1. Traffic Lights Specification Niek

2. Overview Traffic lights are used everywhere Various algorithms: ◦Simple time-based traffic lights ◦Pressure sensors-based systems ◦Connected systems (“green wave”) We don’t bother: ◦This is not an algorithm specification ◦We’ll define a spec to check whether a traffic lights system is correct ◦Not whether it’s the most optimal solution

3. Assumptions Placed at intersections ◦Typically T-type or cross-type intersections ◦Assumption: dedicated light/lane per direction ◦No deadlocks shall occur ◦Example:

4. Definition of an intersection type Intersection := {Road} type Road := {Lane} There is a relation between the number of roads at an intersection and the number of lanes per road:

5. Definition of lanes type Lane := (label :: Label, conflicts :: {Label}, color :: Color) type Color := Green | Red The label of a lane is unique in the scope of the intersection The conflicts labels are the lanes which cannot be green at the same time as the current lane

6. Conflicting lanes example Crossing lanes are not allowed Neither are lanes which arrive on the same road

7. Enforcing unique labels Labels are unique in the scope of the intersection: type Intersection := {Road} type Road := {Lane} type Lane := (label :: Label, conflicts :: {Label}, color :: Color)

8. Conflicting lanes If conflicting labels are specified, they must be part of the same intersection: type Intersection := {Road} type Road := {Lane} type Lane := (label :: Label, conflicts :: {Label}, color :: Color)

9. Correctness Conflicting lanes may not have a green light at the same time: type Intersection := {Road} type Road := {Lane} type Lane := (label :: Label, conflicts :: {Label}, color :: Color)

10. Correctness (cont’d) Intuitively: if a lane has a green light, all non-conflicting lanes should be green as well: This is not the case, example on next slide. type Intersection := {Road} type Road := {Lane} type Lane := (label :: Label, conflicts :: {Label}, color :: Color)

11. Correctness (cont’d) Example: In other words: an algorithm is required to select non-conflicting lanes with the highest “priority” ◦E.g. Longest wait time/most cars

12. Conclusion If a traffic lights system confirms this spec, no accidents will occur ◦At least not due to conflicting lights Whether or not the system is optimized, depends on the algorithm Real-life issues which make the problem harder: ◦Pedestrian paths/lights crossing the lanes ◦Merged lanes/lights

13. Conclusion (cont’d) 0101 5454 2 3 Intersection = { {(0, {2, 4, 3}, Green), (1, {4}, Green)}, {(2, {0, 4, 5}, Red), (3, {0}, Red)}, {(4, {0, 1, 3}, Red), (5, {2}, Green)} }