Two names that add to nothing nameable

The rung below measures the reach of the named vocabulary and reports a collapse: every game born by day two can be written without opening a brace, and sixty-three of the fourteen hundred and seventy-four born by day three can — four point three per cent.

It then defends the number, and the defence is right:

There is a temptation to read 4.3 per cent as a failure of the vocabulary and it is the opposite. A vocabulary that named all 1,474 would have 1,474 words in it and would be a list, not a language.

That is a defence against the wrong charge. Coverage is not the property a vocabulary needs, and a vocabulary can have terrible coverage and be perfectly good. What it cannot have is a failure at the operation it was invented for.

Coverage against closure #

The rung two below makes the case that {L | R} is not shorthand. Up, star and the brace form are the claim that these objects add, and no table of outcomes can replace them, because two positions with identical outcomes can have different sums.

So the notation exists in order that positions can be added. Which means the question to ask of the named half of it is not how many games does it name but:

Add two things it can say. Can it say the answer?

That is closure, it is a different measurement from coverage, and it is the one that decides whether a reader can work in the vocabulary or merely quote from it.

The counts #

Day one has four values, every one of them named, and ten sums of pairs. All ten are named. The vocabulary is closed and the closure is trivial: four values with names, and their sums are among the small integers and nimbers that also have names.

Day two has twenty-two values, fifteen of them named, and a hundred and twenty sums. A hundred and sixteen are named. Four escape. That is 96.7 per cent, and the four are the first sign of anything.

Day three has fourteen hundred and seventy-four values, forty-two of them named, and nine hundred and three sums. Seven hundred and seventy-one are named. A hundred and thirty-two escape — 85.4 per cent, or roughly one sum in seven.

One in seven is not a rounding. It means a reader working in the named vocabulary, adding two things they can pronounce, has to reach for a brace expression about as often as a card in a hand is a heart.

The first one to go #

The escape that matters is the first, and it is not an exotic pair.

At day two, the earliest sum to leave the vocabulary is

↑ + ∗2 = {0 | ∗3}

Up and star-two are two of the six symbols anybody learns first. Both are born on day two, both are named, both turn up constantly in real games. Their sum is a game born on day three and it has no name — it is written as a brace expression and always will be.

Nothing about that pair is contrived. Up is the smallest positive thing there is, star-two is the Nim heap of two, and a position containing both is what a Domineering board with a two-heap Nim game beside it looks like. The vocabulary can name each half of that board and cannot name the board.

What “named” means, exactly #

The counts depend on a decision about what counts as a name, and the decision is worth stating because a looser one would inflate every number.

An expression is named here when it can be written without opening a brace. That admits three kinds of thing:

A glyph of its own — 0, ∗, ↑, ⇓, and the numbers, which are written in the ordinary way.

An expression in named parts — ∗3, 1/2, 2·↑∗, −1∗: several named things composed by the site’s writing conventions, which are fixed and short.

And nothing else. Anything containing a brace or a bar is the brace form and counts as an escape, however short it happens to be.

That is a strict reading and it is the right one, because the distinction the measurement is about is exactly whether a reader has to stop reasoning in symbols and start reading a game tree. {0 | ∗3} is eight characters and is a tree; 2·↑∗ is four characters and is a value with a name.

A looser reading — counting anything under some length as named — would report a higher pass rate and would be measuring typography rather than vocabulary.

What the escapes look like #

Sorting the nine hundred and three sums by what kind of expression the answer needs is more informative than the pass rate.

Two hundred and nineteen land on a glyph of their own — a number, a star, an up, one of the six symbols with a character to themselves.

Five hundred and fifty-two land on an expression in named parts — things like 1∗, 2·↑∗, −1/2∗: several named things written together, which is still inside the vocabulary because the parts are named and the writing convention is fixed.

A hundred and thirty-two need braces.

That middle category is what keeps the number as high as 85 per cent, and it is worth noticing that it is doing most of the work. The vocabulary’s usable range is not the six glyphs; it is the six glyphs plus the convention for writing a number next to an infinitesimal, and without that convention the closure rate would collapse.

So the honest description of the named vocabulary is: a handful of symbols and one rule for combining them, and it is the rule rather than the symbols that gives it reach.

The escape rate rises with the day, and the reason is not size #

The three rates — a hundred per cent, ninety-seven, eighty-five — look like a curve heading somewhere, and it is worth asking what they would do at day four if anybody could compute it.

The mechanism is visible in what changes between the days. At day one the four values are 0, 1, −1 and ∗, and their sums are among the small integers and nimbers, which is a set the vocabulary covers completely. At day two the named values include the first infinitesimals, and adding an infinitesimal to a nimber produces something the conventions were not written for — which is exactly the first escape.

At day three the named set has forty-two members and they are drawn from further apart: numbers of several denominators, nimbers up to a few, ups in several multiples, and combinations of those. The further apart the summands, the less likely the sum lands on anything the conventions can write.

So the escape rate is driven by the diversity of the named set rather than by the size of the universe. Day four’s universe is unimaginably larger and its named set is not — the same six glyphs, the same conventions, a few more combinations — so a reader should expect the rate to fall further, and to fall because the named values are being drawn from a wider spread rather than because there are more games to miss.

That is a prediction and it is offered as one. Nothing on this page computes day four and nothing will.

Walking out of the vocabulary #

Pairwise closure is the sharp measurement and a chain is the readable one: start from a named value, add another one over and over, and watch the names run out.

↑ + ∗2 leaves at the first step, as above, and then oscillates — the next addition of star-two brings it back to up, and the sequence alternates between a name and a brace expression for ever.

↑ + ↑ + ↑ + … never leaves. The multiples of up all have names: ↑, ⇑, 3·↑, 4·↑, 5·↑, and so on for as far as anybody writes.

1/2 + ↑ + ↑ + … survives twice and leaves at the third: a half, then 1/2↑, then 1/2⇑, and then {1/2 | {1/2 | 1/2↑}}.

∗ + ↑ + ↑ + … never leaves either: star, up-star, two-up-star, and onward.

Those are four sequences of the same kind and two of them stay inside the vocabulary for ever while two do not. What separates them is not depth and not size; it is whether the family being generated is one the naming conventions were built for. Multiples of up were, so they have names to any multiple. A number with a growing pile of ups on it was not.

Why the vocabulary is not closed, and could not be #

There is a reason this is not a defect anybody could have avoided, and it is worth being clear about it before the finding is read as a complaint.

The named values are the ones that turn up often, and turning up often is a fact about the games people play rather than about the arithmetic. Numbers, stars, ups and their small combinations are what real positions are worth, because real positions are sums of simple things.

Closure is a fact about the arithmetic, and the arithmetic does not care what turns up often. Add two frequently-occurring values and the answer is whatever it is, and there is no reason for it to be frequently occurring — the sum of two common things is an uncommon thing about one time in seven, and no choice of which things to name changes that. Which values a real game actually produces is the empirical half of the same question, and the answer there is that games produce a narrow band and the arithmetic does not respect it.

A vocabulary closed under addition would have to be closed under the operation that generates the whole universe, and the whole universe at day three is 1,474 games. So closure and brevity are in direct opposition, and the vocabulary chose brevity, which was the right choice.

What the measurement adds is the price of the choice, and the price is a number rather than a warning. The birthday of a sum is where the same arithmetic is measured on the other axis — how far into the universe an addition takes a reader — and the two findings are the same one seen from two sides.

Four of one and none of the other #

The day-two escapes are few enough to list, and listing them says something the rate does not.

All four involve an infinitesimal added to a nimber. Up with star-two, down with star-two, and the two variants with a star already attached. Every other pair of day-two named values — number with number, number with nimber, nimber with nimber, infinitesimal with infinitesimal — lands somewhere the vocabulary can write.

So the vocabulary’s first failure is at a single junction, and the junction is between two of its own families. Numbers compose with numbers because arithmetic; nimbers compose with nimbers because exclusive-or; ups compose with ups because the multiples are named. It is the cross terms that escape, and they escape at the earliest opportunity.

That says where a reader should expect trouble without computing anything. A sum whose parts come from one family of the vocabulary is very likely to stay inside it. A sum crossing two families is where the conventions have nothing to say, and crossing families is what happens whenever two different kinds of game are added — which is most of what a disjunctive sum is for.

What it costs a reader #

The practical consequence is worth stating, because it is what the number means in use.

A reader following an argument in the vocabulary — this position is ↑, that one is ∗2, together they are — hits a brace expression one time in seven and has to change register. Not a hard change, and not an error, but a break: the reasoning up to that point was symbolic and the next step is a game tree.

That is why when the ups add is a question with an answer worth having, and why atomic weight exists at all. Both are machinery for staying inside a vocabulary that addition keeps pushing a reader out of — a way of computing about infinitesimals without writing them down. Reduced canonical form is the same instinct one field over: throw away the part of a value the vocabulary cannot hold and keep the part it can.

And it is why the brace form is not a fallback that could be dispensed with. The rung below establishes that it writes every well-founded game exactly, and the closure measurement says how often a reader working in symbols actually needs it: not rarely, and not at the edges.

What the picture cannot show #

Only pairs are added. A sum of three named values escapes more often than a sum of two, and the sweep does not measure it — the pairwise number is a floor on how leaky the vocabulary is under repeated addition, not a description of it.

And only days nought to three exist to measure over. Day four holds something past 10³⁸ games and nobody has enumerated it. Every count here is about the last day anybody can look at, which is the same limit the rung below runs into and for the same reason.

Nor does the sweep weight by frequency. Every pair of named values counts once, and in practice some pairs turn up constantly and others never. A frequency-weighted closure rate would be a better number and would need a corpus of real positions, which is a different measurement and a harder one.

The convention, named #

Normal play, and everything in the vocabulary is defined under it.

That is worth saying here rather than assumed because the vocabulary is where the convention is least visible. ↑ does not look like a statement about who moves last; it looks like a symbol, and a reader who has learned that ↑ is positive and smaller than every number has learned an ordering fact with no convention attached to it.

The convention is in the definition underneath — ↑ is {0 | ∗}, and 0 is the game with no options, which is a loss for the mover because the mover cannot move. Every name in the vocabulary bottoms out there, and under misère play every one of the equations they support is false.

So the closure measured here is closure under normal-play addition, and the same symbols under the other convention do not add at all.

The surprise: the vocabulary is a language and not a dictionary #

The rung below defends the four point three per cent by observing that a vocabulary naming all 1,474 games would be a list rather than a language. That defence is correct and it understates the case, because the closure measurement shows the vocabulary is doing the thing a language does.

Seven hundred and seventy-one of nine hundred and three sums land back inside the vocabulary. That is not what a list does. A list of names covering four per cent of a universe, with sums landing anywhere, would return a name four per cent of the time — and it returns one eighty-five per cent of the time.

The vocabulary is a hundred and forty times denser under addition than under sampling, and the reason is that it names the values that are closed under the operations people perform, which is a completely different criterion from naming the values that exist.

That inverts what a reader takes from the coverage figure. The four per cent looks like a vocabulary that has given up on the universe. What it actually is is a vocabulary that has abandoned the universe and captured the arithmetic, and the coverage number measures the abandonment while the closure number measures the capture.

The general shape is worth carrying. A notation should be measured against the operations it supports rather than the objects it names, and the two measurements can differ by two orders of magnitude on the same notation. A vocabulary naming half of everything and closed under nothing would be far worse than this one, and the coverage figure would say it was better.

Where the ladder goes next #

notation has three rungs: what the brace form buys, what it costs and where it stops, and how far the named half of it survives the operation it exists for.

The rung above is a rival. A thermograph summarises a position in two numbers — where the fight settles and how much is at stake — and written out that is four or five characters against the twenty-two a day-three brace expression takes at the median. It is readable in a way the expression is not, it is what every practical account of an endgame is written in, and it is not exact. How much it is not exact by is a count of collisions, and the count is large.