The Coffee Automaton: Quantifying the Rise and Fall of Complexity in Closed Systems

Scott Aaronson (MIT)Joint work with Lauren Ouellette and Sean Carroll

It all started with a talk Sean Carroll gave last

summer on a cruise from Norway to Denmark…

Proposed system: the “coffee automaton”

Our modest goal: Understand the rise and fall of complexity quantitatively in some

simple model system

nn grid (n even), initially in the configuration to the right(Half coffee and half cream)

At each time step, choose 2 horizontally or vertically adjacent squares uniformly at random and swap them if they’re colored differently

“Control Experiment”: The Non-Interacting Coffee Automaton

The starting configuration is the same, but now we let an arbitrary number of cream particles occupy the same square (and treat the coffee as just an inert background)

Dynamics: Each cream particle follows an independent random walk

Intuitively, because of the lack of interaction, complexity should never become large in this system

Fundamental requirement: Need to assign a value near 0 to both “completely ordered” states and “completely random” states, but assign large values to other states (the “complex” states)

Also, should be possible to compute or approximate the measure efficiently in cases of interest

How to Quantify “Complexity”? Lots of Approaches in the Santa Fe Community

Warmup: How to Quantify Entropy

K(x) = Kolmogorov complexity of x = length of the shortest program that outputs x

Old, well-understood connection between K and H: 1E

~ODHxK n

Dx n

K(x) is uncomputable—worse than SZK-complete!! But in some (not all) situations, one can approximate K(x) by G(x)K(x), where G(x) is the gzip file size of x

x x

xx pppH

1log2

Problem: Approximating H is SZK-complete!

Let f(x) be a function that outputs only the “important, macroscopic” information in a state x, washing or averaging out the “random fluctuations”

Approach 1 to Quantifying Complexity: Coarse-Grained Entropy

Advantage of coarse-graining: Something physicists do all the time in practice

Disadvantage: Seems to some like a “human” notion—who decides which variables are important or unimportant?

Then look at H(f(x)) H(x). Intuitively, H(f(x)) should be maximized when there’s “interesting structure”

Then consider the expected mutual information between the configurations in P and F:

Approach 2: “Causal Complexity”(Shalizi et al. 2004)

FPHFHPHFPI ,E,E

Given a point (x,t) in a cellular automaton’s spacetime history, let P and F be its past and future lightcones respectively:

F

P(x,t) Time t

Advantages of causal complexity: Has an operational meaningDepends only on causal structure, not on arbitrary choices of how to coarse-grain

Disadvantages: Not a function of the current state onlyRequires going arbitrarily far into past and futureI(P,F) can be large simply because not much is changing

Intuition: If dynamics are “simple” then I(P,F)0 since H(P)H(F)0If dynamics are “random” then I(P,F)0 since H(F|P)H(F)In “intermediate” cases, I(P,F) can be large since the past has nontrivial correlations with the future

Approach 3: Logical Depth(Bennett 1988)

Depth(x) = Running time of the shortest program that outputs x

Depth(0n) = Depth(random string) = nBut there must exist very deep strings, since otherwise Kolmogorov complexity would become computable!

Advantage: Connects “Santa Fe” and computational complexity

Disadvantages: There are intuitively complex patterns that aren’t deepComputability properties are terrible

Given a set S{0,1}n, let K(S) be the length of the shortest program that lists the elements of S

Given x{0,1}n, let Sophc(x) be the minimum of K(S), over all S{0,1}n such that xS and K(S)+log2|S|K(x)+c

In a near-minimal program for x, the smallest number of bits that need to be “code” rather than “random data”

Approach 4: Sophistication(Kolmogorov 1983, Koppel 1987)

Sophistication is often thought of in terms of a “two-part code”:

Program for S Incompressible index of x in S

Sophc(x) = size of this part

Sophc(0n)=O(1), for take S={0n}Sophc(random string)=O(1), for take S={0,1}n On the other hand, one can show that there exist x withSophc(x)n-O(log n)

Theorem (far from tight; follows Antunes-Fortnow): Let c=o(n); then there exists a string z{0,1}n with Sophc(z)n/3

Proof: Let A = {x{0,1}n : x belongs to some set S{0,1}n with K(S)n/3 and K(S)+log2|S|2n/3 }

Let z be the lexicographically-first n-bit string not in A (such a z must exist by counting)

K(z)n/3+o(n), since among all programs that define a set S with K(S)n/3, we simply need to specify which one runs for the longest time

Suppose Sophc(z)n/3. Then there’s a set S containing z such that K(S)n/3 and K(S)+log2|S| K(z)+c n/3+o(n). But that means zA, contradiction.

Special Proof Slide!

Problem: Sophc(x) is tiny for typical states x of the coffee automaton! Why? Because we can let S be the ensemble of sampled states at time t; then x is almost certainly an incompressible element of S

Advantage of resource-bounded sophistication: The two-part code “picks out a coarse-graining for free” without our needing to put it in by hand

Disadvantages: Hard to compute; approximations to Sophc

efficient(x) didn’t work well in experiments

Solution: Could use resource-bounded sophistication, e.g., minimal length of p in a minimal 2-part code consisting of (polytime program p outputting AC0 circuit C, input to C)

Let I = coffee-cup bitmap (n2 bits)

Let C(I) = coarse-graining of I. Each pixel gets colored by the mean of the surrounding LL block (with, say, L~n), rounded to one of (say) 10 creaminess levels

Our “Complextropy” Measure

Complextropy := K(C(I))

G(C(I)) K(C(I)): gzip file size of C(I); approximation to complextropy that we’re able to compute

Complextropy’s connection to sophistication and two-part codes:Compressed coarse-grained image Remaining info in image

K(C(I)) = size of this part

Complextropy’s connection to causal complexity:

The regions over which we coarse-grain aren’t totally arbitrary! They can be derived from the coffee automaton’s causal structure

Complextropy can be seen as an extremely resource-bounded type of sophistication!

Even in the non-interacting case, rounding effects cause a “random” pattern in the coarse-grained image, at the border between the cream and the coffee

Makes K(C(I)) artificially large

The Border Pixels Problem

Hacky Solution: Allow rounding 1 to the most common color in each row. That gets rid of the border pixel artifacts, while hopefully still preserving structure in the interacting case

Behavior of G(I) and G(C(I)) in Interacting Case

Behavior of G(I) and G(C(I)) in Non-Interacting Case

Qualitative Pattern Doesn’t Depend on Compression Program

Dependence on the Grid Size n

Maximum entropy G(I) increases like ~n2 for

an nn coffee cup

Maximum coarse-grained entropy G(C(I)) increases like ~n

We can give a surprisingly clean proof that K(C(I)) never becomes large in the non-interacting case

Analytic Understanding?

Let at(x,y) be the number of cream particles at point (x,y) at step t

Claim: E[at(x,y)]1 for all x,y,t

Proof: True when t=0; apply induction on t

Now let at(B) = (x,y)B at(x,y) be the number of cream particles in an LL square B after t steps

Clearly E[at(B)]L2 for all t,B by linearity

By a Chernoff bound,

300

exp21.0EPr2

2 LLBaBa tt

So by a union bound, provided ,lognL

111.0EPr 2 oBLBaBa tt

If the above happens, then by symmetry, each row of C(I) will be a uniform color, depending only on the height of the row and t

Hence K(C(I)) log2n + log2t + O(1)

Prove that, in the interacting case, K(C(I)) does indeed become (n) (or even (log n))

Requires understanding detailed behavior of a Markov chain prior to mixing—not so obvious what tools to use

Maybe the 1D case is a good starting point?

Open Problems

Clarify relations among coarse-grained entropy, causal complexity, logical depth, and sophistication

Find better methods to approximate entropy and to deal with border pixel artifacts

Long-range ambition: “Laws” that, given any mixing process, let us predict whether or not coarse-grained entropy or other types of complex organization will form on the way to equilibrium

Theorem: In a “gas” of non-interacting particles, no nontrivial complextropy ever forms

Numerically-supported conjecture: In a “liquid” of mutually-repelling particles, some nontrivial complextropy does form

Effects of gravity / viscosity / other more realistic physics?

So far…