Happy families

Tolstoy opens Anna Karenina with a line so famous it has worn smooth: happy families are all alike; every unhappy family is unhappy in its own way. It sounds like a remark about marriages. For us it turned out to be the deepest fact we know about machine reasoning.

Our machine reads math word problems written in ordinary English and turns them into a diagram — a small graph of quantities and the arithmetic that connects them, the kind of thing you’d sketch on scratch paper before actually solving anything. When it reads a problem correctly, the diagram it produces is always the same object, no matter how the sentence was worded. There is exactly one correct diagram for “Maria has three times as many apples as Ben,” and there is exactly one correct diagram for “the reservoir holds triple what the tank does,” and — because both sentences describe the identical underlying arithmetic — those two diagrams are the same diagram. Every correct reading is alike.

But when the machine misreads a problem, the ways it can go wrong are not so cooperative. One misreading swaps which quantity is three times which. Another invents a quantity that was never mentioned. Another wires a number to the wrong role in the arithmetic — treats a rate as a total, say. Each of these failures is broken in its own particular way, and there is no small number of ways to be broken. Wrongness has infinite variety. Rightness has exactly one shape.

This is not a cute observation to open an essay with. It is the engine of the entire safety design.

Here is the move it enables. Suppose you don’t trust any single reading of a problem — you shouldn’t, since any one reading might be one of the infinite ways to be wrong. So instead of reading it once, read it several times, independently, in ways that shouldn’t matter to the answer: reorder the sentences, shuffle which details come first, change nothing about the arithmetic itself. If the readings are genuinely independent — if they don’t share a blind spot — then their wrong answers will scatter. A misreading caused by getting confused about sentence order in one pass has no reason to produce the same wrong diagram as a misreading caused by mixing up two quantities in another pass. Each broken reading is unhappy in its own way, so different broken readings land in different places.

But the right answers don’t scatter. They can’t. There is only one happy family — one correct diagram — for a given problem, so every reading that happens to get it right lands in exactly the same spot. Agreement across independent readings is therefore not just a vote of confidence. It is close to a proof. If five readings of the same problem, shuffled five different ways, all converge on the identical diagram, the chance that five different mistakes all happened to collide by accident is small — much smaller than the chance that they converged because they were all correct.

This is why our machine doesn’t answer a problem, on the strength of a single pass, and hope. It re-reads. It re-reads with the sentences permuted, and only certifies an answer when independent readings land in the same place — a whole wall of witnesses built out of this one asymmetry, layered deeper still with readers trained separately, from different lineages, so that even a shared blind spot in one lineage doesn’t get mistaken for agreement. When the readings scatter, the machine has learned something too: not the answer, but the fact that this problem is not yet safe to answer. It stays silent, and the silence itself is information, earned the same way the confidence would have been — by counting who agrees with whom.

None of this works if wrongness and rightness are symmetric — if mistakes cluster the way correct answers do. We built the entire certification wall on a bet that they don’t, that Tolstoy had mathematics right by accident. So far, the bet holds: it is far easier to be wrong in your own particular way than to be right in someone else’s.

Happy families are all alike. That is not a fact about families. It turns out to be a fact about truth.