Reactions and compatibility classes: consequences on the knowledge system

Let us assume that a system’s reaction is the combination of a given set of elementary basic ones. We identify this set by R. Not all reactions can be contemporarily applied. So, not all the combinations of the elements of R are legal.

So R will be subdivided into a finite number C of compatibility classes C_i .

Each compatibility class will have a given cardinality # C_i.

Therefore the number of different reactions produced by a single class will be given by:

(1)

So, the total number of reactions available will be:

which is a relatively large number also for small values of #R.

If we order the basic elements of each C_i, each of the possible reactions will corresponds to a binary string. Since now we represent complex reactions by binary strings. Unfortunately there are many arbitrary possibilities of ordering. For the moment we assume that each of the elementary reactions of a compatibility class have been added in time, and that will be considered their “natural” ordering.

Within a class C_i all binary strings of length #C_i will represent legal reactions. So we will have 2^#Ci possible elements, coherently with (1).

We can therefore limit the conditional complexity of the reaction strings by :

(2)

The total number of available complex reactions AR is given by

(3)

Assuming that the system is operating in an environment where it must survive, one can try to investigate what is, if there exists, a logic for adding elementary reactions by some evolutionary process.

By (3), one observes that an optimal increase of AR is obtained by adding an elementary reaction to the class with the largest #C_i .

It seems however that, while generically advantageous for the formal increasing of RA, such a move will in principle be also more complex, as the compatibility check with the rest of the elements of C_iwill impose more constraints. More precisely there will be #C_i constraints relations, that is 2^#Ci .

Let us consider a new arbitrary elementary action x, its probability to fit a compatibility class can be approximately estimated. If we adopt a simplifying but still sensible model, by adopting the hypothesis that the average compatibility probability between two arbitrary elementary actions is P₀, the probability to fit a class C_i is given by:

(4)

By combining (3) and (4), one can evaluate the probability of growth of AR, P_g(AR).

(5)

In this simple model, the growth probability is dominated, as it is obvious, by the possible insertion of elementary actions to low cardinality compatibility classes. This is due to the low probability of bypassing the compatibility constraints for C_i with high cardinality.

Complexity concepts can be introduced in different ways. It is interesting, for example to consider the complexity of compatibility tests. Suppose a compatibility test can be represented by a program of average length T . Then, given two elementary reactions x and y, the test program t, can be represented as:

(x, y) -> t -> (0,1)

The complexity of a class compatibility test is therefore of the order of T #C_i . The basic hypothesis here is that compatibility is not transitive.

We can conclude that, under the above condition, the probability of a randomly chosen new basic reaction x has a probability P_F(x,C_i) to fit a compatibility class C_i , for which the following relation holds:

(6)

Equation (6) has quite serious evolutionary consequences. It puts into evidence the low probability of increasing the cardinality of most complex compatibility classes by the introduction of randomly chosen new elementary reactions. Alternatively, it shows the algorithmic complexity of a test program for large cardinality compatibility classes. These results show that, under the above conditions, complex reactions have a very low probability.

We can investigate other alternatives. Until now we have considered each elementary reaction can be represented by a bit in a complex reaction string, corresponding to a yes-no logic. The above results are somehow due to such a choice. A possible evolution is however represented by the refining of the elementary reaction intensity. We can approach the problem in two ways. On one hand we can increase the number of elementary actions in the string by representing different intensities of a single elementary action by different independent bits.

So, given an elementary action x, represented by a single bit (0 or 1) in a yes-no logic, we can substitute it by a longer binary schemata as, for example, (x₁,x₂,x_3,…,x_s), where the intensity is increasing while the 1 moves from the first to the last position. The element of the schemata cannot switched on at the same time; so they would represent incompatible elements. To capture such kind of evolution, while maintaining the concept of compatibility classes, one is led to consider the possibility of increase of the independent evolution of the vocabulary for each element of a compatibility class.

In this case the compatibility test is avoided while the possible scenarios reactions are increased. This suggests that, once a compatibility class has reached a limit cardinality, the best strategy of evolution seems to be that of increasing the elementary actions vocabulary. This is however probably true in principle, while a corresponding genetic mechanism seems not immediately available.

Let us consider the probability that a randomly chosen elementary action corresponds to the increase of vocabulary of an already existing elementary action in some compatibility class. As, in principle, such a probability increases with the number of already existing elementary actions contained in R. One can therefore write:

(7)

By combining (6) and (7) for a single compatibility class, we obtain:

(8)

for the intersection between the two probabilities. As, after equation (7), a is very small, equation (8) suggests that the compatibility complexity of a class plays a major role in defining whether the incorporation of a randomly chosen elementary is more probable than that of a vocabulary increase for one of the already existing elementary actions. Equation (8) defines indirectly the average cardinality of the compatibility classes.

The above considerations have some consequence on the characteristics of the corresponding knowledge system. If we assume, for simplicity, that each elementary action corresponds to a concept in the knowledge system managing the reactions, we conclude that, above values of #C_i identified by (8), the knowledge system must evolve taking into account limits imposed by (8) as well to improve the reaction repertoire of the evolving system where it is stored. Here again the major problem is the identification of some deterministic mechanism capable of implementing such a strategy while totally ignoring (8) or similar equations.

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