What two numbers cannot tell apart

Two rungs below, the brace form is defended against the notations it replaced. Grundy’s numbers were dropped on coverage — they have nowhere to put the fact that the two players have different moves. The outcome table was dropped on arithmetic — it covers everything and cannot be added.

There is a third rival and it is the one people actually use. Every practical account of an endgame is written in it, it is shorter than the brace form by a factor of two, and it fails on a third axis the other two do not: it is not exact.

The rival #

A thermograph summarises a position in two numbers: the mean, where the fight settles once the tax on moving is high enough, and the temperature, how much is at stake. Written out that is m ± t — a fraction, a sign and a fraction.

At day three the median brace expression is twenty-two characters and the median pair is nine. Less than half the length, and the half that remains is readable: a person can compare two positions by their means at a glance, and cannot compare two brace expressions by anything.

That is not a small advantage. Playing the hottest is a rule stated entirely in temperatures; the orthodox account of an endgame adds means and subtracts temperatures in order; and neither is expressible in brace expressions without first computing the pair. The pair is what the practical theory is written in.

The collisions #

Compute the mean and the temperature of every game born by day three and count how many distinct pairs come back.

Fourteen hundred and seventy-four games. Forty-three pairs.

That is the whole finding, and it is worth sitting with before any of the details. The brace form produces 1,474 distinct expressions for 1,474 games — the rung below checks it rather than assuming it, because an expression naming two games would make every equation on this site ambiguous. The pair produces forty-three.

Twenty-three of the forty-three pairs are shared by more than one game, and between them those twenty-three account for 1,454 of the 1,474 — so all but twenty of the games born by day three share their pair with something.

The largest collision is 0 ± 0, which 291 games are written as. Two hundred and ninety-one distinct objects, one label.

What the collisions cost #

A notation that cannot tell two things apart is a problem only if the two things behave differently, so the question is whether the collisions are collisions of things that matter.

They are, and the demonstration is the same test the outcome table fails one rung down: take two games sharing a pair, add an ordinary small position to each, and see whether the winners differ.

{∗2 | 0} and {↑∗ | {∗ | −1}} are both written 0 ± 0. Add a star to each. The first becomes a position Right wins whoever moves; the second becomes a position whoever moves wins. Different outcome classes, from two games the notation writes identically, separated by adding the smallest object in the subject.

That is not a contrived probe. A star is what a Nim heap of one is worth, and adding one to a position is what happens whenever a game with a spare move in it is sitting beside something else. The pair notation cannot see the difference and a board can.

What the two numbers throw away #

Naming what is lost is more useful than counting it, and the loss has a shape.

A thermograph is built from the whole game tree — each wall is an extremum over the options’ opposite walls, shifted by the tax — so the construction sees everything. What it hands back is where the two walls meet and where the mast stands, and everything else about the diagram is discarded.

Two things live in the discarded part.

The infinitesimal residue. Cool a hot position by exactly its temperature and what is left is its mean plus something smaller than every positive number — a star very often, an up sometimes, and never nothing at all. That residue is invisible in the pair and it is what decides close games.

The shape of the walls. A wall can bend, and where it bends is the temperature of the option holding it up. Two positions with the same mean and temperature can have completely different bends, which is to say completely different follow-ups, which is to say they behave differently as soon as the ambient temperature moves.

So the collisions are not arbitrary merges. They are exactly the pairs that agree about where the fight settles and how big it is, and disagree about what happens on the way — and the way is where a game is played.

Which is the outcome table’s failure, one level up #

The structure of that argument is exactly the rung two below’s argument against the outcome table, and the repetition is worth naming because it is the same defect at a different resolution.

An outcome table assigns one of four labels per position, and two positions with identical outcomes can sit inside sums with different outcomes. Four labels for everything, and the labels do not add.

A thermograph pair assigns two rational numbers per position, which is a great deal more information than four labels, and two positions with identical pairs can sit inside sums with different outcomes. Forty-three labels for everything, and the labels do not add.

The pair is coarser than the brace form and vastly finer than the outcome table, and it fails the same test. The failure is not about how much information the label carries; it is about whether the label is a complete description of the game. Anything short of the whole tree can be defeated by a well-chosen companion, and the reason is the same one the rung two below gives: a value is defined by how a position behaves in sums, and a summary is not a value.

Why it is used anyway, and why that is correct #

None of this is an argument against thermographs, and it would be a poor reading to take it as one.

The pair is not a rival notation that lost. It is an abbreviation with a stated domain, and inside its domain it is exactly right. The domain is: positions being compared for how urgent they are, in a game being played move by move, where what a player needs is a decision about where to play next.

For that question the collisions do not matter. A player choosing between two fights wants to know which is hotter and by how much; two positions with the same pair are equally urgent and equally valuable in the long run, and choosing between them is a decision with no consequence for the question being asked.

The collisions matter when the pair is asked to do the brace form’s job — to say what a position is, to be added to something, to decide an equality. And that is a different question, asked by a different reader, in a different part of the subject.

So the finding is not that the pair is a bad notation. It is that the pair is a notation for a question, and the question is not what is this position but how much does moving here matter.

Forty-three is a small number and it is the wrong small number #

There is a way of reading forty-three as a success, and it is worth confronting because it is the reading a reader sympathetic to thermographs will reach for.

Forty-three distinct pairs over a universe of fourteen hundred and seventy-four is a compression of thirty-four to one, and compression is what a summary is for. Canonical form compresses too — thousands of Kōnane arrangements onto dozens of values — and nobody calls that a loss.

The difference is what the compression preserves. Canonical form merges positions that are the same game: swapping one for the other inside any sum whatsoever changes nothing, which is what a value means and is why the merge costs nothing at all. The pair merges positions that are different games and only look alike from one angle.

So the two compressions are not the same operation with different ratios. One is an identification and the other is a projection, and a projection can always be defeated by looking from somewhere else — which is what adding a star does.

That distinction is worth carrying whenever a number of classes is quoted. This many things collapse onto this many is a statement about a map, and the question is whether the map is injective on the thing being asked about. Canonical form’s is; the pair’s is not; and both quote impressive ratios.

Where the twenty go #

Twenty games born by day three have a pair nothing else shares, and they are worth a moment because they are what an exact pair would look like.

They are the games at the extremes: the largest means, the highest temperatures, the values that are furthest from everything else. A game with a mean of two and a temperature of two is not going to collide with anything, because very few day-three games are anywhere near it.

That is the ordinary shape of a lossy summary. The extremes are distinguished and the middle is not, and the middle is where almost everything is — 291 games at 0 ± 0, which is the pair every cold position worth nothing has, and cold positions worth nothing are what most of a universe consists of.

It also says why the collisions are worst exactly where the notation is most used. An endgame being priced is a collection of positions near the middle of the range, and near the middle is where a pair says least.

The three rivals, scored on three axes #

The rung two below sets out two notations that were dropped and the axis each was dropped on, and the pair supplies the third, so it is worth putting all four side by side.

Grundy’s numbers. Exact on impartial games, add perfectly, and have no form at all for a partizan position. Dropped on coverage.

The outcome table. Covers every game, four labels, and cannot be added — two positions with the same label sit in sums with different labels. Dropped on arithmetic.

Mean and temperature. Covers every game, adds in one respect — means add exactly and temperatures do not — and is not exact. Kept, for the questions where exactness is not what is wanted. Half an arithmetic is what a projection can offer: the mean is a homomorphism and the temperature is not, so a reader adding two pairs gets one of the two numbers right and a bound on the other.

The brace form. Covers every well-founded game, exact, adds completely, and is fifty characters at day three. Kept for everything else.

Four notations, four different things given up, and the one that gave up exactness is the only one still in daily use beside the winner. That is worth noticing: the two that were dropped gave up coverage and arithmetic, which are properties a notation must have, and the one that survived gave up something a notation can do without as long as everybody knows it has.

What the picture cannot show #

The probes are a stated handful. Six small positions are added to each colliding pair to look for a separation, and a pair the probes fail to separate has not been shown to be inseparable — it has been shown that these six do not do it. The count of collisions is exact; the claim that collisions cost something is established by exhibiting one that does.

And day four is not measured. The universe past day three is unenumerated, so the ratio of games to pairs at the next day is unknown. What can be said is the direction: the pairs are rationals with bounded denominators at each day and the games roughly square, so the collisions can only get worse.

Nor is the length comparison a comparison of what people write. Nine characters against twenty-two are the medians of the two notations’ own output. In practice a Go player writes neither — they write a number of points, having already decided which fight they are pricing — and the notational question this page is about does not arise for them at all.

The convention, named #

Normal play, and it is buried deeper here than in any other rung of this ladder.

A thermograph is built by cooling: charging a tax on every move and watching the position freeze into a number. The construction is a recursion over options, the recursion bottoms out at positions with no options, and a position with no options is worth nought because the mover has lost. So the mean and the temperature are as convention-dependent as anything else, and there is no misère thermograph. Cooling and heating is where the construction is set out, and every step of it reads a value that a last-move convention produced.

That is worth saying because the pair is the notation that looks least like a claim about who moves last. A mean and a temperature read like measurements of a physical quantity — how much is at stake, where it settles — and a reader could work with them for a long time without meeting the convention at all. It is at the bottom of the construction, as it always is, and under the other convention the whole apparatus has nothing to attach to.

The surprise: the useful notation is the inexact one #

The rung two below tells a story with a clean shape. Two notations were tried, each lost something a notation must have, and the brace form is the one that keeps everything and pays in length.

The pair breaks the shape, because it loses something a notation must have — exactness — and is used constantly anyway, by everybody, in preference to the exact one.

And it is used because it is inexact. Nine characters instead of twenty-two is not a saving on typing; it is the difference between a label a person can hold in mind while comparing four fights and a label they cannot. The collisions are what buys the brevity, and the brevity is what makes the notation usable for the thing it is used for.

That inverts the usual reading of a lossy summary. A summary is normally defended as good enough — nearly the whole thing, at a fraction of the cost. This one is not nearly the whole thing: it merges 1,454 of 1,474 games into twenty-three groups, and one of those groups has 291 members in it. It is defended instead as the right thing for a different question, and the different question is the one a player asks.

The general shape, and it is the one to carry off this ladder: a notation is a choice of which question to make easy, and the choices are not ranked. The brace form makes what is this easy and which is hotter impossible; the pair does the reverse; and a subject with two questions needs two notations and has to keep saying which one is in force.

Where the ladder goes next #

notation has four rungs: what the brace form buys, where it stops, how far the named vocabulary survives addition, and what a two-number summary cannot say.

The rung above is the one all four have been circling and none has taken. Every notation here is written for a position, and what a reader actually manipulates is a sum — a board with several parts on it, each with its own expression, being added. The brace form’s length was measured for single games and a sum’s expression is built from its parts’, so the cost of writing a real position is a compounding of numbers this ladder has measured one at a time. What that costs, and at what size a real endgame stops being writable at all, is a measurement nobody here has made.