Finite automaton
Named by 7 essays across 3 fields — each of them below, with the objects they name alongside it.
A language that can name a set
Allow a sentence to quantify over sets of positions as well as positions, and on words the answer changes completely: the sets buy exactly the languages a finite automaton recognises. Whether the number of letters is even is the smallest example of what the sets are for.
Two patterns, one chance, different waits
HTH and HTT are equally likely in any given window of three tosses. Waiting for HTH takes ten tosses on average and waiting for HTT takes eight, and the difference is not about probability at all — it is about what a failed attempt leaves behind.
A coin that lets the first player win
On a fair coin the second player in Penney's game always has a better pattern than the first, and the first can hold them to no worse than two to one. Bend the coin and every overlap is paid for in the letters it uses: the replies change, the first player's share swings between a third and a half, and past a heads chance of 1/∛2 the first player simply names HHH and wins.
Three patterns in a circle
Race three coin patterns at once and the gamblers' accounting still gives each one's chance of arriving first — one fairness equation per pattern. What it does not give is any way to read the three-way result off the two-way ones. HHHT, TTHH and HTTH beat one another in a circle, and HHH loses both its head-to-head races and still finishes ahead of one of the patterns that beat it.
A machine that carries one bit
Write three numbers in base two, one above another, and read the columns from the right. Whether the bottom number is the sum of the top two can be checked with one bit of memory — the carry. That two-state machine is the whole of addition, and machines can be combined, negated and made to guess a missing number. So every sentence about whole numbers built from addition can be decided by building a machine for it, and the same machines can also handle 'x is a power of two', which addition alone cannot say.
What a missing string of digits leaves behind
Strike from the harmonic series every term whose denominator contains 42 and the rest adds up to 228.45. Strike out 99 instead and it adds up to 253.28. The difference is not about which numbers are lost but about waiting: 99 overlaps itself, so it takes 110 random digits on average to turn up where 42 takes 100, and the sum is almost exactly ln 10 times that wait.
One block of quantifiers, read off an algebra
Which properties of words can be stated with existential quantifiers alone — 'there are positions, in this order, carrying these letters' — and Boolean combinations of such statements? Imre Simon answered in 1975: exactly those whose syntactic monoid is J-trivial. Of the 57,068 languages that automata of up to four states recognise, 848 pass the test, and every one of them is decided by which words of length three or less it contains as scattered subwords.
Named alongside it
The objects these essays reach for when they reach for this one.
ExpectationConditional probabilityDecidabilityExhaustive searchMarkov chainMartingaleQuantifierAssociativityBinaryClassificationConvergence rateDecision procedure