Complexity Science
"The study of emergent behaviour exhibited by interacting systems
operating at the threshold of stability and chaos."William H. Roetzheim, Enter the Complexity Lab, 1994
Artificial Life
"A synthetic approach to studying Life-as-it-could-be
and viewing Life-as-it-is within a larger context."Chris G. Langton, Artificial Life, 1989
Of course definitions are never meaningful unless you know what they mean anyway. In this paper I outline the background to these definitions and explain how they relate to conventional ideas.
First a light-hearted story of why Complexity Science is needed at all (if you don't like satire, stories, science or maths then just skip this section ).
In the beginning there was Newton, and he said "let there be order" and there was order, glorious regular clockwork order, and the Philosophers were pleased. Formulae ruled the deep and all was determined and right for ever more in infinite Euclidean space and time. All were particles and moved in straight lines, unless they were pushed by man to serve his Glorious purpose in God's image.
But wait, we have a problem..., Mercury won't behave ! Yes, it was true, the predictions didn't fit - Mercury did wobble !! Horrors...
In the middle there was Einstein, and he said "time is relative" and it was relative, and space was curved and Mercury was still and all was now well again. And the Philosophers rejoiced, "Lo, The Second Coming". And formulae still ruled the deep and all was determined and linear for ever more in relativistic 4 dimensional space/time. All were waves and propagated at the speed of light to the ends of eternity, unless they were bent by man to serve his glorious purpose in his own image.
But wait, we have a problem..., when I launch a wave at this foil I get a particle ! Yes, it was true waves were particles and particles were waves !! Horrors...
In the end there was Bohr, and he said "all is complementary" and it was complementary. And what you saw depended on what you did, and neither the particles nor waves really existed and oh, it was so confusing. So the Philosophers got together and said "If we ignore the whole thing it will go away" so they called it the Copenhagen Interpretation and it did go away (provided you didn't try to predict anything too small). And it was "The Third Coming" and formulae still ruled the deep, and all was still determined (sort of) for ever more. But now a Priest sect was required as the laity could no longer understand the "Word", and they called themselves Quantum Physicists and hid themselves from the world.
And, Lo, the age of Enlightenment was over and the dark ages returned to the earth.
Huddled in their caves, the Mathematicians rejoiced, for theirs was the kingdom of light, and darkness could never cross their threshold. For didn't the blessed Euclid say "The angles of a triangle add up to 180 degrees" and holy Pythagoras say "The square of the hypotenuse is equal to the sum of the squares of the other two sides", such wisdom !
But wait, we have a problem, in Einstein's curved space angles never add up to 180 degrees and Pythagoras is incorrect ! Yes, the old Axioms were wrong !! Horrors...
In the beginning was Riemann, and he said "axioms are relative" and they were relative. And a new geometry was born, and it was good. And the mathematicians rejoiced, for all was once more consistent and complete. And now they could surely derive all things from first principles. Hallelujah !
But wait, we have a problem, some functions will not behave, they cannot be solved !! Horrors...
In the middle was Poincaré, and he calmed the masses, "Fear not", he cried, "These functions are just pathological and can be ignored", and they were ignored and the mathematicians rejoiced and cried together "Lo, The Second Coming"
And so it remained until....
But wait, we have a problem, Godel has proved that powerful mathematical systems are always inconsistent and incomplete !! Horrors...
We have another problem, those pathological functions are more numerous than orderly ones and cannot be ignored !! Horrors of Horrors...
And a third problem, Aspect has shown that distant objects can communicate faster than light can travel, all things affect each other and both relativistic and quantum physics are wrong !! Oh, woe is us, the end of the world is nigh !!!!!!
And it was true, and darkness entered the caves and the universe of the intellect crumbled unto dust, waiting the Third Coming. But we didn't tell anyone....
Thus endeth the first lesson, thanks be to Woodstock !
The above was a potted history of science and mathematics over the last 2000 years - irreverent version (with apologies to any historians present...)
OK so far ?
What we have at this point is a deterministic science (e.g. Newtonian Mechanics) saying give us a formula and values and we'll give you the answer. For 1 particle ? Easy; for 2 ? yes; for 3 ? well..; how about 500 ? Ah, come on now we can't solve the equations !
And a statistical science (e.g. Quantum Theory) saying give us a system and we will give you the probabilities. For 1 trial ? No, that's impossible; for 2 ? silly..; 500 ? well...; a billion ? Now we're talking, yes !
Now for the Complexity bit...
It's the bit in the middle ! A very new science that takes over where
both the previous ones were inadequate, in the realm of systems having
many interacting parts - too complex for deterministic mathematical solutions,
too simple for averaging by statistics. Sounds easy put
that way doesn't it ?
There's only one minor snag - nobody really knows how to do it yet ! That's
why the problem has been ignored since the beginning of the 20th century,
nothing conspiratorial, just that there were few means available to even
approach it in the past.
What does this area cover ? Well, basically anything complex and having parts
that interact with each other.
Researchers from many areas are involved in this science; including from
physics, biology, economics, medicine, philosophy, sociology, computing,
engineering, game theory, chemistry, mathematics, AI and many others.
It is a true transdisciplinary endeavour.
Is this any different than standard, say psychological, theories you may
ask ? Yes, what we are looking for isn't explanations for any
particular phenomenon (like aggression say) but global properties
shared by many or all classes of complex systems, whether microscopic
or cosmic, real or virtual - in other words universal laws.
The tools we use are largely computers (hence the impossibility before
the cheap computer revolution) - running simulations, but of a very
special sort - we don't write any programs ! Whoa, don't lose it...
But more of that in the ALife bit (every good story needs suspense...).
Back to the plot, remember Poincaré's pathological functions ? Come and
meet one... Easter Bunny version. A simplified population model.
Seems innocuous enough, what would you expect to happen ? "Population
will grow until it reaches the maximum that the available food will
support and then stay there perhaps ?". It almost does, sometimes...
These formulae are common, K times X squared minus one, trig functions,
almost everything you can think of. Mathematics is now a very unruly
place...
So what does it all mean ?
Deterministic/Newtonian science relates to the static low growth rates,
Statistical/Quantum to the chaotic high,
in the middle where dynamic order appears, is our Complexity Science. What
we are doing is linking together both old sciences into a holistic whole
covering all of reality. By changing our parameters we can move between
the disciplines at will (now regarding them as just different views - Deterministic
short term, Statistical long term, Complexity in the middle). We
just change our spatial and temporal reference frames - relativity !
Why does this pathology happen ? It's to do with linearity, if you push a ball
then once it has left your hand it behaves independently of you, that's
why linear Newtonian science generally works so well. Cause and Effect are
separate.
The population model is different, what happens tomorrow contains the
same elements as today, the output is coupled back to the input. Cause
and Effect are the same items ! Positive nonlinear feedback.
So, if that's the case, from a wider viewpoint, aren't...? Exactly - Life, the Universe and
Everything are coupled in just the same way. The reason little
progress has been made in Social Sciences etc. may well be because they
are trying to use an invalid model - linear, instead of non-linear ! Complex
science applies to turbulence (weather, rivers, taps), economies,
heartbeats, brains, molecular biology, evolution and much more...
At both conventional ends of the spectrum - laws and statistics, linear
approximations work very well indeed. In the middle they break down,
because of this nonlinearity, and just give the wrong answers - over-simplifications
which people often take too much on trust.
You know that if you lift a stone a few worms can crawl out ? Well, in
the case of Complexity Science we have found a worm hole into a whole new
Universe !
Pause for breath, have a drink, rest or whatever, then read on...
The above may sound like a scene from 'Alice in Wonderland', but now we
will take a step 'Through the Looking Glass' into that Universe, the
wonderful world of Artificial Life...
We started with a computer model if you recall. You may think of
computers as running programs, laboriously crafted by programmers over
many years to eliminate bugs and performing absolutely predictably.
Normally that is the case (in your dreams...). Simulations of that type take known formulae
and incorporate them into a program, running it to see what happens over
time (e.g. a flight simulator). In this field of ALife we, of course, are different !
The Artificial Intelligence people often try to model the natural brain as a computer
program, instead we do the opposite - modelling computer programs on nature...
What does that mean ? You may well ask - essentially we evolve programs,
as in Darwinian evolution.
Diversity in nature occurs due to genetic variations, produced by sex
and mutation. So we create random strings of computer bits, allow them
to breed (using simplified biological crossover and mutation),
evaluate how fit they are for the task, discard the less fit and then
do it again - unnatural selection ! This is called Evolutionary Computation.
Think of these strings of bits as primitive creatures, suppose we put
them into their own world and leave them to it, what would happen ?
To do this needs a world, our computer. All we need to supply is what
nature itself supplies - energy and space. That is CPU time and
Memory.... They are simplified creatures, so we'll give them a God (a supervisory
program that evaluates fitness and sends in the grim reaper...).
Think of the bits as functions, like genes. A one, say, in the first
position may mean 'move left', in the second 'eat', the third 'look
ahead' and so on. So our creatures can react to their environment and to
each other. Remember complexity, hey - here it is !
O.K. ready to go, so what does happen ? They evolve, just like us,
developing behaviours - some may become successful foragers, some
parasites, others join to form symbiotic groups, others are predators,
some can even evolve simple logic capabilities - like in Avida here !
Populations rise and fall, species become extinct, new ones arise. All
the creatures may die out, they may settle down into an ecosystem, or
become chaotic (see the three stages - low, middle, high rates ?).
You may have come across a problem in Game Theory called The Prisoner's Dilemma,
where intelligent agents need to choose between options. In ALife simulations
solutions to this problem can arise automatically. A strategy called 'Tit for
Tat' can come into being where an agent (bit string) co-operates on the
next turn if the opponent co-operates this turn and defects next turn if
the opponent defects this turn. Justice. Ethics from computer bits ?
We can perturb these systems and see what happens (say introduce a lot
of mutations, or a change in available energy or space). We find the
effects can vary from minor changes in populations, to massive
catastrophes and recoveries (like the Cambrian Explosion in history).
These echo the effects seen in Earthquake activity also. Sometimes such
events occur spontaneously, a chaos related 'butterfly effect'.
Randomly started systems force themselves, by system dynamics, to the
parameters giving the greatest diversity and emergent order - we call it
the 'Edge of Chaos'. Strangely it seems to be the place all similar
systems end up, alive or dead. We believe (or rather I do, not many in
the field would be so bold) that all the life we see around us is in
this middle state, water (and all liquids) are potentially in a state of Complexity,
solids are Newtonian stability, gases Statistical chaos. A new concept of
Phase Transitions. We self-organise our world in our interactions with
it, at all scales, down perhaps to the very quanta of atoms.
Could this explain life on earth ? Perhaps. Models of interacting molecules
when they become sufficiently complex, self-organise
into autocatalytic sets able to reproduce each other and maintain a stable inorganic ecosystem,
(the molecules have been easily created in primordial soup experiments).
These groups can mutate, interact and grow becoming cells. Symbiosis can
incorporate sub-assemblies (like the mitochondria we have in our own cells).
The experiment has not yet been done for real, but biologists are considering it
I believe - ALife techniques possibly begetting new forms of real BLife !
One interesting thing about all this is that the results cling to an
extremely narrow region of state space (that is, the total possibilities
open to the system), like throwing a million sided die and always getting six.
We hope to find out why...
This is a very new field, started in 1987, so results are fairly sparse as yet.
But these areas are exciting enough for many people, including myself, to jump into full time
research on these subjects. For an alternative overview on these themes with a short bibliography, see Complexity and Artificial Life. Current applications of these
techniques include adaptive programs that can learn user preferences, computer game
characters that evolve better responses, more realistic film animations, alien artforms and
web robots that intelligently surf the internet.
Complexity Science
Depends on one thing, 'reproduction rate'. At low growth rates the current population
dies out (fewer births than deaths), at higher rates it does stabilise at a constant value
(the start of this graph). As the rate increases further it suddenly starts cycling between
two values in alternate periods, then between 4 values, then 8,
bifurcating at regular intervals, eventually
becoming totally chaotic never having the same population twice ! Then it can
settle to a rhythm again and so one. And this is the formula itself,
nothing to do with real life complications. Wow...Pathological indeed !!
Artificial Life
Page Version 4.83 February 2007 (Paper V1.7 August 2000, original December 1996)