Structural Operational Semantics for continuous state probabilistic - PowerPoint PPT Presentation

Structural Operational Semantics for continuous state probabilistic processes* Giorgio Bacci (joint with Marino Miculan) Dept. of Mathematics and Computer Science University of Udine, Italy Breakfast Talk 24 May, Aalborg 1 / 1

Structural Operational Semantics for CCS Syntax: P ::= nil | a . P | a . P | τ. P | P + P | P � P ( a ∈ A ) x � x nil a . x a . x τ. x x + x ���� � �� � � �� � ���� � �� � � �� � A × X + A × X + X × X + X × X Σ X = 1 + X + α α − − → x ′ − − → y ′ x y α α α α. x − − → x x + y − − → x ′ x + y − − → y ′ α α x − − → x ′ y − − → y ′ → x ′ � y α α x � y − − x � y − − → x � y ′ a a a a − → x ′ − → y ′ − → x ′ − → y ′ x y x y → x ′ � y ′ → x ′ � y ′ τ τ x � y − x � y − λ : Σ( Id × ( P fin ) L ) ⇒ ( P fin T Σ ) L This corresponds to: 2 / 1

Abstract Structural Operational Semantics (distributing syntax over behaviours: λ : Σ B ⇒ B Σ) denotations operations α β Σ X X BX λ -bialgebra Σ β B α Σ BX B Σ X λ X 3 / 1

λ A Σ BA B Σ A Σ β λ initial λ -bialgebra Ba a β λ initial Σ A A BA Σ-algebra u Σ u Bu final Σ Z Z BZ B -coalgebra z α λ final λ -bialgebra B α λ Σ z Σ BZ B Σ Z λ Z 4 / 1

Benefits of the bialgebraic framework [Turi-Plotkin’97] + denotational model on the final B -coalgebra (by co-induction) + operational model on the initial Σ-algebra (by induction) + universal semantics (full-abstaction) initial algebra semantics = final coalgebra semantics + B -behavioural equivalence is a Σ-congruence + B -bisimilarity is a Σ-congruence (if B pres. weak pullbacks) 5 / 1

Congruential Rule Formats [Turi-Plotkin’97] Distributive laws can be specified as sets of derivation rules � � � � 1 ≤ j ≤ m a � � b ∈ B i a � b → y a − − → x i i x i ij 1 ≤ i ≤ n , a ∈ A i 1 ≤ i ≤ n (GSOS) c f ( x 1 , . . . , x n ) − → t image finite corresponds to. . . λ : Σ( Id × ( P fin ) L ) ⇒ ( P fin T Σ ) L 6 / 1

Discrete state sub-Probabilistic Systems . . . hence labelled sub-probabilistic Markov chains • a [ 1 a [ 2 3 ] 3 ] • • X → ( D fin X ) L in Set where D fin : Set → Set (sub-probability distribution functor) � D fin X = { ϕ : X → [0 , 1] | ϕ ( x ) ≤ 1 , | supp ( ϕ ) | < ∞} x ∈ X 7 / 1

� Rule Formats for Probabilistic Systems [Bartels’04]   a − → a ∈ A i , 1 ≤ i ≤ n x i       b   x i − → b ∈ B i , 1 ≤ i ≤ n   l j [ p j ] − − − → y j 1 ≤ i ≤ J x a j          c [ w · p 1 · ... · p J ]  f ( x 1 , . . . , x n ) − − − − − − − − → t image finite corresponds to. . . λ : Σ( Id × ( D fin ) L ) ⇒ ( D fin T Σ ) L where D fin : Set → Set (sub-probability distribution functor) � D fin X = { ϕ : X → [0 , 1] | ϕ ( x ) ≤ 1 , | supp ( ϕ ) | < ∞} x ∈ X 8 / 1

What if the state space is continuous? (example) Let us extend CCS with a quantitative operator P ::= nil | ( c of α ) . P | P + P | P � P ( c ∈ R ≥ 0 ) α ::= a | a | τ ( a ∈ A ) ( c of a ) . P a [?] a [?] (0 of a ) . P ( c of a ) . P . . . ideally we want that the outcomes are uniformly distributed. . . � b 1 � �� � { ( i of a ) . P | i ∈ [ a , b ] } (0 ≤ a ≤ b ≤ c ) U ( c of a ) . P = c dx a 9 / 1

Discrete state Continuous state (labelled Markov chains) (labelled Markov processes) • • a [ 1 a [ 2 a [ 1 a [ 2 3 ] 3 ] 3 ] 3 ] • • {• · · · •} {• · · · •} 10 / 1

Discrete state Continuous state (labelled Markov chains) (labelled Markov processes) • • a [ 1 a [ 2 a [ 1 a [ 2 3 ] 3 ] 3 ] 3 ] • • {• · · · •} {• · · · •} X → ( D fin X ) L in Set D fin : Set → Set (sub-probability distribution functor) � D fin X = { ϕ : X → [0 , 1] | ϕ ( x ) ≤ 1 , | supp ( ϕ ) | < ∞} x ∈ X 10 / 1

Discrete state Continuous state (labelled Markov chains) (labelled Markov processes) • • a [ 1 a [ 2 a [ 1 a [ 2 3 ] 3 ] 3 ] 3 ] • • {• · · · •} {• · · · •} X → ( D fin X ) L X → (∆ X ) L in Set in Meas D fin : Set → Set (sub-probability distribution functor) � D fin X = { ϕ : X → [0 , 1] | ϕ ( x ) ≤ 1 , | supp ( ϕ ) | < ∞} x ∈ X ∆: Meas → Meas (Giry functor) ∆ X = { µ : Σ X → [0 , 1] | µ sub-probability measure } 10 / 1

Aim: Congruential Rule Formats for Probabilistic Processes with Continuous State Spaces . . . hence, inducing distributive laws of type λ : Σ( Id × ∆ L ) ⇒ (∆ T Σ ) L 11 / 1

The shape of transitions The behaviour functor suggests the shape of transitions. . . Discrete state Continuous state a [ p ] a t ′ t t µ measure Σ-term on Σ-terms 12 / 1

Earlier attempts. . . [Cardelli-Mardare’10] 13 / 1

Earlier attempts. . . [Cardelli-Mardare’10] 13 / 1

Earlier attempts. . . [Cardelli-Mardare’10] rather ad (no general 13 / 1

Measure terms We adopt a new syntax to handle measures syntactically Σ: Meas → Meas (process syntax) M : Meas → Meas (measure syntax) a t µ it’s a M -term! 14 / 1

� Measure GSOS rule format a ij � � 1 ≤ j ≤ m i � � b ∈ B i b − − → µ ij − → x i x i 1 ≤ i ≤ n , a ij ∈ A i 1 ≤ i ≤ n c f ( x 1 , . . . , x n ) − → µ (MGSOS) where + f ∈ Σ with ar ( f ) = n ; + { x 1 , . . . , x n } and { µ ij | 1 ≤ i ≤ n , 1 ≤ j ≤ m i } are pairwise distinct process and measure variables ; + A i ∩ B i = ∅ are disjoint subsets of labels in L , and c ∈ L ; + µ is a M -term with variables in { x 1 , . . . , x n } and { µ ij | 1 ≤ i ≤ n , 1 ≤ j ≤ m i } . 15 / 1

Measure GSOS specification systems An MGSOS specification system consists of Set of MGSOS rules:   a ij � � 1 ≤ j ≤ m i � � b � b ∈ B i x i − − → µ ij x i − →   1 ≤ i ≤ n , a ij ∈ A i 1 ≤ i ≤ n R = c f ( x 1 , . . . , x n ) − → µ   image finte Measure terms interpretation: � | · | � : T M ∆ ⇒ ∆ T Σ 16 / 1

From MGSOS to labelled Markov processes We can obtain a ∆ L -coalgebra on the set of closed Σ-terms γ : T Σ 0 → ∆ L T Σ 0 as � � α γ ( t )( α ) = ⊕ T Σ 0 {� | µ | � T Σ 0 | t − → µ } where, for a finite set of U = { µ 1 , . . . , µ n } of sub-probability measures over X , ⊕ X ( { µ 1 , . . . , µ n } )( E ) = µ 1 ( E ) + · · · + µ n ( E ) µ 1 ( X ) + · · · + µ n ( X ) (weighted sum of sub-probability measures) 17 / 1

Example: Quantitative CCS Measure terms syntax: ( c , c ′ ∈ R ≥ 0 ) µ ::= U α c [ P ] | D [ P ] | µ | µ | µ � c , c ′ µ Measure GSOS Rules*: α, c τ − → D [ x ] ( c of α ) . x − − − → U α c [ x ] (0 of α ) . x α, c α, c − − − → µ − − − → µ x x α, c α, c x + x ′ x � x ′ → µ | D [ x ′ ] − − − → µ − − − α, c α, c ′ a , c a , c ′ x ′ → µ ′ x ′ → µ ′ − − − → µ − − − − − → µ − − − x x τ α, c + c ′ x � x ′ → µ � c , c ′ µ ′ − x � x ′ → µ | µ ′ − − − − − (*) dual rules for + and � are omitted 18 / 1

Example: Quantitative CCS Measure term interpretation: � | · | � X : T M ∆ X ⇒ ∆ T Σ X � 1 where E ′ = [0 , c ] ∩ ( λǫ. ( ǫ of α ) . x ) − 1 ( E ) � | U α c [ x ] | � X ( E ) = c dy E ′ � if x ∈ E 1 � | D [ x ] | � X ( E ) = 0 otherwise � X ) ◦ ( λ ( x , x ′ ) . x � x ′ ) − 1 ( E ) � | µ | µ ′ | � X ( E ) = ( � | µ | � X ⊗ � | µ ′ |  � X ( A 1 ) = c ′ · � | µ ′ | 1 if c · � | µ | � X ( A 2 ) ,   for A i = π i (( λ ( x , x ′ ) . x � x ′ ) − 1 ( E )) � | µ � c , c ′ µ ′ | � X ( E ) =   0 otherwise 19 / 1

From MGSOS to distributive laws Σ( Id × ∆ L ) how do we get the distributive law λ λ out of an MGSOS specification systems? (∆ T Σ ) L 20 / 1

From MGSOS to distributive laws Σ( Id × ∆ L ) 1. define the natural transformation � R � � R � from the image finite set MGSOS rules ( P fin T M ∆) L (∆ T Σ ) L 20 / 1

From MGSOS to distributive laws Σ( Id × ∆ L ) 1. define the natural transformation � R � � R � from the image finite set MGSOS rules ( P fin T M ∆) L � ) L 2. apply the measure terms interpretation ( P fin � | · | � | · | � : T M ∆ ⇒ ∆ T Σ ( P fin ∆ T Σ ) L (∆ T Σ ) L 20 / 1

From MGSOS to distributive laws Σ( Id × ∆ L ) 1. define the natural transformation � R � � R � from the image finite set MGSOS rules ( P fin T M ∆) L � ) L 2. apply the measure terms interpretation ( P fin � | · | � | · | � : T M ∆ ⇒ ∆ T Σ ( P fin ∆ T Σ ) L 3. obtain the actual measure by averaging ( ⊕ T Σ ) L ⊕ X ( { µ 1 , . . . , µ n } )( E ) = µ 1 ( E )+ ··· + µ n ( E ) µ 1 ( X )+ ··· + µ n ( X ) (∆ T Σ ) L 20 / 1