On Semantic Relations: From probabilistic systems to coalgebras and - PowerPoint PPT Presentation

On Semantic Relations: From probabilistic systems to coalgebras and back Ana Sokolova SOS group, Radboud University Nijmegen GEOCAL ’06, Probabilistic Systems Workshop, AS – p.1/30

Outline • Introduction - probabilistic systems and coalgebras GEOCAL ’06, Probabilistic Systems Workshop, AS – p.2/30

Outline • Introduction - probabilistic systems and coalgebras • Bisimilarity - the strong end of the spectrum GEOCAL ’06, Probabilistic Systems Workshop, AS – p.2/30

Outline • Introduction - probabilistic systems and coalgebras • Bisimilarity - the strong end of the spectrum • Application - expressiveness hierarchy (older result) GEOCAL ’06, Probabilistic Systems Workshop, AS – p.2/30

Outline • Introduction - probabilistic systems and coalgebras • Bisimilarity - the strong end of the spectrum • Application - expressiveness hierarchy (older result) • Trace semantics - the weak end of the spectrum (new !) GEOCAL ’06, Probabilistic Systems Workshop, AS – p.2/30

Systems are formal objects, transition systems (e.g. LTS), that serve as models of real (software, hardware,...) systems GEOCAL ’06, Probabilistic Systems Workshop, AS – p.3/30

Systems are formal objects, transition systems (e.g. LTS), that serve as models of real (software, hardware,...) systems • • • • • • GEOCAL ’06, Probabilistic Systems Workshop, AS – p.3/30

� � � Systems are formal objects, transition systems (e.g. LTS), that serve as models of real (software, hardware,...) systems • � � a � � � � � � � � • � • b c • � ���������� a • • c GEOCAL ’06, Probabilistic Systems Workshop, AS – p.3/30

Probabilistic systems arise by enriching transition systems with (discrete) probabilities as labels on the transitions. Examples: GEOCAL ’06, Probabilistic Systems Workshop, AS – p.4/30

� � � � � � Probabilistic systems arise by enriching transition systems with (discrete) probabilities as labels on the transitions. Examples: • • � 1 2 � ����� � � � � � � 3 3 � � ⇒ • • • • 1 � �� • • GEOCAL ’06, Probabilistic Systems Workshop, AS – p.4/30

� � � � � � Probabilistic systems arise by enriching transition systems with (discrete) probabilities as labels on the transitions. Examples: • • � a [ 1 b [ 2 � ����� � 3 ] 3 ] � � � � � � � ⇒ • • • • a [1] � �� • • GEOCAL ’06, Probabilistic Systems Workshop, AS – p.4/30

� � � � � � � Probabilistic systems arise by enriching transition systems with (discrete) probabilities as labels on the transitions. Examples: • ♦ � � � ������ 1 2 � � � � � 3 3 � � � ⇒ • • • • a • • GEOCAL ’06, Probabilistic Systems Workshop, AS – p.4/30

� �� � � � Probabilistic systems arise by enriching transition systems with (discrete) probabilities as labels on the transitions. Examples: • • � � � � � ������ � ����� � � a b � � � � � � ⇒ • • • ♦ 1 • • GEOCAL ’06, Probabilistic Systems Workshop, AS – p.4/30

� �� � � �� � � Probabilistic systems arise by enriching transition systems with (discrete) probabilities as labels on the transitions. Examples: • • ⇒ � � ����� � � 1 � � • • � � � ������ � a b � � � • • • 1 a � • GEOCAL ’06, Probabilistic Systems Workshop, AS – p.4/30

� � � � � � � � Probabilistic systems arise by enriching transition systems with (discrete) probabilities as labels on the transitions. Examples: • • ⇒ � � ����� � � a � � • • 1 2 � � � � 3 3 • • • b 1 � �� • GEOCAL ’06, Probabilistic Systems Workshop, AS – p.4/30

Coalgebras are an elegant generalization of transition systems with states + transitions GEOCAL ’06, Probabilistic Systems Workshop, AS – p.5/30

Coalgebras are an elegant generalization of transition systems with states + transitions as pairs � S, α : S → F S � , for F a functor GEOCAL ’06, Probabilistic Systems Workshop, AS – p.5/30

Coalgebras are an elegant generalization of transition systems with states + transitions as pairs � S, α : S → F S � , for F a functor • based on category theory • provide a uniform way of treating transition systems • provide general notions and results e.g. a generic notion of bisimulation GEOCAL ’06, Probabilistic Systems Workshop, AS – p.5/30

Examples A TS is a pair � S, α : S → P S � !! coalgebra of the powerset functor P GEOCAL ’06, Probabilistic Systems Workshop, AS – p.6/30

Examples A TS is a pair � S, α : S → P S � !! coalgebra of the powerset functor P An LTS is a pair � S, α : S → P S A � !!! coalgebra of the functor P A Note: P A ∼ = P ( A × ) GEOCAL ’06, Probabilistic Systems Workshop, AS – p.6/30

More examples Thanks to the probability distribution functor D D S = { µ : S → [0 , 1] , µ [ S ] = 1 } , µ [ X ] = � s ∈ X µ ( x ) D f ( µ )( t ) = µ [ f − 1 ( { t } )] D f : D S → D T, the probabilistic systems are also coalgebras GEOCAL ’06, Probabilistic Systems Workshop, AS – p.7/30

More examples Thanks to the probability distribution functor D D S = { µ : S → [0 , 1] , µ [ S ] = 1 } , µ [ X ] = � s ∈ X µ ( x ) D f ( µ )( t ) = µ [ f − 1 ( { t } )] D f : D S → D T, the probabilistic systems are also coalgebras ... of functors built by the following syntax | A | P | D | G + H | G × H | G A | G ◦ H F ::= GEOCAL ’06, Probabilistic Systems Workshop, AS – p.7/30

reactive, generative A ) ∼ evolve from LTS - functor P ( A × = P GEOCAL ’06, Probabilistic Systems Workshop, AS – p.8/30

reactive, generative A ) ∼ evolve from LTS - functor P ( A × = P reactive systems: functor ( D + 1) A GEOCAL ’06, Probabilistic Systems Workshop, AS – p.8/30

reactive, generative A ) ∼ evolve from LTS - functor P ( A × = P reactive systems: functor ( D + 1) A generative systems: functor ( D + 1)( A × ) = D ( A × ) + 1 GEOCAL ’06, Probabilistic Systems Workshop, AS – p.8/30

reactive, generative A ) ∼ evolve from LTS - functor P ( A × = P reactive systems: functor ( D + 1) A generative systems: functor ( D + 1)( A × ) = D ( A × ) + 1 note: in the probabilistic case ( D + 1) A �∼ = D ( A × ) + 1 GEOCAL ’06, Probabilistic Systems Workshop, AS – p.8/30

Probabilistic system types D MC + 1) A ( DLTS ) ∼ = P A P ( A × LTS ( D + 1) A React D ( A × ) + 1 Gen D + ( A × ) + 1 Str D + P ( A × ) Alt D ( A × ) + P ( A × ) Var P ( A × D ) SSeg PD ( A × ) Seg . . . . . . GEOCAL ’06, Probabilistic Systems Workshop, AS – p.9/30

� � � � � � � � � � ���� � ���� Probabilistic system types D MC + 1) A ( DLTS ) ∼ = P A P ( A × LTS • ( D + 1) A React � � � � � � a [ 1 3 ] b [1] a [ 2 3 ] D ( A × ) + 1 Gen • • • D + ( A × ) + 1 Str D + P ( A × ) Alt b [1] a [1] D ( A × ) + P ( A × ) Var • • P ( A × D ) SSeg PD ( A × ) Seg . . . . . . GEOCAL ’06, Probabilistic Systems Workshop, AS – p.9/30

� ���� � � ���� � � � � � � � � Probabilistic system types D MC + 1) A ( DLTS ) ∼ = P A P ( A × LTS • ( D + 1) A React � � � � � � a [ 1 b [ 1 2 ] 4 ] a [ 1 4 ] D ( A × ) + 1 Gen • • • D + ( A × ) + 1 Str D + P ( A × ) Alt c [1] c [1] D ( A × ) + P ( A × ) Var • • P ( A × D ) SSeg PD ( A × ) Seg . . . . . . GEOCAL ’06, Probabilistic Systems Workshop, AS – p.9/30

� � � � � � � � � Probabilistic system types D MC + 1) A ( DLTS ) ∼ = P A P ( A × LTS ( D + 1) A React ♦ � � � � � 3 1 D ( A × ) + 1 Gen 4 4 D + ( A × ) + 1 Str • ♦ � ������ 1 a 2 D + P ( A × ) Alt 1 b 2 � �� • • • • D ( A × ) + P ( A × ) Var P ( A × D ) SSeg PD ( A × ) Seg . . . . . . GEOCAL ’06, Probabilistic Systems Workshop, AS – p.9/30

� � � � � �� � � Probabilistic system types D MC + 1) A ( DLTS ) ∼ = P A P ( A × LTS ( D + 1) A React • � � � a b D ( A × ) + 1 Gen � � a � � � � � � � � D + ( A × ) + 1 Str 3 2 � � � � 4 3 1 1 1 � �� D + P ( A × ) 4 � �� Alt 3 • • • • • D ( A × ) + P ( A × ) Var P ( A × D ) SSeg PD ( A × ) Seg . . . . . . GEOCAL ’06, Probabilistic Systems Workshop, AS – p.9/30