Termination — the series · Combinatorial Game Theory

Poker Nim from 3, 5, 7, with reserves of 4 and 4. Nim with one extra kind of move: a player may put any number of counters back onto a heap from a private reserve. It looks as though a losing player could stall for ever. They cannot, and the winner is decided by exactly the same nim-sum as ordinary Nim — checked here over every position within a stated range rather than argued.

The condition the recursion rests on

Not that the moves run out, and not that the options are few. Poker Nim's heaps can grow without bound and it ends; the game called `on` has one option and never does. What every game value needs is that no infinite run of moves exists — and there are three separate ways to fail it.

part 1 · limits

Hydras, and how long each takes to kill. Six small hydras with the ordinal the termination proof assigns to each and the exact number of chops it takes to finish it. Two of them are not finished here: the fight is guaranteed to end and the machine runs out of memory long before it does, which is the gap between a termination proof and a bound.

It ends, and nothing says when

The recursion this site runs needs every line of play to reach a position with no moves, and the condition is usually met by an obvious decreasing quantity. The hydra meets it with no such quantity anywhere: the tree grows at nearly every step and the fight ends regardless, because the only thing that decreases is an ordinal. A four-node hydra dies in twenty chops; one level deeper and 279 chops reach forty thousand nodes with no end in sight.

part 2 · limits

Two ways to be certain and ignorant at once. Ten positions from two games that both terminate for reasons no bound comes out of. Sylver Coinage's proof counts something that goes down and can be counted; the hydra's counts an ordinal, which cannot, and the last column shows what that difference is worth.

Two ways to end with no bound

Sylver Coinage and the hydra are both guaranteed to finish and neither will say when. The difference is that one of them carries its own bound: every move in Sylver removes at least one gap, the gaps can be counted in a moment, and over ten openings the longest play uses every single one. The hydra has no decreasing quantity a solver can hold — three hydras of five nodes each take seven chops, twenty-one, and a number past two hundred and seventy-nine that this machine never reaches.

part 3 · limits

Which games end at which level

Between a game that ends within a computable bound and one that ends with no bound at all there are levels, each corresponding to a strength of induction. This site's games sit at three of them, and which level a game is at is decided by exhibiting its termination measure and checking that every move lowers it.

part 4 · limits

Heights from ω² down to nought. Thirteen Sylver Coinage positions with the gcd of the named numbers, its prime factors counted with repeats, the gaps of the semigroup scaled by the gcd, and the ordinal height those give: ω² for the empty position, ω·4 for {16}, ω for {2}, 3 for {3, 5} and 0 for {2, 3}.

No number bounds it

Sylver Coinage cannot be given a termination measure in the natural numbers: from {2} the game can last one move, two, or any number the second player chooses. The measure proposed for it — the gap count beside the number of distinct primes the named numbers share — rises on {4} + 6. Count the primes with their repeats and it falls on every one of 42,519 moves checked, and it is exact: the game tree is ω squared tall, and the height of a position is ω times the prime factors of its gcd plus the gaps that are left.

part 5 · limits