D: Daddy, what’s a black box?

F: A “black box” is a conventional agreement between scientists to stop trying to explain things at a certain point. I guess it’s usually a temporary agreement.

D: But that doesn’t sound like a black box.

F: No—but that’s what it’s called. Things often don’t sound like their names.

D: No.

F: It’s a word that comes from the engineers. When they draw a diagram of a complicated machine, they use a sort of shorthand. Instead of drawing all the details, they put a box to stand for a whole bunch of parts and label the box with what that bunch of parts is supposed to do.

D: So a “black box” is a label for what a bunch of things are supposed to do….

F: That’s right. But it’s not an explanation of how the bunch works.

D: And gravity?

F: Is a label for what gravity is supposed to do. It’s not an explanation of how it does it.

D: Oh.

Metalogue: "What is an instinct?"

Two are the methods that classical science has used with great success for simple and complicated systems. The question is wheter these methods can also be used for complex systems.

The first, reductionism and its inverse, constructionism, was rated inadequate by the work of Anderson.

The second is the black box method, historically derived from the theory of electronic systems, but widely studied and generalized in the first cybernetics, in particolare da Ross Ashby.

In the approach of the black box we know nothing about the interior of the system, we can only access it through the entrances where the introduction of in-put tand observe the outputs where to gather the out-put. This is a very common situation in physics, electronics, systems theory but also in biology, medicine and psychology, where the system undergoes a cause and effects are observed, reactions to a stimulus etc.. For example, an entire approach to psychology, the behaviorism, is entirely based on the black box, as well as medical drug testing procedures.

x	f	y
input	operation	output
indipendent variable	function	dipendent variable
cause	natural law	effect
minor premise	major premise	conclusion
stimulus	organism	reaction
reason	character	action
goal	system	behavior

A black box system is characterized by a input-output relation through an operator f which to any input x provides the output y: y=f(x).

trivial determined machine

If the operator f is known, by form or function or of the output/input relation for all inputs of interest, then the system is completely determined. Conversely, if f is unknown, and this applies in almost all cases, the way to determine it is to introduce input values to the inputs and observe the corresponding output values on output.

VonFoerster calls the black box "machine" and divides the machines into two categories, those without internal states and those with internal states, defining the first trivial and the second non-trivial machines.
In trivial machines one introduces a certain number of input values x1 x2 x3 ... and will collect the corresponding output values y1 y2 y3 ... If at an input value xn is always and only associated an output value yn, then the machine is history independent (it is a system without memory), is determined in a synthetic way, that is that machine has been designed to operate in that specific way and only in that way, it is analytically determinable, that is given a certain number of input values the machine is fully known and describable, and finally is predictable; that's why Von Foerster defines it trivial. In the example of figure if we give an input value A we get an output value B, if we give B we get C and so on, if we supply again on the first input A we'll get again B etc. In general, the machine is described by tables which associate with each xi of interest the corresponding yi.

If the machine has only one input and one output, then given an input value x1 with the corresponding value y1=f(x1) the machine is fully described for that value. If we have two inputs and two input values x1 and x2 to completely describe the machine is necessary to observe the outputs with the permutations x1x1, x1x2, x2x1, x2x2. In general, if N is the number of inputs (assumed equal to that of outputs) the number of input configuration to be calculated to completely describe the machine is N^N. For a machine with 4 inputs and 4 outputs there are 4⁴=256 different configurations to compute, for a one with 10 input s and 10 outputs there are 10¹⁰=10 billions configurations; assuming that to compute one it is necessary one microsecond, one millionth of second, namely that any configuration is computable at any MIPS (million instructions per second), a 70s computer at 1 MIPS took 10000 seconds, namely little less than three hours to compute all the possible configurations, one of the 80s at 10 MIPS about 16 minutes, a Pentium III less than two minutes and a Pentium 4 about 1 second. The trivial machine seems to confirm the statement of Atlan that for a complicated system, having sufficient resources of time and money, you can come to have complete knowledge.

As Von Foerster tells when we buy a any machine - a car, a blender, a washing machine - we expect, and the seller is ready to confirm, it's a trivial machine; then one day you turn the key and nothing happens, or press a button and floods the house, then go to a mechanic or technician and ask him to make the machine again trivial.

The black box model with trivial machine is and has been a successful tool in the analysis of simple and complicated systems, and its natural extrapolation is to add internal states in the machine to try to use it for complex systems. The presence of internal states within the system (with only one internal state we have the trivial machine) is one of the features of complex systems.

non-trivial internal states machine

In this case the black box system has no more one but two input/output operators: ithe first y=f(x,z) gives the output value as a function of the x and of the internal state z, the second one z=g(x,z1) gives the new internal state of the machine as a function of input x and of the previous internal state z1.

Proceeding as before with the trivial machine, providing an input x1 does not mean that the output is always y1, but can be any output y1z depending on the internal state z which at that time has the machine, state that can be modified by any of the input.

The number of configurations that must be calculated to provide a complete knowledge of the internal state machine is not great as in the trivial machines, neither astronomical, it is meta-astronomical. In the case of 2 inputs, 2 outputs and 2 internal states the possible configurations are 2¹⁶=65536, a number which is easily computable, but at the next step with 4 inputs, 4 outputs and 2 internal states the configurations become 2⁸¹⁹²=10²⁴⁶⁶; to try to evaluate this number considering that the age of universe (around 13 billions years) expressed in seconds is about 10¹⁷ seconds and that to 2010 the most powerful supercomputer with parallel multi-processors (Tianhe-1A with 186368 core processors Intel EM64T Xeon X56xx at 2930 MHz with a 229376 GB memory) had a computing speed of about petaFLOPS, namely about 10¹⁸floating point operations per second and assuming that an input configuration can be computed with 10 FLOPS we have that to compute the total number of 10²⁴⁶⁶configurations is required a time 10²⁴³²times the age of the universe.

As Von Foerster says: "... I highly recommend you not to undertake such an enterprise ... you'll loose time, money ... everything".
The non-trivial machine is determined in a synthetic way as the trivial ones, it is dependent from history due to its internal states, it is analytically indeterminable, as clear from the example, and finally it is umpredictable, like all complex systems.

TRIVIAL MACHINE	NON TRIVIAL MACHINE
input-output stimulus-response	It has internal states that influence what it will do. When acting on it, it can change its internal state. With the same previous imput, not necessarily it will act in the same way.
It is determined in a synthetic way: when it was built, it was determined the way in which it was working.	Determined in a synthetic way: we can build it as we want
Analytically determinable: if analyzed, it produces a determined result.	Analythically indeterminable
Independent from history: whatever is the applied input, it does not remember it following the previous laws.	Dependent from history
Predictable: when given a certain input, we know wht the machine will do.	Unpredictable

As Von Foerster tells Ross Ashby had built a machine of this type and used it to assess students who applied to work with him. It was a machine with two inputs, two outputs and four internal states and was presented as an innocuous small metal box with two switches (inputs), two light bulbs (the output) and was powered by batteries.

During the interview, Ashby casually suggested to the student: "Why not see to understand how this box works?". The student was not very impressed - two switches, two lamps - and began to work and take down input /output tables. Let's say this happened at three in the afternoon. Ashby came home about six o'clock and returned to the office the next morning at seven. If the student was not there, was then asked "What happened with that problem?" and if the student told him he worked until midnight did not understand anything and was gone bored Ashby thought "It's not my man". If the student was there exhausted, sleepless, with a greenish color, surrounded by hundreds of tables he would say, 'You're tough, interested, but let me give you a hint: you'll never solve this problem ".

The simple Ashby machine is perhaps the most concrete physical object that can represent the abstract mathematical concept of infinity. In general, the definition of infinity is represented through the concept of limit, but in the physical reality, even considering astronomical numbers such, for example, the estimated (uncertain) total number of atoms in the visible universe, that is 10⁷⁹- 10⁸⁵, or the estimation of the computational capacity of the entire universe calculated by Lloyd in 10¹²⁰ elementary logical operations with an accumulation of information as to 10⁹⁰ bits, one does not get the meta-stronomical numbers of the simple Ashby non-trivial machine.

Computational capacity of the universe

Seth Lloyd

Phys.Rev.Lett. 88 (2002)

Merely by existing, all physical systems register information. And by evolving dynamically in time, they transform and process that information.The laws of physics determine the amount of information that a physical system can register (number of bits) and the number of elementary logic operations that a system can perform (number of ops). The universe is a physical system. This paper quantifies the amount of information that the universe can register and the number of elementary operations that it can have performed over its history. The universe can have performed no more than 10^120 ops on 10^90 bits