Complete Axiomatization for the Bisimilarity Distance on MCs - PowerPoint PPT Presentation

Complete Axiomatization for the Bisimilarity Distance on MCs Giorgio Bacci , Giovanni Bacci, Kim G. Larsen, and Radu Mardare Dept. of Computer Science, Aalborg University, DK CONCUR 2016

Introduction • Kleene’s Theorem : fundamental correspondence between regular expressions and DFAs • Salomaa‘66 , Kozen‘91 : complete axiomatization for proving equivalence of regular expressions • Milner‘84 : applied the above program on process behaviors and LTSs • Many variations of the above schema 2/25

Example: Markov chains Expressions: t,s := X | a .t | t + e s | rec X.t 3/25

Example: Markov chains names X ∈ 𝕐 Expressions: t,s := X | a .t | t + e s | rec X.t 3/25

Example: Markov chains names X ∈ 𝕐 Expressions: t,s := X | a .t | t + e s | rec X.t action-prefix a ∈ 𝔹 3/25

Example: Markov chains names probabilistic X ∈ 𝕐 choice Expressions: t,s := X | a .t | t + e s | rec X.t action-prefix a ∈ 𝔹 3/25

Example: Markov chains names probabilistic X ∈ 𝕐 choice Expressions: t,s := X | a .t | t + e s | rec X.t action-prefix recursion a ∈ 𝔹 3/25

Example: Markov chains names probabilistic X ∈ 𝕐 choice Expressions: t,s := X | a .t | t + e s | rec X.t action-prefix recursion a ∈ 𝔹 Kleene’s theorem for MCs a ,1/3 c ,1 t = rec X.( a .X + 1/3 b .s) b ,2/3 t s s = rec Y.( c .Y) 3/25

Example: Markov chains names probabilistic X ∈ 𝕐 choice Expressions: t,s := X | a .t | t + e s | rec X.t action-prefix recursion a ∈ 𝔹 finite MCs Kleene’s theorem for MCs a ,1/3 c ,1 t = rec X.( a .X + 1/3 b .s) b ,2/3 t s s = rec Y.( c .Y) 3/25

Example: Markov chains (B1) ⊢ t + 1 s = t (B2) ⊢ t + e t = t (SC) ⊢ t + e s = s + 1-e t (SA) ⊢ (t + e s) + e’ u = t + ee’ (s + e’-ee’ u) — for e,e’ ∈ [0,1) 1-ee’ (Unfold) ⊢ rec X.t = t[rec X.t / X] (Fix) {t = s[t / X]} ⊢ t = rec X.s — for X guarded in t (Unguard) ⊢ rec X.(t + e X) = rec X.t 4/25

Example: Markov chains Stark-Smolka axiomatization (B1) ⊢ t + 1 s = t (B2) ⊢ t + e t = t (SC) ⊢ t + e s = s + 1-e t (SA) ⊢ (t + e s) + e’ u = t + ee’ (s + e’-ee’ u) — for e,e’ ∈ [0,1) 1-ee’ (Unfold) ⊢ rec X.t = t[rec X.t / X] (Fix) {t = s[t / X]} ⊢ t = rec X.s — for X guarded in t (Unguard) ⊢ rec X.(t + e X) = rec X.t 4/25

Example: Markov chains Stark-Smolka axiomatization Stone’s barycentric axioms (B1) ⊢ t + 1 s = t (B2) ⊢ t + e t = t (SC) ⊢ t + e s = s + 1-e t (SA) ⊢ (t + e s) + e’ u = t + ee’ (s + e’-ee’ u) — for e,e’ ∈ [0,1) 1-ee’ (Unfold) ⊢ rec X.t = t[rec X.t / X] (Fix) {t = s[t / X]} ⊢ t = rec X.s — for X guarded in t (Unguard) ⊢ rec X.(t + e X) = rec X.t 4/25

Example: Markov chains Stark-Smolka axiomatization Stone’s barycentric axioms (B1) ⊢ t + 1 s = t (B2) ⊢ t + e t = t (SC) ⊢ t + e s = s + 1-e t (SA) ⊢ (t + e s) + e’ u = t + ee’ (s + e’-ee’ u) — for e,e’ ∈ [0,1) 1-ee’ Milner’s recursion axioms (Unfold) ⊢ rec X.t = t[rec X.t / X] (Fix) {t = s[t / X]} ⊢ t = rec X.s — for X guarded in t (Unguard) ⊢ rec X.(t + e X) = rec X.t 4/25

…for probabilistic systems • Generative Markov chains: 
 Baeten-Bergstra-Smolka‘95 & Stark-Smolka‘00 • Simple Probabilistic Automata: 
 Bandini-Segala‘01 • (fully) Probabilistic Automata: 
 Mislove-Ouaknine-Worrell‘04 (strong-bisimulation) 
 Deng-Palamidessi‘07 (weak-bisimulation & behavioral eq.) • Quantitative Kleene Coalgebras: 
 Silva-Bonchi-Bonsangue-Rutten‘11 (coagebraic bisim.) 5/25

In this talk… 6/25

In this talk… • From equivalences to distances: we present 6/25

In this talk… • From equivalences to distances: we present • a sound & complete axiomatization for the bisimilarity distance of Desharnais et al. 6/25

In this talk… • From equivalences to distances: we present • a sound & complete axiomatization for the bisimilarity distance of Desharnais et al. • and a quantitative Kleene’s Theorem 
 for generative Markov chains 6/25

In this talk… • From equivalences to distances: we present • a sound & complete axiomatization for the bisimilarity distance of Desharnais et al. • and a quantitative Kleene’s Theorem 
 for generative Markov chains • How do we do it? 
 By using Quantitative Equational Theories* of Mardare-Panangaden-Plotkin (LICS’16) 6/25

In this talk… • From equivalences to distances: we present • a sound & complete axiomatization for the bisimilarity distance of Desharnais et al. • and a quantitative Kleene’s Theorem 
 for generative Markov chains • How do we do it? 
 By using Quantitative Equational Theories* of Mardare-Panangaden-Plotkin (LICS’16) s = t s = ε t 6/25

In this talk… • From equivalences to distances: we present • a sound & complete axiomatization for the bisimilarity distance of Desharnais et al. • and a quantitative Kleene’s Theorem 
 for generative Markov chains • How do we do it? 
 By using Quantitative Equational Theories* of Mardare-Panangaden-Plotkin (LICS’16) s s e n e t e l p m o c s = t s = ε t ! e e r f r o f t s o m l a 6/25

Equational Theories {t i = s i | i ∈ I} ⊢ t = s inference 7/25

Equational Theories {t i = s i | i ∈ I} ⊢ t = s inference (Refl) ⊢ t = t (Symm) {t = s} ⊢ s = t (Trans) {t = u, u = s} ⊢ t = s (Cong) {t 1 = s 1 ,…,t n = s n } ⊢ f(t 1 ,…,t n ) = f(s 1 ,…s n ) — for f ∈ Σ 7/25

Quantitative Theories Mardare-Panangaden-Plotkin (LICS’16) {t i = ε s i | i ∈ I} ⊢ t = ε s ε i quantitative inference (Refl) ⊢ t = 0 t (Symm) {t = ε s} ⊢ s = ε t (Triang) {t = ε u, u = δ s} ⊢ t = ε + δ s (NExp) {t 1 = ε s 1 ,…,t n = ε s n } ⊢ f(t 1 ,…,t n ) = ε f(s 1 ,…s n ) — for f ∈ Σ (Max) {t = ε s} ⊢ t = ε + δ s — for δ >0 (Arch) {t = δ s | δ > ε } ⊢ t = ε s 8/25

Quantitative Semantics Quantitative Algebra (A, Σ A ) — Universal algebra 𝓑 = (A, Σ A ,d A ) (A,d A ) — metric space Satisfiability ( ) 𝓑 ⊨ {t i = ε s i | i ∈ I} ⊢ t = ε s ε i iff for all i ∈ I. d A ( ⟦ t i ⟧ , ⟦ s i ⟧ ) ≤ ε i implies d A ( ⟦ t ⟧ , ⟦ s ⟧ ) ≤ ε 9/25

completeness quantitative quantitative algebra theory ( ) ( ) 𝓑 ⊨ ⊢ t = ε s ⊢ t = ε s ∈ 𝓥 soundness 10/25

completeness quantitative quantitative algebra theory ( ) ( ) 𝓑 ⊨ ⊢ t = ε s ⊢ t = ε s ∈ 𝓥 𝓑 MC 𝓥 MC soundness 10/25

The Quantitative Universal Algebra 11/25

Universal Algebra of MCs Signature: X : 0 | a.- : 1 | + e : 2 | rec X : 1 12/25

Universal Algebra of MCs Signature: X : 0 | a.- : 1 | + e : 2 | rec X : 1 (X) MC = X 12/25

Universal Algebra of MCs Signature: X : 0 | a.- : 1 | + e : 2 | rec X : 1 (X) MC = X a.m a ,1 m ( a . ) MC = ℳ m ℳ 12/25

Universal Algebra of MCs Signature: X : 0 | a.- : 1 | + e : 2 | rec X : 1 m+ e n n m 𝜉 (X) MC = ( + e ) MC = 𝜈 X e 𝜈 +(1-e) 𝜉 ℳ 𝒪 + ℳ 𝒪 a.m a ,1 m ( a . ) MC = ℳ m ℳ 12/25

Universal Algebra of MCs Signature: X : 0 | a.- : 1 | + e : 2 | rec X : 1 m+ e n n m 𝜉 (X) MC = ( + e ) MC = 𝜈 X e 𝜈 +(1-e) 𝜉 ℳ 𝒪 + ℳ 𝒪 m recX.m a.m 𝜈 𝜈 a ,1 m ( a . ) MC = (rec X. ) MC = ℳ e 𝜈 m ℳ ℳ ℳ e 1-e 1-e X 12/25

Bisimilarity distance for MCs (Desharnais et al. TCS’04) it is the least 1-bounded pseudometric satisfying m n d MC ( , ) = min { ∫ Λ (d MC ) d ω | ω ∈ Ω ( 𝜈 , 𝜉 ) } 𝜉 𝜈 ℳ 𝒪 13/25

Bisimilarity distance for MCs (Desharnais et al. TCS’04) it is the least 1-bounded pseudometric satisfying m n d MC ( , ) = min { ∫ Λ (d MC ) d ω | ω ∈ Ω ( 𝜈 , 𝜉 ) } 𝜉 𝜈 ℳ 𝒪 Kantorovich lifting 13/25

Bisimilarity distance for MCs (Desharnais et al. TCS’04) couplings it is the least 1-bounded pseudometric satisfying = probabilistic “relations” m n d MC ( , ) = min { ∫ Λ (d MC ) d ω | ω ∈ Ω ( 𝜈 , 𝜉 ) } 𝜉 𝜈 ℳ 𝒪 Kantorovich lifting 13/25

Bisimilarity distance for MCs (Desharnais et al. TCS’04) couplings it is the least 1-bounded pseudometric satisfying = probabilistic “relations” m n d MC ( , ) = min { ∫ Λ (d MC ) d ω | ω ∈ Ω ( 𝜈 , 𝜉 ) } 𝜉 𝜈 ℳ 𝒪 Kantorovich lifting Λ (d MC ) — greatest 1-bounded pseudometric on ( 𝔹 × MC) ∪ 𝕐 m n m n s.t, for all a ∈ 𝔹 , Λ (d MC )(( a , ),( a , )) = d MC ( , ) ℳ ℳ 𝒪 𝒪 13/25

Running example a ,1/2 a ,1/3 d MC ? m n 1/2 2/3 Z Z m = rec X. ( a .X + 1/2 Z) n = rec Y. ( a .Y + 1/3 Z) 14/25