The Power of Convex Algebra Ana Sokolova CONCUR 17 Filippo Bonchi - PowerPoint PPT Presentation

The Power of Convex Algebra Ana Sokolova CONCUR ’17 Filippo Bonchi Alexandra Silva NII Shonan Meeting “Enhanced Coinduction” 15.11.17

probabilistic automata The true nature of PA as transformers of belief states Ana Sokolova Shonan 15-11-17

g k ✏ " ✏ | Determinisations NFA X ➝ 2 x ( P (X)) A x 1 x 1 a a a ✏ b / x 3 b x 2 , x 3 x 2 x 3 ˚ b ˚ [Silva, Bonchi, Bonsangue, Rutten, FSTTCS’10] Ana Sokolova Ana Sokolova Shonan 15-11-17

✏ ✏ ✏ } x ! & Determinisations Generative PTS X ➝ D (1 + A x X) x 1 a, 1 x 1 a, 1 a ✏ 2 2 1 2 x 2 ` 1 2 x 3 x 2 x 3 b c b, 1 ✏ c, 1 1 1 x 4 x 5 2 x 4 2 x 5 1 ✏ 1 1 1 2 ✏ 2 ˚ ˚ ˚ ˚ [Silva, S. MFPS’11] c X Ñ FX [Jacobs, Silva, S. JCSS’15] Ana Sokolova Shonan 15-11-17

u " k 3 | k � | | Determinisations PA X ➝ ( PD (X)) A b x 1 a a 2 1 1 1 belief-state 3 3 " 2 2 " transformer x 2 x 3 x 4 belief state a b b 1 3 x 1 ` 2 3 x 2 . . . + c ⌘ a ( a X Ñ FX 2 3 x 2 ` 1 6 x 3 ` 1 9 x 2 ` 1 8 6 x 4 . . . 9 x 3 Ana Sokolova Shonan 15-11-17

k " ) v k 3 | | | � Belief-state transformer PA foundation ? X ➝ ( PD (X)) A how does it emerge? b x 1 a a 2 1 1 1 3 3 " 2 2 " what is it? x 2 x 3 x 4 a b b 1 3 x 1 ` 2 3 x 2 . . . - ◆ a a c X Ñ FX 8 9 x 2 ` 1 2 3 x 2 ` 1 6 x 3 ` 1 9 x 3 . . . 6 x 4 Ana Sokolova Shonan 15-11-17

| | ) " � v | k k 3 Belief-state transformer „ „ „ d b x 1 a a 2 1 1 1 3 3 " 2 2 " x 2 x 3 x 4 a b b 1 3 x 1 ` 2 3 x 2 . . . - ◆ a a 8 9 x 2 ` 1 2 3 x 2 ` 1 6 x 3 ` 1 ˆ 2 ˙ 9 x 3 . . . 6 x 4 1 3 x 2 ` 1 ` 2 3 p 1 x 2 q 3 x 3 3 Ana Sokolova Shonan 15-11-17

| | ) " � v | k k 3 Belief-state transformer „ „ „ d b x 1 a a 2 1 1 1 3 3 " 2 2 " x 2 x 3 x 4 a b b very infinite LTS on belief states 3 x 1 ` 2 1 3 x 2 . . . - ◆ a a 9 x 2 ` 1 8 3 x 2 ` 1 2 6 x 3 ` 1 ˆ 1 ˙ 1 2 x 3 ` 1 ` 2 9 x 3 . . . 6 x 4 3 p 1 x 2 q 2 x 4 3 Ana Sokolova Shonan 15-11-17

Probabilistic Automata Can be given different semantics: strong 1. Bisimilarity 
 probabilistic / bisimilarity combined 2. Convex bisimilarity 
 bisimilarity belief-state 3. Distribution bisimilarity bisimilarity Ana Sokolova Shonan 15-11-17

Bisimilarity largest „ bisimulation „ bisimulation R a a μ ৵ ” R lifting of R to distributions assign the same probability to “R-classes” Ana Sokolova Shonan 15-11-17

Convex bisimilarity „ „ „ c largest convex bisimulation convex bisimulation combined R transition a a μ ৵ ” R convex combination of a -steps Ana Sokolova Shonan 15-11-17

Distribution bisimilarity „ „ „ d largest distribution bisimulation distribution bisimulation μ ৵ R transition in a a the belief-state „ d transformer μ ’ R ৵ ’ is LTS bisimilarity on the belief-state transformer [Hermanns, Krcal, Kretinsky CONCUR’13] ˆ 1 ˙ 1 2 x 3 ` 1 ` 2 3 p 1 x 2 q 2 x 4 3 Ana Sokolova Shonan 15-11-17

k " ) v k 3 | | | � Belief-state transformer PA foundation ? X ➝ ( PD (X)) A how does it emerge? b x 1 a a 2 1 1 1 3 3 " 2 2 " what is it? x 2 x 3 x 4 a b b 1 3 x 1 ` 2 3 x 2 . . . - ◆ a a c X Ñ FX 8 9 x 2 ` 1 2 3 x 2 ` 1 6 x 3 ` 1 9 x 3 . . . 6 x 4 Ana Sokolova Shonan 15-11-17

Coalgebras Uniform framework for dynamic transition systems, based on category theory. c X Ñ FX generic notion of behavioural equivalence « behaviour form a states type category too functor on the object in the base base category C category C CoAlg C p F q Ana Sokolova Shonan 15-11-17

! ✏ 3 k � k " } | ✏ | ✏ g | " ✏ k | Examples Generative PTS NFA X ➝ D (1 + A x X) X ➝ 2 x ( P (X)) A a, 1 x 1 a, 1 x 1 x 1 2 2 a a a A a A } A } ~ } x 2 x 3 x 2 x 3 b x 2 x 3 b, 1 ✏ c, 1 PA x 4 x 5 b ˚ 1 ✏ 1 X ➝ ( PD (X)) A ˚ ˚ b all on x 1 Sets a a c X Ñ FX 2 1 1 1 3 3 " 2 2 " x 2 x 3 x 4 a b b Ana Sokolova Shonan 15-11-17

v 3 3 k | k k � " ) | | k | " � | | PA coalgebraically X ➝ ( P D (X)) A X ➝ ( C (X)) A on and all convex Sets combinations b b x 1 x 1 „ c “ « … „ “ « a a a a 2 1 1 1 2 1 1 1 3 3 " 2 2 " 3 3 " 2 2 " x 2 x 3 x 4 x 2 x 3 x 4 Mio a a b b b b FoSSaCS ’14 X ➝ ( Pc (X)+1) A on convex algebras EM p D q 3 x 1 ` 2 1 3 x 2 . . . - ◆ a a „ d “ « 9 x 2 ` 1 8 2 3 x 2 ` 1 6 x 3 ` 1 9 x 3 . . . 6 x 4 Ana Sokolova Shonan 15-11-17

Convex Algebras binary ones infinitely many “suffice” convex finitary operations combinations n n ÿ ÿ p A, p i p´q i q p i P r 0 , 1 s , p i “ 1 • algebras 
 i “ 1 i “ 1 • convex (affine) maps satisfying ˜ n ¸ n n ÿ ÿ ÿ “ p i h p a i q Projection h p i a i p k “ 1 • p i a i “ a k , i “ 1 i “ 1 i “ 1 ˜ m ˜ n ¸ ¸ n m ÿ ÿ ÿ ÿ Barycenter • p i p i,j a j p i p i,j a j “ i “ 1 j “ 1 j “ 1 i “ 1 Ana Sokolova Shonan 15-11-17

✏ 
 ✏ ✏ ✏ ✏ ✏ Eilenberg-Moore Algebras EM p D q convex algebras abstractly • objects 
 satisfying D A η / D A µ / D A a A DD A A a D a ✏ a a / A a A D A • morphisms D A D h / D B D B D A h b a ✏ a b h / B A A B Ana Sokolova Shonan 15-11-17

✏ Belief-state transformers DD X coalgebras on D X “ µ free convex algebras D X convex combinations free convex constant exponent D S Ñ p P c p D S q ` 1 q A algebra termination nonempty convex powerset Minkowski sum pA 1 ` p 1 ´ p q A 2 “ t pa 1 ` p 1 ´ p q a 2 | a 1 P A 1 , a 2 P A 2 u Ana Sokolova Shonan 15-11-17

k " ) v k 3 | | | � Belief-state transformer PA foundation ? X ➝ ( PD (X)) A how does it emerge? b x 1 a a 2 1 1 1 3 3 " 2 2 " what is it? x 2 x 3 x 4 a b b 1 3 x 1 ` 2 3 x 2 . . . - ◆ a a c X Ñ FX 8 9 x 2 ` 1 2 3 x 2 ` 1 6 x 3 ` 1 9 x 3 . . . 6 x 4 Ana Sokolova Shonan 15-11-17

k " ) v k 3 | | | � Belief-state transformer PA foundation ? X ➝ ( PD (X)) A how does it emerge? b x 1 a a 2 1 1 1 3 3 " 2 2 " coalgebra over free x 2 x 3 x 4 convex algebra a b b 1 3 x 1 ` 2 3 x 2 . . . - ◆ a a c X Ñ FX 8 9 x 2 ` 1 2 3 x 2 ` 1 6 x 3 ` 1 9 x 3 . . . 6 x 4 Ana Sokolova Shonan 15-11-17

3 + Determinisations I forget determinise CoAlg EM p T q p F q CoAlg C p FT q CoAlg C p F q needed: a lifting works for NFA not for generative PTS not for PA / belief-state [Silva, Bonchi, Bonsangue, Rutten, FSTTCS’10] transformer Ana Sokolova Shonan 15-11-17

+ 3 Determinisations II forget determinise CoAlg EM p T q p G q CoAlg C p TF q CoAlg C p G q needed: a lifting works for generative PTS [Silva, S. MFPS’11] not for PA / belief-state [Jacobs, Silva, S JCSS’15] transformer Ana Sokolova Shonan 15-11-17

3 + Determinisations III forget determinise CoAlg EM p T q p H q CoAlg C p F q CoAlg C p G q needed: a quasi lifting and a lax lifting ( Pc +1) A works for PA /belief-state on EM p D q transformer is a quasi lifting and lax lifting of C A P A Sets and on Ana Sokolova Shonan 15-11-17

k " ) v k 3 | | | � Belief-state transformer PA foundation ? X ➝ ( PD (X)) A how does it emerge? b x 1 a a 2 1 1 1 3 3 " 2 2 " coalgebra over free x 2 x 3 x 4 convex algebra a b b 1 3 x 1 ` 2 3 x 2 . . . - ◆ a a c X Ñ FX 8 9 x 2 ` 1 2 3 x 2 ` 1 6 x 3 ` 1 9 x 3 . . . 6 x 4 Ana Sokolova Shonan 15-11-17

k " ) v k 3 | | | � Belief-state transformer PA foundation ? X ➝ ( PD (X)) A via a generalised 3 determinisation b x 1 a a 2 1 1 1 3 3 " 2 2 " coalgebra over free x 2 x 3 x 4 convex algebra a b b 1 3 x 1 ` 2 3 x 2 . . . - ◆ a a c X Ñ FX 8 9 x 2 ` 1 2 3 x 2 ` 1 6 x 3 ` 1 9 x 3 . . . 6 x 4 Ana Sokolova Shonan 15-11-17

k " ) v k 3 | | | � Belief-state transformer PA are natural indeed X ➝ ( PD (X)) A via a generalised 3 determinisation b x 1 a a 2 1 1 1 3 3 " 2 2 " coalgebra over free x 2 x 3 x 4 convex algebra a b b 1 3 x 1 ` 2 3 x 2 . . . - ◆ a a c X Ñ FX 8 9 x 2 ` 1 2 3 x 2 ` 1 6 x 3 ` 1 9 x 3 . . . 6 x 4 Ana Sokolova Shonan 15-11-17

Coinductive proof method „ for distribution bisimilarity „ bisimulation up-to convex hull to prove μ ~ d ৵ it suffices to find a μ R ৵ bisimulation up-to convex hull R a a with μ R ৵ f.p. = f.g. conv-con p R q there always μ ’ ৵ ’ for (positive) exists a finite convex algebras one! convex and equivalence [S., Woracek JPAA’15] closure of R Ana Sokolova Shonan 15-11-17

k " ) v k 3 | | | � Belief-state transformer PA are natural indeed X ➝ ( PD (X)) A via a generalised 3 determinisation b x 1 a a Thank You! 2 1 1 1 3 3 " 2 2 " a coalgebra over x 2 x 3 x 4 free convex algebra a b b 1 3 x 1 ` 2 sound proof 3 x 2 . . . - ◆ method for a a c X Ñ FX distribution 8 9 x 2 ` 1 2 3 x 2 ` 1 6 x 3 ` 1 9 x 3 . . . 6 x 4 bisimilarity Ana Sokolova Shonan 15-11-17