Independence is a claim

Every essay on this site that adds two positions together begins the same way: the board falls into independent parts, so its value is the sum of theirs.

The second half of that sentence is a theorem. The first half is a claim about the position, and it is the half nothing checks.

A boundary drawn and a boundary there #

The clearest way to see the difference is to make it one column wide.

Nothing about the two pictures says which one is legitimate. They differ by whether a column of squares is there, and the difference between “there is a boundary here” and “treat this as a boundary” is the whole of the essay.

It is worth saying why this is not a pedantic point. Splitting is not an occasional convenience in this subject — it is the method. A position of any interesting size is beyond direct evaluation, so every value anybody computes for a real board is computed by splitting it, and the splitting step is done by eye. An error there is not an error in one figure; it is an error in the way the answer was obtained, and it propagates to everything downstream with no symptom at all.

What a legitimate split is for is worth keeping in view while the failures are counted. Independent components, each evaluated once and each with a value, and a total that is the sum of those values: that is the operation everything on this site is built on, and the reason a decomposition is worth having in the first place. Nothing below is an argument against it. It is an argument that the word independent in that sentence has to be earned.

The failures that get caught, and the ones that do not #

Cut every small Domineering board at every column and the equation fails more often than it holds: over thirty cuts, eight agree and twenty-two do not.

That much is unsurprising. Cutting through the middle of a connected board obviously loses the dominoes that would have crossed. What is worth counting is how the failures divide.

Eight of them change the winner. A cut board and a whole board can be in different outcome classes, and that is a failure the next paragraph of any essay would catch, because the essay would go on to say who wins and be wrong.

Fourteen of them do not. The value is different and the winner is the same, and nothing downstream reports it. Those are the ones this essay is about.

The general shape is worth stating plainly: checking the outcome is not checking the value. An outcome class is one of four labels and a value is an object, so the outcome can agree by coincidence and does, on roughly two failures in three here.

A game where no split works #

Domineering at least has legitimate splits: block a column and the halves really are independent, because a domino needs two free squares and the blocked column supplies neither.

Kōnane has none.

Over every row of seven squares, cutting at a single empty square gives the right answer 842 times out of 1,266 — 66.5%. Widen the gap to two and it is 382 of 600, which is 63.7%; to three and it is 108 of 162, which is 66.7%.

The rate does not improve. It sits at about two thirds and dips in the middle, and a reader expecting a widening gap to work its way towards a wall gets no encouragement at all from the first three widths.

That is worth dwelling on, because the expected shape would have been the more dangerous finding. A rate climbing steadily towards one is what a heuristic looks like when it is about to be mistaken for a rule — it persuades somebody it is converging on a truth, and the remaining failures start to look like small cases rather than counterexamples. A rate that is simply flat offers nothing to extrapolate from, and the honest reading is the blunt one: a stone can always enter a gap, whatever the gap’s width, so the two halves are never independent and widening the gap does not make them any more so.

The width at which it does work, and why that is worthless #

Run the same sweep two widths further and the rate finally moves: 26 of 36 at a gap of four, and 6 of 6 at a gap of five. So there is a width at which cutting a seven-square Kōnane row is exact, and quoting it would be the most misleading true sentence on this page.

The reason is the row rather than the gap. A seven-square row with five empty squares in the middle has one square on each side, and a single stone with a single stone opposite it has almost nothing it can do — there are six such rows in total, and the sweep is reporting that a game with no moves in it decomposes correctly. Nothing about the gap has become a barrier; the board has run out of pieces to put on either side of it.

That is a window artefact, and it is the reason the sweep runs at a fixed row length. Every one of these censuses holds the row at seven and varies the gap, so the wider the gap, the fewer squares are left to hold stones and the smaller and more degenerate the population being measured. A reader who saw only the last two rows would conclude the wall exists at width four or five; a reader who saw the whole sequence sees a flat two thirds followed by a population collapsing to nothing.

The general caution generalises past this game. A rate measured over a shrinking population is not a trend, and any sweep that varies one parameter inside a fixed budget is measuring the budget as well as the parameter. The first three widths are the informative ones here precisely because they are the ones with enough room left over to be a test.

What would settle it, and what could not #

That leaves the negative claim resting on three flat numbers, which is weaker evidence than a converging sequence would have been — so it is worth being clear about where the actual confidence comes from.

Not from the census. From the rule: a Kōnane move is a hop over an adjacent enemy stone into the space beyond, so a stone lands in a gap, and a later hop leaves from where it landed. Two moves therefore cross any gap the first move can reach into, and no width of gap forbids the first move — it forbids only the crossing being done in one. That is an argument about the rules and it holds for every row length and every gap.

So the sweep is a check on the argument rather than the evidence for it, which is the right way round and is not how the numbers were originally read. The measurement’s job was to find the threshold the argument says does not exist, and its job is done by not finding one — the flat two thirds being a more useful non-finding than a climb would have been, because a climb would have had to be argued away.

The function that produced those numbers was written expecting to find the threshold — the width at which a gap becomes a wall — and its docstring now records that there is none. That is the more interesting finding and it is the reason the Kōnane essay states its census and stops: a game whose positions do not decompose has no route to the values of large positions, however well its small ones are understood, which is why the one-dimensional game is still unsolved.

Reading the smallest failure #

The one-row example is worth reading in full, because it is four squares and the whole mechanism is visible.

A 1 × 4 Domineering strip: Left plays vertically and has no move at all on a single row, Right plays horizontally and can take any two adjacent squares. Right has three moves available. Taking a domino at one end leaves two adjacent squares and a second move; taking the middle one leaves two isolated squares and no second move. Right will take an end, so the strip is worth −2 — two free moves for Right and none for Left.

Cut it after the first square. The left piece is one square, worth 0 — nobody can move there. The right piece is three squares in a row, worth −1, because Right can take two of them and then nothing is left adjacent. The sum is −1.

The missing move is the domino spanning squares one and two, which the cut has abolished. It is a real option of Right’s, worth exactly one free move, and it is the entire difference.

Right wins either way, so the outcome check passes. The value is wrong by one, which in a sum with anything else is the difference between winning and losing — and that is why a value being wrong matters even when the winner is right. A value’s whole purpose is to be added to something, and an error survives the addition.

What independence actually requires #

The condition is precise and it is worth stating in the form that can be checked.

Two parts of a position are independent when no move in either changes the options available in the other. Not “they are far apart”, not “they look separate”, and not “a piece cannot get from one to the other in a single move”.

Domineering satisfies it when a blocked column separates the halves, because a domino occupies two adjacent free squares and no domino can occupy one square on each side. Kōnane fails it because a hop lands in the gap and a later hop leaves from there, so a two-move sequence crosses what a one-move check would call a wall.

The word “options” in that condition is doing real work and it is easy to read past. It is not enough that a move in one part leaves the pieces of the other alone; it must leave the other part’s whole set of legal moves alone, at every depth. A move that fills a square nobody can use has changed nothing; a move that fills a square somebody could have landed on in three moves’ time has changed a great deal, and neither is visible in a picture of the position after one move.

The move that does the damage is the horizontal domino. It spans two columns, so a boundary between columns is a boundary a move can cross, and imagining it away removes real options from Right and none from Left. How many it removes is the number of rows, because one such domino sits in every row of the board.

That asymmetry is the mechanism behind the quiet failures. Cutting a Domineering board removes horizontal dominoes only, so it takes options from one player and not the other, and the value moves in a direction. A cut that removed options symmetrically would be less dangerous and less common.

What a real decomposition buys #

None of this is an argument against decomposing. It is an argument for checking.

That refusal is the practical form of this essay’s argument. A figure that draws a decomposition and asserts the product identity has made the independence claim testable, and a figure that draws two halves and adds their values has not.

That is the temptation in its sharpest form. Small Domineering boards have known values, so splitting a large board into small ones and reading the answers off a table is exactly the shortcut anybody would want; it is legitimate when the small boards are really there and wrong when they have been imagined, and the second case does not announce itself.

How a real game supplies the walls #

The games in the applied field are instructive here, because in two of them the decomposition is produced by the play rather than asserted by an author.

In Amazons the arrows burn squares, and after enough of them the board is cut by barriers no queen can cross — a decomposition arrived at by the players. In Dots and Boxes the position drifts toward a state where every coin has two strings, and a graph with every vertex of degree two is a union of paths and cycles, so the endgame is a multiset of chains whether anybody wanted it or not.

Both of those are genuine independence, and in both cases the reason is the same: the rules have made the crossing move illegal, rather than a drawing having omitted it.

What the theory then does with the parts is downstream of all of this. Once the split is real, choosing where to move is choosing a component, and that choice is its own problem — one the values do not solve and the option lists do. Every step of it computes nonsense if the independence claim underneath is false, and none of the steps has any way of noticing.

The same trap in three other places #

Once the shape is recognised it turns up repeatedly, and naming the instances is worth more than the general principle.

A cut through a Go position. Go players routinely analyse a corner as though the rest of the board were not there, and it is a decomposition claim of exactly this kind — legitimate when the groups are settled and false when a move in the corner has a follow-up elsewhere. The whole apparatus of ko threats is a statement that Go positions are not independent.

A sum with a loopy component. A drawn component poisons a sum in a way no value arithmetic anticipates, because the drawn part has no value to contribute. That is a failure of the theorem rather than of the split, and it looks the same from the outside: two parts, one answer, and it is wrong.

A misère sum of genuinely independent parts. Here the split is perfect and the arithmetic still fails, because misère play has no value per position. Independence is necessary and not sufficient.

Three different failures with one symptom. The discipline that separates them is to say, before adding, which claim is being made: that the parts do not interact, that each part has a value, and that values of that kind compose. All three are needed and only the first is about the picture.

What the picture cannot show #

The boards here are at most fifteen squares, which is where the whole board’s value can still be computed for comparison.

That is the essay’s own limitation and it is a real one: the interesting cases are large boards, where nobody can compute the whole for comparison, and where the temptation to split is strongest precisely because the alternative is impossible. Everything above is measured where the check is available, and the check is unavailable exactly where the practice matters.

The second thing not shown: a partial dependence. The figures divide cleanly into legitimate splits and illegitimate ones, and a real position can be nearly independent — two regions joined by one square, say, so that most lines of play never cross. The value of such a position is not the sum of its parts and may be very close, and nothing here measures how close. That would be a genuinely useful quantity for a player and this site has no way of computing it.

Why the check is cheap and nobody runs it #

The check that would have caught every failure above is one line: evaluate the whole and evaluate the sum, and compare.

It is cheap exactly when the decomposition is unnecessary — on a board small enough to evaluate whole — and impossible exactly when the decomposition is load-bearing. That is an awkward shape and it is why the practice is to reason about independence rather than to test it.

What can be done instead, and what the machinery here does, is to assert a consequence of independence that is cheaper than the value. The position graph of a sum is the product of the graphs of its parts, so counting positions in the whole and multiplying the counts in the parts is a test that needs no game values at all — and it is the test the decomposition figure runs before it is published, refusing to draw when it fails.

That is the general recipe and it is worth naming: where the direct check is unaffordable, find a necessary condition that is cheap and check that instead. It cannot prove independence and it can catch a great many failures, which is the ordinary bargain a gate makes.

The convention, named #

All of this is normal play, and the independence question is convention-dependent in a way worth noticing.

Under normal play, the value of a sum is determined by the values of the parts — that is the theorem the whole subject rests on. Under misère play it is false even for genuinely independent parts: two positions can agree in isolation and differ inside a sum, which is why the misère theory needs a monoid per universe rather than a value per position.

So there are two separate ways the sum can fail. The parts can fail to be independent, which is this essay. Or the parts can be independent and the theory can still not compose, which is the misère field. Confusing the two is easy and the difference is total.

Where the ladder goes next #

disjunctive-sum reaches four rungs: the sum as the object, which part to move in, the other ways to add, and now the question of whether the parts are parts.

The rung above is the quantitative one the last section named — how wrong a nearly-independent split is, and whether the error can be bounded. That is a question with the same shape as a bound instead of an answer, and this site has the machinery to ask it and has not.