Expected Reachability-Time Games

Q: What is the first technical property of quasi-simple functions?

The first technical property of quasi-simple functions will allow us to establish that when the authors take a delay so that a boundary of a region is reached, quasi-simplicity is preserved.

Q: what is the k-shift of a fractional signature?

To see that, notice that the να in the definition of p̂ (Definition 15) satisfies that 〈να〉 is a k-shift of 〈ν〉 = ( f0, . . . fm) for k chosen so that fm is the fractional part of clocks that have integer value in να .

Q: What is the way to prove that the limit of the step-bounded value functions is?

The authors first inductively show that, when the number of steps is bounded, then the value functions in timed automata and boundary region graph coincide and are non-expansive within a region.

Q: What is the motivation for studying simpler properties?

Studying simpler properties is thus motivated by the desire to obtain decidable properties while still being able to study sufficiently complex class of properties.

Q: What is the proof direction for Kleene fixpoint theorem?

Another possible proof direction is Kleene fixpoint theorem, which requires Scott-continuity on the value functions, which again is a property that is difficult to establish in their setting.

Question

1. What contributions have the authors mentioned in the paper "Expected reachability-time games" ?

2. What have the authors stated for future works in "Expected reachability-time games" ?

3. What is the importance of probabilistic behaviour in controller synthesis?

4. What is the demanding step of the proof of the theorem 18?

Accepted Answer

In this paper, the authors consider a more complex setting and study reachability-time time objectives, which express the expected time to reach a given target set.

Accepted Answer

A natural future direction is to investigate the possibility of devising similar algorithms for probabilistic timed games.. On the theoretical level, the authors plan to study if their approach can be utilised for extensions of reachability-time objectives by considering an appropriate class of reward-based properties.

Accepted Answer

Probabilistic behaviour is important in modelling, e.g., faulty or unreliable components, the random coin flips of distributed communication and security protocols, and performance characteristics.

Accepted Answer

The crucial, and most demanding, step of the proof of Theorem 18 is proving that the problems on PTGAs can be reduced to problems on BRAs.

Accepted Answer

Using standard results from probability theory [33] the authors can construct a probability space ( μ,χ (s), Fμ,χ (s), Prμ,χs ) where Fμ,χ (s) is the smallest σ -algebra generated by the basic cylinder sets and Prμ,χs : F → [0, 1] is the unique probability measure such that for any finite play ρ = 〈s0, (α1, β1), s1, . . . , sk−1, (αk, βk), sk〉 ∈ μ,χf (s):Prμ,χs (Cyl(ρ)) = ∏ki=1 p(si−1,αi, βi)(si) where pre(ρ, i) = 〈s0, (α1, β1), s1, . . . , si−1, (αi, βi), si〉 for all i < k.

Accepted Answer

The first technical property of quasi-simple functions will allow us to establish that when the authors take a delay so that a boundary of a region is reached, quasi-simplicity is preserved.

Accepted Answer

To see that, notice that the να in the definition of p̂ (Definition 15) satisfies that 〈να〉 is a k-shift of 〈ν〉 = ( f0, . . . fm) for k chosen so that fm is the fractional part of clocks that have integer value in να .

Accepted Answer

The authors first inductively show that, when the number of steps is bounded, then the value functions in timed automata and boundary region graph coincide and are non-expansive within a region.

Accepted Answer

Studying simpler properties is thus motivated by the desire to obtain decidable properties while still being able to study sufficiently complex class of properties.

Accepted Answer

Another possible proof direction is Kleene fixpoint theorem, which requires Scott-continuity on the value functions, which again is a property that is difficult to establish in their setting.

Accepted Answer

The works of [17] and [14] show that the decision problem for such games over timed automata with at least two clocks is EXPTIME-complete.

Accepted Answer

To their best knowledge, this is the first decidability result for stochastic games on timed automata in which the objective concerns a random variable that takes non-binary values.

Accepted Answer

Although the computational complexity of solving games on timed automata is high, UPPAAL Tiga [6] is able to solve practical reachability and safety properties for timed games by using efficient symbolic zone-based algorithms [7,11].

Accepted Answer

The authors consider an extension of probabilistic time automata (PTA) [8–10], a model for real-time systems exhibiting nondeterministic and probabilistic behaviour.

Accepted Answer

The correctness of these equations can be easily obtained from the fact that for any n 0, s /∈ F , path ρ with last(ρ) = s and strategies μ and χ , where α = win(μ(ρ), χ(ρ)), by definition of Eμ,χρ the authors have:E μ,χ ρ (Reach n+1 F ) = E μρ,χρ s (Reach n+1 F ) = ∫ρ ′∈ μρ,χρ (s) Reachn+1F (ρ ′)dPrμρ,χρs (by definition of expectation)= ∑ s′∈S ( ∫ ρ̄∈ μραs′ ,χραs′ (s′) p(s,μρ(s),χρ(s))(s ′)· ( τ (s,μρ(s),χρ(s)) + ReachnF (ρ̄) ) dPr μ ραs′ ,χραs′ s′ (ρ̄) ) (by definition of μρ,χρ (s), Pr μρ,χρ s and Reach n+1 F )= τ (s,μρ(s),χρ(s))+ ∑ s′∈S p(s,μρ(s),χρ(s))(s ′) · ⎛⎜⎜⎝ ∫ ρ̄∈ μραs′ ,χραs′ (s′) ReachnF (ρ̄)dPr μ ραs′ ,χραs′ s′ (ρ̄) ⎞⎟⎟⎠ (rearranging) = τ (s,μ(ρ),χ(ρ)) +∑ s′∈Sp(s,μ(ρ),χ(ρ))(s′) ·Eμ,χρ α s′(ReachnF ) (by properties of μρ , χρ and definition of expectation)Let us now turn to the equations for infinite-horizon objectives.

Accepted Answer

A natural extension of games with reachability-time objectives are games on priced timed automata where the objective concerns the cumulated price of reaching a target.

Expected Reachability-Time Games

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Most frequently asked questions

1. What contributions have the authors mentioned in the paper "Expected reachability-time games" ?

2. What have the authors stated for future works in "Expected reachability-time games" ?

3. What is the importance of probabilistic behaviour in controller synthesis?

4. What is the demanding step of the proof of the theorem 18?

5. what is the smallest -algebra for a finite play?

6. What is the first technical property of quasi-simple functions?

7. what is the k-shift of a fractional signature?

8. What is the way to prove that the limit of the step-bounded value functions is?

9. What is the motivation for studying simpler properties?

10. What is the proof direction for Kleene fixpoint theorem?

11. What is the decision problem for games with at least two clocks?

12. What is the first decidability result for stochastic games on timed automata?

13. How can the authors solve reachability and safety properties for timed games?

14. What is the definition of a probabilistic time automata?

15. What is the correctness of the equations for infinite-horizon objectives?

16. What is the definition of a natural extension of games with reachability-time objectives?

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Citations

Game-based Abstraction and Controller Synthesis for Probabilistic Hybrid Systems

Discrete-Time Verification and Control for Probabilistic Rectangular Hybrid Automata

Verification and Control of Turn-Based Probabilistic Real-Time Games.

Expected Reachability-Time Games

Expected reachability-time games

References

Lecture Notes in Computer Science 2382

A theory of timed automata

The control of discrete event systems

Competitive Markov decision processes

The complexity of stochastic games

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