In Teaser (I) and Teaser (II) I discussed about the possibility to construct neural networks with controlled connections, resembling the B-type neural networks of Turing, in the following sense:

- they are formed by “neurons”, which in Turing’s description are boolean logic gates, and here they are graphs from graphic lambda calculus which correspond to lambda calculus terms. But there’s no real restriction on that,
, one may consider “neurons” which are outside the lambda calculus sector. The fact is that a “neuron” here should be interpreted as any graph in with at least one input but only one output. Later, when we shall speak about the order of performing moves in such neural networks, it will be seen that each “neuron” is a bag containing graphs which are modified (or reduced, as people say) independently one from another. Neurons are packing conventions from a formal viewpoint, but this viewpoint may not be the best one, a better viewpoint would be to see neurons as well as synapses, described further, as real constructs, like in the related project of the chemical concrete machine,**graphic lambda calculus being more general and powerful than untyped lambda calculus**.**In particular, neurons may do other things than computing in the lambda calculus sense (classical computation), they may do some more geometrical or physical or robot-like activities, not usually considered computations** - the connections between neurons are controlled in a B-type NN, by TRUE – FALSE control wires or inputs. Turing explains that controls themselves can be realized as constructions with boolean neurons (i.e. in his language B-type NNs are particular cases of his A-type NNs). In Teaser (II) is explained how the connections between graphic lambda calculus neurons can be constructed by using a 2-zipper and a switch, such that
(by the same TRUE-FALSE mechanism, but this time TRUE and FALSE are literary the graphs of the corresponding terms in lambda calculus)**a connection is controlled**.**by another connection**

There is an important difference, besides the power of the calculi used (boolean vs. graphic lambda), namely that * there is no signal transmitted through the wires of the network*. That is because graphic lambda calculus does not need variable names. I shall come back to this very important point (which is a big advantage for implementing such networks in reality) in a future post, but let me mention two simple facts:

- an analogy: think about hardware and software, a difference between them is that software may be associated to the pattern of signals which circulates in the hardware. The hardware is the machine inhabited by the software, they are not in the same plane of physical reality, right? Well, in this implementation there is literary no difference between those, exactly because there are no signals (software) transmitted through the hardware.
- this use of names, software, variable and signals is a pain for those trying to make sense how to implement in reality computing constructs. But, if you think, the need to name that stuff “x” and that other stuff “y”, to make alpha conversions in order to avoid name clashes, or even to describe computations in a linear string of symbols, all this are in no relation with physical reality, they are only cultural constraints of humans (and their artificial constructs). It all has to do with the fact that humans communicate science through writing, and of course, it has to do with the enumeration techniques of the cartesian method , which “is designed as a technique for understanding performed by one mind in isolation, severely handicapped by the bad capacity of communication with other isolated minds”. Molecules in a cell or in a solution do not align in a string according to the experimenter writing habits, from left to right, right to left, or vertical. They just don’t wait for an external clock to rhythm their ballet, nor for the experimenter to draw it’s familiar coordinate system. These are all limitations of the method, not of nature.

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Further, in this post, I shall use “synapse” instead “connection”. Let’s see, now that we know we can implement synapses by a TRUE-FALSE mechanism, let’s see if we can do it simpler. Also, if it is possible to make the “synapses control other synapses” concept more symmetric.

Instead of the little magenta triangles used in Teaser I, II posts (which are not the same as the magenta triangles used in the chemical concrete machine post) , I shall use a more clear notation:

The design of a synapse proposed in Teaser (II) involved two switches and a zipper, justified by the direct translation of TRUE-FALSE lambda calculus terms in the freedom sector of graphic lambda calculus. Think now that in fact we have there two synapses, one controlling the other. The zipper between them is only a form of binding them together in unambiguous way.

A simplified form of this idea of a pair of synapses is the following:

In the upper part of the figure you see the pair of synapses, taking the a form like this: . There are two synapses there, one is , which is controlled by the other . The connection between the blue 1 and the red 3′ is controlled by the other synapse. In order to see how, let’s perform a graphic beta move and obtain the graph from the lower part of the figure.

The red 3 can connect, in the sense explained by the switch mechanism from Teaser (II) with blue 1′ or blue 2′, all three of them belonging to the controlling synapse. Say red 3 connects with blue 1′ and look what happens:

OK, so we obtain a connection between blue 1 and red 3′. Suppose now that red 3 connects with blue 2′, look:

Yes, blue 1 does not connect with red 3′, they just exchanged a pair of termination gates. That’s how the pair of synapses work.

Finally, let me remark that the use of termination gates is a bit ugly, it breaks the symmetry, is not really needed.

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