There are many variants of the counter machine, among them those of Hermes, Ershov, Péter, Minsky, Lambek, Shepherdson and Sturgis, and Schönhage.
Shepherdson & Sturgis (1963) observe that "the proof of this universality [of digital computers to Turing machines] ... seems to have been first written down by Hermes, who showed in [7--their reference number] how an idealized computer could be programmed to duplicate the behavior of any Turing machine", and: "Kaphengst's approach is interesting in that it gives a direct proof of the universality of present-day digital computers, at least when idealized to the extent of admitting an infinity of storage registers each capable of storing arbitrarily long words".
Kaphengst's paper is written in German; Sheperdson and Sturgis's translation uses terms such as "mill" and "orders".
And, although not clear from Sheperdson and Sturgis's exposition, the model contains an "extension register" designated by Kaphengst "infinity-prime"; we will use "E".
Shepherdson & Sturgis (1963) observe that Ersov's model allows for storage of the program in the registers.
If we use the context of his model, "keeping tally" means "adding by successive increments" (throwing a pebbles into) or "subtracting by successive decrements"; transferring means moving (not copying) the contents from hole A to hole B, and comparing numbers is self-evident.
The instruction is a single "ternary operation" he calls "XYZ": Of all the possible operations, some are not allowed, as shown in the table below: Some observations about the Melzak model: Original "abacus" model of Lambek (1962): Lambek references Melzak's paper.
He atomizes Melzak's single 3-parameter operation (really 4 if we count the instruction addresses) into a 2-parameter increment "X+" and 3-parameter decrement "X-".
This form is virtually identical to the Minsky (1961) model, and has been adopted by Boolos, Burgess & Jeffrey (2007, p. 45, Abacus Computability).
Observe, however, that B-B and B-B-J do not use a variable "X" in the mnemonics with a specifying parameter (as shown in the Lambek version) --i.e. "X+" and "X-" – but rather the instruction mnemonics specifies the registers themselves, e.g. "2+", or "3-": Shepherdson & Sturgis (1963) reference Minsky (1961) as it appeared for them in the form of an MIT Lincoln Laboratory report: In Section 10 we show that theorems (including Minsky's results [21, their reference]) on the computation of partial recursive functions by one or two tapes can be obtained rather easily from one of our intermediate forms.Their model is strongly influenced by the model and the spirit of Hao Wang (1957) and his Wang B-machine (also see Post–Turing machine).
Single-Register Machine SRM: Here they are implementing the tag system of Emil Post and thereby allow only writing to the end of the string and erasing from the beginning.
While peculiar, Schönhage's model shows how the conventional counter-machine's "register-to-register" or "read-modify-write" instruction set can be atomized to its simplest 0-parameter form.