Naked Science Forum
General Science => General Science => Topic started by: NakedScientist on 22/05/2003 19:13:41
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I saw this amusing press release today and thought you lot might appreciate it !
Anyone else got any pets with uncanny abilities or daft habits ?
TNS
Follows...
"You may not be able to teach an old dog new tricks, but you can get him to solve a calculus problem.
That's what Dr. Tim Pennings, associate professor of mathematics at Hope College in Holland, Mich. realized when he took his Welsh Corgi, Elvis, for an outing to Lake Michigan to play fetch with his favorite tennis ball.
"Most calculus students are familiar with the problem of finding the optimal path from point A to point B," he says. "Standing on the water's edge at Point A, I throw the ball into the water at Point B. Elvis runs along the shore a portion of the way, then plunges into the lake at a point of his choosing and swims diagonally to the ball. By the look in Elvis's eyes, it seems clear that his objective is to retrieve it as quickly as possible."
Pennings assumed, then, that Elvis unconsciously attempts to find a path that minimizes retrieval time. But he wanted to test his theory.
So, he clocked Elvis's running and swimming speeds and spent three hours collecting data measuring the distance between the ball and the shore and where Elvis jumped in the water to retrieve it. Plotting the results revealed that, in most cases, Elvis chose a path which is in close agreement with the optimal path calculated mathematically.
In fact, given complicating factors -- waves, the movement of the ball in the water, and others -- Pennings suggests that dogs may choose a path that is actually better than the calculated ideal.
"Although he made good choices, Elvis does not know calculus," admits Pennings, who published his findings in the May 2003 issue of The College Mathematics Journal. "Though he does not do the calculations, Elvis's behavior is an example of the uncanny way in which nature often finds optimal solutions."
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This is one of the few things I actually know a little about.
The real question is one of optimization, and whether or not the problem is convex or not. Convex optimization is like finding the lowest point in an empty swimming pool by throwing a tennis ball in it. The tennis ball will finally find the drain. Mathematically, the pool problem is "convex", meaning in broad terms that all the tennis ball has to do is to respond to its immediate surroundings, and it will find the global optimum.
A near convex problem is finding the lowest spot on earth by walking downhill all of the time. Most of the time you will wind up at sea level, which is within a couple of thousand feet of the true answer.
A non-convex problem is one of finding the highest spot on earth by walking uphill all of the time. The chances of even winding up in Nepal are almost zero. The chances of winding up on a minor hill a continent away are about 100%.
The point being that convex, or near convex optimization problems are easy to solve by any number of means, and that fundamentally non-convex problems are essentially impossible by any means. The only guaranteed way to solve non-convex problem is by exhaustive enumeration of the possible choices, which is usually so many that it is physically impossible.
Regardless of calculus, the question is : is the dog solving a convex or non-convex optimization problem? I would guess it is convex.The proof would be to start with any path, and then try to improve it a small step at a time. If that solves it, then the problem is convex. It would be quicker to solve the underlying calculus problem, but a simple iterative improvement method would get the same answer quick enough. The dog doesn't have to solve a calculus problem from scratch every time. It just has to have figured out the general routine once by gettting better and better at it, and then recognize the same situation again.
Computers, simple physical circuits, animals, and even humans can all solve (by arguably the same methods) convex optimization problems. Start doing it one way, and then keep improving it. The inherent ability of natural systemsm to solve convex problems (like a tennis ball fiding the drain in a pool) is an important feature of reality, but I would not call it "uncanny".
It could be argued that intelligence is the ability to solve non-convex problems. An example is that a 747 is not simply an improved Wright Flyer. Design jumps were made on the way. The real question is, can the dog, or the owner, solve a problem that does not simply involve successivlly taking the easier and easier way? I don't know.
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The cerebellum (part of the hind brain) has been proposed to function as a neural integrator. When a tennis player reacts to a ball coming over the net the brain's motor system has to calculate the trajectory of the ball in order to place the player and the racket at the right spot, at the right time, in order to return the ball successfully. Playing tennis is not a conscious skill. With serves running at over 100 mph the reactions required are too fast for conscious processing.
Chris
"I never forget a face, but in your case I'll make an exception"
- Groucho Marx
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Perhaps Elvis's tennis ball fetching skill and the elegant mathematical explanation of the steps necessary to describe them, are quite common to both man and beast. Dealing with speed, time and angles while using binocular vision is certainly one of those traits that would be important in an evolutionary capacity. An animal's ability to chase down pray, which is randomly dodging about, would be a useful trait in the natural selection of most carnivores. And it's an evolved skill which is basic to life in general. (Not to mention all successful athletes.) It's certainly improved through practice! I never got much better solving calculus problems, no matter how many I worked.
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"Pennings suggests that dogs may choose a path that is actually better than the calculated ideal. "
That would prove that they are not using calculus.
The problem is equivalent to the problem a photon faces when it sets out from a light source, moves through some air then through a piece of glass and ends up at a destination.
Either photons can do calculus or the dogs don't need to.
The answer to the problem is related to Snell's law.
http://en.wikipedia.org/wiki/Snell's_law
The dog doesn't need to understand the derivation of the rule, just the outcome. No calculus needed there.
Also the illustrations on that Wiki page suggest to me that this problem is "convex" as PG describes it.
The whole thing reminds me of the story about someone trying to make a robot that can catch a cricket ball. They had a complicated algorythm that tried to sort out the ball's aceleration, velocity and position and do calculus.
Then someone noticed that you could replace that with
If the ball is to your left move left, if it's to your right move right. If it seems to be going up in your field of view move backwards. If it's moving down go forwards.
Which is a hell of a lot simpler to code and implement and (provided you can move fast enough) will guarantee that the ball hits you in the eye.
My money is on the dogs not bothering with calculus; evolution will have taught them a short cut.
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My dog can do Calculus. I asked him to differentiate a lamp post, so he whizzed all over it.
BTW, Please format the title of this topic in the form of a question.
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I wonder if spiders know calculus too? Some of their webs are very well designed, and they do it really quick which suggests some sort of efficiency and design algorithms...