The Complexity of Computing a Bisimilarity ... - Semantic Scholar
The Complexity of Computing a Bisimilarity Pseudometric on Probabilistic Automata Franck van Breugel Joint work with James Worrell
May 23, 2014
1 / 22
Franck van Breugel
Bisimilarity Pseudometric on Probabilistic Automata
Probabilistic Automaton
3 1 2
1 4
1
2 1 2
3 4
4
A probabilistic automaton contains nondeterministic and probabilistic choices. 2 / 22
Franck van Breugel
Bisimilarity Pseudometric on Probabilistic Automata
Probabilistic Bisimilarity
3 1 2
1 4
1
2 1 2
3 4
4
Probabilistic bisimilarity captures which states of the automaton behave the same. 3 / 22
Franck van Breugel
Bisimilarity Pseudometric on Probabilistic Automata
Robero Segala Roberto Segala, in collaboration with Nancy Lynch, introduced probabilistic bisimilarity for probabilistic automata.
4 / 22
Franck van Breugel
Bisimilarity Pseudometric on Probabilistic Automata
Probabilistic Bisimilarity is not Robust
3 1 2
1 2
+ε
1
2 1 2
1 2
−ε
4
States 1 and 2 are not bisimilar for all ε > 0. 5 / 22
Franck van Breugel
Bisimilarity Pseudometric on Probabilistic Automata
Scott Smolka
Scott Smolka, in collaboration with Alessandro Giacalone and Chi-chang Jou, first suggested to use pseudometrics instead of equivalence relations.
6 / 22
Franck van Breugel
Bisimilarity Pseudometric on Probabilistic Automata
From Equivalence Relations to Pseudometrics
An equivalence relation on a set S can be viewed as function in S×S →B A (1-bounded) pseudometric on a set S is a function in S × S → [0, 1] Equivalence is captured by distance zero.
7 / 22
Franck van Breugel
Bisimilarity Pseudometric on Probabilistic Automata
A Metric for Nondeterministic Choices
Nondeterministic choices can be modelled as subsets of a set. The distance of the subsets A and B is defined by d(A, B) = max max min d(a, b), max min d(b, a) a∈A b∈B
b∈B a∈A
This can be seen as a quantitative generalization of ∧ (∀a∈A ∃b∈B . . . , ∀b∈B ∃a∈A . . .) which should remind you of bisimilarity.
8 / 22
Franck van Breugel
Bisimilarity Pseudometric on Probabilistic Automata
Felix Hausdorff Felix Hausdorff introduced the metric on subsets. This metric is known as the Hausdorff metric.
9 / 22
Franck van Breugel
Bisimilarity Pseudometric on Probabilistic Automata
A Metric for Probabilistic Choices
Probabilistic choices can be modelled as probability distributions on a set. The distance of the probability distributions µ and ν is defined by ( ) X - [0, 1] d(µ, ν) = max f (x)(µ(x) − ν(x)) f ∈ (X , d) -----< x∈X
10 / 22
Franck van Breugel
Bisimilarity Pseudometric on Probabilistic Automata
Leonid Kantorovich Leonid Kantorovich introduced the metric on probability distributions. This metric is known as the Kantorovich metric.
11 / 22
Franck van Breugel
Bisimilarity Pseudometric on Probabilistic Automata
Probabilistic Bisimilarity s 1 n
s1
t 1 n
···
1 n
sn
t1
1 n
···
tn
States s and t are probabilistic bisimilar if and only if ∃π is a permutation ∀1≤i≤n si and tπ(i) are probabilistic bisimilar This is generalized by ( n ) X1 d(s, t) = min · d(si , tπ(i) ) π is a permutation . n i=1
12 / 22
Franck van Breugel
Bisimilarity Pseudometric on Probabilistic Automata
Catuscia Palamidessi Catuscia Palamidessi, in collaboration with Yuxin Deng, Tom Chothia and Jun Pang, combined the Hausdorff metric and the Kantorovich metric to obtain a pseudometric on the state space of a probabilistic automaton and showed States s and t are probabilistic bisimilar if and only if d(s, t) = 0.
13 / 22
Franck van Breugel
Bisimilarity Pseudometric on Probabilistic Automata
Our Main Result
Theorem The problem of computing the bisimilarity pseudometric introduced by Palamidessi et al. is in PPAD. Computing Nash equilibria of two player games is PPAD-complete. Computing values of simple stochastic games is in PPAD. Computing fixed points of discretized Brouwer functions is in PPAD.
14 / 22
Franck van Breugel
Bisimilarity Pseudometric on Probabilistic Automata
Characterizations of Bisimilarity
Bisimilarity for labelled transition systems has been characterized in terms of a logic (Hennessy and Milner, 1980), a fixed point (Milner, 1980), and a game (Stirling, 1993).
15 / 22
Franck van Breugel
Bisimilarity Pseudometric on Probabilistic Automata
Characterizations of Probabilistic Bisimilarity
Probabilistic bisimilarity for probabilistic automata has been characterized in terms of a logic (Parma and Segala, 2007), and a fixed point (Segala, 1995).
16 / 22
Franck van Breugel
Bisimilarity Pseudometric on Probabilistic Automata
Characterizations of Bisimilarity Pseudometric
The bisimilarity pseudometric for probabilistic automata has been characterized in terms of a logic (De Alfaro et al, 2007), a fixed point (Deng et al, 2005).
17 / 22
Franck van Breugel
Bisimilarity Pseudometric on Probabilistic Automata
Another Result
Theorem The bisimilarity distance of two states is the value of a simple stochastic game. This provides a game theoretic characterization of the bisimilarity pseudometric and also of probabilistic bisimilarity.
18 / 22
Franck van Breugel
Bisimilarity Pseudometric on Probabilistic Automata
Simple Stochastic Game
max
19 / 22
avg
min
1
0
Franck van Breugel
Bisimilarity Pseudometric on Probabilistic Automata
A Characterization of the Bisimilarity Pseudometric
d(s, t) = max max min d(µ, ν), max min d(ν, µ) s→µ t→ν
t→ν s→µ
where d(µ, ν) ) - [0, 1] = max f (s)(µ(s) − ν(s)) f ∈ (S, d) -----< s∈S X = min ω(u, v )d(u, v ) ω ∈ Ωµ,ν (
X
u,v ∈S
20 / 22
Franck van Breugel
Bisimilarity Pseudometric on Probabilistic Automata
Couplings
The set Ωµ,ν consists of the couplings of µ and ν. A probability distribution ω on S × S is a coupling of µ and ν if for all u, v ∈ S, X X ω(u, v ) = µ(u) and ω(u, v ) = ν(v ) v ∈S
u∈S
The set Ωµ,ν is a convex polytope. We denote its set of vertices by V (Ωµ,ν ).
21 / 22
Franck van Breugel
Bisimilarity Pseudometric on Probabilistic Automata
A Characterization of the Bisimilarity Pseudometric d(s, t) = max max min d(µ, ν), max min d(ν, µ) s→µ t→ν
t→ν s→µ
where d(µ, ν) ) - [0, 1] = max f (s)(µ(s) − ν(s)) f ∈ (S, d) -----< s∈S X = min ω(u, v )d(u, v ) ω ∈ Ωµ,ν u,v ∈S X = min ω(u, v )d(u, v ) ω ∈ V (Ωµ,ν ) (
X
u,v ∈S
22 / 22
Franck van Breugel
Bisimilarity Pseudometric on Probabilistic Automata
May 23, 2014
1 / 22
Franck van Breugel
Bisimilarity Pseudometric on Probabilistic Automata
Probabilistic Automaton
3 1 2
1 4
1
2 1 2
3 4
4
A probabilistic automaton contains nondeterministic and probabilistic choices. 2 / 22
Franck van Breugel
Bisimilarity Pseudometric on Probabilistic Automata
Probabilistic Bisimilarity
3 1 2
1 4
1
2 1 2
3 4
4
Probabilistic bisimilarity captures which states of the automaton behave the same. 3 / 22
Franck van Breugel
Bisimilarity Pseudometric on Probabilistic Automata
Robero Segala Roberto Segala, in collaboration with Nancy Lynch, introduced probabilistic bisimilarity for probabilistic automata.
4 / 22
Franck van Breugel
Bisimilarity Pseudometric on Probabilistic Automata
Probabilistic Bisimilarity is not Robust
3 1 2
1 2
+ε
1
2 1 2
1 2
−ε
4
States 1 and 2 are not bisimilar for all ε > 0. 5 / 22
Franck van Breugel
Bisimilarity Pseudometric on Probabilistic Automata
Scott Smolka
Scott Smolka, in collaboration with Alessandro Giacalone and Chi-chang Jou, first suggested to use pseudometrics instead of equivalence relations.
6 / 22
Franck van Breugel
Bisimilarity Pseudometric on Probabilistic Automata
From Equivalence Relations to Pseudometrics
An equivalence relation on a set S can be viewed as function in S×S →B A (1-bounded) pseudometric on a set S is a function in S × S → [0, 1] Equivalence is captured by distance zero.
7 / 22
Franck van Breugel
Bisimilarity Pseudometric on Probabilistic Automata
A Metric for Nondeterministic Choices
Nondeterministic choices can be modelled as subsets of a set. The distance of the subsets A and B is defined by d(A, B) = max max min d(a, b), max min d(b, a) a∈A b∈B
b∈B a∈A
This can be seen as a quantitative generalization of ∧ (∀a∈A ∃b∈B . . . , ∀b∈B ∃a∈A . . .) which should remind you of bisimilarity.
8 / 22
Franck van Breugel
Bisimilarity Pseudometric on Probabilistic Automata
Felix Hausdorff Felix Hausdorff introduced the metric on subsets. This metric is known as the Hausdorff metric.
9 / 22
Franck van Breugel
Bisimilarity Pseudometric on Probabilistic Automata
A Metric for Probabilistic Choices
Probabilistic choices can be modelled as probability distributions on a set. The distance of the probability distributions µ and ν is defined by ( ) X - [0, 1] d(µ, ν) = max f (x)(µ(x) − ν(x)) f ∈ (X , d) -----< x∈X
10 / 22
Franck van Breugel
Bisimilarity Pseudometric on Probabilistic Automata
Leonid Kantorovich Leonid Kantorovich introduced the metric on probability distributions. This metric is known as the Kantorovich metric.
11 / 22
Franck van Breugel
Bisimilarity Pseudometric on Probabilistic Automata
Probabilistic Bisimilarity s 1 n
s1
t 1 n
···
1 n
sn
t1
1 n
···
tn
States s and t are probabilistic bisimilar if and only if ∃π is a permutation ∀1≤i≤n si and tπ(i) are probabilistic bisimilar This is generalized by ( n ) X1 d(s, t) = min · d(si , tπ(i) ) π is a permutation . n i=1
12 / 22
Franck van Breugel
Bisimilarity Pseudometric on Probabilistic Automata
Catuscia Palamidessi Catuscia Palamidessi, in collaboration with Yuxin Deng, Tom Chothia and Jun Pang, combined the Hausdorff metric and the Kantorovich metric to obtain a pseudometric on the state space of a probabilistic automaton and showed States s and t are probabilistic bisimilar if and only if d(s, t) = 0.
13 / 22
Franck van Breugel
Bisimilarity Pseudometric on Probabilistic Automata
Our Main Result
Theorem The problem of computing the bisimilarity pseudometric introduced by Palamidessi et al. is in PPAD. Computing Nash equilibria of two player games is PPAD-complete. Computing values of simple stochastic games is in PPAD. Computing fixed points of discretized Brouwer functions is in PPAD.
14 / 22
Franck van Breugel
Bisimilarity Pseudometric on Probabilistic Automata
Characterizations of Bisimilarity
Bisimilarity for labelled transition systems has been characterized in terms of a logic (Hennessy and Milner, 1980), a fixed point (Milner, 1980), and a game (Stirling, 1993).
15 / 22
Franck van Breugel
Bisimilarity Pseudometric on Probabilistic Automata
Characterizations of Probabilistic Bisimilarity
Probabilistic bisimilarity for probabilistic automata has been characterized in terms of a logic (Parma and Segala, 2007), and a fixed point (Segala, 1995).
16 / 22
Franck van Breugel
Bisimilarity Pseudometric on Probabilistic Automata
Characterizations of Bisimilarity Pseudometric
The bisimilarity pseudometric for probabilistic automata has been characterized in terms of a logic (De Alfaro et al, 2007), a fixed point (Deng et al, 2005).
17 / 22
Franck van Breugel
Bisimilarity Pseudometric on Probabilistic Automata
Another Result
Theorem The bisimilarity distance of two states is the value of a simple stochastic game. This provides a game theoretic characterization of the bisimilarity pseudometric and also of probabilistic bisimilarity.
18 / 22
Franck van Breugel
Bisimilarity Pseudometric on Probabilistic Automata
Simple Stochastic Game
max
19 / 22
avg
min
1
0
Franck van Breugel
Bisimilarity Pseudometric on Probabilistic Automata
A Characterization of the Bisimilarity Pseudometric
d(s, t) = max max min d(µ, ν), max min d(ν, µ) s→µ t→ν
t→ν s→µ
where d(µ, ν) ) - [0, 1] = max f (s)(µ(s) − ν(s)) f ∈ (S, d) -----< s∈S X = min ω(u, v )d(u, v ) ω ∈ Ωµ,ν (
X
u,v ∈S
20 / 22
Franck van Breugel
Bisimilarity Pseudometric on Probabilistic Automata
Couplings
The set Ωµ,ν consists of the couplings of µ and ν. A probability distribution ω on S × S is a coupling of µ and ν if for all u, v ∈ S, X X ω(u, v ) = µ(u) and ω(u, v ) = ν(v ) v ∈S
u∈S
The set Ωµ,ν is a convex polytope. We denote its set of vertices by V (Ωµ,ν ).
21 / 22
Franck van Breugel
Bisimilarity Pseudometric on Probabilistic Automata
A Characterization of the Bisimilarity Pseudometric d(s, t) = max max min d(µ, ν), max min d(ν, µ) s→µ t→ν
t→ν s→µ
where d(µ, ν) ) - [0, 1] = max f (s)(µ(s) − ν(s)) f ∈ (S, d) -----< s∈S X = min ω(u, v )d(u, v ) ω ∈ Ωµ,ν u,v ∈S X = min ω(u, v )d(u, v ) ω ∈ V (Ωµ,ν ) (
X
u,v ∈S
22 / 22
Franck van Breugel
Bisimilarity Pseudometric on Probabilistic Automata
