Algorithms, Searches, Dualism and (Possibly) Declarative Computation. Part 2

Here I continue with my analysis and comments in relation to this post by Joe Felsenstein and Tom English (FE for short) on Panda’s thumb. Part 1 of this series can be seen here

A priori sophisticated targeting methods and a distant object: Is this all American paradigm for hitting targets the right one for evolution?

In their paper here North American Intelligent Design ID theorists Dembski, Ewert and Marks (DEM for short) use the idea of an algorithmic search looking for targets. The target configurations at the back of their minds are, no doubt, the configurations of life. However, according to FE DEM use a model where the actual target sought for by the algorithm is immaterial:

DEM have a “target” for which the search is searching. Except that they don’t actually require that the “search” actually search for something that makes sense. The target can be any set of points. If each point is a genotype and each of them has a fitness, the target can be genotypes with unusually high fitnesses, with unusually low fitnesses, mediocre fitnesses, or any mixture of them. They do not have to be points that are “specified” by fitness or by any other criterion. DEM do not require that the “search” even consider the fitnesses. They calculate the fraction of all M points that are in the target. If |T| is the size of the target, for this fraction If we divide that by the number of points in the space, N, we get p = |T|/|N|. This of course is also the probability that a random point drawn uniformly from the space hits the target.

What I guess DEM are saying, then, is that their result is very general: Whatever the target whether, it be the configurations of life or something else, in terms of computational complexity the conclusion is the same for small targets (i.e relatively small |T|); namely, that they are extremely hard to find whatever they may be. Jumping ahead a bit, we might conclude therefore that if an algorithm is to find a specified target in reasonable time and with a reasonable probability that algorithm must be suitably provisioned with the right up front information to do so. Right? Well, that’s one solution to the problem. (More about that question another day). 

One of DEM’s important intermediate conclusions is expressed by FE as follows (My emphasis)

DEM assume that there is a baseline “search” that does not favor any particular “target”. For our space of genotypes, the baseline search generates all outcomes with equal probability. DEM in fact note that on average over all possible searches, the probability of success is the same as if we simply drew randomly (uniformly) from the space of genotypes.

That is, with respect to the class of all possible searches the average search will do no better than random chance. This, it seems, is a consequence of the search concept DEM are using. FE describe this concept as follows:

Searches as distributions on the space of points

DEM consider the probability distribution of all outcomes of a search. Different instances of the search can find different results, either because they choose different starting points, or because of random processes later during the search. They assume very little about the machinery of the search – they simply identify the search with the distribution of results that it gets. Suppose that two searches lead to the same distribution of outcomes, say a probability 0.6 of coming up with point x1, probability 0.4 of being coming up with x12, and probability 0 of everything else. They consider these two processes to be the same identical search. They don’t consider what intermediate steps the searches go through. Correspondingly, two searches that lead to different probability distributions of outcomes are considered to be different searches. All distributions that you can make can apparently be found by one or another of DEM’s search processes. From this point on they talk about the set of possible distributions, which to them represent the set of possible searches.

Note that this means that they are including “searches” that might either fail to be influenced by the fitnesses of the genotypes, and even ones that deliberately move away from highly fit genotypes, and seek out worse ones. Anything that gets results is a “search”, no matter how badly it performs.

On DEM’s definition, then, a search is identified by the probability distribution it returns over the total possible outcomes of the search. If we take the i^th outcome, then for a particular search distribution the probability of the i^th outcome, P_i, will have a value between 0 and 1. Given that the ensemble of distributions is in fact the set of all possible distributions, then selecting a distribution at random will effectively make P_i a random variable. But this random variable is subject to the constraint:

S P_i = 1

Given this constraint it follows that if DEM type searches are selected at random then the varying values of P_i must average out to a value a_i where we require:

S a_i  = N a_i = 1

….and where N is the total number of possible outcomes. From this relation it follows that a_i, the average value of the distribution for the i^th outcome, is 1/N.  This value 1/N is in fact identical to the probability of selecting an outcome at random. Hence, provided there is no particular weighting or bias placed on the way the searches are selected it follows, as FE state, that on average over all possible searches, the probability of success is the same as if we simply drew randomly (uniformly) from the space of genotypes. So this result of DEM looks to be intuitively agreeable.

Although, as FE tell us, DEM make few assumption about the mechanics of the search, typically algorithmic searches walk step by step through configuration space from one configuration to the next, where adjacent configurations in the walk are separated by small differences. This kind of configuration space can be modelled as a network of connected nodes where the walk metric between configurations is the smallest possible change between two configurations. FE approach this walk concept of search by way of a concrete example, namely, a network formed from the possible configurations of a genotype of 1000 bases. Using this model it follows that a particular genotype has as its immediate neighbors all those configurations that can be formed simply by changing any one base on the genotype to any of the three other bases available from the set of four possible bases, A, G, C and T. Therefore using this specific example it follows that each genotype is directly connected to 3000 near neighbors.

With this concrete example in mind a search will consist of a starting genotype plus the subsequent walk through configuration space. If we take a starting genotype and then select a walk at random then the likelihood is that the subsequent walk will conform to random walk statistics and generate a Gaussian probability envelope across the configuration space. It  then follows, of course, that targets which are nearest the starting configuration will likely be found first. However, when searching for a target we must take into account the chances of starting at a configuration in the near vicinity of the target; if the starting configurations are being selected at random it follows that locating stating points near the target are less likely than wider misses. It turns out that the increased chances of finding a target from a nearby starting configuration are exactly offset by the reduced chance in selecting a starting configuration in the near vicinity of the target.

This tradeoff between the chance of finding a search and the chance of the search finding a target is the subject of a general theorem by DEM.  This theorem can be expressed as follows:

Probability of finding a search  = p/q   (p < q)

....where p is the probability of finding a target completely at random and q is the probability of finding a target given the search method in question. It can be seen from this relationship that if we attempt to increase the chance of finding a target by increasing q this simply has the effect of decreasing the probability of finding the search algorithm.  I give my ‘dime store’ proof of this theorem in the first of my Dembski and Felsenstein posts  - see here.

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FE don’t attack DEM’s mathematical model; I don’t think anyone is doing that. What is at stake is the significance of DEM’s work and how they use this work to support their philosophy.  In fact  the mere content of DEM’s theorem is intuitively compelling; systems where organization and constraints are at a minimum - that is, where there is no a priori bias -  will not seek out small targets (such as life) in realistic times. It follows then that if a system is capable of generating life in realistic times it must be provisioned to do so. One (and I repeat one) of those ways of provisioning the system is by ‘front loading’ it – that is, by giving the system what DEM call ‘active information’ from the outset. In fact, I think you will find that FE would agree with this conclusion, although they are unlikely to favor the term ‘active information’.  But it is when the argument passes on to the realm of biased systems that FE come into their own as we shall see.  However, we will also see that the battle ground is philosophical dualism. As good dualists they all see it as a choice between natural forces and God. It is that debate which I will be considering next.  In due time I also hope to show that ‘active information isn't the only way forward.