Convexity Meets Coalgebra in Probabilistic Systems Ana Sokolova - PowerPoint PPT Presentation

Convexity Meets Coalgebra in Probabilistic Systems Ana Sokolova Coalgebra Now @ FloC 2018

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 Coalgebra Now @ FloC 8.7.18 Ana Sokolova

Probabilistic systems coalgebraically a monad Probability distribution functor on Sets ÿ D X “ t ξ : X Ñ r 0 , 1 s | ξ p x q “ 1 , supp p ξ q is finite u x P X for we have D f : D X Ñ D Y by f : X Ñ Y ÿ ξ p x q “ ξ p f ´ 1 p y qq D f p ξ qp y q “ x P f ´ 1 p y q Coalgebra Now @ FloC 8.7.18 Ana Sokolova

Probabilistic systems coalgebraically a monad ! Probability distribution monad on Sets unit η X : X Ñ D X p D X, η , µ q µ X : DD X Ñ D X multiplication Dirac distribution η X p x q “ p x Þ Ñ 1 q ÿ µ X pp ξ i Þ Ñ p i qq “ p i ξ i convex combination Coalgebra Now @ FloC 8.7.18 Ana Sokolova

Probabilistic systems coalgebraically a monad ! Probability distribution monad on Sets unit η X : X Ñ D X p D X, η , µ q µ X : DD X Ñ D X multiplication η X p x q “ 1 x ÿ ÿ µ X p p i ξ i q “ p i ξ i Coalgebra Now @ FloC 8.7.18 Ana Sokolova

| | k " | " ✏ � | 3 3 k k } ! ✏ ✏ k Examples MC Generative PTS X ➝ D (X) X ➝ D (1 + A x X) x 1 1 1 a, 1 a, 1 x 1 3 2 1 2 2 6 x 2 x 3 x 2 x 3 x 4 b, 1 ✏ c, 1 PA 1 1 1 x 4 x 5 1 ✏ 1 X ➝ P ( D (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 Coalgebra Now @ FloC 8.7.18 Ana Sokolova

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 Coalgebra Now @ FloC 8.7.18 Ana Sokolova

✏ 
 ✏ ✏ ✏ ✏ ✏ 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 D A A • morphisms D A D h / D B D B D A h a ✏ b a b h / B A A B Coalgebra Now @ FloC 8.7.18 Ana Sokolova

Free Convex Algebras convex combinations as carried by distributions expected n n ÿ ÿ p i P r 0 , 1 s , p i “ 1 D X “ p D X, p i p´q i q wherever there i “ 1 i “ 1 are distributions, there is convexity ÿ ÿ p i ξ i “ ξ ô @ x P X. ξ p x q “ p i ξ i p x q finitely generated free convex algebras are simplexes Coalgebra Now @ FloC 8.7.18 Ana Sokolova

Convexity in Probabilistic Systems Semantics Coalgebra Now @ FloC 8.7.18 Ana Sokolova

✏ ✏ ! Traces Generative PTS D (1 + A x (-)) tr( x 1 )( ab ) = 1 tr( x 1 )( ac ) = 1 x 1 a, 1 a, 1 C 2 } { 4 C { C { 6 8 x 2 x 3 b, 1 c, 1 3 ✏ 2 x 4 x 5 1 ✏ 1 tr: X → D A ∗ c X Ñ FX Coalgebra Now @ FloC 8.7.18 Ana Sokolova

✏ ✏ ! Traces via determinisation Generative PTS D (1 + A x (-)) tr( x 1 )( ab ) = 1 tr( x 1 )( ac ) = 1 x 1 a, 1 a, 1 C 2 } { 4 C { C { 6 8 x 2 x 3 b, 1 c, 1 3 ✏ 2 x 4 x 5 1 ✏ 1 tr: X → D A ∗ trace = bisimilarity after determinisation c Happens in X Ñ FX convex algebra Coalgebra Now @ FloC 8.7.18 Ana Sokolova

Trace axioms for generative PTS Axioms for bisimilarity p a p 1 E 1 p 2 E 2 p 1 a p E 1 p 2 a p E 2 D c X Ñ FX soundness and Happens in [Silva, S. MFPS’11] completeness convex algebra Coalgebra Now @ FloC 8.7.18 Ana Sokolova

✏ ✏ ✏ � � Trace axioms for generative PTS Generative PTS D (1 + A x (-)) ‚ a, 1 a, 1 ‚ 2 � 4 a, 1 2 ✏ b, 1 c, 1 ‚ ‚ ‚ 3 � 4 b, 1 c, 1 3 ✏ 2 ‚ ‚ ‚ ‚ 1 ✏ 1 ✏ 1 1 ˚ ˚ ˚ ˚ D 1 1 1 1 c 1 c 1 a a 4 2 2 4 1 1 1 1 1 1 1 1 Cong a b 1 a c 1 a b 1 a c 1 2 3 4 2 2 3 2 4 c X Ñ FX 1 1 1 D a b 1 c 1 2 3 4 Coalgebra Now @ FloC 8.7.18 Ana Sokolova


 The quest for completeness Inspired lots of new research: • A. S., H. Woracek Congruences of convex algebras JPAA’15 
 if f.p. = f.g. and • S. Milius Proper functors CALCO’17 then completeness our axiomatisation would does not hold be proven complete if only one particular convex functor were c X Ñ FX proper it works ! [S., Woracek FoSSaCS’18] Coalgebra Now @ FloC 8.7.18 Ana Sokolova

Finitely generated, finitely presentable finitely generated (f.g.) = quotients of free finitely generated ones finitely presentable (f.p.) = quotients of free finitely generated ones under finitely generated congruences smallest congruence containing a finite set of pairs Theorem Every congruence of convex algebras is f.g. [S., Woracek JPAA’15] Hence f.p. = f.g. Coalgebra Now @ FloC 8.7.18 Ana Sokolova

✏ ✏ Proper semirings Ésik&Maletti 2010 A semiring is proper iff for every two equivalent states x ≡ y in WA with f.f.g. carriers, there is a zigzag of WA whose all nodes have f.f.g. carriers that relates them S n S m … free finitely generated S ˆ p S n q A S ˆ p S m q A Coalgebra Now @ FloC 8.7.18 Ana Sokolova

✏ ✏ Proper functors Milius 2017 behaviour F-coalgebras functor F on an equivalence Set T algebraic category A semiring is proper iff for every two equivalent states x ≡ y in WA with f.f.g. carriers, there is a zigzag of WA whose all nodes have f.f.g. carriers that relates them … Tn Tm free finitely generated FTm FTn Coalgebra Now @ FloC 8.7.18 Ana Sokolova

✏ ✏ Proper functors Milius 2017 Set T A functor F on an algebraic category , for a finitary monad T, is proper iff for every two behaviourally equivalent states x ≡ y in F-coalgebras with f.f.g. carriers, there is a zigzag of F-coalgebras whose all nodes have f.f.g. carriers that relates them. … Tn Tm free finitely generated FTm FTn Proper functors enable “easy” completeness proofs of axiomatizations of expression languages… proving properness is difficult Coalgebra Now @ FloC 8.7.18 Ana Sokolova

Previous results Proper: • Boolean semiring 
 Bloom & Ésik ‘93 • Finite commutative ordered semirings 
 Ésik & Kuich ‘01 7 2 • Eucledian domains, skew fields 
 Béal & Lombardy & Sakarovich ‘05 1 Béal & Lombardy & Sakarovich ‘05 • , skew fields 
 N , B , Z • Noetherian semirings, commutative rings, finite semirings 
 1 Ésik & Maletti ‘10 these are all known Improper: c X Ñ FX (im)proper semirings • Tropical semiring 
 Ésik & Maletti ‘10 Coalgebra Now @ FloC 8.7.18 Ana Sokolova

We have [S., Woracek FoSSaCS’18] Framework for proving properness 
 Instantiate it on known semirings Prove new semirings proper Q ` • Noetherian • Non-negative rationals 1 1 1 • Naturals 
 • Non-negative reals 
 N R ` 1 Prove new convex functors proper on positive r 0 , 1 s ˆ p´q A convex • 1 Set D c X Ñ FX algebras • F*, a subfunctor of the above 
 3 Coalgebra Now @ FloC 8.7.18 Ana Sokolova

✏ ! & x } ✏ ✏ Determinisations belief-state transformer Generative PTS X ➝ D (1 + A x X) x 1 a, 1 a, 1 x 1 a ✏ 2 2 2 x 2 ` 1 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] Coalgebra Now @ FloC 8.7.18 Ana Sokolova

✏ k " k 3 | Belief-state transformers MC X ➝ D (X) x 1 1 1 3 2 1 6 belief-state x 2 x 3 x 4 transformer 1 1 1 belief state 3 x 1 ` 2 1 3 x 2 . . . _ a ✏ ˆ 1 ˙ 7 9 x 2 ` 1 18 x 3 ` 1 1 3 x 2 ` 1 6 x 3 ` 1 ` 2 6 x 4 3 p 1 x 2 q 2 x 4 3 Coalgebra Now @ FloC 8.7.18 Ana Sokolova

k " ) v k 3 | | | � Belief-state transformers 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 8 9 x 2 ` 1 3 x 2 ` 1 2 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 Coalgebra Now @ FloC 8.7.18 Ana Sokolova

Probabilistic Automata Can be given different semantics: strong 1. Bisimilarity 
 probabilistic / bisimilarity combined 2. Convex bisimilarity 
 bisimilarity belief-state 3. Distribution bisimilarity bisimilarity Coalgebra Now @ FloC 8.7.18 Ana Sokolova

| | | 3 k 3 v 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 a a b b b b X ➝ ( Pc (X)+1) A on convex algebras 1 3 x 1 ` 2 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 Coalgebra Now @ FloC 8.7.18 Ana Sokolova