Add, then reduce again

The reduced canonical form throws away every infinitesimal: it is what is left when the small change is thrown away, the coarser cousin of the canonical form, and the object a player uses when the board is hot enough that infinitesimals cannot decide anything.

Its one structural property is that it survives addition. The reduced form of a sum is the reduced form of the sum of the reduced forms — so a board may be reduced part by part, which is what makes the whole apparatus usable. How hot a background has to be established when that licence is safe, and closed on the operation the property promises:

The rung above is the second one the rung below named: the sum’s reduced form computed from the parts’. The homomorphism says such an arithmetic exists … and nothing says what it looks like as an operation.

It looks like addition with a second reduction after it, and the second reduction is needed exactly where both parts are hot.

What “an arithmetic on reduced forms” would mean #

Two things have to hold before the phrase means anything, and only one of them was known.

The operation has to be well defined. If two values with the same reduced form could give sums with different reduced forms, there would be no operation on reduced forms at all — only an operation on values that happens to respect them sometimes. That is the homomorphism, and it holds on all 3,600 pairs here.

And it has to be describable. A well-defined operation is not automatically a useful one: add the two originals and reduce is a description that requires the originals, which is precisely what a reduced form is supposed to let a player forget. The question is whether the reduced forms alone determine the answer by some rule shorter than redoing the work.

The answer is that they do, and the rule is short: add the two reduced forms as games, then reduce once more. The second reduction is what this page is about, because if it were never needed the arithmetic would be plain addition and the whole question would be closed.

When the second pass is needed #

Of the 3,600 pairs, 431 need the second pass — one in eight. And the rule for which is a single bit about the parts:

Both parts hot: 431 of 1,849, about a quarter. A cold part: none of 1,751.

That is a clean condition and it is not obvious in advance. A cold part is a number plus an infinitesimal, and the reduction has already turned it into a number; adding a number to a reduced form shifts it and cannot make an option dominated that was not dominated before. Two hot parts is where the second pass can happen, and where it happens a quarter of the time.

So a program carrying reduced forms across a board has a cheap test: if either part is cold, plain addition is enough. That is worth having because temperatures are already computed on such a board — the whole reason for reducing is that the board is hot — so the test costs a comparison rather than an evaluation.

What it does when it happens #

It deletes an option, nearly every time — 260 sums lose one and 94 lose two. The remaining 77 change form without changing width, which is the other reduction at work: a bypass replaces a reversible option with a list, and the list can be the same length as what it replaced.

The worked pair is the mechanism. Each part’s reduced form is as small as it can be on its own; put them together and a Right option of the sum — take the exchange in the first fight — becomes dominated by another, because the second fight has changed what the opponent can do afterwards. Domination is a comparison between options, and a sum has options that neither part had.

That is why no amount of reducing the parts can anticipate it, and why the operation cannot be plain addition however the reduced form is defined.

There is a converse worth stating, because it bounds how bad the situation could have been. The second pass never adds an option: the reduced form of a sum is never wider than the sum of the reduced forms, on any of the 3,600 pairs. So the operation is a contraction, and iterating it across a board of many components cannot make the forms grow — which is exactly the property bypassing a reversible option lacks in the ordinary canonical form, and which makes the reduced form safe to carry across a large board.

What the second pass discards #

Every one of the 431 differences is an infinitesimal — its two stops are both nought, so no number can see it. That is checked rather than assumed, and it is the property the whole construction rests on: a reduction that discarded something a number could see would be discarding something a player could lose by.

It is worth noticing what the check is really testing. The reduction is defined to remove infinitesimals, so the difference between a form and its reduction is infinitesimal by construction — but the difference between the sum of two reductions and the reduction of the sum is a third object, and nothing in the definition says it has to be small. It is, on every pair here.

What it costs to carry #

The practical reading is about where a program spends its time, and it is favourable.

Reducing is a one-off. A component’s reduced form is computed once and used in every sum it appears in, which is the whole point of a normal form. What this page adds is that combining two of them is an addition plus, one time in eight, a reduction of a form that is already small.

And the small forms make the second pass cheap. A reduced form has fewer options than a canonical one — that is what the reduction does — so the domination check the second pass runs is over a smaller list than the same check on the unreduced sum would be. The saving compounds: a board of five reduced components costs four additions and about half a reduction, where the same board evaluated from the positions costs five full canonicalisations and four additions of much larger forms.

That is the ordinary argument for a normal form and it is worth checking rather than assuming, because it fails whenever the normal form is not closed under the operation. Here it does not fail; the closure costs one extra pass, and the pass is on the smallest available object.

Against the thermograph precedent #

The rung below issued a warning worth taking up, because the analogy is close:

It would be the analogue of adding thermographs, and that construction turns out to work only where nothing is at stake.

Thermographs do not add: the diagram of a sum is not built from the diagrams of its parts, except where the parts are cold. The reduced form does better, and the difference between the two cases is worth naming.

A thermograph is a summary and a reduced form is a game. The diagram throws away everything except two walls, so the information a sum needs is simply gone; the reduced form throws away only infinitesimals, and keeps a whole form to add. So the arithmetic exists here and not there, and the price of it is the second reduction — a computation on the sum rather than a lookup from the parts.

That is the general shape of the trade, and it recurs everywhere in this subject: the coarser the summary, the less of the arithmetic survives it. The mean survives every sum, the temperature survives none, the stops survive with a bound, and the reduced form survives with a second pass.

Who built the reduced form, and what for #

The reduced canonical form is Grossman and Siegel’s, and it exists for a practical reason: in a game with a hot board, the infinitesimals a canonical form carefully preserves make no difference to anything, and carrying them makes the forms much larger than they need to be. Throwing them away gives a smaller object that still answers every question a number could ask.

The construction is a good example of a pattern this site keeps meeting: a canonical form is unique because the reductions are confluent, and any weakening of the comparison used in those reductions gives another normal form with its own uniqueness. The reduced form weakens better than to better than, up to an infinitesimal, and everything else follows unchanged.

What is not automatic is that the weakened form still adds. Uniqueness is about one position; the homomorphism is about two, and it is the property that decides whether the object is a curiosity or a tool. Grossman and Siegel prove it; this page measures what it costs to use, which is a different question and one the proof does not raise.

What this does not say #

Four limits.

Sixty values, ordered pairs. The pool is the first sixty values of day three, which is a slice of a day rather than a day, and the pairs are ordered so each is counted twice. Neither affects the shares.

Day three is not a board. These are values of the abstract construction added to one another, not components of any game — a real board’s components are drawn from a much narrower population, and the one-in-eight rate would be different on it.

A second reduction, not a formula. The operation is add and reduce, which is honest and is not the closed form the rung below might have hoped for. What is not ruled out is a rule that says which option the second pass will delete — the census has all 431 cases in hand and this page does not describe them beyond the count.

The hot condition is measured, not proved. A cold part means no second pass holds on 1,751 pairs and has an argument behind it — a number shifts a form without changing which options dominate — but the argument is a paragraph rather than an induction.

And sums of two. A board of three components is two applications of the operation, and whether the second pass is needed more often on the second application than on the first is a question this sweep does not ask. It is the same census with triples and it is affordable.

The convention, named #

Normal play, day-three values in canonical form.

The reduced canonical form applies the two ordinary reductions with comparison weakened to up to an infinitesimal: an option is dominated when it is no better than another once infinitesimals are ignored, and reversible likewise. The result is a form differing from the original by an infinitesimal, and it is unique.

Hot means a temperature above nought; a cold value here is a number or an infinitesimal, whose reduced form is a number.

The second reduction is the one applied to the sum of two already-reduced forms, and needing it means that sum is not already in reduced form. A difference is the plain sum less the reduced one, computed as a game.

Why a second pass is not an admission of failure #

Add and reduce again reads like a workaround, and it is worth saying why it is the right shape of answer rather than a gap where a formula should be.

A homomorphism guarantees that the reduced form of a sum can be computed from the parts’ reduced forms. It does not guarantee that the computation is the sum, and there is no reason it should: the promise is about information, not about arithmetic. What the theorem says is that nothing is lost by reducing first, and what an implementation needs is a procedure — and a procedure with two steps is still a procedure.

The comparison worth drawing is with canonical form itself. Adding two canonical forms does not give a canonical form either: options that were incomparable in each part can become comparable once both fights are on the board, so a sum has to be reduced. Nobody reads that as a defect in canonicalisation. It is the ordinary situation for a normal form under an operation, and the reduced form inherits it.

What is specific here is the second clause and the count that goes with it. The reduced form’s extra rule throws away everything below every number, and putting two positions together can create a comparison that clause did not have before — 431 sums of 3,600 need it, and none of the 1,751 pairs with a cold part does. That last figure is the informative one: a cold part contributes nothing to fight over, so nothing new becomes comparable, and the second pass is idle.

So the shape of the answer is reduce, add, reduce, the second reduction is needed exactly when both parts are hot, and the honest way to describe it is as an operation with a cost rather than a theorem with an exception.

What a player carries #

Two readings for somebody with a board in front of them rather than a census.

A hot board can be summarised component by component. Each component’s reduced form is a smaller object than its canonical form and answers every question a number could ask, so a player who has learnt the reduced forms of the shapes has learnt something that composes. That is a genuinely useful state of affairs and it is what the homomorphism is for.

And the summarising is exact rather than approximate. This is worth separating from the other readings on this site, most of which are bounds: the reduced form is not an estimate of the value, it is a different object which is equal to the value up to something no number can see. On a board where infinitesimals cannot decide the outcome — which is what hot means — that is not a loss at all.

The one caution is the one this page has measured. Two hot components do not simply add, and a player who treats the reduced forms as numbers to be summed will be wrong one time in four on such a pair. What is needed is one more look at the sum, which is exactly the work the second pass does.

One number from the census is worth carrying on its own: the second pass is needed on 23 per cent of the pairs where both parts are hot. That is the figure a program’s author needs, because a board hot enough to be worth reducing is a board whose components are mostly hot, and one sum in four then costs an extra reduction.

Where the ladder goes next #

The reduced-form anchor has three rungs to here: the reduction and its classes, when its licence to substitute is good, and now what the arithmetic on reduced forms actually is.

The rung above asks whether the option the second pass deletes can be named from the parts, and the answer is a warning about how a near-miss should be read. The option nothing names scores eight candidate rules; the best reaches four in five, which sounds like a rule with exceptions — and on a pool closed under negation it falls to under half. So the near-miss was a property of the population rather than of the rule, and the original pool’s asymmetry was doing the work.

What survives is smaller and honest: the deleted option has a recognisable shape, and there is a cheap test that rules it out. Ruling out is not naming, and the anchor says so.

Not a domination in that order then overturns the assumption underneath both rungs. Everyone had taken the second pass to be an ordinary domination — a surviving option at least as good as the deleted one — and on none of the 525 deletions is that true. Read in the order the reduced form actually works in, where one option beats another when both its stops are at least as good, every deletion with a survivor is a domination, the dominator is unique on all but twelve of them, and it always comes from the other part.

That last clause is the one to carry. The option deleted by the second pass is always beaten by something from the other component, which is exactly why no rule stated in terms of one part could name it, and why the second pass exists at all: it is the reduction noticing a comparison that only becomes available once both fights are on the board.