# Answer to A Bongard Problem

As a few people nearly guessed, the left side is “things a linear system can do” and the right side is “(additional) things a nonlinear system can do.”

On the left:

• decaying oscillation
• exponential decay
• simple accumulation
• equilibrium
• exponential growth
• 2nd order goal seeking with damped oscillation

On the right:

Bongard problems test visual pattern recognition, but there’s no reason to be strict about that. Here’s a slightly nontraditional Bongard problem:

The six on the left conform to a pattern or rule, and your task is to discover it. As an aid, the six boxes on the right do not conform to the same pattern. They might conform to a different pattern, or simply reflect the negation of the rule on the left. It’s possible that more than one rule discriminates between the sets, but the one that I have in mind is not strictly visual (that’s a hint).

The original problem was here.

## 4 thoughts on “Answer to A Bongard Problem”

1. Richard Kennaway says:

The top left and bottom right of the right-hand panel can also result from linear processes. The top-left will result from a step change in position filtered through two spring-and-dashpots in series. The bottom-right can result from a linear inverted pendulum controller with the pendulum immersed in water, subjected to a step change in the reference position for the bob. The cart carrying the pendulum pivot must initially move in the opposite direction in order to get the pendulum moving in the right direction. The viscosity of the water makes it unnecessary for the cart to get ahead of the pendulum to bring it to rest.

1. Tom says:

Nice catch! I was afraid there’d be some overlap due to non-simple initial states or inputs that I hadn’t thought of.

2. Quite interesting and I was indeed tempted to think of linear vs. nonlinear systems, but alas looking at pages 108 and 109 of “Business Dynamics” (2000) by John D. Sterman seemed to invalidate that hunch, as S-shaped growth with overshoot and oscillation as given as an example for a nonlinear system’s behavior mode.
(The middle graph in the lower row of Figure 4-1 on page 108 exactly matches the bottom right graph on the left hand side of the Bongard problem here.)

At least considering the comment by @Richard Kennaway I would suggest a correction to Prof. Sterman’s remarks.

I would also like to point out, that the descriptions for the modes on the left hand side falsly state that the top right graph shows exponential decay — it is goal seeking behavior, though (the rate of decay is not growing, but diminishing).

1. Tom says:

Re the last point – I assume you mean that the goal is nonzero? I guess assuming that the box frame is the axis would lead one to that conclusion, though my imperfect drawing probably wasn’t quite like that.

I’d like to do a version 2 that perfects this. Suggestions for other exclusively nonlinear behaviors would be most welcome.

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