Definitions of life and the resulting Draft of the deductive theory of life
Andrzej Gecow� independent researcher, Warsaw, Poland
Agency and Individuality Conference,
UJ Kraków, Poland. April 14-16, 2026.
Definitions of life and the resulting Draft of the deductive theory of life
My presentation is based on the article:
Andrzej Gecow, (2025).
Two coherent definitions of the life process
derived from the half-chaos theory
and the (unintentional) purposeful information theory. BioSystems, Volume 256, 105533, ISSN 0303-2647, https://doi.org/10.1016/j.biosystems.2025.105533 .
Both of these theories were constructed to define terms
used to above life definition formulation.
Definitions of life and the resulting Draft of the deductive theory of life
The book, currently in Polish,
469 pages; 55 complex, typically colour diagrams of simulation results.
English, wider version �is under review.
The book contains much more than the article.
There are e.g. structural tendecies derived from �life definitions.
Node counts logical function c:=f(a,b,...) of K variables. a, b, c = 0 or 1
One logical value c (state of node) is sending in k different directions.
Kauffman uses RBN - Erdős-Rényi Random Boolean Network
fully randomly created, finite, discrete,
directed, deterministic, autonomous,
dynamical, synchronously calculated.
k=3
c:=f(a,b)
K=2 connectivity, typically fixed
<k> = K for autonomous
c
a
b
c
c
node out-degree is variable
N – Number of nodes in the network
t – number of time step in synchronous calculation
A(t) – number of different node states between disturbed and undisturbed networks,
Avalanche size
d(t) = A(t)/N – damage, measure of difference in functioning in effect of disturbation
s – number of equally probable signal variants, s=2 for Boolean networks
half-chaos theory
Two coherent definitions of the life process
derived from the half-chaos theory
and the (unintentional) purposeful information theory.
Kauffman investigated Boolean networks = Kauffman networks.
There he found, that they can be ordered or chaotic, but not simultaneously.
half-chaotic system
The ordered network has one, left peak �for very small damage d.
The chaotic network has one, right peak �of large damage.
But in the half-chaotic network �both of these peaks occur simultaneously �with a similar proportion of cases clearly separated by a large gap.
P(d)
d
s,K = 4,3
gap
Half-chaos occurs in the range of parameters that for a fully random systems give normal, clear chaos.
Attractor is a cyclic trajectory of states in a deterministic network. The length is measured by the number of steps taken to calculate successive network states. Complex networks, with hundreds of nodes, almost always have very long attractors.
A short attractor in a complex network defines half-chaos.
P(A(tmx))*48000
One of simulation result.
c,d without regulation
a,b with negative feedback
A(tmx)
dr
cr
cf
df
ordered
chaotic
definition of the life process �derived from the half-chaos theory
Natural criterion for the identity of an evolving object:
A system, functioning similarly after small damage, remains the same. �Leaving disturbances (evolutionary changes) that produce small damage �does not go out of half-chaos.
Large damage, that is a completely different functioning of the system, �causes it to cease to be the same system and irreversibly falls into �normal chaos with a long attractor. This is a good model for the death �and elimination, freeing the Darwinian definition from tautology.
The essence of the life process—maintaining a system in a state of half-chaos under circumstances of random variability.
(unintentional) purposeful information theory
Two coherent definitions of the life process derived from the half-chaos theory and the
Before Shannon analyzes the transmitting of something, he describes the thing that will be transmitted, and I take this as information.
Information is a choice in a set of possibilities: I = -log p .
Object contains information in his construction on how to respond to given external conditions
Okay!
I'm falling!
But which direction!?
What does
lack of information �look like?
– a ball on top
of the other ball...
Conductor or insulator?
Circuit with �a light bulb :
+ -
Come down finally!
Natural coder
Isolated system э situation , F – physical law as coder
sit2:= F(sit1) sit1=cause; sit2=effect.
If sit1 = A1+B1 then sit2:= (F+A1)(B2)
But F is one: sit2 := A1(B1) then A1 –natural coder
Let’s A =environment. Great and statistically described part, constant in consecutive situation. We use it as natural coder.
B = object. Small and strict described, variable.
ob2 := env(ob1)
Goal - the assumed effect for which the cause is sought.
- an argument in the reverse natural code. cause:=decoder(effect)
Purposeful information - a record of the cause encoding the goal.
The universal decoder (U-decoder)
A human as a U-decoder perform an experiment:
• sets hypotheses; (establishes initial situations)
• allows coding to take place;
• compares the effect with the given effect, i.e. with the goal;
• records the found cause.
Only coding occurs spontaneously, i.e. in the Isolated System, �the remaining activities seem to be external.
So can the accumulation of purposeful information be independent? (i.e. without a human, etc.) That is:�Can the U-decoder fit into the Isolated System?
3 dimensions of purposeful information
• Amount described by the Hartley-Shannon theory, i.e.: I = -log p (p - probability of indicating the cause encoding the purpose), �in other words - the uniqueness of such an object's construction.
• Effectiveness - the probability of achieving the purpose using the indicated cause. It is like fitness.
• Length of the record - complexity. This is the most striking, �but it also includes a ‘dead record of purposeful information’, �i.e. ineffective, because it has failed.
When it is not clear which dimension we are consider, we should use the term ‘size’ - the length of a three-dimensional vector.
definition of the life process �derived from purposeful information theory
Can the U-decoder fit into the Isolated System?
Basing on an unambiguous code (but this unambiguity holds a trap):
Independent U-decoder has to:
• set hypotheses; (establishes initial situations)
• allows coding to take place;
• compares the effect with the given effect, i.e. with the goal;
• records the found cause.
xt=k(xt-1) effect is a next hypothese of goal cause.
coding xt+1=k(xt) takes place in System.
At time t+1 exists only xt+1 , but its cause xt should exist to be recorded, then : xt+1=xt . It means: the only goal can be ‘to continue to exist’ . �Recording by contuation of existence and comparison to the goal are in one. This specific coincidence creates a natural U-decoder, however, �ambiguous code is needed for the sucesive accumulation of significant size of purposeful information.
definition of the life process �derived from purposeful information theory
The only goal can be ‘to continue to exist’ . �Recording by contiuation of existence and comparison to the goal are in one. This specific coincidence creates a natural U-decoder, ambiguous code is needed for the successive accumulation of significant size of purposeful information.
This way we obtain definition of life: an independent, long, and effective process of accumulating purposeful information.
The need to reproduce sufficiently rapidly, is the first feature and purposeful information resulting from this definition.
We have thus obtained the complete basic mechanism of Darwinian natural selection.
Mechanisms of purposeful information growth despite typical entropy increases. 1
horizontal diffusion
A sufficiently rapid multiplication with elimination allows the objects left in the process to resist the increase in entropy. This creates a "biotic level" of sufficient effectiveness of the purposeful information the object possesses.
At this level object can difuse isotropically in amount and complexity dimentions, but at the beginning this space is asymetric - there is nothing to reduce.
After a time objects with great value in these dimentions were encountered, but we have drawn attention to them. There is no mechanism here forcing such an increase for the indicated object, but the average and the maximum values increase.
Mechanisms of purposeful information growth despite typical entropy increases. 2
When effectiveness increasing is allowed, then elimination occurs statistically insignificant - it does not affect the size of purposeful information. Sorting from above during quantitative explosion increases share of objects that reproduce the fastest, thus effectiveness increases.
'Permitted degeneracy' collects significantly larger changes in the 'reserve of permitted degeneracy'. �It is a lifesaver when bypassing the environmental capacity barrier or when emerging from a 'dead end of specialization.'
Mechanisms of purposeful information growth despite typical entropy increases. 3
The quantitative explosion is stopped by the limit of the environmental capacity, which creates competition. Now the normal mechanism of natural selection by elimination works. Bypassing the barier increases the effectiveness and usually the coplexity. However, the object encounters another barrier, and its effectiveness drops to the biotic level, which is due to a change in the environment, not of object. The complexity and uniqueness of structure typically increase. Effectiveness, however, fluctuates around a constant value, but the object increases number of properly functioning, purposeful mechanisms.
This were the most important things I had to show.
Thank you for your attention.
Definitions of life and the resulting Draft of the deductive theory of life
Andrzej Gecow� independent researcher, Warsaw, Poland
Agency and Individuality Conference,
UJ Kraków, Poland. April 14-16, 2026.
Andrzej Gecow, (2025).
Two coherent definitions of the life process
derived from the half-chaos theory and
the (unintentional) purposeful information theory. BioSystems, Volume 256, 105533, ISSN 0303-2647, https://doi.org/10.1016/j.biosystems.2025.105533 .
Simplifications that were too big
1. Living objects in the aspect of stability can be described by a fully random network.
But living objects are the result of natural selection, and this selection is just in the aspect of stability. Then in this aspect they are not random.
2. Since each system can be described by a Boolean network, it is enough to statistically examine the Boolean network to provide general conclusions.
I have shown that even such a simple system as the fridge thermostat described in the Boolean network introduces impossible states, and these are taken into the statistics and lie to the result. For statistical surveys, it is necessary to consider s>2.
3. To use the mathematical theory of chaos, the transition from finite and discrete state space to infinite and continuous is used.
In this step the length of the attractor measured in time steps and the path length to the attractor disappear from the description. But that these two values play a fundamental role in the 'half-chaos'.
Evolution from the point attractor
A(t) d(t)=A(t)/N N=400 s,K=4,3
q(t) = P(A<60|t) , #(A(t)<60)/(s-1) , degree of order
typical half-chaos
P(A=0|t) #(A(t)=0)/(s-1)
in-ice-module
typical chaos
Evolutionary stability
of the half-chaos .
Natural criterion for the identity and elimination of an evolving object.
If only disturbances that cause damage in the ordered peak are left, then evolution does not lead out from the half-chaos state.
Leaving the disturbance that led to great damage from the chaotic peak leads to normal chaos.
The picture is similar to that described by Kauffman, who calls nodes that don’t change their states the ice. A half-chaotic network is usually a set of small modules of activity separated by the ice.
Results
Experiments:
X chaotic
d - 7 half-chaotic
d s,K=2,4 point
4 pattern c attractor
4 - 7 s,K=4,3
5,6,7 - evolution
starts from:
5 – point attractor
6 – small attractor
7 – constructed
in-ice-modularity
Network types:
r – Erdős-Rényi
k=0
f – scale-free
s – single-scale
left ‘ordered’ peak
0
1
0.5
q
degree of order and chaos
coefficient of damage propagation w = <k> (s-1)/s
node
if one
input signal
is changed
convert:
new output signal := f(new input signal)
typically another than
old output signal
Only for <k>=K=2, s=2 (w=1) change doesn’t grow. Therefore typical Boolean networks are extreme and
give other phenomena (special order)
than in the more general case (chaos).
s - equally probable
signal variants
k - node outputs
How many output signals are changed on average?
When should damage be extinguished as in ordered networks and when it should go into an avalanche and end up in a state of equilibrium as in chaotic networks sensitive to the initial state?