Even prior to a definition of "combination", would you have to be thinking about whether combined meant: a) merging populations, b) union of the "capability set" of agents, c) both, d) some other thing (perhaps as well).

Friends!

Very many thanks for your answers to my question. I shall follow up your helpful suggestions.

As regards the particular point(s) of possible complexity raised by Edmund around the notion of combination of ArtSocs – yes, I fully agree…

To illustrate my current approach to mathematical formalization of artificial societies, my own current (inadequate!) attempt at a mathematical proof that the combination of two artificial societies is an artificial society is the following – but there are a multitude of other ways of proceeding -- I feel the following is pretty poor with holes and somewhat misconceived.

A possible “Weak-ArtSoc” definition

An agent is a finite state automaton (possibly with additional specified structure)

A weak-ArtSoc is a set of agents divided into two disjoint sets, (1) a “natural” environment set [in the form of a cellular automaton, say] and (2) a non-environment agent set. Some pairs of agents (of either type] pass output to input (i.e. at least one output of the first agent becomes an input of the second). In the non-environment set of agents it is required that any agent can be reached from any other by a sequence of such output à input directed links.

So a weak-ArtSoc is a 3-tuple as follows

{agent set (divided into environment and non-environment sets – not necessarily disjoint);

set of partitions of output values, one partition for each agent;

set of mappings one for each agent from its output values to  agents}

satisfying the requirement that there is a sequence of directed links between any two non-environment agents (as above a directed link exists between two agents when some particular output value of the first  goes to the second)

Define a combination of two weak-ArtSocs to be the union of their two agent sets (not necessarily disjoint) with N randomly generated directed links added between agents.

Then given two specific weak-ArtSocs and the value of N, the probability that the combination (as just defined) of the two weak-ArtSocs  is itself a weak-ArtSoc is well-defined, depends upon N, and can be determined either to desired accuracy by experimental trials  or,  in principle, mathematically.

A trivial example: two identical weak-ArtSocs, each with a single agent, which are combined with N=1 so that there is a single directed link from one agent (call it A1) to the other (call it A2) in the combined society. Trivially the combined society is NOT a weak-ArtSoc because there is NO sequence of directed links from A2 to A1. (NOTE the natural environment sets in this example are null – the defn allows.)

A less weak notion of an artificial society is a weak-ArtSoc in which all of its non-environmental agents have models (in some mathematically precise sense) of themselves, of >0 other non-environment agents, and of the ArtSoc as a whole. This requires a mathematical definition of a “model” – hard! - not attempted here.[Minsky “Matter Mind and Models” 1965 is a possibly relevant early and influential (in AI) discussion of self-modelling and its use and complexities.]

I shall be grateful for all criticism of the foregoing and all further pointers to attempts already made at a formal mathematical definition, handling and investigation of abstract artificial societies i.e. towards a solid theory of abstract societies. Note -- I assume that any definition of an artificial society is to be broadly in line with our intuitions about society (e.g. as concerns Spartan society or chimpanzee society or as applied towards an emergent society of robots) without being conceived as a model of any specific society.

Surely such attempts are already out there!

Best regards

Jim

This is a very interesting, worthwhile question.   

In addition to the help observations already made, there is a bit more that can be observed.

From a mathematical perspective, each artificial society can be modelled as a nonempty set objects, each having its own location identified by the coordinates of the center of mass of the object.   Let X be such a societal set and let X be equipped with a proximal relator $\mathscr{R}$ (i.e., a set of proximity relations such as Lodato proximity $\delta$, overlap Lodato proximity $\delta_o$ and descriptive proximity $\delta_{Phi}$).   The pair (X,  $\mathscr{R}$) is an example of a societal proximity space.  Further, let (Y, $\mathscr{R}$) be a second societal proximity space.   

Since each object is identified minimally by the location of its center of mass at a point p in a Euclidean space R^3, we can find the convex hull of the set of points that represent societal sets X and Y.   Let convX, convY be the convex hulls for X and Y.   Then we can start considering the potential comination of the societies modelled by convX, convY in terms of

spatial closeness: convX, convY are close spatially, provided convX $\delta_o$ convY, i.e., the intersection of the sets convX, convY is not empty.   In other words, convX, convY have one or more common points.

descriptive closeness: convX, convY are close descriptively, provided convX $\delta_{Phi}$ convY, i.e., convX, convY contain at least one pair of subsets with matching or similar descriptions.

This is a suggested application of a computational proximity approach to combining artificial societies.   For more about computationa proximity, see

https://www.researchgate.net/publication/301790590_Computational_Proximity_Excursions_in_the_Topology_of_Digital_Images

It is also possible to use a blackboard as a means of mathematically modellilng the emergence of new artificial societies from the combination of artificial societies.    See, e.g.,

http://cogprints.org/1477/3/ASIA.pdf

Book Computational Proximity. Excursions in the Topology of Digital Images.

I found that this question is very interesting with many answers.