Markov's Ant


We have ants on our patio. Millions of them. See my entry entitled Avogadro's Ants. In that entry, I advocate the squashing of ants. When we sit and eat breakfast on the patio, the ants sometimes bite us on the feet.

The Lovely One has discovered a potent chemical arsenal against the scourge -- chlorine bleach. When the ants invade the patio stones in large numbers, the chemical warfare against them begins in earnest. A couple of days ago, The Lovely One noticed big ant hills in the joints of the patio stones. I got out the bleach, and dumped a healthy dose in the cracks.

Then I noticed one ant trapped in an area surrounded by bleach. It wanted to go home across the sea of bleach, but couldn't abide walking on it. I watched in fascination to see if the ant would get out. Moreover, I was interested to see how the ant would solve the problem. As the observer, I could see that there was a solution to the problem -- there was a safe passage, but it had to go around the sea of bleach. I eagerly watched to see if the ant could solve the problem.

I wondered if the ant would use randomness to find a safe passage. If the ant used reasoning, it would find the safe passage rather quickly. But if the ant used random tries, then the problem solving process was either a random walk or a Markov Chain. Let me explain.

Quoting Wikipedia: A random walk is a mathematical formalization of a trajectory that consists of taking successive steps in random directions. The results of random walk analysis have been applied to computer science, physics, ecology, economics and a number of other fields as a fundamental model for random processes in time. For example, the path traced by a molecule as it travels in a liquid or a gas, the search path of a foraging animal, the price of a fluctuating stock and the financial status of a gambler can all be modeled as random walks.

A Markov chain is subtly different: In mathematics, a Markov chain, named after Andrey Markov, is a discrete-time random process with the Markov property. Having the Markov property means that, given the present state, future states are independent of the past states. Future states will be reached through a probabilistic process instead of a deterministic one.


A basic example is a coin toss - if you throw heads the first time, the next time you toss the coin you could get heads or tails with 50/50 chance. If you get heads 100 times in a row, the next toss could be equally heads or tails, no exceptions - the past does not determine the future. The present state is that you have a coin, and it has heads and tails with equal weighting, no other information can influence to future toss of the coin (assuming a fair toss and that the coin is equally balanced)
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How would you or I solve a problem of a similar scope? A thinker would walk around the perimeter to see if there was an opening. The ant didn't do that.

What the ant did was to pick a point, walk to it and test if it was bleachy. If it was, it turned around, went a random direction, picked another point and walked until it hit bleach. The ant kept picking points.

After about 15 minutes, it did finally pick a point that was through the safe passage, but after getting out of the bleach enclosure, it decided that it was going the wrong way, and went back into the enclosure to test other points. The ant did not discern that the bleach barrier was continous, except for the back part.

You have to wonder about this. The method seems like a successful strategy, because ants have been around millions of years. Yet it didn't have the brain power to figure out that the barrier was continuous, and it should follow it around.

But it's behaviour begs another question -- how does the ant know that it was going the wrong direction when it did get out? The ant was determinedly trying to go a certain direction. How could it determine the direction. The bleach would have obliterated the scent trails that ants lay. It had to know somehow of the desired direction of travel.

I watched the ant using random processes for a good half an hour. At one point, it found the dead carcass of another ant, and picked it up. It tried walking home with the carcass, using random processes again. Finally it decided to ditch the stiff and try to get out.

Watching this ant was highly instructive. Who says that we can't learn from other species. So how did the ant finally make out? It is a sad story. I squashed it.

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