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Friday, February 22, 2008

IV. Functional Information

The Intelligence Hypothesis suggests that intelligence can produce effects that require a significant amount of functional information. To proceed, we need a method to measure functional information and, second, we need to decide what constitutes a significant level of functional information.

Measuring functional information

A method to measure functional information has recently been published by Hazen et al.
whereby functional information is defined as

I(E.x.) = - log.2.[M(E.x.)/N] (1)

where E.x. is the degree of function x, M(E.x.) is the number of different configurations that achieves or exceeds the specified degree of function x, ≥ E.x., and N is the total number of possible configurations.2 To illustrate, suppose we inherit grandfather's safe that has a combination lock that requires three numbers, each within the range of 0 to 99. Since each number has 100 possibilities, and there are three numbers, N = 100.3. = 1,000,000
possible combinations. Let us suppose that the mechanism has a little slop to it such that one need only get within 1 digit of each of the three numbers. In other words, for each of the three numbers in the combination, there are actually three functional options. Therefore, the total number of functional combinations that will open the safe is M(E.x.) = 3.3. = 27 functional combinations. The amount of functional information required to open grandfather's safe is therefore

I(E.x.) = - log.2.[27/1,000,000] = 15 bits of functional information.

As Hazen et al. point out, 'functional information quantifies the probability that, for a particular system, a configuration with a specified degree of function will emerge', where the probability is denoted by M(E.x.)/N. Strictly speaking, the probability that Hazen et al. speak of is the probability P.f. of achieving the function in a single sampling, or

P.f. = M(E.x.)/N. (2)

As more trials are attempted, the probability of achieving the function improves.

Estimating I.sig.

This raises the second question; what constitutes a significant level of functional information I.sig.? The Intelligence Hypothesis suggests that the attribute that distinguishes intelligence from mindless natural processes, is the ability to produce significant levels of functional information. Mindless natural processes can accidentally produce effects requiring a low level of functional information. For example, if we were to move grandfather's safe down to the riverbank, and attach a water driven turbine to the dial, and install the turbine in a turbulent portion of the current, where the turbine could be turned either direction by the current, it is possible that, after a long enough time, the variable current may actually open grandfather's safe. Of course, the number of trials may vastly exceed 1,000,000 if the same combinations are mindlessly tried more than once.

Recall, as Hazen et al. point out, that probability is at the core of the equation to measure functional information. We must establish a relationship between the number of trials mindless natural processes have for the particular problem, and P.f..

A search by mindless natural processes is essentially a random walk, where the search
proceeds in no set direction and, for any point in the search, it can be returned to any number of times. This is not to be confused with an evolutionary search that is directed by a fitness function or a fitness landscape, which will be discussed later. We must first establish I.sig. for a mindless natural search. In such a search, the probability that a given sampling will not be successful is 1 – P.f.. For a search involving R trials, the probability that it will not be successful is (1 – P.f.).R.. Therefore, the probability that the search will be successful is simply 1 - (1 – P.f.).R.. Let us assume that a search will be successful if the search performs enough trials to raise the probability of success to 0.5, or

0.5 = 1 - (1 – P.f. ).R..

Simplifying, we get

Pf = 1-(1-0.5).1/R. (2)

Eqn. (2) gives us an estimate for the most improbable functional event that a blind search could reasonably expect to find, given R trials. That being the case, the highest level of functional information that natural processes could reasonably be expected to produce for a given function would be the case where only one functional configuration would
reasonably be found in R trials, or

I.nat. = - log.2.[1-(1-0.5).1/R.]. (3)

The requirement for I.sig. is that it must be greater than I.nat.. For example, if the turbine method of trying to open grandfather's safe was capable of 500,000 trials before the system wore out, then the turbine-safe system could reasonably be expected to produce as much as 13 bits of functional information (I.sig. = 13 bits). Since a functional combination requires 15 bits of functional information, one could not reasonably expect the turbine system to open the safe without any intelligent design so far as finding the right combination is concerned. Therefore, if such a system were built and the safe successfully opened, we could on reasonable grounds accuse the engineer of having biased the system to find the right combination, for the physical system was unlikely to have done it without any intelligently designed bias built in. Due to the nature of probability, however, it is possible that the river current could open grandfather's safe on the very first try, or it might never open the safe. It is also possible that the engineer did not build in an intelligently designed bias to find the right combination, we were just fortunate. We could never be absolutely sure, therefore, whether there was a built in intelligently designed bias or not. Since at the core of functional information is probability, we can never arrive at a definitive conclusion, only a likely, probable, or plausible conclusion. This leads to the following considerations.

Probability considerations

1. I.nat. is not a cutoff for the amount of functional information natural processes can produce. Rather, the probability that natural processes can produce x amount of functional information decreases exponentially as the amount of functional information increases beyond I.nat.. For example, if I.nat. = 32 bits of functional information, using Eqn. (1), this corresponds to a probability of approximately 10.-10. whereas 64 bits of functional information corresponds to a probability of approximately 10.-19.. In other words, 64 bits of functional information is only twice as much information as 32 bits, but one billion times more difficult to find in a search.

2. Our observations indicate that there does not seem to be any known limit to the amount of functional information that intelligence can produce. It seems to be capable of producing anywhere from 0 bits and up.

3. In view of the previous two points, we can only speak of the likelihood that an effect required intelligent design, where the greater the difference between the functional information required for the effect and I.nat., the more likely it is that intelligent design was required. This would hold true for SETI, archeology, forensic science, and biological life.

Method to gauge the likelihood of intelligent design

Given that there is no known upper limit for the amount of functional information a mind can produce, for any effect requiring or producing functional information, intelligent design is the more likely explanation if

I(E.x.) > I.nat.. (4)

The greater the difference between I(E.x.) and I.nat., the more likely it is that intelligent design was required. It will be assumed, for simplicity, that the probability that mindless natural processes can achieve I.nat. is 1 and decreases probabilistically for I(E.x.) > I.nat.. The probability that intelligent design can achieve I(Ex) will be assumed to be 1 for any finite amount of functional information. This is a reasonable assumption, given our observations of what intelligence can do and the apparent absence of any upper limit.

This method can, in principle, be applied within the fields of forensic science,
archeology, SETI, and biology, as well as in areas outside of science, such as lottery gaming investigations, plagiarism investigations, and the justice system, to name a few.

Next: V. Application to Biological Life

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