I mentioned before that the Y combinator, seen in chemlambda, is a gun which shoots pair of fanout and application nodes. No more, no less.
The problem is then how to dispense with the Y combinator once we don’t need it any more.
The mentioned solution is to lock it into a quine. Here is how.
First, we put the combinator (in its purest, two nodes form) into a neuron architecture.
What we see: the pale blue dot from the upper right part of the picture is a FROUT (free out) port.
- NEURON AXON: connected to the free out is a pair of nodes which IS the Y combinator as seen in chemlambda (see this old demo of the reduction of the Y combinator to this pair of nodes, which after starts to shoot; I’ll probably add a page for this reduction at the demos site, this time with the new color convention, and for the chemlambda version with FOE nodes, but the old demo explains sufficiently well the phenomenon) .
- NEURON SOMA: in this example the soma is made by one green node (is an application A node), but more sophisticated graphs are acceptable, with the condition to be propagators (i.e. to “propagate” through FO or FOE nodes after a sequence of chemlambda moves; one can transform any lambda term expressed in chemlambda into a propagator)
- NEURON DENDRITES: in this example there are two dendrites, because the soma (A node) has two in ports. The dendrites have FRIN (free in) ports which can be connected to other graphs. In the picture the FRIN nodes are color coded.
- DENDRITES ENDS: at the end of each dendrite there is a small molecule, well chosen.
With the random reduction algorithm the following happen (see the demo in d3.js)
- the NEURON AXON grows and becomes a string of copies of the SOMA with the free in ports attached to it (check out that the colours match!) This is the way we use the Y combinator!
- when the dendrites are consumed, the graph built at the axon detaches from the rest of the neuron
- there are now two graphs, one is the one built at the axon and the other is the one which still contains the Y combinator gun,
- the DENDRITES ENDS, together with the SOMA, lock the Y combinator (i.e. the NEURON AXON) into a quine, i.e. they form together a quine
- the quine is chosen to be one which does not live long (from a probabilistic pov) so it dies after producing some bubbles (i.e, some Arrow elements with the in connected to the out or to other Arrow elements).