Implications for everyday systems
Covering: Natural systems and the advantages of modeling them algorithmically.
[Note: This is a paraphrased summary of the chapter. I've edited to sequence of the argument a little to help the flow.]
The simplest possible model of the system is the most desirable. Simple models are better, whether they're differential equations or algorithms. They're easier to understand, and more likely to be right
[as it's easier for them to be wrong].
Central Discovery Of Book: Simple models can produce Complex behaviour.
This chapter argues the superiority of algorithmic models for understanding complex processes in the natural world. It presents examples to back the claim. Like any other model, the examples are an abstraction, capturing only certain aspects of the real system being studied.
Claim: The examples provided are groundbreaking in being able to actually reproduce the gross behaviour of the systems they model.
Gripe: If this were economics, it'd be heralded as a useful thing.
However, physicists expect to be able to test a model by comparing its detailed output with the real world.
This works for relatively simple systems
[eg?], as their state is relatively easy to characterise. However, as increasingly complex systems are tackled, the amount of information they produce make it counter-productive to validate by testing their detailed output against the real world.
[Presumably because it's there's more potential for noise to get in the way? --- Which makes checking output in detail more error prone, and easier for relatively unimportant aspects of the system to obscure an otherwise good model?]
For such systems, it's better to assess a model by ignoring the detail and instead watching for gross behaviour that looks correct.
The Big Ass Claim: Furthermore: equations are not powerful enough to create simple models of complex systems.
- They result in complex models which are hard to verify and understand.
- Equations are declarative, often giving no guidence for working out what behaviour satisfies their constraints.
- Also, they involve continuous variables, forcing the use of approximation when computing behaviour with no guarantee of continued faithfulness to the original model.
On the other hand, an algorithmic approach:
- [Can presumably be more powerful and] allows for simpler, and therefore better models.
- It is easier to derive predictions from programmatic models as they need only be executed, not solved.
- This means that determining a model's predictions/behaviour is often much more efficient.
The following examples to back the above aren't intended to conform to every detail of the physical system they model. Instead, they aim to
model the basic mechansism responsible for obvious features. The goal
is to explain/understand the principles driving a natural system's
intricacies, not to re-produce the exact details of individual cases.
By the way, we're not suggesting that these processes
are CAs underneath it all, in the same way that we don't claim that the planets orbit due to the existence of differential equations which model their movements.
Questions:
- Do the programs lead to understanding of the system they model?
- Are they simpler than the equivalent equation based approach?
- Are they more tractable?
- Do they give better predictions?
[He follows with examples to back up the algorithmic approach]
Snowflakes
- Captures one process behind this using a CA, whose output is indeed reminiscient of a snowflake.
- Prediction from model of an invariant feature, which is confirmed when looking at real snowflakes.
- Difficult to make a more complete model as there are many physical processes involved, hard to know which ones are important [but if your claim is correct, algorithmic models sould make this easier to determine, no?]
- Literature uses equations which have only been able to deal with simplistic structures and have been (claim) largely unsuccesful.
[More examples follow to try and back up the claim. Interesting, but the above gives the gist of the argument.]
Breaking of materials
Fluid Flow
"Fundamental issues in biology"
Plant Growth/Animal Growth
Financial Markets
...
Richard P
back to
Wolframania
EResearch