Joint work with Lauren Ouellette and Sean Carroll

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Joint work with Lauren Ouellette and Sean Carroll
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…

Our modest goal: Understand the rise and fall of complexity quantitatively in some simple model system Proposed system: the “coffee automaton” nn 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

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

Warmup: How to Quantify Entropy
Problem: Approximating H is SZK-complete! K(x) = Kolmogorov complexity of x = length of the shortest program that outputs x Old, well-understood connection between K and H: 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

Approach 1 to Quantifying Complexity: Coarse-Grained Entropy
Let f(x) be a function that outputs only the “important, macroscopic” information in a state x, washing or averaging out the “random fluctuations” Then look at H(f(x))  H(x). Intuitively, H(f(x)) should be maximized when there’s “interesting structure” 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?

Approach 2: “Causal Complexity” (Shalizi et al. 2004)
Given a point (x,t) in a cellular automaton’s spacetime history, let P and F be its past and future lightcones respectively: F (x,t) Time t P Then consider the expected mutual information between the configurations in P and F:

Intuition: If dynamics are “simple” then I(P,F)0 since H(P)H(F)0 If 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 Advantages of causal complexity: Has an operational meaning Depends only on causal structure, not on arbitrary choices of how to coarse-grain Disadvantages: Not a function of the current state only Requires going arbitrarily far into past and future I(P,F) can be large simply because not much is changing

Approach 3: Logical Depth (Bennett 1988)
Depth(x) = Running time of the shortest program that outputs x Depth(0n) = Depth(random string) = n But 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 deep Computability properties are terrible

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 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 xS 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” 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 with Sophc(x)n-O(log n)

Special Proof Slide for Hebrew U!
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 zA, contradiction.

Problem: Sophc(x) is tiny for typical states x of the coffee automaton
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 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) 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 Sophcefficient(x) didn’t work well in experiments

Our “Complextropy” Measure
Let I = coffee-cup bitmap (n2 bits) Let C(I) = coarse-graining of I. Each pixel gets colored by the mean of the surrounding LL block (with, say, L~n), rounded to one of (say) 10 creaminess levels 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

Compressed coarse-grained image Remaining info in image
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 can be seen as an extremely resource-bounded type of sophistication! 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

The Border Pixels Problem
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 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 nn coffee cup Maximum coarse-grained entropy G(C(I)) increases like ~n

Analytic Understanding?
We can give a surprisingly clean proof that K(C(I)) never becomes large in the non-interacting case 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 LL square B after t steps Clearly E[at(B)]L2 for all t,B by linearity

By a Chernoff bound, So by a union bound, provided 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)

Open Problems 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? 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 So far… 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?

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