Undecidability
Named by 10 essays across 3 fields — each of them below, with the objects they name alongside it.
The rule that computes
One of the 256 elementary rules can run any program. Not simulate one, not approximate one — a machine that can compute anything computable, built from a lookup table with eight rows and nothing else.
The question nobody can answer
Halve it if it is even, triple it and add one if it is odd. Every number anyone has tried comes down to one. Nobody can prove they all do, and the reason is not that the problem is hard to state.
The instance that has to be guessed
Every rule of a propositional tableau replaces a formula by shorter ones, which is why it stops. The rule for a universal claim does not replace it — it keeps it and adds an instance — and one word changing turns a decision procedure into a search that may run forever.
Tiles that never repeat
Two rhombs with angles taken from the pentagon, cut each into halves, and cut each half into smaller copies of the two halves by a fixed rule. Repeat, and the pieces fill the plane with no gaps and no overlaps, in a pattern with five-fold stars everywhere and no period anywhere. The reason it cannot repeat is a single number: thick tiles outnumber thin ones by φ, and a repeating pattern would make that ratio a fraction.
Fourteen fractions that list the primes
Change the Collatz rule so that the multiplier depends on the remainder modulo some other number than two, and the resulting maps can compute anything a computer can. John Conway proved it in 1972, which means no method can decide, for every such map, whether its orbits come down — and fourteen fractions, applied in order, turn out to be enough to print every prime.
No algorithm can read what a program does
Whether a program halts cannot be decided by any algorithm. Henry Rice showed in 1953 that the halting problem is not special: no algorithm can decide any property of what a program does — whether it ever prints a 7, whether it computes the successor function, whether it is a virus — except the two properties that hold of every program or of none. Every one of the 20,736 smallest Turing machines can be checked by hand; the theorem says why that stops.
Dominoes that no algorithm can match
Emil Post's puzzle has no machines and no programs in it: a handful of dominoes, each with a string on top and a string underneath, and the question whether some row of them, repeats allowed, spells the same word along both edges. No algorithm can answer it for every set of dominoes, because a set can be built whose only matches are a machine's run written out twice, one step out of phase — and three dominoes with strings of three characters already need thirty-six in a row before the edges agree.
An ant that builds a highway
Two rules: on a white cell turn right, on a black cell turn left, and flip the colour as you leave. From an empty grid the ant wanders for 9,975 steps without visible order, then falls into a cycle of 104 steps that carries it diagonally for ever. That it always escapes to infinity is proved; that it always ends on a highway is not.
The ash a random soup leaves
Fill a grid at random, apply Conway's rule — three neighbours give birth, two or three keep a cell alive — and the population collapses from half the cells to about three in a hundred, leaving scattered blocks and blinkers. Five cells in the shape of an R take 1,103 generations to do the same and peak at 319 on the way. The Game of Life is a universal computer, so nothing short of running it can say when a pattern will settle, and the figures run it.
The shortest description nobody can certify
A string's complexity is the length of the shortest program that prints it. Counting shows almost every string has no short program, and diagonal arguments show nothing can compute the length, or even prove it large for any single string past a fixed point. All 16,777,216 three-state Turing machines leave only 95 strings on their tapes, and a compressor treats π exactly as it treats random bits.
Named alongside it
The objects these essays reach for when they reach for this one.
Cellular automatonComputationHalting problemTuring machineBusy beaverCollatzExhaustive searchGliderIterationReductionSimulationTermination