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

Behavioural Preorders on Stochastic Systems - Logical, Topological, and Computational Aspects

Thesis defence January 29, 2019 Mathias Ruggaard Pedersen

Department of Computer Science, Aalborg University, Denmark

• M. R. Pedersen | Behavioural Preorders on Stochastic Systems

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Agenda

Introduction Models Contributions Paper A Paper B and Paper C Paper D Conclusion Future work

Introduction

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Introduction

Background

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Introduction

Background

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Introduction

Background

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Introduction

Time is important

◮ Airbag must deploy within a precise time window. ◮ Light must not be red for more than a minute. ◮ A pacemaker must take over quickly and produce a precisely timed pattern.

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Introduction

Time is important

We want to be able to analyse timing aspects of systems.

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Introduction

Model checking

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Introduction

Model checking

Modelling/formalising

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Introduction

Model checking Requirements

"Must complete within two minutes." Modelling/formalising

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Introduction

Model checking

ϕ

Requirements

"Must complete within two minutes." Modelling/formalising Translating to formal specification language

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Introduction

Model checking

| = ϕ

Requirements

"Must complete within two minutes." Modelling/formalising Translating to formal specification language

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Introduction

Model checking

M | = ϕ

The model M satisfies the requirements given by ϕ.

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Introduction

Relations between models

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Introduction

Relations between models

Bisimulation ∼

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Introduction

Relations between models

Bisimulation ∼ Simulation

Models

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Models

Weighted transition systems

Robot vacuum cleaner

✇❛✐t✐♥❣ ❝❧❡❛♥✐♥❣ ❝❤❛r❣✐♥❣ 1 1 2 60 100 5 10 15

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Models

Weighted transition systems

Definition 2.6.1

A weighted transition system (WTS) is a tuple M = (S, →, ℓ), where ◮ S is a set of states, ◮ → ⊆ S × R≥0 × S is the transition relation, and ◮ ℓ : S → 2AP is the labelling function.

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Models

Semi-Markov processes

Intelligent traffic light

0.1 g1, r2 2 g1, r2 0.1 r1, g2 2 r1, g2 ❝❛r1?, st❛②! : 1 ❝❛r1?, ❝❤❛♥❣❡! : 1 ❝❛r1?, st❛②! : 0.9 ❝❛r1?, ❝❤❛♥❣❡! : 0.1 ❝❛r1?, ❝❤❛♥❣❡! : 1 ❝❛r2?, st❛②! : 1 ❝❛r2?, ❝❤❛♥❣❡! : 1 ❝❛r2?, st❛②! : 0.9 ❝❛r2?, ❝❤❛♥❣❡! : 0.1 ❝❛r2?, ❝❤❛♥❣❡✦ : 1

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Models

Semi-Markov processes

Definition 2.6.4

A semi-Markov process (SMP) is a tuple M = (S, τ, ρ, ℓ), where ◮ S is a countable set of states, ◮ τ : S × ■♥ → D(S × ❖✉t) is the transition function, ◮ ρ : S → D(R≥0) is the time-residence function, and ◮ ℓ : S → 2AP is the labelling function.

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Models

Semi-Markov processes

Reactive semi-Markov processes:

τ : S × ■♥ → D(S) input

Generative semi-Markov processes:

τ : S → D(S × ❖✉t)

• utput

Contributions

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Contributions

Papers

◮ Paper A: Reasoning About Bounds in Weighted Transition Systems, published in LMCS.

Co-authors: Mikkel Hansen, Kim Guldstrand Larsen, and Radu Mardare.

◮ Paper B: Timed Comparisons of Semi-Markov Processes, published in LATA ’18.

Co-authors: Nathanaël Fijalkow, Giorgio Bacci, Kim Guldstrand Larsen, and Radu Mardare.

◮ Paper C: A Faster-Than Relation for Semi-Markov Decision Processes, unpublished.

Co-authors: Giorgio Bacci and Kim Guldstrand Larsen.

◮ Paper D: A Hemimetric Extension of Simulation for Semi-Markov Decision Processes, published in QEST ’18.

Co-authors: Giorgio Bacci, Kim Guldstrand Larsen, and Radu Mardare.

Paper A

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Contributions

Paper A

Contribution 1

We present a language for reasoning about lower and upper bounds in weighted transition systems and we show that this language characterises exactly those systems that have the same kind of behaviour.

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Contributions

Paper A

Weighted logic with bounds (WLWB):

ϕ, ψ ::= p | ¬ϕ | ϕ ∧ ψ | Lrϕ | Mrϕ

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Contributions

Paper A

Weighted logic with bounds (WLWB):

ϕ, ψ ::= p | ¬ϕ | ϕ ∧ ψ | Lrϕ | Mrϕ Lrϕ: a transition with at least weight r can be taken to where ϕ holds. Mrϕ: a transition with at most weight r can be taken to where ϕ holds.

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Contributions

Paper A

✇❛✐t✐♥❣ s1 ❝❧❡❛♥✐♥❣ s2 ❝❤❛r❣✐♥❣ s3 1 1 2 60 100 5 10 15

s1 | = M2❝❤❛r❣✐♥❣ ❝❤❛r❣✐♥❣ ✇❛✐t✐♥❣ ✇❛✐t✐♥❣

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Contributions

Paper A

✇❛✐t✐♥❣ s1 ❝❧❡❛♥✐♥❣ s2 ❝❤❛r❣✐♥❣ s3 1 1 2 60 100 5 10 15

s1 | = M2❝❤❛r❣✐♥❣ s1 | = M1❝❤❛r❣✐♥❣ ✇❛✐t✐♥❣ ✇❛✐t✐♥❣

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Contributions

Paper A

✇❛✐t✐♥❣ s1 ❝❧❡❛♥✐♥❣ s2 ❝❤❛r❣✐♥❣ s3 1 1 2 60 100 5 10 15

s1 | = M2❝❤❛r❣✐♥❣ s1 | = M1❝❤❛r❣✐♥❣ s2 | = L2✇❛✐t✐♥❣ ✇❛✐t✐♥❣

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Contributions

Paper A

✇❛✐t✐♥❣ s1 ❝❧❡❛♥✐♥❣ s2 ❝❤❛r❣✐♥❣ s3 1 1 2 60 100 5 10 15

s1 | = M2❝❤❛r❣✐♥❣ s1 | = M1❝❤❛r❣✐♥❣ s2 | = L2✇❛✐t✐♥❣ s2 | = L7✇❛✐t✐♥❣

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Contributions

Paper A

Theorem A.2.5

For image-finite WTSs, we have s ∼ t if and only if for all ϕ, s | = ϕ if and only if t | = ϕ.

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Contributions

Paper A

Contribution 2

We provide a complete axiomatisation of the logical specification language, and give an algorithm for deciding the model checking problem and an algorithm for deciding satisfiability of a formula.

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Contributions

Paper A

(A1): ⊢ ¬L0⊥ (A2): ⊢ Lr+qϕ → Lrϕ if q > 0 (A2′): ⊢ Mrϕ → Mr+qϕ if q > 0 (A3): ⊢ Lrϕ ∧ Lqψ → Lmin{r,q}(ϕ ∨ ψ) (A3′): ⊢ Mrϕ ∧ Mqψ → Mmax{r,q}(ϕ ∨ ψ) (A4): ⊢ Lr(ϕ ∨ ψ) → Lrϕ ∨ Lrψ (A5): ⊢ ¬L0ψ → (Lrϕ → Lr(ϕ ∨ ψ)) (A5′): ⊢ ¬L0ψ → (Mrϕ → Mr(ϕ ∨ ψ)) (A6): ⊢ Lr+qϕ → ¬Mrϕ if q > 0 (A7): ⊢ Mrϕ → L0ϕ (R1): ⊢ ϕ → ψ = ⇒ ⊢ (Lrψ ∧ L0ϕ) → Lrϕ (R1′): ⊢ ϕ → ψ = ⇒ ⊢ (Mrψ ∧ L0ϕ) → Mrϕ (R2): ⊢ ϕ → ψ = ⇒ ⊢ L0ϕ → L0ψ + axioms for propositional logic.

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Contributions

Paper A

Soundness and completeness Theorem A.4.2 and A.4.10

⊢ ϕ if and only if | = ϕ

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Contributions

Paper A

Model checking: Does a given model M satisfy a given formula ϕ? Theorem A.5.4

The model checking problem for WLWB is decidable.

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Contributions

Paper A

Model checking: Does a given model M satisfy a given formula ϕ? Theorem A.5.4

The model checking problem for WLWB is decidable.

Satisfiability: Does there exist a model which satisfies a given formula ϕ? Theorem A.5.11

The satisfiability problem for WLWB is decidable.

Paper B and Paper C

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Contributions

Paper B and Paper C 2 s1 1 s2 a : 1 a : 1

s1 s2

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Contributions

Paper B and Paper C 2 s1 1 s′

1

1 s2 2 s′

2

a : 1 a : 1 a : 1 a : 1

s1 s2

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Contributions

Paper B and Paper C 2 s1 1 s′

1

1 s′′

1

1 s2 2 s′

2

1 s′′

2

a : 1 a : 1 a : 1 a : 1 a : 1 a : 1

s1 s2

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Contributions

Paper B and Paper C

Generative: Definition B.2.3

s1 is faster than s2 (s1 s2) if for all a1 . . . an and t we have P(s1)(a1 . . . an, t) ≥ P(s2)(a1 . . . an, t).

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Contributions

Paper B and Paper C

Generative: Definition B.2.3

s1 is faster than s2 (s1 s2) if for all a1 . . . an and t we have P(s1)(a1 . . . an, t) ≥ P(s2)(a1 . . . an, t).

Reactive: Definition C.4.3

s1 is faster than s2 (s1 s2) if for all schedulers σ, a1 . . . an, and t there exists a scheduler σ′ such that Pσ′(s1)(a1 . . . an, t) ≥ Pσ(s2)(a1 . . . an, t).

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Contributions

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.

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Contributions

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.

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Contributions

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.

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Contributions

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.

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Contributions

Paper B and Paper C

Contribution 5

We give an algorithm for unambiguous processes which can decide whether one process is faster than another.

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Contributions

Paper B and Paper C

A SMP is unambiguous if every output label leads to a unique successor state.

s1 s2 s3 a : 1

2

a : 1

2

a : 1 a : 1

3

b : 2

3

Figure 1: Ambiguous

s1 s2 s3 a : 1

2

b : 1

2

a : 1 a : 1

3

b : 2

3

Figure 2: Unambiguous

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Contributions

Paper B and Paper C

A SMP is unambiguous if every output label leads to a unique successor state.

s1 s2 s3 a : 1

2

a : 1

2

a : 1 a : 1

3

b : 2

3

Figure 1: Ambiguous

s1 s2 s3 a : 1

2

b : 1

2

a : 1 a : 1

3

b : 2

3

Figure 2: Unambiguous

Theorem B.5.2

For unambiguous SMPs, the faster-than problem is decidable.

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Contributions

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.

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Contributions

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.

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Contributions

Paper B and Paper C Context Component Component

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Contributions

Paper B and Paper C Context Component Component

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Contributions

Paper B and Paper C Context Component Component

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Contributions

Paper B and Paper C Context Component Component

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Contributions

Paper B and Paper C Context Component Component

Timing anomaly

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Contributions

Paper B and Paper C

Theorem C.6.15

There exist decidable conditions that guarantee the absence of timing anomalies.

Paper D

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Contributions

Paper D

Reactive processes

Simulation

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Contributions

Paper D

Reactive processes

Simulation

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Contributions

Paper D

Reactive processes

Simulation

• But how close is the

process to simulating the other process?

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Contributions

Paper D

Reactive processes

Simulation

• But how close is the

process to simulating the other process? Quantitative measure of distance

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Contributions

Paper D

Definition D.2.2

s2 simulates s1, written s1 s2, if . . . ◮ Fs1(t) ≤ Fs2(t) for all t ∈ R≥0 . . .

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Contributions

Paper D

Exponential distribution

2 4 6 0.2 0.4 0.6 0.8 1 time probability F(x) F(2 · x) F(4 · x)

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Contributions

Paper D

Definition D.2.2

s2 simulates s1, written s1 s2, if . . . ◮ Fs1(t) ≤ Fs2(t) for all t ∈ R≥0 . . .

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Contributions

Paper D

Definition D.2.2

s2 ε-simulates s1, written s1 ε s2, if . . . ◮ Fs1(t) ≤ Fs2(ε · t) for all t ∈ R≥0 . . .

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Contributions

Paper D

Definition D.2.2

s2 ε-simulates s1, written s1 ε s2, if . . . ◮ Fs1(t) ≤ Fs2(ε · t) for all t ∈ R≥0 . . .

Definition D.4.5

d(s1, s2) = inf{ε ≥ 1 | s1 ε s2}

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Contributions

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.

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Contributions

Paper D

Contribution 9

We show that, under mild assumptions, composition is non-expansive with respect to the distance between semi-Markov processes.

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Contributions

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.

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Contributions

Paper D

Timed Markovian logic

❚▼▲ : ϕ, ϕ′ ::= α | ¬α | ℓpt | mpt | La

pϕ | Ma pϕ | ϕ ∧ ϕ′ | ϕ ∨ ϕ′

❚▼▲ ❚▼▲

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Contributions

Paper D

Timed Markovian logic

❚▼▲ : ϕ, ϕ′ ::= α | ¬α | ℓpt | mpt | La

pϕ | Ma pϕ | ϕ ∧ ϕ′ | ϕ ∨ ϕ′

La

pϕ: probability of going with an a to where ϕ holds is at least p.

Ma

pϕ: probability of going with an a to where ϕ holds is at most p.

❚▼▲ ❚▼▲

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Contributions

Paper D

Timed Markovian logic

❚▼▲ : ϕ, ϕ′ ::= α | ¬α | ℓpt | mpt | La

pϕ | Ma pϕ | ϕ ∧ ϕ′ | ϕ ∨ ϕ′

La

pϕ: probability of going with an a to where ϕ holds is at least p.

Ma

pϕ: probability of going with an a to where ϕ holds is at most p.

ℓpt: probability of leaving state before time t is at least p. mpt: probability of leaving state before time t is at most p. ❚▼▲ ❚▼▲

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Contributions

Paper D

Timed Markovian logic

❚▼▲ : ϕ, ϕ′ ::= α | ¬α | ℓpt | mpt | La

pϕ | Ma pϕ | ϕ ∧ ϕ′ | ϕ ∨ ϕ′

La

pϕ: probability of going with an a to where ϕ holds is at least p.

Ma

pϕ: probability of going with an a to where ϕ holds is at most p.

ℓpt: probability of leaving state before time t is at least p. mpt: probability of leaving state before time t is at most p. ❚▼▲≥ : ϕ ::= α | ¬α | ℓpt | La

pϕ | ϕ ∧ ϕ′ | ϕ ∨ ϕ′

❚▼▲≤ : ϕ ::= α | ¬α | mpt | Ma

pϕ | ϕ ∧ ϕ′ | ϕ ∨ ϕ′

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Contributions

Paper D

Perturbation (ϕ)ε: ◮ (ℓpt)ε = ℓpε · t ◮ (mpt)ε = mpε · t ❚▼▲ ❚▼▲

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Contributions

Paper D

Perturbation (ϕ)ε: ◮ (ℓpt)ε = ℓpε · t ◮ (mpt)ε = mpε · t

Theorem D.7.2

For finite SMPs we have ◮ d(s1, s2) ≤ ε if and only if for all ϕ ∈ ❚▼▲≥, s1 | = ϕ implies s2 | = (ϕ)ε ◮ d(s2, s1) ≤ ε if and only if for all ϕ ∈ ❚▼▲≤, s2 | = (ϕ)ε implies s1 | = ϕ

Conclusion

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Conclusion

Summary

◮ Formalisms for specifying, comparing, and reasoning about properties involving time. ◮ Algorithms enabling use of these formalisms in practice.

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Conclusion

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.

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Conclusion

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.

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Conclusion

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

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Future work

Strong completeness

Weak completeness

| = ϕ implies ⊢ ϕ

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Future work

Strong completeness

Strong completeness

Φ | = ϕ implies Φ ⊢ ϕ

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Future work

Timing anomalies Context Component Component

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Future work

Timing anomalies Context Component Component

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Future work

Timing anomalies Context Component Component

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Future work

Timing anomalies Context Component Component

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Future work

Timing anomalies Context Component Component

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Future work

Branching in simulation distance Exp[1] s1 Dirac[0] s′

1

Exp[2] s′′

1

a : 0.5 a : 1 a : 0.5 a : 1 Exp[1] s2 Dirac[0] s′

2

Exp[2] s′′

2

a : 0.49 a : 1 a : 0.51 a : 1

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Future work

Branching in simulation distance Exp[1] s1 Dirac[0] s′

1

Exp[2] s′′

1

a : 0.5 a : 1 a : 0.5 a : 1 Exp[1] s2 Dirac[0] s′

2

Exp[2] s′′

2

a : 0.49 a : 1 a : 0.51 a : 1

d(s1, s2) = ∞

Thank you!

Exp[3] s1 Exp[θ] s2 a : 1 a : 1 Exp[4] t1 Exp[5] t2 Exp[9] t3 a : 0.1 a : 1 a : 0.9 a : 1

Figure 3: A semi-Markov process where s1 t1 if θ ≤ 5 and s1 t1 if θ > 5.

Time-bounded approximation

b m m m m m n times

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

Tableau {¬(¬(L2p1 ∧ M5L1p1) ∧ M2p2)}, [0, 0], [0, 0] (¬∧) {¬¬(L2p1 ∧ M5L1p1)}, [0, 0], [0, 0] (¬¬) {L2p1 ∧ M5L1p1}, [0, 0], [0, 0] (∧) {L2p1, M5L1p1}, [0, 0], [0, 0] (mod) {p1, L1p1}, [2, ∞), [5, ∞) (mod) {p1}, [1, ∞), [0, ∞) {¬¬M2p2}, [0, 0], [0, 0] (¬¬) {M2p2}, [0, 0], [0, 0] (mod) {p2}, [0, ∞), [0, 2]

sT {} s1 {p1} s2 {p1} 5 2 1 1

Image-finite counterexample

ω

. . .

n

. . .

2 1 s t

. . . . . .

2 1 4 1 4 1 4 3 1 4 1 4 1 4

Figure 4: s and t satisfy the same logical formulas, but s ∼ t.

Kantorovich counterexample

ui vi εi 1 − εi u v 1

Figure 5: A Markov process with states ui and vi for each i ∈ N.

New axioms {Lqϕ | q < r} ⊢ Lrϕ and {Mqϕ | q < r} ⊢ Mrϕ

Generative composition – synchronous

s1 s2 s3 t1 t2 t3 a : 1

2

b : 1

2

a : 1

3

c : 2

3

s1 t1 s2 t2 s3 t2 s2 t3 s3 t3 aa : 1

6

ba : 1

6 ac : 1 3

bc : 1

3

Example from Ana Sokolova and Erik P . de Vink, Probabilistic Automata: System Types, Parallel Composition and Comparison, in Validation of Stochastic Systems - A Guide to Current Research, Lecture Notes in Computer Science volume 2925, pp. 1–43, 2004