Complexity Explorer Santa Few Institute

Introduction to Renormalization

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3.3 Networks of Renormalization » Quiz Solution

1. Which rule is fixed point of the Israeli-Goldenfeld renormalization procedure?

A. 105

B. 150

C. 110

D. all of them


Answer: (B). While we found that rule 105 coarse-grained to rule 150 (given an appropriate projection operator, in this case the "edge detector"), we also noted that rule 150 could (given an appropriate projection operator) coarse-grain into itself.


2. What is a "garden of Eden" supercell state for a cellular automaton rule?

A. a supercell that resets the system to a standard position.

B. a supercell that forces neighbouring supercells to convert to its value.

C. a supercell that could not emerge from the operation of the rule.

D. a supercell that once produced, never goes away.


Answer: (C). A garden of Eden supercell immediately disappears under the evolution operation. You can put one in as an initial condition, but it is never produced naturally through the course of the rule's evolution. Garden of Eden supercells are a sort of exception case that make the coarse-graining and evolution commute, but not in an interesting way.


Rule 110, for example, has garden of Eden states. So we can say that 110 coarse grains into the trivial evolution rule 0 (everything goes to white), if the projection operator maps garden of Eden states to black, and everything else to white. For Rule 110, Israeli and Goldenfeld could only find projections that relied on this garden of Eden state trick. This is sad because if they had found non-trivial ones that did something else, we would be able to talk about renormalization group flows on Turing-complete systems.


3. What's the problem with picking the Alice coarse-graining rule, and looking for an evolution operator?

A. it's no longer possible to make the diagram commute in general, no matter how many time-steps you coarse-grain

B. the evolution becomes non-local

C. an event at some time-step t is no longer conditioned entirely by the state of the system at time t-1

D. all of the above.


Answer: (D). By picking a coarse-graining ahead of time, we've set ourselves an impossible problem. Not only can we not find a cellular automaton that works, we can't find any kind of Markovian process. For any constant C, we can always make a point at time t dependent on event that happen before time t-C.