R � FA × FB A × B α × β Simulations in Coalgebra – p.6/16

� � � � � Excursion: bisimulations A bisimulation over F -coalgebras � A, α � and � B, β � is a Rel ( F ) -coalgebra: Rel ( F )( R ) R � FA × FB A × B α × β It is a relation R such that ⇒ α ( a ) Rel ( F )( R ) β ( b ) . aRb Simulations in Coalgebra – p.6/16

� � � Lax relation liftings Rel ( F )( − ) Rel Rel � Set × Set Set × Set F × F Simulations in Coalgebra – p.7/16

� � � Lax relation liftings An order ⊑ : Set � PreOrd induces a lax relation lifting via composition. ⊑◦ Rel ( F )( − ) ◦⊑ Rel Rel � Set × Set Set × Set F × F Simulations in Coalgebra – p.7/16

� � � Lax relation liftings An order ⊑ : Set � PreOrd induces a lax relation lifting via composition. ⊑◦ Rel ( F )( − ) ◦⊑ Rel Rel � Set × Set Set × Set F × F We write x Rel ⊑ ( F )( R ) y just in case x ( ⊑◦ Rel ( F )( R ) ◦⊑ ) y Simulations in Coalgebra – p.7/16

� � � Lax relation liftings An order ⊑ : Set � PreOrd induces a lax relation lifting via composition. ⊑◦ Rel ( F )( − ) ◦⊑ Rel Rel � Set × Set Set × Set F × F We write x Rel ⊑ ( F )( R ) y just in case x ( ⊑◦ Rel ( F )( R ) ◦⊑ ) y ∃ x ′ , y ′ . x ⊑ x ′ Rel ( F )( R ) y ′ ⊑ y. Simulations in Coalgebra – p.7/16

� � � � � Simulations A simulation on � A, α � and � B, β � is a Rel ⊑ ( F ) -coalgebra over α × β . Rel ⊑ ( F )( R ) R � FA × FB A × B α × β Simulations in Coalgebra – p.8/16

� � � � � Simulations A simulation on � A, α � and � B, β � is a Rel ⊑ ( F ) -coalgebra over α × β . Rel ⊑ ( F )( R ) R � FA × FB A × B α × β It is a relation R on A × B such that ∃ x ′ , y ′ . α ( a ) ⊑ x ′ Rel ( F )( R ) y ′ ⊑ β ( b ) . ⇒ aRb Simulations in Coalgebra – p.8/16

� � � � � Simulations A simulation on � A, α � and � B, β � is a Rel ⊑ ( F ) -coalgebra over α × β . Rel ⊑ ( F )( R ) R � FA × FB A × B α × β This definition includes all of the common notions of coalgebraic simulation. Simulations in Coalgebra – p.8/16

� � � � � Simulations A simulation on � A, α � and � B, β � is a Rel ⊑ ( F ) -coalgebra over α × β . Rel ⊑ ( F )( R ) R � FA × FB A × B α × β This definition includes all of the common notions of coalgebraic simulation. For any pair of coalgebras, the greatest simulation exists. Simulations in Coalgebra – p.8/16

Examples Consider FX = 1 + × X . � 1 � 2 � 3 � 5 � 8 � . . . 1 Simulations in Coalgebra – p.9/16