A board is written as a sum

Four measurements of the brace notation have been made, and every one of them was of a single position. Where the braces stop wrote out every game born by day three and found the middle one twenty-two characters long. Two names that add to nothing nameable added pairs of the named values and found a sixth of them escaping the vocabulary. What two numbers cannot tell apart set the pair of a mean and a temperature against the brace form, again one game at a time.

That table is the notation’s cost for one position, and it is a cost that grows fast with the day a position is born on. It was the natural thing to measure first, because the brace form is defined position by position and its length is a property of each one.

And a reader of this subject almost never writes one game at a time. A Domineering board in its middle game is four or five regions. A Go endgame is a list of local fights. The object being written is a sum, and a sum has two ways to be written that a single position does not: as its parts, joined by plus signs, or as the one value the parts add up to.

The obvious expectation is that the second is much longer than the first — a value built from several positions ought to carry all of them, and then some. On most boards it is the other way round.

Every sum of day-two games, written both ways #

There are twenty-two games born by day two, and a board of k parts drawn from them is a multiset of k of the twenty-two. For every such board with two, three or four parts, the sum is computed, reduced to its canonical form, and written out; beside it the parts are written with plus signs between them.

At four parts the middle board is eleven characters as one value and twenty-three as a sum. At three parts it is nine against sixteen; at two, five against ten. The value is roughly half the length of its sum in the middle of every row, and that is not what compounding looks like.

The top of each row goes the other way. The longest single value at two parts is 45 characters, at three parts 95, at four 133 — against sums of those same boards between a quarter and two fifths that length. And the share of boards whose value cannot be written without opening a brace climbs from 43 per cent at two parts to 57 at three and 68 at four.

So the notation does two opposite things to a board at once. It shrinks the typical one, and it swells the dearest, and the more parts a board has the more often it lands among the dear ones.

Where the short values come from #

The shrinking has a mechanism, and it is the arithmetic the notation was built to support.

Numbers add to numbers. A board of 1, −1/2 and 1/2 is the number 1, and a board of any numbers at all is written as a single fraction however many parts it has. Stars cancel in pairs: ∗ + ∗ is nought, so a board with two independent star-valued regions is written as nothing. Infinitesimals of opposite sign cancel. Every one of those identities is what the notation was the argument means by saying the brace form is a claim that positions add — and every time one applies, a board written as a sum of parts collapses to something shorter than any of its parts.

The table shows those collapses thinning as parts are added. At two parts about one board in six is a plain number; at four parts it is one in fifteen, because a board of four random day-two games usually has something hot in it, and one hot part is enough to stop the whole board being a number.

Many boards, few values #

The collapse has a second measure, and it is the one that says what kind of object a board’s value is.

At two parts, 253 boards have 184 distinct values. At three parts, 2,024 boards have 866. At four parts, 12,650 boards have 2,783 — so on average a single value stands for between four and five different boards, and the ratio grows with every part added. The value of a board is a many-to-one name for it, and the more parts a board has, the more other boards share its name.

That is exactly the property that makes the value worth having and the sum worth keeping. Two boards with the same value are interchangeable in any company whatever — that is what equal values mean — so the value throws away nothing a player could ever need. What two numbers cannot tell apart draws the distinction carefully: canonical form merges positions that are the same game, and a lossy summary merges positions that only look alike. The merge here is the first kind, and it is large. But it also means the value cannot be used to recover the board, and a record of a game that kept only values would not say which regions were where.

The same climb is visible from the side of the names. Sums of two named values escape the vocabulary one time in seven at day three, and the day-two sums here escape it more often the more parts they have: the share of boards whose value can be written without a brace falls from 57 per cent at two parts to 43 at three and 32 at four. A vocabulary that is not closed under addition loses a little at each addition, and a board is several additions.

The counts also say something uncomfortable about the sum form, and it is the price of the plus sign’s brevity. A sum written as parts does not show when two boards are equal. Twelve thousand six hundred and fifty boards of four parts have 2,783 values between them, so at least 9,867 of those boards share their value with some other board written with different parts — and nothing in the two sums says so. A reader handed 1 + ∗ + ∗ and 1/2 + 1/2 has two boards that are exactly the same game and two expressions that share no symbol. The value is the only form in which equality is visible, which is why equality is where the notation is forced to expand.

Where the long values come from #

The dearest boards are the ones with several fights in them. A switch is short to write — 0 | −1 is six characters — and a sum of switches is not, because the value has to say what happens when either player moves in either fight first. The board in the figure is two switches and a small hot game; its parts are twenty-six characters and its value is ninety-five, with every option of the value itself a game with options.

This is the part of the notation’s cost the single-position measurements could not see. A position born by day two is at most a few characters, whatever it is. A sum of positions born by day two is not born by day two; its birthday can be as late as the sum of the parts’ birthdays, and the notation’s length grows with the birthday. The parts carry the birthday in small pieces and the value carries it all at once.

It is also why temperatures do not add is the natural companion to this measurement. A sum of hot parts is hot in a way no part is, and the brace form records that interaction in full — which is exactly what makes it long.

The same measurement on a real board #

Day-two games are an abstract sample, and it is fair to ask whether a real board breaks into parts that behave like them.

Domineering is the game a board that is a sum of its regions is about, and its regions of up to six squares come in 104 shapes. Their values are short — the middle one is three characters — and a board of several regions is exactly the kind of sum this page is about.

The pattern repeats. At four regions the middle board is eleven characters as one value and twenty-four as a sum; the share needing a brace climbs from a third at two regions to more than half at three and four; and the longest single values run past 130 characters. Domineering regions are cooler than random day-two games — many of them are numbers, or numbers with a star — so fewer boards need braces at two regions. But the shape of the result does not depend on which games were chosen.

What the plus sign is for #

The notation’s plus sign turns out to be doing a job nothing else in it does, and it is worth saying what the job is.

A sum written with plus signs is exact, composable, and never longer than its parts. It names the board precisely — the value of a sum is a function of the parts, so the parts determine it — and it costs exactly the lengths of the parts plus a few characters. What it does not do is say what the board is worth: whether 0 | −1 + {∗ | −1} is positive, whether it is hot, who wins it.

A single value is exact and says what the board is worth, and its length is unbounded in the parts. It is the form that can be compared with nought, read for its temperature, or checked against another board.

So the notation offers a choice, and the choice is between two costs. Writing a board as a sum defers the arithmetic; writing it as a value does the arithmetic and pays for it in length. Comparison is a search is the reason the deferral is not free: to decide anything about a board written as a sum, the parts have to be added eventually, or at least their difference with something else has to be searched. But for recording a board — for writing it down so that it can be reasoned about later, or passed to someone else — the sum is the form that stays short.

Players have always known this, and the notation they use for a real endgame is the evidence. The orthodox account of an endgame writes a board as a list of its regions’ means and temperatures, adds the means and works through the temperatures one at a time; it never writes the board’s brace form at all. That is a sum written in a lossy notation for each part, and it survives in practice for exactly the reason measured here — the parts are short, and the one value that would replace them is not reliably short.

That is the resolution of the worry the single-position measurements raised. A day-three expression is twenty-two characters in the middle and a board has several parts, and it looked as though a real board would stop being writable at a modest size. It does not, because nobody has to write the board’s value. The plus sign keeps the board as short as its parts, and the value is computed only when a question needs it.

Why most boards shrink, stated as a count #

The middle-of-the-row result deserves a sentence about what it depends on, because it would be easy to over-read.

The sums here are of random parts, and random parts cancel a great deal. Out of twenty-two day-two games, several are numbers and several are infinitesimals, and a random multiset of four has a good chance of containing pairs that cancel. The 12,650 boards of four parts have only 2,783 distinct values between them — on average one value stands for between four and five boards — and a value that stands for many boards is usually a short one.

A board assembled by play is not random. Regions that are left on a board late in a game are the regions nobody wanted to move in, which skews them toward numbers and cold positions and makes them cancel more, not less. Regions that are hot get played. So the Domineering sample, drawn uniformly from shapes, is likely to overstate the brace share for a late middle game and understate the collapse. What it cannot do is reverse the direction, and the direction is what this page reports.

What the measurement cannot say #

Two samples, both small. Day-two games are twenty-two values; Domineering regions of up to six squares are 104 shapes, sampled four hundred boards a row from one seed. A board from a real game has parts born later than day two and regions larger than six squares, and every length here would grow with them.

Length is counted in characters of the printed form. The printer writes named values as their names — ∗, ↑, a fraction — wherever it can, and opens a brace only where no name applies; a different printer, or a different vocabulary, moves every number in the tables. Every value measured was checked to print in full rather than with an ellipsis for depth.

Nor are the boards weighted by how often they occur. Every multiset of day-two games counts once, so a board of four identical switches — which a real game almost never produces — weighs as much as a board of four different regions. A sample weighted by play would move the middle of every row, and the direction is not obvious: play leaves cold regions behind, which shortens values, and it also produces many copies of the commonest shapes, which is exactly where stars and switches pile up and stop cancelling.

And nothing here measures the cost of computing a value, only of writing it. A board of four parts whose value is eleven characters may have taken a great deal of comparison to reduce; the catalogue a strong player needs is where the cost of computing region values is priced.

The convention the lengths depend on #

Normal play, and canonical forms. The collapses that make most boards short — numbers adding, stars cancelling, dominated and reversible options disappearing — are identities of normal-play values, and they hold because a game plus its negative is nought. Under misère play, where that equation fails on every game, nothing cancels, a sum has no shorter form than its parts in general, and the one-value column of every table here would not exist.

The surprise: the value is shorter than the sum #

The expectation going in was compounding — a notation already twenty-two characters per position, applied to a board of several positions, producing expressions no reader could hold. The measurement says compounding happens only at the top of the distribution. In the middle, the one value is about half the length of the sum it replaces, because a sum of random positions is mostly a sum of things that cancel.

The practical reading is the one a player already follows without knowing it. Write the board as its parts. Add the parts only when a question requires it, and when the answer turns out to be a number, write the number. The brace form is the notation for what a position is; the plus sign is the notation for a board, and it was never the expensive half.

Still open: a board with a cycle in it #

Every sum here is of finite games, and every one of them has a value. Where the braces stop found the first place the brace form does not reach at all: a game with a cycle, which has no finite expression. A board with one loopy region and several ordinary ones is a sum of that kind, and it is the board a Go player with a ko in one corner is writing. Whether such a board can be written as a sum of its parts with the loopy part named — and what adding it to the rest would even mean — is the next question this notation poses.