Sieves for Selection of Genes According to Desired Behaviors - Systematic Design for Emergence in Cellular Nonlinear Networks - page 93

A precise definition of the emergent behavior is usually not possible, therefore
we shall connect the notion of a “sieve” with ideas bore from the field of Fuzzy
Logic [83,84]. In fuzzy logic there is no pure truth like “a cell ID gives precisely
the emergence of gliders”. Instead one can define (including some human subjec-
tivity) functions called membership functions in the form

P where x is associated
with one or more precisely determined measures of complexity, and A represents a
linguistic concept (phrase) like “this CA belongs to Wolfram's class III” or “this
CA behaves like a perfect random number generator”. Such concepts are actually
the linguistic expression of the “design for emergence” objectives, while x (a vector

x

of complexity measures) is a measurable value obtained using algorithmic process
as described in Chaps. 4 and 5. More advanced analytic techniques as described
in Chap. 7 allow the determination of x using well defined mathematical formulae.

Here we will give some examples and hints in defining the functional expression

1

meaning a degree of truth for statement A. It is imposed that

P
where 1 represents the absolute truth of statement A, and 0 its absolute falsity.
If x is collected from a pool X of genes as explained above, by assuming a truth
threshold T it is possible to filter-out from X only those cells with desirable prop-
erties (expressed by the statement A ). This process can be described as using a
sieve

P

x 0 x

A S applied to the pool of genes X . A cell (here defined either structurally,
by its ID but also behaviorally, by its associated vector of complexity measures x )
belongs to the filtered-out set,

X

if and only if T

x A .
By extension, in the next we will also call a sieve [85] the mathematical expression
of the function

x S X P !

A

P . Its definition is subjective in nature, capturing the uncertainties
related to the definition of emergent behaviors. Therefore its shape can be itera-
tively adjusted by applying the sieve and changing it until the desired objectives
are best fulfilled. Once sieves are defined one can explicitly use them to locate
genes of interest assuming that a database of emergence measures (the set X
calculated for a pool of genes) is available. If such a set is not available, the sieve
can still be used as an objective function to guide a search process regarded as an
optimization algorithm within the space of all possible cell IDs. In the next we
will give examples in the first category, namely when a set X of measurements is
already available as a mapping from the ID space (using algorithms exposed in
Chaps. 4 and 5) and the sieve is used to select from it only those cells with desired
properties. Such a process is depicted in Fig. 6.1, where two “sieves” are succes-
sively applied to locate CA genes giving interesting behaviors.

x

6.2 Defining Sieves

Given a signal processing problem to be solved with cellular arrays starts with a
“fuzzy” definition of the objective function. Let consider as a first example,