Abstract
We develop a new method for determining the consistency of timed scenarios. If the scenario is consistent, we obtain a canonical representation for the entire class of equivalent scenarios. This allows us to optimise a scenario according to various criteria. In particular, we are able to minimize the largest constant in the scenario’s set of constraints: this technique is directly relevant to decreasing the costs of verification for timed automata synthesized from timed scenarios.
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Notes
- 1.
Most model-checking tools for timed automata (e.g., UPPAAL [7] and KRONOS [9]) use region-based and zone-based abtraction methods in order to make verification possible in spite of the infinite state spaces of timed automata. It is well-known that both of these abstraction methods depend on the number of clocks and on the size of the constants that appear in constraints. In fact, the size of the region graph is exponential in the number of clocks and the (encoding of) constants [3].
- 2.
Behaviours are essentially the “timed words” of Alur [2].
- 3.
To keep the presentation compact, we do not allow sharp inequalities: allowing them would complicate our definitions and proofs, without affecting the general principles. Notice that sharp inequalities are of mainly theoretical interest: in practice we can measure time only with some finite granularity \(\gamma \), so \(x < c\) is for all practical purposes equivalent to \(x \le c - \gamma \).
- 4.
There is nothing new in the intuition that a scenario describes a set of behaviours: see, e.g., the paper by Somé et al. [20].
- 5.
The other way is to increase \(M_{0 1}\) to 4, but that would introduce new behaviours.
- 6.
As elsewhere, \(t_{ij} = t_j - t_i\).
- 7.
If \(h_{ij} = \infty \) then \(t_{ij}^{\mathcal {B^H}}\) can be an arbitrary number not smaller than \(l_{ij}\).
- 8.
We are only proving the existence of such a behaviour. A method of actually constructing it is discussed in Sect. 3.3.
- 9.
The values in the table are rational numbers, but they have a least common denominator. Adding or subtracting two such values cannot produce a result that does not share that common denominator. So our algorithm cannot decrease the difference between two such values indefinitely without making it negative.
- 10.
For example, (R5) depends on \(h_{ik}\) and \(l_{jk}\), which can be changed by (R6) and (R3). But if (R5) is applicable, i.e., if \(h_{ij} + l_{jk} > h_{ik}\) holds, then (R6) is not applicable: we cannot have \(h_{ik} > h_{ij} + h_{jk}\), because \(l_{jk} \le h_{jk}\) (the table is valid); similarly, (R3) is not applicable: \(l_{ik} > h_{ij} + l_{jk}\) would imply \(l_{ik} > h_{ik}\).
- 11.
In practice it is often convenient to pinpoint the first “offending” constraint that makes a scenario inconsistent. This is easily done by making sure that the order of constraints in \(\varPsi \) corresponds to the textual order of constraints in the scenario, and attempting to stabilise the table each time a new constraint is added to it. If the number of constraints is proportional to n, the cost becomes \(O(n^4)\).
- 12.
If the attempt to do so fails because of the inconsistency of \(\xi \), then producing an equivalent scenario with smaller constants is trivial (and probably pointless).
- 13.
More formally, we create a finite sequence of scenarios, \(\eta ^0 \eta ^1 \ldots \) and a corresponding sequence of tables, \(\mathcal {D}^{\eta ^0}\mathcal {D}^{\eta ^1}\ldots \). We felt that a more pedantic presentation would be harder to follow.
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Saeedloei, N., Kluźniak, F. (2018). Timed Scenarios: Consistency, Equivalence and Optimization. In: Massoni, T., Mousavi, M. (eds) Formal Methods: Foundations and Applications. SBMF 2018. Lecture Notes in Computer Science(), vol 11254. Springer, Cham. https://doi.org/10.1007/978-3-030-03044-5_14
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