Behavioural Preorders on Stochastic Systems - Logical, Topological, - PowerPoint PPT Presentation

M. R. Pedersen | Behavioural Preorders on Stochastic Systems Contributions 22 Paper B and Paper C 2 a : 1 1 a : 1 s 1 s ′ 1 s 1 �� s 2 1 a : 1 2 a : 1 s 2 s ′ 2

M. R. Pedersen | Behavioural Preorders on Stochastic Systems Contributions 22 Paper B and Paper C 2 a : 1 1 a : 1 1 a : 1 s 1 s ′ s ′′ 1 1 s 1 � s 2 1 a : 1 2 a : 1 1 a : 1 s 2 s ′ s ′′ 2 2

M. R. Pedersen | Behavioural Preorders on Stochastic Systems Contributions 23 Paper B and Paper C Generative: Definition B.2.3 s 1 is faster than s 2 ( s 1 � s 2 ) if for all a 1 . . . a n and t we have P ( s 1 )( a 1 . . . a n , t ) ≥ P ( s 2 )( a 1 . . . a n , t ) .

M. R. Pedersen | Behavioural Preorders on Stochastic Systems Contributions 23 Paper B and Paper C Generative: Definition B.2.3 s 1 is faster than s 2 ( s 1 � s 2 ) if for all a 1 . . . a n and t we have P ( s 1 )( a 1 . . . a n , t ) ≥ P ( s 2 )( a 1 . . . a n , t ) . Reactive: Definition C.4.3 s 1 is faster than s 2 ( s 1 � s 2 ) if for all schedulers σ , a 1 . . . a n , and t there exists a scheduler σ ′ such that P σ ′ ( s 1 )( a 1 . . . a n , t ) ≥ P σ ( s 2 )( a 1 . . . a n , t ) .

M. R. Pedersen | Behavioural Preorders on Stochastic Systems Contributions 24 Paper B and Paper C Contribution 3 We show that deciding the faster-than relation is a difficult problem. In particular, the relation is undecidable and approximating it up to a multiplicative constant is impossible.

M. R. Pedersen | Behavioural Preorders on Stochastic Systems Contributions 25 Paper B and Paper C Contribution 4 We give an algorithm for approximating a time-bounded version of the faster-than relation up to an additive constant for slow processes.

M. R. Pedersen | Behavioural Preorders on Stochastic Systems Contributions 26 Paper B and Paper C Assumptions: ◮ Time-bounded: We only look at behaviours up to a given time bound. ◮ Slow residence-time functions: all transitions take some time to fire.

M. R. Pedersen | Behavioural Preorders on Stochastic Systems Contributions 26 Paper B and Paper C Assumptions: ◮ Time-bounded: We only look at behaviours up to a given time bound. ◮ Slow residence-time functions: all transitions take some time to fire. Theorem B.4.3 and C.5.6 The time-bounded approximation problem is decidable.

M. R. Pedersen | Behavioural Preorders on Stochastic Systems Contributions 27 Paper B and Paper C Contribution 5 We give an algorithm for unambiguous processes which can decide whether one process is faster than another.

M. R. Pedersen | Behavioural Preorders on Stochastic Systems Contributions 28 Paper B and Paper C A SMP is unambiguous if every output label leads to a unique successor state. s 1 s 1 a : 1 b : 2 a : 1 b : 2 3 3 3 3 a : 1 a : 1 a : 1 a : 1 2 2 s 3 s 2 s 3 s 2 a : 1 b : 1 2 2 Figure 1: Ambiguous Figure 2: Unambiguous

M. R. Pedersen | Behavioural Preorders on Stochastic Systems Contributions 28 Paper B and Paper C A SMP is unambiguous if every output label leads to a unique successor state. s 1 s 1 a : 1 b : 2 a : 1 b : 2 3 3 3 3 a : 1 a : 1 a : 1 a : 1 2 2 s 3 s 2 s 3 s 2 a : 1 b : 1 2 2 Figure 1: Ambiguous Figure 2: Unambiguous Theorem B.5.2 For unambiguous SMPs, the faster-than problem is decidable.

M. R. Pedersen | Behavioural Preorders on Stochastic Systems Contributions 29 Paper B and Paper C Contribution 6 We introduce a logical language which characterises the faster-than relation and we show that both the satisfiability problem and the model checking problem for this language are decidable.

M. R. Pedersen | Behavioural Preorders on Stochastic Systems Contributions 30 Paper B and Paper C Contribution 7 We give examples of parallel timing anomalies occuring for the faster-than relation. However, we also describe some conditions under which parallel timing anomalies can not occur, and we develop an algorithm for checking whether these conditions are met.

M. R. Pedersen | Behavioural Preorders on Stochastic Systems Contributions 31 Paper B and Paper C Component Context Component

M. R. Pedersen | Behavioural Preorders on Stochastic Systems Contributions 31 Paper B and Paper C Component Context Component

M. R. Pedersen | Behavioural Preorders on Stochastic Systems Contributions 31 Paper B and Paper C Component Component Context

M. R. Pedersen | Behavioural Preorders on Stochastic Systems Contributions 31 Paper B and Paper C Component Component Context

M. R. Pedersen | Behavioural Preorders on Stochastic Systems Contributions 31 Paper B and Paper C Component Component Context Timing anomaly

M. R. Pedersen | Behavioural Preorders on Stochastic Systems Contributions 32 Paper B and Paper C Theorem C.6.15 There exist decidable conditions that guarantee the absence of timing anomalies.

Paper D

M. R. Pedersen | Behavioural Preorders on Stochastic Systems Contributions 33 Paper D Reactive processes Simulation �

M. R. Pedersen | Behavioural Preorders on Stochastic Systems Contributions 33 Paper D Reactive processes Simulation � �

M. R. Pedersen | Behavioural Preorders on Stochastic Systems Contributions 33 Paper D Reactive processes Simulation � � But how close is the process to simulating the other process?

M. R. Pedersen | Behavioural Preorders on Stochastic Systems Contributions 33 Paper D Reactive processes Simulation � � But how close is the process to simulating the other process? Quantitative measure of distance

M. R. Pedersen | Behavioural Preorders on Stochastic Systems Contributions 34 Paper D Definition D.2.2 s 2 simulates s 1 , written s 1 � s 2 , if . . . ◮ F s 1 ( t ) ≤ F s 2 ( t ) for all t ∈ R ≥ 0 . . .

M. R. Pedersen | Behavioural Preorders on Stochastic Systems Contributions 35 Paper D Exponential distribution 1 0 . 8 probability 0 . 6 0 . 4 F ( x ) 0 . 2 F ( 2 · x ) F ( 4 · x ) 0 0 2 4 6 time

M. R. Pedersen | Behavioural Preorders on Stochastic Systems Contributions 36 Paper D Definition D.2.2 s 2 simulates s 1 , written s 1 � s 2 , if . . . ◮ F s 1 ( t ) ≤ F s 2 ( t ) for all t ∈ R ≥ 0 . . .

M. R. Pedersen | Behavioural Preorders on Stochastic Systems Contributions 36 Paper D Definition D.2.2 s 2 ε -simulates s 1 , written s 1 � ε s 2 , if . . . ◮ F s 1 ( t ) ≤ F s 2 ( ε · t ) for all t ∈ R ≥ 0 . . .

M. R. Pedersen | Behavioural Preorders on Stochastic Systems Contributions 36 Paper D Definition D.2.2 s 2 ε -simulates s 1 , written s 1 � ε s 2 , if . . . ◮ F s 1 ( t ) ≤ F s 2 ( ε · t ) for all t ∈ R ≥ 0 . . . Definition D.4.5 d ( s 1 , s 2 ) = inf { ε ≥ 1 | s 1 � ε s 2 }

M. R. Pedersen | Behavioural Preorders on Stochastic Systems Contributions 37 Paper D Contribution 8 We describe an algorithm for computing the distance from one process to another. This algorithm runs in polynomial time using known techniques, making it relevant for use and implementation in practice.

M. R. Pedersen | Behavioural Preorders on Stochastic Systems Contributions 38 Paper D Contribution 9 We show that, under mild assumptions, composition is non-expansive with respect to the distance between semi-Markov processes.

M. R. Pedersen | Behavioural Preorders on Stochastic Systems Contributions 39 Paper D Contribution 10 We introduce a logical specification language called timed Markovian logic and show that this language characterises both the ε -simulation relation and the distance between semi-Markov processes.

❚▼▲ ❚▼▲ M. R. Pedersen | Behavioural Preorders on Stochastic Systems Contributions 40 Paper D Timed Markovian logic ϕ, ϕ ′ ::= α | ¬ α | ℓ p t | m p t | L a p ϕ | ϕ ∧ ϕ ′ | ϕ ∨ ϕ ′ p ϕ | M a ❚▼▲ :

❚▼▲ ❚▼▲ M. R. Pedersen | Behavioural Preorders on Stochastic Systems Contributions 40 Paper D Timed Markovian logic ϕ, ϕ ′ ::= α | ¬ α | ℓ p t | m p t | L a p ϕ | ϕ ∧ ϕ ′ | ϕ ∨ ϕ ′ p ϕ | M a ❚▼▲ : L a p ϕ : probability of going with an a to where ϕ holds is at least p . M a p ϕ : probability of going with an a to where ϕ holds is at most p .

❚▼▲ ❚▼▲ M. R. Pedersen | Behavioural Preorders on Stochastic Systems Contributions 40 Paper D Timed Markovian logic ϕ, ϕ ′ ::= α | ¬ α | ℓ p t | m p t | L a p ϕ | ϕ ∧ ϕ ′ | ϕ ∨ ϕ ′ p ϕ | M a ❚▼▲ : L a p ϕ : probability of going with an a to where ϕ holds is at least p . M a p ϕ : probability of going with an a to where ϕ holds is at most p . ℓ p t : probability of leaving state before time t is at least p . m p t : probability of leaving state before time t is at most p .

M. R. Pedersen | Behavioural Preorders on Stochastic Systems Contributions 40 Paper D Timed Markovian logic ϕ, ϕ ′ ::= α | ¬ α | ℓ p t | m p t | L a p ϕ | ϕ ∧ ϕ ′ | ϕ ∨ ϕ ′ p ϕ | M a ❚▼▲ : L a p ϕ : probability of going with an a to where ϕ holds is at least p . M a p ϕ : probability of going with an a to where ϕ holds is at most p . ℓ p t : probability of leaving state before time t is at least p . m p t : probability of leaving state before time t is at most p . ❚▼▲ ≥ : p ϕ | ϕ ∧ ϕ ′ | ϕ ∨ ϕ ′ ϕ ::= α | ¬ α | ℓ p t | L a ❚▼▲ ≤ : p ϕ | ϕ ∧ ϕ ′ | ϕ ∨ ϕ ′ ϕ ::= α | ¬ α | m p t | M a

❚▼▲ ❚▼▲ M. R. Pedersen | Behavioural Preorders on Stochastic Systems Contributions 41 Paper D Perturbation ( ϕ ) ε : ◮ ( ℓ p t ) ε = ℓ p ε · t ◮ ( m p t ) ε = m p ε · t

M. R. Pedersen | Behavioural Preorders on Stochastic Systems Contributions 41 Paper D Perturbation ( ϕ ) ε : ◮ ( ℓ p t ) ε = ℓ p ε · t ◮ ( m p t ) ε = m p ε · t Theorem D.7.2 For finite SMPs we have ◮ d ( s 1 , s 2 ) ≤ ε if and only if for all ϕ ∈ ❚▼▲ ≥ , s 1 | = ϕ implies s 2 | = ( ϕ ) ε ◮ d ( s 2 , s 1 ) ≤ ε if and only if for all ϕ ∈ ❚▼▲ ≤ , s 2 | = ( ϕ ) ε implies s 1 | = ϕ

Conclusion

M. R. Pedersen | Behavioural Preorders on Stochastic Systems Conclusion 42 Summary ◮ Formalisms for specifying, comparing, and reasoning about properties involving time. ◮ Algorithms enabling use of these formalisms in practice.

M. R. Pedersen | Behavioural Preorders on Stochastic Systems Conclusion 42 Summary ◮ Formalisms for specifying, comparing, and reasoning about properties involving time. ◮ Algorithms enabling use of these formalisms in practice. ◮ Weighted logic with bounds allows reasoning about upper and lower bounds on time.

M. R. Pedersen | Behavioural Preorders on Stochastic Systems Conclusion 42 Summary ◮ Formalisms for specifying, comparing, and reasoning about properties involving time. ◮ Algorithms enabling use of these formalisms in practice. ◮ Weighted logic with bounds allows reasoning about upper and lower bounds on time. ◮ Faster-than relation allows qualitative comparison of time behaviour of different systems.

M. R. Pedersen | Behavioural Preorders on Stochastic Systems Conclusion 42 Summary ◮ Formalisms for specifying, comparing, and reasoning about properties involving time. ◮ Algorithms enabling use of these formalisms in practice. ◮ Weighted logic with bounds allows reasoning about upper and lower bounds on time. ◮ Faster-than relation allows qualitative comparison of time behaviour of different systems. ◮ ε -simulation allows quantitative comparison of time behaviour of different systems.

Future work

M. R. Pedersen | Behavioural Preorders on Stochastic Systems Future work 43 Strong completeness Weak completeness | = ϕ implies ⊢ ϕ

M. R. Pedersen | Behavioural Preorders on Stochastic Systems Future work 43 Strong completeness Strong completeness Φ | = ϕ implies Φ ⊢ ϕ

M. R. Pedersen | Behavioural Preorders on Stochastic Systems Future work 44 Timing anomalies Component Context Component

M. R. Pedersen | Behavioural Preorders on Stochastic Systems Future work 44 Timing anomalies Component Context Component

M. R. Pedersen | Behavioural Preorders on Stochastic Systems Future work 44 Timing anomalies Component Context Component

M. R. Pedersen | Behavioural Preorders on Stochastic Systems Future work 44 Timing anomalies Component Context Component

M. R. Pedersen | Behavioural Preorders on Stochastic Systems Future work 44 Timing anomalies Component Component Context

M. R. Pedersen | Behavioural Preorders on Stochastic Systems Future work 45 Branching in simulation distance Exp [ 1 ] Exp [ 1 ] s 1 s 2 a : 0 . 5 a : 0 . 49 a : 0 . 5 a : 0 . 51 Dirac [ 0 ] Exp [ 2 ] Dirac [ 0 ] Exp [ 2 ] s ′ s ′′ s ′ s ′′ 1 1 2 2 a : 1 a : 1 a : 1 a : 1

M. R. Pedersen | Behavioural Preorders on Stochastic Systems Future work 45 Branching in simulation distance Exp [ 1 ] Exp [ 1 ] s 1 s 2 a : 0 . 5 a : 0 . 49 a : 0 . 5 a : 0 . 51 Dirac [ 0 ] Exp [ 2 ] Dirac [ 0 ] Exp [ 2 ] s ′ s ′′ s ′ s ′′ 1 1 2 2 a : 1 a : 1 a : 1 a : 1 d ( s 1 , s 2 ) = ∞

Thank you!

Exp [ 3 ] Exp [ 4 ] s 1 t 1 a : 0 . 1 a : 0 . 9 a : 1 Exp [ θ ] Exp [ 5 ] Exp [ 9 ] s 2 t 2 t 3 a : 1 a : 1 a : 1 Figure 3: A semi-Markov process where s 1 � t 1 if θ ≤ 5 and s 1 � � t 1 if θ > 5.

Time-bounded approximation 0 b m m m m m n times ◮ P ( s , a n , b ) → 0 as n → ∞ . ◮ Hence we can find N such that P ( s , a n , b ) ≤ ε for all n ≥ N . ◮ We only need to consider words of length ≤ N .

Tableau �{¬ ( ¬ ( L 2 p 1 ∧ M 5 L 1 p 1 ) ∧ M 2 p 2 ) } , [ 0 , 0 ] , [ 0 , 0 ] � ( ¬∧ ) �{¬¬ ( L 2 p 1 ∧ M 5 L 1 p 1 ) } , [ 0 , 0 ] , [ 0 , 0 ] � �{¬¬ M 2 p 2 } , [ 0 , 0 ] , [ 0 , 0 ] � ( ¬¬ ) ( ¬¬ ) �{ L 2 p 1 ∧ M 5 L 1 p 1 } , [ 0 , 0 ] , [ 0 , 0 ] � �{ M 2 p 2 } , [ 0 , 0 ] , [ 0 , 0 ] � ( ∧ ) (mod) �{ L 2 p 1 , M 5 L 1 p 1 } , [ 0 , 0 ] , [ 0 , 0 ] � �{ p 2 } , [ 0 , ∞ ) , [ 0 , 2 ] � (mod) �{ p 1 , L 1 p 1 } , [ 2 , ∞ ) , [ 5 , ∞ ) � (mod) �{ p 1 } , [ 1 , ∞ ) , [ 0 , ∞ ) � 2 1 {} { p 1 } { p 1 } s T s 1 s 2 5 1

Image-finite counterexample 0 ω . . . 0 2 3 n 0 . . . 1 1 4 4 0 . . . . . . 1 1 s 2 t 4 4 1 1 0 4 4 1 Figure 4: s and t satisfy the same logical formulas, but s �∼ t .

Kantorovich counterexample ε i u i u 1 1 − ε i 0 v i v Figure 5: A Markov process with states u i and v i for each i ∈ N .