Computation
Named by 7 essays across 2 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.
No local rule can count the votes
A ring of cells, each holding 0 or 1, has to agree on whichever value is in the majority — every cell seeing only its neighbours. The best-known rule gets it right most of the time and wrong near a tie; no rule of any radius gets it right always. Yet two rules run one after the other do, on every ring, and the first of them is the traffic rule.
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.
The program that prints itself
The diagonal argument has always been used to destroy — to show that a list misses something, that a sentence cannot be proved. Run the same move the other way and it builds. Kleene's recursion theorem says every program can be given its own text to work with, and the proof is a program that prints itself, thirty-two characters long, which can be run and checked.
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.
What a proof about numbers can compute
The proofs of (o → o) → o → o without detours are the whole numbers, one for each count of how often the assumption is used. A proof of the implication from that formula to itself therefore takes a number and returns one, and removing its detours computes the answer. Enumerating all 14,659 such proofs up to size 16 finds 299 different functions — sums, products, squares, cubes, tests on zero — and not one of them subtracts. The predecessor, n − 1, is not among them, and a single line about polynomials says it never will be.
![Raising the input's type, and what becomes computable. N[o]: 299 functions, 0 not extended polynomials; N[o → o]: 163 functions, 42 not extended polynomials; N[(o → o) → o]: 128 functions, 46 not extended polynomials; N[o → o → o]: 72 functions, 33 not extended polynomials.](/f/raising-the-input-s-type-and-what-becomes-computable-671114cd.svg)
One type up
A numeral is an iterator, and the formula it proves fixes what it may iterate. At the lowest type it can only repeat the successor, and every function it computes is a polynomial — no predecessor. Let the same numeral iterate a function on functions and 2ⁿ appears; let it iterate a function that takes a function and returns a value, and the predecessor appears at size 15; let it iterate a pair of values, and the predecessor, halving and parity all arrive at size 16. The trick is Kleene's, from 1932, and an exhaustive search finds it without being told.
Named alongside it
The objects these essays reach for when they reach for this one.
UndecidabilityCellular automatonExhaustive searchHalting problemLocalityNormal formPolynomialProof systemTerminationUniversalityBusy beaverCollatz