I described in my previous post a possible mathematical way to set up an Boolean sphere. I showed how the Boolean sphere could represent a discreet inner and outer coordinate system and how it also could store memories, do arithmetic and represent Boolean functions within the sphere – properties that, at least in theory, could be used for computations. I also showed that the Boolean sphere’s symmetrical properties made it possible to easily self replicate the Boolean sphere starting from the 2 dimensional “mother system” as a blueprint and how this eventually could be used in an attempt to create a self replicating machine.
In this post I would like to examine how these properties perhaps can bridge the gap between Von Neumanns Universal constructor and Cellular Automata (CA). Von Neuman had the living cell in mind when he developed his ideas so I will make comparison to how well a binary sphere would be able to mimic some of the properties of a living cell (note the word mimic – I don’t say that living cells are Boolean spheres)
Von Neumann’s idea about an Universal Constructor was based on the presumption that the information as well as the operations of the self replicating machine are located within the machine itself – mimicking the way living cells work. I think this is a very sound and necessary approach, not least from a philosophical and religious perspective – it’s hard to overcome the fact that some divine intervention has to occur if the self replicating construction is dependent of instructions and constructions outside the construction itself.
Von Neumann found it hard to implement a 3 dimensional UC in a mathematical rigorous way. Instead he concentrated his attention on 2 dimensional discreet Cellular Automaton that entered different states. Cellular Automata (CA) has been proven to describe many biological and non-biological phenomena. But CA has limitations in describing a Universal Constructor like a living cell in that the operations are not carried out within the cells – it’s the computer that perform the computations when it interpret a number of states – in other words – the computer’s action could by itself be seen as an divine intervention since its occurring outside the cells. The same can be said about the grid (coordinate system) that CA operates within – if the cell doesn’t create it’s own coordinate system some sort of overlord (the programmer of the computer) has to set it up.
So, as I see it, a Universal Constructor has to be able to create it’s own coordinate system and be able to store as well as carry out instruction within the UC itself in order to overcome the “divine intervention” problem. The model I propose would be able to create it’s own discreet coordinate system where the coordinates even could be interpreted interchangeable as Boolean representations or/and as memory.
A Boolean sphere would be able to mimic some of the properties living cells have. The Boolean sphere can divide through making “buds” using a central “mother plane” similar to the way living cells divide on opposite sides of the metaphase plate. Living cells divides in mitosis into two equal cells where the division occur in a plane in the equatorial region of the cell where to distinct poles polarize the cell (the metaphase plate, see figure 4 and 5 below) ).
When the Boolean sphere divide in a similar way it would also, at the same time, make a replica of the external discreet coordinate system .The division will occur perpendicular on either side of the central plane – simulating the metaphase plate. The division will also occur where two poles “polarize” the Boolean sphere (see the 3D animation below).
The internal discreet coordinate system would also at the same time replicate itself using the central “mother”coordinate system (quasi metaphase plate) as a “blueprint” (see 3D animation below):
The Boolean sphere would also be able to mimic the asymmetric division that occur in living cells when a stem cell give “birth” to a differentiated cell. During asymmetrical division the division doesn’t occur at the central plane of the mother cell – the division plane is displaced from the central plane giving rise to a smaller daughter cell as shown below (picture from wikipedia):
The daughter cell in a living cell will be differentiated an have less abilities then the stem cell.
The Boolean can also split up into a mother cell and a daughter cell where a displaced division occur:
The interior Boolean coordinate system will then also reproduce itself:
The daughter Boolean sphere would be differentiated in the sense that it would have less “Boolean space” then the mother binary sphere.
Coordinate system and pattern formation
“The concept of positional information proposes that cells acquire positional values as in a coordinate system, which they interpret by developing in particular ways to give rise to spatial patterns”
Above is quote from the famous biologist Lewis Wolpert. He highlight “the central problems” in this paper, quote:
“Central problems are how positional information is set up, how it is recorded and then how it is interpreted by the cells.”
The surface of a Boolean sphere area is divided into parts where each surface area describe a certain value the can be unambiguously described by a string of ones and zeros. A certain surface area will also be part of a Boolean function (see my previous post). Hence the Boolean sphere can then use both the positional information and at the same time use it for making decisions according to Boolean logic.
The surface area below is represented by the binary string (0010)starting from the pole:
The binary surface areas can have different shape in different areas but the logical string will always be unambiguously defined. The Boolean sphere may even be deformed but the logical string defining the surface area will still be unambiguously defined, see picture below:
Any given surface area defined by the logical string can be used as an anchor point for other Boolean spheres. The anchor point will have a certain degree of freedom within it’s domain, see below:
A Boolean sphere can divide and the offspring can attach it’s anchor point to another Boolean sphere’s anchor point, see animation below:
The shape of the constellation above would also have a rotational freedom in space (as well as “floating” anchor points), See animation below:
The shape of the constellation can be locked by attaching additional Boolean spheres to each other, restricting the degree of freedom, see below:
To conclude: there’s no need to use a fixed predefined grid for a CA/UC based on a Boolean sphere – a new sphere will recreate a coordinate system that can be attached to previous created Boolean spheres. The binary string defining the surface area can simultaneously be part of a Boolean function.
Von Neumann’s envisioned a physical self-replicating machine using a “sea” of spare parts as its source of raw materials. The machine had a program stored on a memory tape that directed it to retrieve parts from this “sea” using a manipulator, assemble them into a duplicate of itself, and then copy the contents of its memory tape into the empty duplicate’s
Lets say that a Boolean sphere is placed in water where spare parts are drifting by. Lets say that a given surface area have a sensor that recognize a given spare part and is able to bind it to the surface are. The sensor could then trigger the binary signal representing the surface. The Boolean sphere could then use this information for making a decision since the binary pathway to the surface area also represent part of a Boolean function. The Boolean sphere could for instance decide to open a gateway, dragging the spare part into the interior of the sphere for further processing based om the memory stored within the Boolean sphere (see previous post). The memory could be copied to the offspring when the Boolean sphere had collected enough spare parts to divide into two Boolean spheres. There would be no need for an outside divine interpreter and instructor (the computer and programmer as in for instance Conways “Game of life”) . The Boolean sphere should be autonomous by itself and be able to interact with other Boolean spheres using the Boolean surface areas as communication channels as well as anchor points and coordinate system.