Ordinary bisimulation Bisimulation relations Define a (note the indefinite article) bisimulation relation R to be an equivalence relation on S such that a a → s ′ ⇒ ∃ t ′ , t → t ′ with s ′ Rt ′ sRt means ∀ a , s − − and vice versa. This is not circular; it is a condition on R . We define s ∼ t if there is some bisimulation relation R with sRt . This is the version that is used most often. Panangaden Bisimulation, Logics and Metrics for Labelled Markov Processes Turing June 2018 7 / 39
� � � � � � � � � � Ordinary bisimulation An example s 0 t 0 a a a a s 1 s 2 s 3 t 2 c b c c b b s 4 s 5 t 4 t 5 P 1 P 2 Panangaden Bisimulation, Logics and Metrics for Labelled Markov Processes Turing June 2018 8 / 39
� � � � � � � � � � Ordinary bisimulation An example s 0 t 0 a a a a s 1 s 2 s 3 t 2 c b c c b b s 4 s 5 t 4 t 5 P 1 P 2 Here s 0 and t 0 are not bisimilar. Panangaden Bisimulation, Logics and Metrics for Labelled Markov Processes Turing June 2018 8 / 39
� � � � � � � � � � Ordinary bisimulation An example s 0 t 0 a a a a s 1 s 2 s 3 t 2 c b c c b b s 4 s 5 t 4 t 5 P 1 P 2 Here s 0 and t 0 are not bisimilar. However s 0 and t 0 can simulate each other ! Panangaden Bisimulation, Logics and Metrics for Labelled Markov Processes Turing June 2018 8 / 39
Ordinary bisimulation How do we know that two processes are not bisimilar? Define a logic as follows: Panangaden Bisimulation, Logics and Metrics for Labelled Markov Processes Turing June 2018 9 / 39
Ordinary bisimulation How do we know that two processes are not bisimilar? Define a logic as follows: φ ::== T |¬ φ | φ 1 ∧ φ 2 | � a � φ Panangaden Bisimulation, Logics and Metrics for Labelled Markov Processes Turing June 2018 9 / 39
Ordinary bisimulation How do we know that two processes are not bisimilar? Define a logic as follows: φ ::== T |¬ φ | φ 1 ∧ φ 2 | � a � φ a → s ′ and t | s | = � a � φ means that s − = φ . Panangaden Bisimulation, Logics and Metrics for Labelled Markov Processes Turing June 2018 9 / 39
Ordinary bisimulation How do we know that two processes are not bisimilar? Define a logic as follows: φ ::== T |¬ φ | φ 1 ∧ φ 2 | � a � φ a → s ′ and t | s | = � a � φ means that s − = φ . We can define a dual to �� (written [] ) by using negation. Panangaden Bisimulation, Logics and Metrics for Labelled Markov Processes Turing June 2018 9 / 39
Ordinary bisimulation How do we know that two processes are not bisimilar? Define a logic as follows: φ ::== T |¬ φ | φ 1 ∧ φ 2 | � a � φ a → s ′ and t | s | = � a � φ means that s − = φ . We can define a dual to �� (written [] ) by using negation. s | = [ a ] φ means that if s can do an a the resulting state must satisfy φ . Panangaden Bisimulation, Logics and Metrics for Labelled Markov Processes Turing June 2018 9 / 39
Ordinary bisimulation Examples of HM Logic T is satisfied by any process, F is not satisfied by any process. Panangaden Bisimulation, Logics and Metrics for Labelled Markov Processes Turing June 2018 10 / 39
Ordinary bisimulation Examples of HM Logic T is satisfied by any process, F is not satisfied by any process. s | = � a � T means s can do an a action. Panangaden Bisimulation, Logics and Metrics for Labelled Markov Processes Turing June 2018 10 / 39
Ordinary bisimulation Examples of HM Logic T is satisfied by any process, F is not satisfied by any process. s | = � a � T means s can do an a action. s | = ¬ � a � T or s | = [ a ] F means s cannot do an a action. Panangaden Bisimulation, Logics and Metrics for Labelled Markov Processes Turing June 2018 10 / 39
Ordinary bisimulation Examples of HM Logic T is satisfied by any process, F is not satisfied by any process. s | = � a � T means s can do an a action. s | = ¬ � a � T or s | = [ a ] F means s cannot do an a action. s | = � a � ( � b � T ) means that s can do an a and then do a b . Panangaden Bisimulation, Logics and Metrics for Labelled Markov Processes Turing June 2018 10 / 39
Ordinary bisimulation The logical characterization theorem Two processes are bisimilar if and only if they satisfy the same formulas of HM logic. Panangaden Bisimulation, Logics and Metrics for Labelled Markov Processes Turing June 2018 11 / 39
Ordinary bisimulation The logical characterization theorem Two processes are bisimilar if and only if they satisfy the same formulas of HM logic. Basic assumption: the processes are finitely-branching (otherwise you need infinitary conjunctions). Panangaden Bisimulation, Logics and Metrics for Labelled Markov Processes Turing June 2018 11 / 39
Ordinary bisimulation The logical characterization theorem Two processes are bisimilar if and only if they satisfy the same formulas of HM logic. Basic assumption: the processes are finitely-branching (otherwise you need infinitary conjunctions). To show that two processes are not bisimilar find a formula on which they disagree. Panangaden Bisimulation, Logics and Metrics for Labelled Markov Processes Turing June 2018 11 / 39
� � � � � Ordinary bisimulation The role of negation Consider the processes below: s 0 t 0 a a a s 1 s 2 t 2 b b s 3 t 3 Panangaden Bisimulation, Logics and Metrics for Labelled Markov Processes Turing June 2018 12 / 39
� � � � � Ordinary bisimulation The role of negation Consider the processes below: s 0 t 0 a a a s 1 s 2 t 2 b b s 3 t 3 s 0 | = � a � ¬ � b � T but t 0 does not. Panangaden Bisimulation, Logics and Metrics for Labelled Markov Processes Turing June 2018 12 / 39
� � � � � Ordinary bisimulation The role of negation Consider the processes below: s 0 t 0 a a a s 1 s 2 t 2 b b s 3 t 3 s 0 | = � a � ¬ � b � T but t 0 does not. s 0 and t 0 agree on all formulas without negation . Panangaden Bisimulation, Logics and Metrics for Labelled Markov Processes Turing June 2018 12 / 39
� � � � � Ordinary bisimulation The role of negation Consider the processes below: s 0 t 0 a a a s 1 s 2 t 2 b b s 3 t 3 s 0 | = � a � ¬ � b � T but t 0 does not. s 0 and t 0 agree on all formulas without negation . Note that [ a ] has an implicit negation. Panangaden Bisimulation, Logics and Metrics for Labelled Markov Processes Turing June 2018 12 / 39
Discrete probabilistic transition systems Discrete probabilistic transition systems Just like a labelled transition system with probabilities associated with the transitions. Panangaden Bisimulation, Logics and Metrics for Labelled Markov Processes Turing June 2018 13 / 39
Discrete probabilistic transition systems Discrete probabilistic transition systems Just like a labelled transition system with probabilities associated with the transitions. ( S , L , ∀ a ∈ L T a : S × S − → [ 0 , 1 ]) Panangaden Bisimulation, Logics and Metrics for Labelled Markov Processes Turing June 2018 13 / 39
Discrete probabilistic transition systems Discrete probabilistic transition systems Just like a labelled transition system with probabilities associated with the transitions. ( S , L , ∀ a ∈ L T a : S × S − → [ 0 , 1 ]) The model is reactive : All probabilistic data is internal - no probabilities associated with environment behaviour. Panangaden Bisimulation, Logics and Metrics for Labelled Markov Processes Turing June 2018 13 / 39
� � � � � � � � Discrete probabilistic transition systems Bisimulation for PTS: Larsen and Skou Consider t 0 s 0 a [ 1 a [ 1 a [ 1 a [ 2 3 ] 3 ] 3 ] 3 ] a [ 1 3 ] s 1 s 2 s 3 t 1 t 2 b [ 1 ] b [ 1 ] b [ 1 ] s 4 t 3 P 1 P 2 Panangaden Bisimulation, Logics and Metrics for Labelled Markov Processes Turing June 2018 14 / 39
� � � � � � � � Discrete probabilistic transition systems Bisimulation for PTS: Larsen and Skou Consider t 0 s 0 a [ 1 a [ 1 a [ 1 a [ 2 3 ] 3 ] 3 ] 3 ] a [ 1 3 ] s 1 s 2 s 3 t 1 t 2 b [ 1 ] b [ 1 ] b [ 1 ] s 4 t 3 P 1 P 2 Should s 0 and t 0 be bisimilar? Panangaden Bisimulation, Logics and Metrics for Labelled Markov Processes Turing June 2018 14 / 39
� � � � � � � � Discrete probabilistic transition systems Bisimulation for PTS: Larsen and Skou Consider t 0 s 0 a [ 1 a [ 1 a [ 1 a [ 2 3 ] 3 ] 3 ] 3 ] a [ 1 3 ] s 1 s 2 s 3 t 1 t 2 b [ 1 ] b [ 1 ] b [ 1 ] s 4 t 3 P 1 P 2 Should s 0 and t 0 be bisimilar? Yes, but we need to add the probabilities. Panangaden Bisimulation, Logics and Metrics for Labelled Markov Processes Turing June 2018 14 / 39
Discrete probabilistic transition systems The Official Definition Let S = ( S , L , T a ) be a PTS. An equivalence relation R on S is a bisimulation if whenever sRs ′ , with s , s ′ ∈ S , we have that for all a ∈ A and every R -equivalence class, A , T a ( s , A ) = T a ( s ′ , A ) . Panangaden Bisimulation, Logics and Metrics for Labelled Markov Processes Turing June 2018 15 / 39
Discrete probabilistic transition systems The Official Definition Let S = ( S , L , T a ) be a PTS. An equivalence relation R on S is a bisimulation if whenever sRs ′ , with s , s ′ ∈ S , we have that for all a ∈ A and every R -equivalence class, A , T a ( s , A ) = T a ( s ′ , A ) . The notation T a ( s , A ) means “the probability of starting from s and jumping to a state in the set A .” Panangaden Bisimulation, Logics and Metrics for Labelled Markov Processes Turing June 2018 15 / 39
Discrete probabilistic transition systems The Official Definition Let S = ( S , L , T a ) be a PTS. An equivalence relation R on S is a bisimulation if whenever sRs ′ , with s , s ′ ∈ S , we have that for all a ∈ A and every R -equivalence class, A , T a ( s , A ) = T a ( s ′ , A ) . The notation T a ( s , A ) means “the probability of starting from s and jumping to a state in the set A .” Two states are bisimilar if there is some bisimulation relation R relating them. Panangaden Bisimulation, Logics and Metrics for Labelled Markov Processes Turing June 2018 15 / 39
Labelled Markov processes What are labelled Markov processes? Labelled Markov processes are probabilistic versions of labelled transition systems. Labelled transition systems where the final state is governed by a probability distribution - no other indeterminacy. Panangaden Bisimulation, Logics and Metrics for Labelled Markov Processes Turing June 2018 16 / 39
Labelled Markov processes What are labelled Markov processes? Labelled Markov processes are probabilistic versions of labelled transition systems. Labelled transition systems where the final state is governed by a probability distribution - no other indeterminacy. All probabilistic data is internal - no probabilities associated with environment behaviour. Panangaden Bisimulation, Logics and Metrics for Labelled Markov Processes Turing June 2018 16 / 39
Labelled Markov processes What are labelled Markov processes? Labelled Markov processes are probabilistic versions of labelled transition systems. Labelled transition systems where the final state is governed by a probability distribution - no other indeterminacy. All probabilistic data is internal - no probabilities associated with environment behaviour. We observe the interactions - not the internal states. Panangaden Bisimulation, Logics and Metrics for Labelled Markov Processes Turing June 2018 16 / 39
Labelled Markov processes What are labelled Markov processes? Labelled Markov processes are probabilistic versions of labelled transition systems. Labelled transition systems where the final state is governed by a probability distribution - no other indeterminacy. All probabilistic data is internal - no probabilities associated with environment behaviour. We observe the interactions - not the internal states. In general, the state space of a labelled Markov process may be a continuum . Panangaden Bisimulation, Logics and Metrics for Labelled Markov Processes Turing June 2018 16 / 39
Labelled Markov processes Motivation Model and reason about systems with continuous state spaces or continuous time evolution or both. hybrid control systems; e.g. flight management systems. Panangaden Bisimulation, Logics and Metrics for Labelled Markov Processes Turing June 2018 17 / 39
Labelled Markov processes Motivation Model and reason about systems with continuous state spaces or continuous time evolution or both. hybrid control systems; e.g. flight management systems. telecommunication systems with spatial variation; e.g. cell phones Panangaden Bisimulation, Logics and Metrics for Labelled Markov Processes Turing June 2018 17 / 39
Labelled Markov processes Motivation Model and reason about systems with continuous state spaces or continuous time evolution or both. hybrid control systems; e.g. flight management systems. telecommunication systems with spatial variation; e.g. cell phones performance modelling, Panangaden Bisimulation, Logics and Metrics for Labelled Markov Processes Turing June 2018 17 / 39
Labelled Markov processes Motivation Model and reason about systems with continuous state spaces or continuous time evolution or both. hybrid control systems; e.g. flight management systems. telecommunication systems with spatial variation; e.g. cell phones performance modelling, continuous time systems, Panangaden Bisimulation, Logics and Metrics for Labelled Markov Processes Turing June 2018 17 / 39
Labelled Markov processes Motivation Model and reason about systems with continuous state spaces or continuous time evolution or both. hybrid control systems; e.g. flight management systems. telecommunication systems with spatial variation; e.g. cell phones performance modelling, continuous time systems, probabilistic process algebra with recursion. Panangaden Bisimulation, Logics and Metrics for Labelled Markov Processes Turing June 2018 17 / 39
Labelled Markov processes The Need for Measure Theory Basic fact: There are subsets of R for which no sensible notion of size can be defined. Panangaden Bisimulation, Logics and Metrics for Labelled Markov Processes Turing June 2018 18 / 39
Labelled Markov processes The Need for Measure Theory Basic fact: There are subsets of R for which no sensible notion of size can be defined. More precisely, there is no translation-invariant measure defined on all the subsets of the reals. Panangaden Bisimulation, Logics and Metrics for Labelled Markov Processes Turing June 2018 18 / 39
Labelled Markov processes The Need for Measure Theory Basic fact: There are subsets of R for which no sensible notion of size can be defined. More precisely, there is no translation-invariant measure defined on all the subsets of the reals. Actually there is if you only require finite additivity. Panangaden Bisimulation, Logics and Metrics for Labelled Markov Processes Turing June 2018 18 / 39
Labelled Markov processes Stochastic Kernels A stochastic kernel (Markov kernel) is a function h : S × Σ − → [ 0 , 1 ] with (a) h ( s , · ) : Σ − → [ 0 , 1 ] a (sub)probability measure and (b) h ( · , A ) : X − → [ 0 , 1 ] a measurable function. Panangaden Bisimulation, Logics and Metrics for Labelled Markov Processes Turing June 2018 19 / 39
Labelled Markov processes Stochastic Kernels A stochastic kernel (Markov kernel) is a function h : S × Σ − → [ 0 , 1 ] with (a) h ( s , · ) : Σ − → [ 0 , 1 ] a (sub)probability measure and (b) h ( · , A ) : X − → [ 0 , 1 ] a measurable function. Though apparantly asymmetric, these are the stochastic analogues of binary relations Panangaden Bisimulation, Logics and Metrics for Labelled Markov Processes Turing June 2018 19 / 39
Labelled Markov processes Stochastic Kernels A stochastic kernel (Markov kernel) is a function h : S × Σ − → [ 0 , 1 ] with (a) h ( s , · ) : Σ − → [ 0 , 1 ] a (sub)probability measure and (b) h ( · , A ) : X − → [ 0 , 1 ] a measurable function. Though apparantly asymmetric, these are the stochastic analogues of binary relations and the uncountable generalization of a matrix. Panangaden Bisimulation, Logics and Metrics for Labelled Markov Processes Turing June 2018 19 / 39
Labelled Markov processes Formal Definition of LMPs An LMP is a tuple ( S , Σ , L , ∀ α ∈ L . τ α ) where τ α : S × Σ − → [ 0 , 1 ] is a transition probability function such that Panangaden Bisimulation, Logics and Metrics for Labelled Markov Processes Turing June 2018 20 / 39
Labelled Markov processes Formal Definition of LMPs An LMP is a tuple ( S , Σ , L , ∀ α ∈ L . τ α ) where τ α : S × Σ − → [ 0 , 1 ] is a transition probability function such that ∀ s : S . λ A : Σ . τ α ( s , A ) is a subprobability measure and ∀ A : Σ . λ s : S . τ α ( s , A ) is a measurable function. Panangaden Bisimulation, Logics and Metrics for Labelled Markov Processes Turing June 2018 20 / 39
Probabilistic bisimulation Larsen-Skou Bisimulation Let S = ( S , i , Σ , τ ) be a labelled Markov process. An equivalence relation R on S is a bisimulation if whenever sRs ′ , with s , s ′ ∈ S , we have that for all a ∈ A and every R -closed measurable set A ∈ Σ , τ a ( s , A ) = τ a ( s ′ , A ) . Two states are bisimilar if they are related by a bisimulation relation. Panangaden Bisimulation, Logics and Metrics for Labelled Markov Processes Turing June 2018 21 / 39
Probabilistic bisimulation Larsen-Skou Bisimulation Let S = ( S , i , Σ , τ ) be a labelled Markov process. An equivalence relation R on S is a bisimulation if whenever sRs ′ , with s , s ′ ∈ S , we have that for all a ∈ A and every R -closed measurable set A ∈ Σ , τ a ( s , A ) = τ a ( s ′ , A ) . Two states are bisimilar if they are related by a bisimulation relation. Can be extended to bisimulation between two different LMP s. Panangaden Bisimulation, Logics and Metrics for Labelled Markov Processes Turing June 2018 21 / 39
Probabilistic bisimulation A game for bisimulation Two players: spoiler (S) and duplicator (D). Panangaden Bisimulation, Logics and Metrics for Labelled Markov Processes Turing June 2018 22 / 39
Probabilistic bisimulation A game for bisimulation Two players: spoiler (S) and duplicator (D). Duplicator claims x , y are bisimilar. Panangaden Bisimulation, Logics and Metrics for Labelled Markov Processes Turing June 2018 22 / 39
Probabilistic bisimulation A game for bisimulation Two players: spoiler (S) and duplicator (D). Duplicator claims x , y are bisimilar. Spoiler exhibits a set C and says C is bisimulation-closed and that τ ( x , C ) � = τ ( y , C ) . Assume that the inequality holds; it is easy to check. Panangaden Bisimulation, Logics and Metrics for Labelled Markov Processes Turing June 2018 22 / 39
Probabilistic bisimulation A game for bisimulation Two players: spoiler (S) and duplicator (D). Duplicator claims x , y are bisimilar. Spoiler exhibits a set C and says C is bisimulation-closed and that τ ( x , C ) � = τ ( y , C ) . Assume that the inequality holds; it is easy to check. Duplicator responds by saying that C is not bisimulation-closed and that exhibits x ′ ∈ C and y ′ �∈ C and claims that x ′ , y ′ are bisimilar. Panangaden Bisimulation, Logics and Metrics for Labelled Markov Processes Turing June 2018 22 / 39
Probabilistic bisimulation A game for bisimulation Two players: spoiler (S) and duplicator (D). Duplicator claims x , y are bisimilar. Spoiler exhibits a set C and says C is bisimulation-closed and that τ ( x , C ) � = τ ( y , C ) . Assume that the inequality holds; it is easy to check. Duplicator responds by saying that C is not bisimulation-closed and that exhibits x ′ ∈ C and y ′ �∈ C and claims that x ′ , y ′ are bisimilar. A player loses when he or she cannot make a move. Note that if C is all of the state space, duplicator loses. Duplicator wins if she can play forever. Panangaden Bisimulation, Logics and Metrics for Labelled Markov Processes Turing June 2018 22 / 39
Probabilistic bisimulation A game for bisimulation Two players: spoiler (S) and duplicator (D). Duplicator claims x , y are bisimilar. Spoiler exhibits a set C and says C is bisimulation-closed and that τ ( x , C ) � = τ ( y , C ) . Assume that the inequality holds; it is easy to check. Duplicator responds by saying that C is not bisimulation-closed and that exhibits x ′ ∈ C and y ′ �∈ C and claims that x ′ , y ′ are bisimilar. A player loses when he or she cannot make a move. Note that if C is all of the state space, duplicator loses. Duplicator wins if she can play forever. We prove that x is bisimilar to y iff Duplicator has a winning strategy starting from ( x , y ) . Panangaden Bisimulation, Logics and Metrics for Labelled Markov Processes Turing June 2018 22 / 39
Probabilistic bisimulation Logical Characterization L ::== T | φ 1 ∧ φ 2 | � a � q φ Panangaden Bisimulation, Logics and Metrics for Labelled Markov Processes Turing June 2018 23 / 39
Probabilistic bisimulation Logical Characterization L ::== T | φ 1 ∧ φ 2 | � a � q φ We say s | = � a � q φ iff ∃ A ∈ Σ . ( ∀ s ′ ∈ A . s ′ | = φ ) ∧ ( τ a ( s , A ) > q ) . Panangaden Bisimulation, Logics and Metrics for Labelled Markov Processes Turing June 2018 23 / 39
Probabilistic bisimulation Logical Characterization L ::== T | φ 1 ∧ φ 2 | � a � q φ We say s | = � a � q φ iff ∃ A ∈ Σ . ( ∀ s ′ ∈ A . s ′ | = φ ) ∧ ( τ a ( s , A ) > q ) . Two systems are bisimilar iff they obey the same formulas of L . [DEP 1998 LICS, I and C 2002] Panangaden Bisimulation, Logics and Metrics for Labelled Markov Processes Turing June 2018 23 / 39
� � � � � Probabilistic bisimulation That cannot be right? s 0 t 0 a a a s 1 s 2 t 1 b b s 3 t 2 Two processes that cannot be distinguished without negation. The formula that distinguishes them is � a � ( ¬ � b �⊤ ) . Panangaden Bisimulation, Logics and Metrics for Labelled Markov Processes Turing June 2018 24 / 39
� � � � � Probabilistic bisimulation But it is! s 0 t 0 a [ p ] a [ q ] a [ r ] s 1 s 2 t 1 b b s 3 t 2 We add probabilities to the transitions. Panangaden Bisimulation, Logics and Metrics for Labelled Markov Processes Turing June 2018 25 / 39
� � � � � Probabilistic bisimulation But it is! s 0 t 0 a [ p ] a [ q ] a [ r ] s 1 s 2 t 1 b b s 3 t 2 We add probabilities to the transitions. If p + q < r or p + q > r we can easily distinguish them. Panangaden Bisimulation, Logics and Metrics for Labelled Markov Processes Turing June 2018 25 / 39
� � � � � Probabilistic bisimulation But it is! s 0 t 0 a [ p ] a [ q ] a [ r ] s 1 s 2 t 1 b b s 3 t 2 We add probabilities to the transitions. If p + q < r or p + q > r we can easily distinguish them. If p + q = r and p > 0 then q < r so � a � r � b � 1 ⊤ distinguishes them. Panangaden Bisimulation, Logics and Metrics for Labelled Markov Processes Turing June 2018 25 / 39
Metrics A metric-based approximate viewpoint Move from equality between processes to distances between processes (Jou and Smolka 1990). Panangaden Bisimulation, Logics and Metrics for Labelled Markov Processes Turing June 2018 26 / 39
Metrics A metric-based approximate viewpoint Move from equality between processes to distances between processes (Jou and Smolka 1990). Formalize distance as a metric: d ( s , s ) = 0 , d ( s , t ) = d ( t , s ) , d ( s , u ) � d ( s , t ) + d ( t , u ) . Quantitative analogue of an equivalence relation. Panangaden Bisimulation, Logics and Metrics for Labelled Markov Processes Turing June 2018 26 / 39
Metrics A metric-based approximate viewpoint Move from equality between processes to distances between processes (Jou and Smolka 1990). Formalize distance as a metric: d ( s , s ) = 0 , d ( s , t ) = d ( t , s ) , d ( s , u ) � d ( s , t ) + d ( t , u ) . Quantitative analogue of an equivalence relation. Quantitative measurement of the distinction between processes. Panangaden Bisimulation, Logics and Metrics for Labelled Markov Processes Turing June 2018 26 / 39
Metrics Criteria on Metrics Soundness: d ( s , t ) = 0 ⇔ s , t are bisimilar Panangaden Bisimulation, Logics and Metrics for Labelled Markov Processes Turing June 2018 27 / 39
Metrics Criteria on Metrics Soundness: d ( s , t ) = 0 ⇔ s , t are bisimilar Stability of distance under temporal evolution:“Nearby states stay close forever .” Panangaden Bisimulation, Logics and Metrics for Labelled Markov Processes Turing June 2018 27 / 39
Metrics Criteria on Metrics Soundness: d ( s , t ) = 0 ⇔ s , t are bisimilar Stability of distance under temporal evolution:“Nearby states stay close forever .” Metrics should be computable (efficiently?). Panangaden Bisimulation, Logics and Metrics for Labelled Markov Processes Turing June 2018 27 / 39
Metrics Bisimulation Recalled Let R be an equivalence relation. R is a bisimulation if: s R t if: ( s − → P ) ⇒ [ t − → Q , P = R Q ] ( t − → Q ) ⇒ [ s − → P , P = R Q ] where P = R Q if ( ∀ R − closed E ) P ( E ) = Q ( E ) Panangaden Bisimulation, Logics and Metrics for Labelled Markov Processes Turing June 2018 28 / 39
Metrics A putative definition of a metric analogue of bisimulation m is a metric-bisimulation if: m ( s , t ) < ǫ ⇒ : s − → P ⇒ t − → Q , m ( P , Q ) < ǫ t − → Q ⇒ s − → P , m ( P , Q ) < ǫ Panangaden Bisimulation, Logics and Metrics for Labelled Markov Processes Turing June 2018 29 / 39
Metrics A putative definition of a metric analogue of bisimulation m is a metric-bisimulation if: m ( s , t ) < ǫ ⇒ : s − → P ⇒ t − → Q , m ( P , Q ) < ǫ t − → Q ⇒ s − → P , m ( P , Q ) < ǫ Problem: what is m ( P , Q ) ? — Type mismatch!! Panangaden Bisimulation, Logics and Metrics for Labelled Markov Processes Turing June 2018 29 / 39
Metrics A putative definition of a metric analogue of bisimulation m is a metric-bisimulation if: m ( s , t ) < ǫ ⇒ : s − → P ⇒ t − → Q , m ( P , Q ) < ǫ t − → Q ⇒ s − → P , m ( P , Q ) < ǫ Problem: what is m ( P , Q ) ? — Type mismatch!! Need a way to lift distances from states to a distances on distributions of states. Panangaden Bisimulation, Logics and Metrics for Labelled Markov Processes Turing June 2018 29 / 39
Metrics A detour: Kantorovich metric Metrics on probability measures on metric spaces. Panangaden Bisimulation, Logics and Metrics for Labelled Markov Processes Turing June 2018 30 / 39
Metrics A detour: Kantorovich metric Metrics on probability measures on metric spaces. M : 1 -bounded pseudometrics on states. Panangaden Bisimulation, Logics and Metrics for Labelled Markov Processes Turing June 2018 30 / 39
Metrics A detour: Kantorovich metric Metrics on probability measures on metric spaces. M : 1 -bounded pseudometrics on states. � � d ( µ , ν ) = sup | fd µ − fd ν | , f 1-Lipschitz f Panangaden Bisimulation, Logics and Metrics for Labelled Markov Processes Turing June 2018 30 / 39
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