Dear Prof Doran,

I do not have a concrete answer, but I have done a cursory reading of the "holonic agents" article of Gerber et al in 1999 titled "Holonic Multi-Agent Systems":

http://scidok.sulb.uni-saarland.de/volltexte/2011/3775/pdf/RR_99_03.pdf

Perhaps this article is of interest to you.   All the best with your research.

Best regards

I think that one would need to characterise the agents as members of an algebraic structure of some sort. Once such a characterisation is established you can then start looking at mappings of sets of agents (with associated state) to their image under a projection through time.  Dealing with infinite numbers of agents would need results on the closure of the space, and some justification that an analogue of sequences sets of increasing numbers of agents remain within the space.  Apologies ... This is pretty cryptic, but I am on my phone with *teensy* writing.

I can elaborate later if anyone is really interested.

James,

I guess an *infinite* hierarchy is rather coalgebraic then algebraic. The construction of an infinite hierarchy of mutual expectations (Harsanyi-"type space") is already rather involved, see Moss/Viglizzo: Harsanyi Type Spaces and Final Coalgebras Constructed from Satisfied Theories, 2004, Electronic Notes in Theoretical Computer Science, Vol. 106.

If you want multi-agents you may start with my paper with Achim Blumensath: "A coalgebraic framework for games in economics". Here we show how to construct games aka multi-agent systems as coalgebras and natural transformations as the "glueing". You may then want to prove whatever you want to prove by some modal logics, as this is the internal logic of the category of coalgebras.

If you want to do that you may want to join us, as me with some other coalgebra researchers (mostly UK, Scotland, Netherlands) are about to restart this work (the paper with Achim is from 2013) again and combine it with our higher-order decision and game theory (2015): 

http://arxiv.org/abs/1506.01003

http://arxiv.org/abs/1506.01002

Also some thoughts related to your list of topics to your question: you mention "emergence". With a semantics based on coalgebras = behavioral = observational versus algebraic = constructive  approach there is no need for "emergence" (reductionism needs to construct emergent properties) since the behaviors of all structured subparts of the system are mapped into the final coalgebra which contains the behavior of all coalgebras, ie also of the whole system. Hence, semantics via coalgebras or behavioural approaches as often done in computer science is "holistic" (by construction or by observation :-) without the need to construct emergent properties, which take into account the relations of the subparts, which in semantics is given by the syntax.

Or even more clearly in bialgebras:

SBX -> X -> BSX

the structured (by syntax) behaviours of subparts SB (a functor) is mapped via the state space X (coalgebra SBX -> X) into the behaviour of the structure (the whole thing)   (algebra X -> BSX). Think about that via the functional language known as Excel: the known behaviour of the cells (simple functions) are combined in a certain structure or syntax (the rectangular cells with links among them) and the question is what is the behavior of the whole thing, the Excel sheet, ie what does the sheet calculate (in the final cells which are not referred to by other cells)?

James, what do you want to do with that?

cheers

Viktor

Dear Alwyn, Randall and Victor

Many thanks for these helpful answers -- I shall study and respond to them as soon as I have finished the text I am currently struggling to write for the MoHAT project (That will take a couple of weeks).

By the way -- note that my question involved COGNITIVE agents...

Thanks again

Best

Jim

I have seen that. Our framework has what we have called econometric agents. Ie they observe and maintain an internal state. So you could make them into cognitive ones, however you want them to model.

Victor

Thanks -- In your earlier answer you asked what I wanted from this ...

Can you answer the following two questions with your machinery?

(i) Under what conditions is the cognitive effectiveness of a finite hierarchy always less than the cognitive effectiveness of an infinite hierarchy that extends it?

(ii) Under what conditions does the cognitive effectiveness of an infinite hierarchy always increase with time?

Obviously an acceptable and tractable precise definition of cognitive effectiveness is required -- not that different from intelligence perhaps -- and more by way of definition.

Jim

This is an interesting question and definitely answerable with our machinery and definitely a coalgebraic problem in that you want explicit characterization of finite versus infinite structures. Both, finite and infinite sequences are in the final coalgebra hence you can discuss that properly. But yes, all that depends on the definition of cognitive effectiveness.

3 JOTTINGS re AGENTS

An agent is an FSA

“An agent is a multi-agent system” means that EVERY agent is a set of (>0) intercommunicating agents. Recursion. It follows that every agent is an infinite set of agents and can be viewed as a hierarchy.

A “cognitively effective agent” can be interpreted in two ways. EITHER such an agent solves (?) problems (?) successfully (?) OR it has discernible internal processes (?) of the cognitive type (?) which work well (?).  I prefer the latter but find the former more tractable.

[NB (?) means mathematically precise definition is required.]

Dear All

Again thanks for helpful answers to my question re infinite hierarchies of cognitive agents. The answers are all certainly relevant, but seemingly “only” as start points for an investigation. Unless I’ve missed something no one has yet pointed to work on infinite hierarchies of agents actually done. I guess Koestler only wrote about infinite hierarchies.

I am a touch sceptical about Victor’s algebraic approach (connecting to decision theory and game theory) – mainly because AI people (amongst whom I count myself – though long, long ago I did do a first degree in mathematics) have typically found that abstract mathematical ideas and game theory formulations tend to block rather than help the engineering of AI programs – because to gain mathematical traction the actual AI content/requirement  tends to get watered down or discarded – obvious examples are the AI chess playing programs that are very successful but without non-trivial mathematical ideas. But maybe that experience doesn’t really apply in this case, as the immediate aim is NOT to write a program but to prove emergent properties. (I should say that I haven’t yet read the key paper that Victor recommends...)

So now I shall hang out a tiny, tiny sprat in hope of getting back a big mackerel!

                Assume a hierarchy of agents with the top agent called BOSS

                Assume each agent in the hierarchy has none, one or two immediate subordinates all initially alive

                Assume that an infinite series of identical messages is sent to BOSS. Each message we may understand as saying: YOU FAILED AGAIN, YOU IDIOT - KILL YOURSELF!

                Every agent, including BOSS, on receiving this message passes it to a live immediate subordinate (arbitrarily chosen) if it has one – if not it kills itself

CONJECTURE – BOSS survives indefinitely if and only if the initial hierarchy is infinite (i.e. contains an infinite number of agents)

Proof ??

A mackerel?

Best regards

JIm

I guess, you can formulate this question like this: "We are applying depth-first post-order traversal in a binary tree. The root node is not visited during such a traversal if and only if the tree has infinite depth."

Many thanks for more helpful, interesting and informative answers.

But perhaps nothing that fully responds to my original question yet, I think.

I like Hosny Abbas' remark that "for infinite hierarchies to survive, agents must be simple and react to their environment changes under predefined laws. Intelligence of agents is not necessary in large-scale systems or infinite systems...........................!"  I assume that this referts to MATH/COMPUTATIONAL agents. There seem to be two linked conjectures in Hosny's remark. Can anyone (Hosny?) make either (or both) of these conjectures precise enough for mathematical proof or disproof to be attempted?

I am still hoping to see a big mackerel for my little sprat!

Cheers

Jim

PS For clarification, in my original question the idea is that every agent in the infinite hierarchy is a multi-agent system. In particular, every agent in any multi-agent system is itself a multi-agent system. Recursion!

I second Alwyn Husselmann's suggestion about the Gerber et al. article on holonic agents. A more recent description of the recursive structure of hierarchies of agents can be found in:

Cossentino, Massimo, et al. "ASPECS: an agent-oriented software process for engineering complex systems." Autonomous Agents and Multi-Agent Systems 20.2 (2010): 260-304. See http://www.aspecs.org/images/Jaamas10.pdf

The structures mentioned there are not infinite, but you might be able to generalize.

As to the cognitive aspect, on a slightly different tangent, you might find the work of my PhD student Harmen de Weerd interesting. He has implemented agents that have "k+1-st order" theory of mind by putting themselves in the perspective of their "k-order" colleagues. See, for example:

De Weerd, Harmen, Rineke Verbrugge, and Bart Verheij. "How much does it help to know what she knows you know? An agent-based simulation study." Artificial Intelligence 199 (2013): 67-92. Downloadable at http://www.rinekeverbrugge.nl/PDF/Articles%20in%20refereed%20journal/WeerdVerbruggeVerheij-AI.pdf