Abstract
The concept of weak emergence is a refinement or specification of the intuitive, general notion of emergence. Basically, a fact about a system is said to be weakly emergent if its holding both (i) is derivable from the fundamental laws of the system together with some set of basic (non-emergent) facts about it, and yet (ii) is only derivable in a particular manner, called “simulation.” This essay analyzes the application of this notion Conway’s Game of Life, and concludes that a modification of the notion would provide a better refinement of the general notion of emergence. It is proposed that emergence be taken as a matter of degree, defined in terms of the amount of simulation required to derive a fact.
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Notes
Bedau (2003) suggests that there is a good notion of weak emergence as coming in degrees.
Hereafter, ‘GOL’ means Game of Life on an infinite grid.
One could also think of things a little differently; if S is the set of at-a-time states (basically the set of all functions \(s: Z \times Z \to\{0,1\}),\)we could think of GOL as a function \(g: S\to S.\) Even farther out, we could simply think of GOL (via an encoding) as a function from the real interval [0,1] to itself. These are equivalent, but, in our first-order setting, it is better to focus on the fairly natural way of thinking of GOL in the main text.
There are uncountably many worlds since there is exactly one for each possible initial state; and there are uncountably many initial states, since for each of (countably) infinitely many cells, it may be either on or off.
In fact, we could generalize our discussion to include some states in which infinitely many cells are alive—any state that is describable in \({\mathcal{L}}.\) But this requires addressing some delicacies about what to count as a description; e.g., do we want to count as a state-description “For all x and y, cell (x,y) is alive iff x and y are relatively prime”? The bounded states include enough to make GOL very interesting, and they have a nice canonical representation in \({\mathcal{L}}.\) Further, the standard models of computation, e.g., Turing machines, assume a similar kind of boundedness.
Of course, many of these clauses are “redundant,” and, indeed, R is derivably equivalent with much shorter sentences.
Q should be something in the neighborhood of Robinson arithmetic, or the system Q of Boolos et al. (2002), but expanded to deal with negative integers.
Technically, we intend \(\forall w(World(w)\to(\phi_0({\bf 0},w)\ensuremath {\mathrel{\to}}\phi_t({\bf T},w)));\) we will continue to use the variable ‘w’ as implicitly restricted to worlds.
One does not need mathematical induction in the formal system in order to derive and make use of this fact.
It depends on the details of our choice of Q, but, presumably Q is minimal enough that (A) will not be derivable in D. Q does not include mathematical induction, so it will likely have non-standard models in which instances of induction fail; it is highly plausible that instances involving predicates like State can fail. Thus we get models that verify \(Q\wedge R\wedge L\) , along with the negation of (A).
In fact, one might plausibly allege that nothing about GOL on an infinite plane can, in the strictest sense, be known by mere simulation, since one cannot simulate the infinitely many applications of the evolution rule to each cell. But we could reconceive the GOL on bounded states so as to handle this issue, e.g., as a function from the natural numbers (codes of bounded states) to the natural numbers. This corresponds to the caveat a few paragraphs back.
Where the path is described finitely, relative to the initial position of the glider.
Another possibility is that the measure of “amount of simulation used” by a derivation can be adapted to alternate systems, in some independently motivated way, with the result that different systems yield the same comparative ranking of amount required.
The notion of complexity I use in this paragraph is an intuitive one, not meant to be identified with (but not unrelated to) any of the formal notions (e.g. Kolmogorov complexity) studied in “complexity theory.”
It is unclear how deep the relativity (to a derivation system) is, especially in the case of this notion of maximal emergence. There is the Big Question of whether human reasoning is machine-like enough to be captured in some in-principle (but not in-practice, at least not human practice) specifiable algorithm; if so, then there is a class of GOL facts that are maximally emergent for us.
References
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Acknowledgement
This paper benefitted from discussions with Mark Bedau.
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Hovda, P. Quantifying Weak Emergence. Minds & Machines 18, 461–473 (2008). https://doi.org/10.1007/s11023-008-9123-5
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DOI: https://doi.org/10.1007/s11023-008-9123-5