Probabilities in Higher-Order Languages Valeria Vignudelli University of Bologna/Inria FOCUS (Joint work with Marco Bernardo, University of Urbino and Davide Sangiorgi, University of Bologna/Inria FOCUS) June 21, 2014 Vignudelli (Bologna-Inria) Probabilities in H-O June 21, 2014 1 / 24
Higher-order languages Variables may be instantiated with terms of the language itself (e.g. terms can be copied) functions ( λ -calculus) → M { N / ( λ x . M ) N − x } higher-order communication (Higher-Order π -calculus) → M { N / a ( x ) . M | a � N � . R − x } | R action on locations (kells), as with passivation → N { M / [ [ M ] ] l | pass l ( x ) . N − x } M is a running term [Schmitt,Stefani GC’04, Lenglet et al. Inf.Comp.’11, Pi´ erard,Sumii FOSSACS’12, Koutavas,Hennessy CONCUR’13] Vignudelli (Bologna-Inria) Probabilities in H-O June 21, 2014 2 / 24
Probabilistic processes nondeterministic (LTSs) Vs. reactive probabilistic (RPLTSs) P P a a a ∆ 1 2 • • 3 3 b d b • • b d • • • b • • • c c • • external & internal nondeterminism only external nondeterminism [Larsen,Skou POPL’89, van Breugel et al. Theor.Comp.Sci.’05] Vignudelli (Bologna-Inria) Probabilities in H-O June 21, 2014 3 / 24
A contextual/testing approach Discriminating power of higher-order languages and interplay between probabilities, concurrency, higher-order operators Plan of the talk 1 main ingredients 2 results in the nondeterministic setting 3 results in the probabilistic setting [Bernardo,Sangiorgi,Vignudelli CSL-LICS’14] Vignudelli (Bologna-Inria) Probabilities in H-O June 21, 2014 4 / 24
The discriminating power of a language A testing language L , a set of tested terms P , Q . . . � P ≃ L Q C [ P ] and C [ Q ] ‘equally successful’ , ∀ contexts C of L Otherwise: P , Q are discriminated by L Two classes of first-order (CCS-like) processes as tested terms P , Q : nondeterministic reactive probabilistic Comparison of testing languages L with different constructs: sequential higher-order ( λ -calculus) higher-order communication (HO π ) ordinary first-order concurrency (CCS) passivation no probabilities in L refusal Vignudelli (Bologna-Inria) Probabilities in H-O June 21, 2014 5 / 24
Higher-order sequential languages: Kell λ -calculus ( K Λ) M is evaluated in kell N (of process type) → � N ′ ; M ′ � � N ; M � − Contexts interact with the kell by: testing actions → � P ′ ; true � a → P ′ � P ; a ? � − if P − � a � P ; a ? � − → � P ; false � if P − → reading (passivating) and rewriting the kell � P ; � P ′ ; M �� − → � P ′ ; M � � P ; pass � − → � P ; P � Variants: - call by name ( K Λ N ) and call by value ( K Λ V ) - refusal free ( K Λ N − ref , K Λ V − ref ) Vignudelli (Bologna-Inria) Probabilities in H-O June 21, 2014 6 / 24
Concurrent languages Interactions between context and process: action synchronization (CCS) P ′ | M a → P ′ P | a . M − → if P − higher-order communication (HO π ) → M { P / a ( x ) . M | a � P � . N − x } | N refusal on kells (CCS ref , HO π ref ) � a [ [ P ] ] l | � a l . M − → [ [ P ] ] l | M if P − → passivation of kells (HO π pass , HO π pass , ref ) → M { P / [ [ P ] ] l | pass l ( x ) . M − x } Vignudelli (Bologna-Inria) Probabilities in H-O June 21, 2014 7 / 24
Testing nondeterministic processes Vignudelli (Bologna-Inria) Probabilities in H-O June 21, 2014 8 / 24
‘Success’ on nondeterministic processes � P ≃ L Q C [ P ] and C [ Q ] ‘equally successful’ , ∀ contexts C of L C [ P ] P is an LTS, C [ P ] ⇓ • • Success states • are: ω − → in CCS , HO π M • • • � N ; true � in K Λ • Success is ⇓ ( = a success state • is reachable) [may success] Vignudelli (Bologna-Inria) Probabilities in H-O June 21, 2014 9 / 24
The Spectrum on LTSs ready simulation equivalence K Λ V = HO π ref , pass simulation equivalence failure trace equivalence K Λ V − ref = HO π pass K Λ N = HO π ref = CCS ref K Λ N − ref = HO π = CCS trace equivalence CBV = passivation in HO π sequential = concurrent first-order communication = higher-order communication Vignudelli (Bologna-Inria) Probabilities in H-O June 21, 2014 10 / 24
Call By Name & Call By Value P does a a T a � a ? � P ; a ? � ⇓ − → iff P P refuses a � a T ¬ a � if a ? then false else true � P ; T ¬ a � ⇓ iff P − → P passes T 1 and T 2 in sequence ⇒� P ′ ; true � ∧ � P ′ ; T 2 �⇓ � P ; Seq ( λ. T 1 )( λ. T 2 ) �⇓ iff � P ; T 1 � = � for Seq λ x .λ y . if x ⋆ then y ⋆ else false λ. M thunking M ⋆ unthunking Vignudelli (Bologna-Inria) Probabilities in H-O June 21, 2014 11 / 24
Example: discriminating LTSs Q P a a a P 1 P 2 Q 1 b c b b c P 3 P 4 P 5 Q 2 Q 3 C [ P ] ⇓ C [ Q ] �⇓ C = � · ; Seq T a T ¬ c � completed trace/simulation equivalent but not failure equivalent processes [see van Glabbeek ’90 spectrum] Vignudelli (Bologna-Inria) Probabilities in H-O June 21, 2014 12 / 24
Call By Value P passes both T 1 and T 2 � P ; And ( λ. T 1 )( λ. T 2 ) � ⇓ iff � P ; T 1 � ⇓ ∧ � P ; T 2 � ⇓ � for λ x .λ y . (( λ z . if x ⋆ then � z ; y ⋆ � else false) pass ) And a b C [ P ] ⇓ iff P − → ∧ P − → � P ; ( λ x . if a ? then � x ; b ? � else false) pass � − → � P ; ( λ x . if a ? then � x ; b ? � else false) P � − → � P ; if a ? then � P ; b ? � else false � → � P ′ ; if true then � P ; b ? � else false � − → � P ′ ; � P ; b ? �� − − → � P ; b ? � ⇓ Vignudelli (Bologna-Inria) Probabilities in H-O June 21, 2014 13 / 24
Call By Value P passes both T 1 and T 2 � P ; And ( λ. T 1 )( λ. T 2 ) � ⇓ iff � P ; T 1 � ⇓ ∧ � P ; T 2 � ⇓ � for λ x .λ y . (( λ z . if x ⋆ then � z ; y ⋆ � else false) pass ) And a b C [ P ] ⇓ iff P − → ∧ P − → � P ; ( λ x . if a ? then � x ; b ? � else false) pass � − → � P ; ( λ x . if a ? then � x ; b ? � else false) P � − → � P ; if a ? then � P ; b ? � else false � → � P ′ ; if true then � P ; b ? � else false � − → � P ′ ; � P ; b ? �� − − → � P ; b ? � ⇓ Vignudelli (Bologna-Inria) Probabilities in H-O June 21, 2014 13 / 24
Call By Value P passes both T 1 and T 2 � P ; And ( λ. T 1 )( λ. T 2 ) � ⇓ iff � P ; T 1 � ⇓ ∧ � P ; T 2 � ⇓ � for λ x .λ y . (( λ z . if x ⋆ then � z ; y ⋆ � else false) pass ) And a b C [ P ] ⇓ iff P − → ∧ P − → � P ; ( λ x . if a ? then � x ; b ? � else false) pass � − → � P ; ( λ x . if a ? then � x ; b ? � else false) P � − → � P ; if a ? then � P ; b ? � else false � → � P ′ ; if true then � P ; b ? � else false � − → � P ′ ; � P ; b ? �� − − → � P ; b ? � ⇓ Vignudelli (Bologna-Inria) Probabilities in H-O June 21, 2014 13 / 24
Call By Value P passes both T 1 and T 2 � P ; And ( λ. T 1 )( λ. T 2 ) � ⇓ iff � P ; T 1 � ⇓ ∧ � P ; T 2 � ⇓ � for λ x .λ y . (( λ z . if x ⋆ then � z ; y ⋆ � else false) pass ) And a b C [ P ] ⇓ iff P − → ∧ P − → � P ; ( λ x . if a ? then � x ; b ? � else false) pass � − → � P ; ( λ x . if a ? then � x ; b ? � else false) P � − → � P ; if a ? then � P ; b ? � else false � → � P ′ ; if true then � P ; b ? � else false � − → � P ′ ; � P ; b ? �� − − → � P ; b ? � ⇓ Vignudelli (Bologna-Inria) Probabilities in H-O June 21, 2014 13 / 24
Call By Value P passes both T 1 and T 2 � P ; And ( λ. T 1 )( λ. T 2 ) � ⇓ iff � P ; T 1 � ⇓ ∧ � P ; T 2 � ⇓ � for λ x .λ y . (( λ z . if x ⋆ then � z ; y ⋆ � else false) pass ) And a b C [ P ] ⇓ iff P − → ∧ P − → � P ; ( λ x . if a ? then � x ; b ? � else false) pass � − → � P ; ( λ x . if a ? then � x ; b ? � else false) P � − → � P ; if a ? then � P ; b ? � else false � → � P ′ ; if true then � P ; b ? � else false � − → � P ′ ; � P ; b ? �� − − → � P ; b ? � ⇓ Vignudelli (Bologna-Inria) Probabilities in H-O June 21, 2014 13 / 24
Call By Value P passes both T 1 and T 2 � P ; And ( λ. T 1 )( λ. T 2 ) � ⇓ iff � P ; T 1 � ⇓ ∧ � P ; T 2 � ⇓ � for λ x .λ y . (( λ z . if x ⋆ then � z ; y ⋆ � else false) pass ) And a b C [ P ] ⇓ iff P − → ∧ P − → � P ; ( λ x . if a ? then � x ; b ? � else false) pass � − → � P ; ( λ x . if a ? then � x ; b ? � else false) P � − → � P ; if a ? then � P ; b ? � else false � → � P ′ ; if true then � P ; b ? � else false � − → � P ′ ; � P ; b ? �� − − → � P ; b ? � ⇓ Vignudelli (Bologna-Inria) Probabilities in H-O June 21, 2014 13 / 24
Example: discriminating LTSs Q P a a a P 1 P 2 Q 1 b b b b P 3 P 4 Q 2 Q 3 c c d d P 5 P 6 Q 4 Q 5 C [ P ] �⇓ C [ Q ] ⇓ C = � · ; Seq T a ( And ( Seq T b T c )( Seq T b T d )) � ready trace equivalent but not simulation equivalent processes Vignudelli (Bologna-Inria) Probabilities in H-O June 21, 2014 14 / 24
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