8
A. LOCKER
tions; this latter has been contrived for purely mathematical purposes so
as to prove the computability of certain functions. The TURING machine is
capable of simulating other machines since code instructions may be
developed by means of which the first machine can be induced to imitate
another, and, as a matter of fact, any other machine. Also, a third type of
TURING machine may be envisaged, the so-called universal TURING machine
which imitates the action of a TURING automaton [13].
Growing automata, from the viewpoint of TURING machines, are
therefore those which can write symbols on their own tapes and thus
induce their enlargement and growth. Self-reproducing automata, in their
simplest form, are those which write down their own coding programs,
provided a universal TURING-machine is at hand.
Examples of more elaborate self-reproducing automata may be presented: (1) the kinematic model, (2) the tesselation model (both of them
proposed by VON NEUMANN) and (3) the string-processing automaton
proposed by STAHL [13]. The kinematic model, however, may be shown to
contain an intrinsic paradox [11]. The tesselation model, in which selfreproduction is related to a configuration (and not to a concrete entity) has
not been successfully solved to this day and, therefore, there remain only
a few words to say on the string-processing automaton which is not merely
a product of the mind but has actually been realized by a computer [13].
Herein the cell is considered as a string-processing enzyme automaton
which accepts or reads not only single symbols but whole strings of
symbols and may process such strings in a flexible manner, e.g. carry out
enzyme synthesis, etc.
It should be mentioned that the application of the automata theory to
biology introduces into the latter certain paradoxes and logical limitations
such as those arising in a given axiomatic system of mathematics. Examples
of unsolvable cases have been reported, e.g. if a cellular computational
automaton (like a self-organizing one) is designed to synthesize a new
gene, then the automaton is unable to determine whether the newly formed
cell with the new gene can undergo self-reproduction or not. Several other
paradoxes are associated with the problems of self-applicability, selfreferencing and self-describing (cf. [13]).
Concluding Remarks
In this brief and rather sketchy outline I have tried to trace a few
possible aspects of the nature of models in general, of the application of
models to science, and biology in particular, and I have also tried to throw
some light on the importance of the gradually maturing theory of models
for the accomplishment of which the fundamental investigations of STACHOWIAK [12J have been valuable contributions. Since the constructing of
A. LOCKER
tions; this latter has been contrived for purely mathematical purposes so
as to prove the computability of certain functions. The TURING machine is
capable of simulating other machines since code instructions may be
developed by means of which the first machine can be induced to imitate
another, and, as a matter of fact, any other machine. Also, a third type of
TURING machine may be envisaged, the so-called universal TURING machine
which imitates the action of a TURING automaton [13].
Growing automata, from the viewpoint of TURING machines, are
therefore those which can write symbols on their own tapes and thus
induce their enlargement and growth. Self-reproducing automata, in their
simplest form, are those which write down their own coding programs,
provided a universal TURING-machine is at hand.
Examples of more elaborate self-reproducing automata may be presented: (1) the kinematic model, (2) the tesselation model (both of them
proposed by VON NEUMANN) and (3) the string-processing automaton
proposed by STAHL [13]. The kinematic model, however, may be shown to
contain an intrinsic paradox [11]. The tesselation model, in which selfreproduction is related to a configuration (and not to a concrete entity) has
not been successfully solved to this day and, therefore, there remain only
a few words to say on the string-processing automaton which is not merely
a product of the mind but has actually been realized by a computer [13].
Herein the cell is considered as a string-processing enzyme automaton
which accepts or reads not only single symbols but whole strings of
symbols and may process such strings in a flexible manner, e.g. carry out
enzyme synthesis, etc.
It should be mentioned that the application of the automata theory to
biology introduces into the latter certain paradoxes and logical limitations
such as those arising in a given axiomatic system of mathematics. Examples
of unsolvable cases have been reported, e.g. if a cellular computational
automaton (like a self-organizing one) is designed to synthesize a new
gene, then the automaton is unable to determine whether the newly formed
cell with the new gene can undergo self-reproduction or not. Several other
paradoxes are associated with the problems of self-applicability, selfreferencing and self-describing (cf. [13]).
Concluding Remarks
In this brief and rather sketchy outline I have tried to trace a few
possible aspects of the nature of models in general, of the application of
models to science, and biology in particular, and I have also tried to throw
some light on the importance of the gradually maturing theory of models
for the accomplishment of which the fundamental investigations of STACHOWIAK [12J have been valuable contributions. Since the constructing of