register

Part I, The Sentence and the Kernel · defined in Chapter I, The Founding Sentence

A record-keeping unit. Its state is its past, up to what no future can reveal.

Plate W.10The minimal register’s four states, its reports

(s,\ell)

, each with the shortest word that reaches it:

a

b

ba

and

bb

. No letters are drawn yet.

Two histories are future-indistinguishable for a register when every admitted continuation gives both the same readings, and a register is a record-holding unit whose states are its histories modulo that relation: the Myhill–Nerode construction in its response-function form. Its state is derived, and it holds exactly the distinctions some future can reveal. The smallest register whose records matter carries one blind sign and remembers the last symbol written, and its histories fall into exactly four classes, the reports.

An observer is a register: what it holds is its log, a word of the program’s tower, and what it knows of the world is its completion, the identification of all world histories that leave it the same marked log. In the world of growing records a register has an anchor and a word; it appends a letter equal or adjacent to its top by itself, and it receives the antipode only in an exchange with a partner.

Definition(Occurrence, record, register)

(1) An occurrence is one application of a law to named inputs. It makes one distinction and leaves records. A history is a finite admitted sequence of occurrences. (2) A record is a mark left by an occurrence that later occurrences can read. The records held by one register form its word, or log, ordered by age; the newest is its top. (3) Two histories are future-indistinguishable for a register when every admitted continuation gives both the same readings. A register is a record-holding unit whose states are its histories modulo future indistinguishability.

As mathematics

A past modulo the Myhill–Nerode congruence: a state of the minimal automaton. The general fact is part of the program’s forcing theorem: future indistinguishability is the largest congruence contained in equality of present readings, and every deterministic realization of the readings maps onto its classes.

The minimal consequential register has alphabet $\{a,b,p\}$ acting on a pair of signs, $a:(s,\ell)\mapsto(s,+1)$ , $b:(s,\ell)\mapsto(-s,-1)$ , $p:(s,\ell)\mapsto(s\ell,+1)$ , from $(+1,+1)$ , and reads $s$ . Its states are the four pairs $(s,\ell)$ , reached by the words $a$ , $b$ , $ba$ and $bb$ .

Its name in another field	Bridge
a state of the minimal automaton	built
a causal state of computational mechanics	type
an observability quotient of realization theory	type
a record-keeping unit	a reading