Wednesday, 31 December 2014

A set theoretic interpretation of design detection

A configuration of elements in a given system is its spatial-functional structure, i.e. an arrangement of its elements one relative to another that also reflects connections between the elements. E.g. molecules of water can take different configurations depending on the ambient conditions; in particular, at temperatures below 0 oC molecules are arranged in crystal lattice. Below I do not consider variability of configurations over time and I assume they are static.

Configurations of elements can be produced by the system itself as well as by intelligent or non-intelligent factors in its environment. In cases where factors of intelligent nature are involved directly or indirectly, configurations are said to be artificial. A design is any configuration of artificial origin. E.g. ripples on the water surface are not a design if they are naturally caused by the wind. However, they are a design if masses of air causing the ripples to appear are themselves artificially generated. This said, it does not appear interesting to distinguish non-intended designs from non-designs. Therefore in the balance of this note I am narrowing down the definition of designs to denote any configuration of artificial purposeful origin.

Factors of intelligent nature are viewed as a separate category in order to develop understanding of whether it is possible and if so, under what circumstances, to classify a given configuration as a design.

What is presented below is the argument of design detection in general form using a set theoretic notation.

1. Given is a system S. There is a set X = {x1, ..., xN} of designs in S. In other words, artificial origins of the elements of X is a given established fact. Denote this as D(X). Here D(Θ) is a Boolean function over a set of configurations Θ that evaluates to 1 for designs and to 0 otherwise.

Denote U the set of all possible configurations in S: X ⊂ U. Conventionally, we use big letters to denote sets and the respective small letters to denote their elements.

Define X to contain only those designs that are characterized by a set of common properties P:

∀i ∈ {1,..,N} ∃ P = {p1,..,pK}: ∀j ∈ {1,..,K} pj(xi),

or simply P(X).

One of such distinguishing properties could be that every element of X is a symbol system [1-3].

2. There is another set of configurations Y = {y1,..., yM} ⊂ U such that Y ∩ X = ∅ and (X ∪ Y) ⊂ U.

It is not known if Y contains designs. Nonetheless, assume that from observations we know that every element of Y is characterized by the properties in P:

∀i ∈ {1,..,M} ∀j ∈ {1,..,K} pj(yi),

or simply P(Y). Since P(X), we now have P(X ∪ Y).

3. The properties P, apart from the elements of X or Y, have never been observed elsewhere so far. That they cannot be observed other than in X or Y is a null hypothesis:

¬P(U \ (X ∪ Y)).

4. Now, from 1-3 we abductively infer that every element of Y is a design:

P(X ∪ Y) ∧ D(X) ∧ ¬P(U \ (X ∪ Y)) → D(Y).

This abductive inference stands until natural phenomena are observed that are guaranteed to give rise without recourse to intelligent causation (in particular to the interference of an experimenter) to configurations of matter that have at least one property from P. Here for simplicity we assume that it would be enough to establish such natural phenomena considering them unintelligent and not dealing with their own origins.

So it is possible to refute implication 4 by means of refuting null hypothesis 3. The refutation of null hypothesis 3, if it exists, can be represented as:

∃ Z ⊂ (U \ (X ∪ Y)): ∀i ∈ {1,..,L} ∃ j ∈ {1, .., K}: pj(zi) ∧ ¬D(zi).

Implication 4 about the artificial origins of the elements of Y is equivalent to the following statement: P(X ∪ Y) ↔ D(X ∪ Y).

As an example, X can represent artificial information processing systems and Y — the set of all biosystems. As noted above, a property p that the former share with the latter is that they are semiotic systems i.e. they function based on a realization of the sign-denoted relation. This relation is present wherever one element of a system represents another.

  1. David L. Abel, The First Gene, 2011.
  2. Болотова Л.С., "Системы искусственного интеллекта", М., Финансы и статистика, 2011 (in Russian).
  3. свящ. Евгений Селенский, "Живые организмы как системы принятия решений", Православие.Ру (in Russian).

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