Three-way roundabout logic

Edward Tsang 2015.04.11

This is a mind exercise to analyse the dynamics in a British roundabout. It explains how deadlocks sometimes occur. Deadlocks can also be broken by deceiving other drivers (not advised).

Rules and notations:

Cars in Britain drive on the left. They must drive clockwise in a roundable. Consider the roundabout on the right. If a car entering the roundabout from road A wants to exit at road C, it must drive (clockwise) pass road B before it can turn into road C.

To keep description simple, we shall use A, B and C to refer to roads that lead to the roundabout, as well as the car on those roads. When there are no cars at road x, all propositions that refer to x will be treated as false.

Let x be a car that has arrived at a roundabout. We use left(x) and right(x) to denote the entrances on the left and right of x respectively. For example, left(A) is B, left(B) is C, which is also right(A). We use SR(x) to state that the car at road x signals right. We use Move(x) to state that the car at road x is moving into the roundabout. Stop(x) means the car at road x is stationary before entering the roundabout.

Rules that drivers tend to follow:

The car at entrance x will act according to the following rules:

  1. If the car on the right signals right and is moving, then x remains stationary
    SR(right(x)) AND Move(right(x) --> Stop(x)
    We call the condition Stop Condition 1.

    Example: In the above figure, if x=A, then A will stop if C (= right(A)) signals to turn right (i.e. to turn into B), and C is already moving into the roundabout.

  2. If (a) it is not true that there is a car on the left which is signalling right and moving, and (b) there exists a car on the right which signals right, then x remains stationary.
    (NOT (SR(left(x)) AND Moving(left(x)))) AND SR(right(x)) --> Stop(x)
    We call the condition Stop Condition 2.

    Example: In the above figure, A will stop if car C signals to turn right, and there is no moving car at B which is signalling to turn into A.

    This is a reasonable rule for a driver to follow, because a driver should give priority to the car on the right.

  3. If neither conditions in 1 nor 2 is true, then x can move into the roundabout.
    [NOT (SR(right(x)) AND Move(right(x)))]
    AND
    [NOT ((NOT (SR(left(x)) AND Moving(left(x)))) AND SR(right(x)))]
    --> NOT Stop(x)

Simplification of Rule 3

The condition of Rule 3 can be simplified in the following steps:

[NOT (SR(right(x)) AND Move(right(x)))]
AND
[(SR(left(x)) AND Moving(left(x))) OR (NOT SR(right(x)))]

[NOT (SR(right(x)) AND Move(right(x))) AND (SR(left(x)) AND Moving(left(x)))]
OR
[NOT (SR(right(x)) AND Move(right(x))) AND (NOT SR(right(x)))]

[NOT (SR(right(x)) AND Move(right(x))) AND (SR(left(x)) AND Moving(left(x)))]
OR
(NOT SR(right(x)))

We call the first term Move Condition 1:
[NOT (SR(right(x)) AND Move(right(x))) AND (SR(left(x)) AND Moving(left(x)))]
This means the car on the left signals right has already started moving into the roundabout, while the car on the right is not (a) signalling right and (b) already moved into the roundabout. This is a sufficient condition for car x to move into the roundabout.

We call the second term Move Condition 2:
(NOT SR(right(x)))
This means if there is no car on the right that signals to turn right, then car x can move into the roundabout.

Scenario 1 - Deadlock:

Let us assume that a car is waiting to enter the roundabout from each of the three roads. Let us assume that all of them are signalling to turn right, and all of them have stopped. In this case, they will be in a deadlock.

SR(A) AND NOT Moving(A)
AND SR(B) AND NOT Moving(B)
AND SR(C) AND NOT Moving(C)

This is a deadlock because all three cars will wait for each other (due to Stop Condition 2). Such deadlocks can easily be broken by any car taking initiative.

Scenario 2 - Deceiving signal:

Suppose we are in a deadlock as described above. Although A's intention is to turn right, it may benefit from signalling left (to turn into road B). Here is the logic: By signalling left, A will encourage B to enter the roundabout (due to Move Condition 2). This will stop C moving into the roundabout (due to Stop Condition 1), which would allow A to enter the roundabout (due to Move Condition 1).

The change of states is described below:

NOT SR(A) AND NOT Moving(A)
AND SR(B) AND NOT Moving(B)
AND SR(C) AND NOT Moving(C)
-->
NOT SR(A) AND NOT Moving(A)
AND SR(B) AND Moving(B) {due to NOT SR(A) AND Move Condition 2}
AND SR(C) AND NOT Moving(C)
-->
NOT SR(A) AND Moving(A) {due to Move Condition 1}
AND SR(B) AND Moving(B)
AND SR(C) AND NOT Moving(C)

In other words, by disguising its intention, A will gain access to the roundabout before B and C.

Note that as soon as A moves, it may immediately correct its signal "right" to avoid accidents. This is because when A turns towards C, if it continues to signal left, cars from road B may enter the roundabout when they expect A to turn into B.

What if all three cars attempt to deceive others?

This will be left as an exercise for interested readers.

One transportation expert told me that dynamics in small roundabouts are complicated because they require users to anticipate actions by others and there is not much time to react to unexpected actions.

Basic rules for reference:

Disclaimers:

  1. This is just a mind exercise. I have never used the tactic in Scenario 2 to gain priority in using a roundabout. Neither would I encourage anyone to do so.
  2. This analysis applies to small towns in UK. It may not work in big cities or in other countries.

[End]

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