The Impossible Line

Jvelthra was an agricultural inspector responsible for grain stores across twelve villages. Mould and fungal contamination were destroying stockpiles, and she needed to predict which stores were at risk. She had records of two hundred and thirty grain stores, with two key factors: humidity level and temperature.

The Easy Problem

Trviksha started with Jvelthra's first dataset: mould contamination. She plotted the data on a stone grid — humidity along one axis, temperature along the other. Each grain store was a pebble: black for mouldy, white for clean.

The pattern was straightforward. High humidity and high temperature together produced mould. Low humidity or low temperature (or both) did not. The black pebbles clustered in one corner. The white pebbles filled the rest.

A single perceptron — Drysska, with two inputs and a threshold — drew a line across the grid. On one side: predict mould. On the other: predict clean. After training, the perceptron achieved 91% accuracy. The line separated the clusters neatly.

Trviksha: One line, two factors, clean separation. This is the kind of problem Drysska was built for.

Then Jvelthra brought a second dataset.

The Strange Pattern

This contamination was different — a fungus that behaved oddly. It attacked grain stores that were either hot-and-dry or cold-and-wet. Stores that were hot-and-wet were fine (the heat killed the fungus when moisture was present). Stores that were cold-and-dry were also fine (the fungus needed some moisture to activate in cold conditions).

Trviksha plotted the data. Black pebbles clustered in two opposite corners: upper-left (cold, wet) and lower-right (hot, dry). White pebbles clustered in the other two corners: upper-right (hot, wet) and lower-left (cold, dry).

She tried to place a line separating black from white. She could not. Any line that captured the upper-left cluster also swept in either the upper-right or lower-left white cluster. The contaminated stores were not on one side of any line. They were on opposite corners.

A stone grid with humidity on one axis and temperature on the other. Black pebbles (contaminated stores) are in the upper-left and lower-right corners. White pebbles (clean stores) are in the upper-right and lower-left corners. A frustrated Trviksha tries to position a straight stick that separates black from white, but no position works

The Systematic Failure

Trviksha ran the weight adjustment anyway. Perhaps she was wrong about what was possible, and the numbers would find a solution her eyes had missed.

The perceptron settled on weights that produced 51% accuracy — essentially random. No matter how many passes she ran, no matter how she started the weights, the perceptron could not do better than a coin flip on this data.

Trviksha: I tried every starting position. I ran the adjustment hundreds of times. It cannot solve this.

Drysska: That is because you are asking me to do something that cannot be done with the tools I have. I compute a sum and compare it to a threshold. The answer is not a sum-and-threshold kind of answer.

Trviksha: Since when do you have opinions about what kind of answer things are?

Drysska: I do not have opinions. I have operational constraints.

The failure was not a matter of insufficient training or bad starting weights. It was structural. A perceptron partitioned the input space with a single straight line. If the true boundary between the classes was not a straight line — if the classes were tangled in a way that no single cut could untangle — then no set of weights would work. Not because the right weights had not been found, but because the right weights did not exist.

The Name

Blortz, who had been watching the stone grid, offered a characterisation.

Blortz: The mould problem is linearly separable. You can separate the two groups with a straight line. The fungus problem is not linearly separable. The groups are tangled in a way that no straight line can untangle.

Trviksha: Can anything untangle them?

Blortz: Two lines could. One line to isolate the cold-wet corner. Another to isolate the hot-dry corner. A store is at risk if it is on the flagged side of either line.

Trviksha: But Drysska draws one line. That is all she can do. One weighted sum, one threshold, one boundary.

Jvelthra: So your system cannot solve my problem?

Trviksha: This system cannot. I need a different system. One that can draw more than one line.

Jvelthra: How different?

Trviksha: More velociraptors.

Trviksha has discovered one of the most famous limitations in the history of artificial intelligence. A single perceptron can only solve linearly separable problems — problems where a straight line (or flat surface in higher dimensions) divides the two classes. Many real-world problems are not linearly separable. The classic example is XOR (exclusive or): a function that is true when exactly one of two inputs is true, but false when both are true or both are false. If you plot XOR on a grid, the "true" cases land on opposite corners — just like Jvelthra's fungus. No single straight line can separate them. This limitation was identified by Marvin Minsky and Seymour Papert in 1969, and it temporarily discouraged neural network research for over a decade. The solution, as Blortz suggests, requires more than one line — which means more than one computing unit. Try drawing a grid with two axes, each running from 0 to 1. Mark the corners (0,1) and (1,0) as "true" and the corners (0,0) and (1,1) as "false." Can you separate the marked corners from the unmarked ones with a single straight line?