A simple tought experiment

Suppose that you have a robot that is positioned at an arbitrary location in Central Park (NY) and must reach an also arbitrary goal in the park. The robot is not able to sense the environment and is always executing the same action, going forward at a constant velocity. Finally, the robot is able to announce, in a textual manner, the goal location to surrounding humans. Do you think our little robot can ever reach the goal position in finite time ? Check the correct answer here.

Now, most of you will argue that it is not truly a robot since it cannot sense the environment. Well, suppose that the critter a) is equipped with a GPS which just happens to loose the position fix the instant the experiment starts and b) the program is such that in the absence of GPS information the robot moves forward at a constant velocity. Formally, both cases are identical.

Some amongst you, might cry foul. The robots, that is the objects sensing the environment and taking actions according to a list of instructions, are in fact the people in the street. It just happens that given the input symbol "smiling card box with written text", the people/robots react with the action "point smiling card box in correct direction". There is some research which suggests humans internally represent maps in Euclidean form, among other formats.

But ok, these are very reasonable objections. So now let the robot have an always working GPS receiver, it can announce verbally and visually the direction in which it wants to travel but, no free lunches, has no locomotion capabilities. Just to make it clear. The robot is able to sense the environment, compute a path to the goal position but has only one action: announce the desired direction of travel to surrounding humans. Can this second robot reach the desired goal ? Of course it can, here and here are many commercial examples of such robots.