Artificial Life 8, Sydney, December 2002

Self-Assembling

Dynamical Hierarchies

Alan Dorin & Jon McCormack

http://www.csse.monash.edu.au/~{aland, jonmc}

Centre for Electronic Media Art
School of Computer Science & Software Engineering
Monash University, Melbourne, Australia

Outline

Motivation

• The synthesis of virtual, multi-levelled, dynamical hierarchies exhibiting emergent properties at each level is an open problem in Artificial Life research.
  
  
• In this context, there is no consensus on the meaning of the terms:
  

• Debate continues about the possibility of building such systems and the conditions under which it might (not) be possible.
  
  
  
  
• THIS PAPER...
  
  
  • Presents a trivial, virtual, infinitely-levelled dynamical system with emergent properties occurring at each level, with base units of fixed complexity.
    
    
  • Demonstrates one way in which more rigour in measuring hierarchical organization may be introduced.
    
    
  • Highlights the problems when discussing emergent properties of representations.
    
    

Previous work

• Papers which sparked this exercise were:
  
  
  • Rasmussen et al. 2001, "Ansatz for Dynamical Hierarchies"
    
    
  • Gross & McMullin 2001, "Is it the Right Ansatz?"
    
    

  ...but there are many others which examine these issues also.
  
  
  
  

The model

• A discretized space, tessellated with triangles (or squares).
  
  
• Elements which move randomly through the space.
  
  
• Bonds which may form and break between neighbouring elements to assemble aggregates.
  
  

A screen grab is available from software which runs this model (using squares).

Some sample structures

Hierarchies

• What is a hierarchical object? (E.g. Baas '94)
  

Division/Nested hierarchy

• This hierarchy may self-assemble in the above system.
  
  
• No additional information needs to be added to the base units for the assembly of this trivial but nevertheless, infinitely-levelled hierarchy.

A different hierarchy
(Differently shaped at each level)

• Are these hierarchies dynamical?
  
  
  • The structures move around the grid.
    
    
  • Structural elements dissociate and associate in a constant stream.
    
    
  • Internal bonds are constantly being broken and rebuilt.
    
    

What exactly do we mean by "dynamical"!?

As well as this hierarchy seen earlier,

This is also a hierarchy,

• How can we measure the difference between them?

Measuring & Comparing Hierarchies

• In the digital realm we can easily measure the redundancy in the specification of a structure.

Consider a structure X of order n, written Xn.

• Composed of 4^(n-1) X1 primary elements.
  
  
• If each primary element X1 requires p bits to specify position and orientation,
  
  
  Xn requires p*4^(n-1) bits to specify as an aggregate of X1’s.

But...

• if Xn can be described in terms of the position and orientation of the 4 lower level elements Xn-1 that compose it,
  
  
  Xn requires p*4 bits to specify in terms of Xn-1.

So, for n levels, Xn can be specified hierarchically in: p*4*(n-1) bits.

Since...

• p*4*(n-1) < p*4^(n-1), if n > 2
  

The hierarchical description is measurably more efficient than the non-hierarchical one.

Properties

Properties are observed by people.

• We do not recognize/observe everything.
  
• We overlook, ignore, are fascinated and captivated.

In a simulation, properties may be distinguished where the machine has states to represent them.

• Observers may assign meanings to these states.
  
  
• Changes between machine states (and their meanings) are governed by logical rules, not by chemistry or physics.
  

Trivial Properties in the Model

• A single element has the property "I may move as an aggregate of 1 unit".
  
  
• A pair of elements has the property "I may move as an aggregate of 2 units".
  
  

Each level in the hierarchy has a property not found at lower levels.

This sounds like Bedau's "Nominal Emergence".

Not So Trivial Properties in the Model

• Higher levels possess internal-bonding possibilities not present at lower levels.
  

• Higher levels possess external-bonding possibilities not present at lower levels.
  
  
• If a glider in The Game of Life is emergent, surely so are the structures in this model.
  
  
• This sounds like Bedau's "Weak Emergence" (Nominal Emergence + complex derivation).
  
  
  "How complex is a complex derivation?" (Now emergence is subjective once again)
  

We have built an infinitely-levelled dynamical hierarchy which exhibits emergent properties at each level and has base units of fixed complexity!

So why are we not excited?

The Artificial Life mantra:

"We seek interesting behaviour"

Multiple-levels of trivial behaviour may produce a level of interesting behaviour.

• Is the interaction of atoms as "interesting" as the interaction of molecules?
  
  
• Is the interaction of molecules as "interesting" as the interaction of cells?
  
  

Perhaps the stuff of our simulations is just too boring to ever produce anything interesting.

Questions.

• What do we mean by dynamical hierarchies and emergent properties?
  (Does it matter and why?)
  
  
• When is something "interesting"?
  
  
• How subjective/objective can we be about this problem?

Conclusions.

• Is this model what we are looking for?
  
  I doubt it.
  

• What is missing from this model?
  
  It isn't interesting.
  

• Can we measure hierarchical structure in simulations?
  
  Certainly.
  

• Can we measure properties in simulations?
  
  Certainly.
  

• Can we tell when a property is emergent?
  
  It seems so... but it might be completely subjective... I'll know it when I see it!

The issue hinges on the Artificial Life mantra.

©Copyright Alan Dorin/Animaland 2002