Wasting Tax-Dollars on Beg-Buttons

At most intersections, you can usually find a pair of “beg-buttons” for pedestrians to push to tell the traffic-control system that they are there and want to cross. There are usually separate buttons for each direction (one for crossing north-south and another for crossing east-west). This results in up to eight buttons at each intersection. However, this is wasteful over-engineering. In fact, with proper software design, only a single button is ever necessary at each corner of a standard 4-way intersection. Can you figure out the logic to prove this?

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Intersection button design options
Intersections can have either one or two crossing buttons at each corner

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