Robotic Paradox?

[display name]

I beleive I breathed life into an old concept with the following thought experiment problem.

 

Ok, take a robotic vehicle traveling at one km per hour towards a distant wall just over one km away. The machine is aproacing a starting line drawn on the ground that marks the 1000 meters to the far wall.

 

While the robot is aproaching at a steady crawl, it's powerfull onboard computer is preparing its plan of action to be excuted the instant it reaches the starting line.

 

The plan that is being prepaired is as follows and is about to be exicuted is as follows and will be repeated in a loop as soon as its mission is acheived;

 

a- before reaching the starting point, calculate the distance from the starting point to the wall.

 

b- divide by 2 and set a virtual destination point half way to the wall from the starting point.

 

c- travel to the virtual mid way destination at a constant one km per hour.

 

d-before reaching the half way goal, prepare what is to occour once the destination is acheived. which will be to repeat steps a through d. With the only alteration that the definition of the 'starting point' will be replaced with the most recent 'virtual' half way destination.

 

e-roll over virtual destinations, do not stop. (this is the purpose of pre-calculating the next leg of the journey in advance, to allow a rolling start. Maintain a constant forward velosity of one km per hour.

 

What will happen?

 

Will the robot reach the wall in one hour?

 

Will the robot never get there because the computer is only allowed to set a destination goal of half way to the wall from what ever its last position was?

 

I do not know what to think about this one, Is this a perfect paradox? What do you say and why?

 

It is interesting to imagine what would happen with a real-world test beyond the thought experiment only realm.

 

 

 

 

 

 

 

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Edited January 24, 2011 by display name

[tomgwyther]

i suppose it would get infinately close to the wall but would never actually touch it. i.e it would go:

500m then recalculate and go

250m, then go

125m then

62.5m

31.25m

15.625m

7.8125m

3.90625m

 

Etc.

[Trog]

This sounds like a variation on one of Zeno's Paradox. I thought the standard solution to this lay in the mathematical idea of convergence. Your description of the problem can be described my a mathematical series (I forget which one). The question is whether that series converges or diverges. The standar Zeno paradaox on motion is a converent series.

Edited January 24, 2011 by Trog

[ewmon]

In this variation of the Dichotomy Paradox first described by Zeno of Elea 2,500 years ago, it takes time to measure and compute the next distance to travel. Thus, the robot slows as it approaches the wall.

 

The situation here whether to command the robot: 1) to travel at 1 km/hr (in which case it reaches the wall in 1 hour by starting 1 km away from it), or 2) to command the robot to measure, divide and move. It cannot do both indefinitely. No paradox exists.

Edited January 24, 2011 by ewmon

[TonyMcC]

Toward the end of its journey the steps will be so short that there will not be time to precalculate the next step. It depends on what the robot is progammed to do in this situation. If it is told to stop,make its calculation and move to the next point and keep repeating the process then it should never get to the wall. If it's told to carry on moving at the initial speed while attempting to calculate the next step and keep repeating this process then it will travel at its initial speed until it reaches the wall. It just depends on the programmer! (IMO)

Just had a good look at ewmon's post - I see I have pretty much said the same thing!

Edited January 24, 2011 by TonyMcC

[Mr Skeptic]

There's no paradox, the robot will do as it was programmed to do. Since a programmer dumb enough to write such a program with the intention of the robot reaching the wall probably won't worry about the capacity of its numbers, it is likely that due to either measurement error or rounding error the robot will reach the wall eventually.

[TonyMcC]

it is likely that due to either measurement error or rounding error the robot will reach the wall eventually.

A debatable point. On the basis of a rounding error it might decide it has to move something like 0.000000000001 units of measurement which the computer might take as zero. In which case it might stop an almost indescribable amount short of the wall unable to calculate the last step. Alternatively the computer might crash trying to do a "divide by zero" just before reaching the wall. On the other hand the slightest tremor might make you right!

[display name]

OK step two of this possible paradox.

 

1 Would you agree that there is a paradox if the computer was infinitely fast and capable?

 

2 The vehicle would hit the wall in one hour if it did not need to slow down to give time to calculate if the computer was infinitely fast. So it would both travel the 1000 meters in one hour and at the same time never get more than half way to the wall from any point during the journey it happens to be at any given moment.

 

3 Einstein is allowed to have imaginary tools in his thought experiments so why not me. For example the time dilation thought experiment that is comparing the relative motions of an atomic clocks aboard a jet to one on the surface of the earth to an imaginary clock in the center of the earth.

 

4 And this question which seems to be a paradox just in asking. Would a computer need to be infinitely fast and capable or just good enough to keep up for one hour? If it can do its job without requiring the wheels to slow for only one hour it will hit the wall. And one hour is not infinite -hence the computer would not have to be infinitely capable...

 

 

 

 

5 Just for fun, those who believe in god then consider this variation. God takes the place of the robot and the will of god takes the place of the computer! What then, will you say? Will god slow down because he cant keep up with the demanding math!

 

 

 

3

Edited January 24, 2011 by display name

[TonyMcC]

If you set theoretical conditions that basically "tell" the robot not to slow down then it must arrive at the wall in an hour. The fact that this takes an infinite number of steps is a theoretical condition set by you that is unrealistic in real life.(imo)

[display name]

If you set theoretical conditions that basically "tell" the robot not to slow down then it must arrive at the wall in an hour. The fact that this takes an infinite number of steps is a theoretical condition set by you that is unrealistic in real life.(imo)

 

 

 

You may be missing the subtly or the essence of this. You said 'it must arrive' but why so confident? In fact there is significant reasonable doubt here resulting from the fact that nowhere in the programming will the computer ever 'lock on' to the wall as its actual destination. Its ironic really, the computer program will allow it to go anywhere EXCEPT the wall.

 

I am not arguing that it will hit the wall, just that it is no more likely to make it to the wall than not to. Paradox. At least in the perfect computer version and especially for the god one.

 

This sounds like a variation on one of Zeno's Paradox. I thought the standard solution to this lay in the mathematical idea of convergence. Your description of the problem can be described my a mathematical series (I forget which one). The question is whether that series converges or diverges. The standar Zeno paradox on motion is a convergent series.

 

I am unfamiliar with convergent or divergent series. What they are and how they may apply here and to Zeno's Paradox. Any links or clarification please?

 

There's no paradox, the robot will do as it was programmed to do. Since a programmer dumb enough to write such a program with the intention of the robot reaching the wall probably won't worry about the capacity of its numbers, it is likely that due to either measurement error or rounding error the robot will reach the wall eventually.

 

The purpose of the program was not to reach the wall so the programmer of the computer or designer of the thought experiment is not 'dumb'. The purpose was to decribe a paradox where a vehicle traveling at a constant velosity will NOT hit the wall in one hour as it must, or not...?

 

Most of the world's populations believe in god, so what would they say if it was god instead of the robot doing this experiment, then it would really be a paradox unless you were willing to admit that god was limited in how fast he can divide by the number 2!

 

Toward the end of its journey the steps will be so short that there will not be time to recalculate the next step. It depends on what the robot is programmed to do in this situation. If it is told to stop,make its calculation and move to the next point and keep repeating the process then it should never get to the wall. If it's told to carry on moving at the initial speed while attempting to calculate the next step and keep repeating this process then it will travel at its initial speed until it reaches the wall. It just depends on the programmer! (IMO)

Just had a good look at Eamon's post - I see I have pretty much said the same thing!

 

Ya you first three respondents have it right, and it would not be possible because of the limited commuting power of the machine. Put its is still a great one with the god version! In fact I should post it over there except they might not care much about a scientific paradox...

[Mr Skeptic]

Well first you should look up the word "paradox".

[lemur]

5 Just for fun, those who believe in god then consider this variation. God takes the place of the robot and the will of god takes the place of the computer! What then, will you say? Will god slow down because he cant keep up with the demanding math!

I think God would notice the number approaching zero and then consider allowing the infinite approach to converge with the endpoint. Maybe God would elevate the discrepancy between the eternally nearing point and the eternally unreachable endpoint to infinity despite infinite nearness. I think God could multiply this question in ways that we haven't even thought of, yet, just because God is the superset with all possible human interpretations as a subset. God could probably continue coming up with multiple interpretations that make it half way to totally solving the paradox without every reaching a true explanation, just for the sake of generating an infinitude of explanations AND THEN transcend the problem by coming up with a single simple reductive solution that defies the paradox. God is the subjective idea of limitless power so however far you can subjectively imagine power to deal with this paradox, your subjective concept of God transcends that - if only by virtue of the fact that God is an idea that any and all limits can be transcended.