There are many variants of the counter machine, among them those of Hermes, Ershov, Péter, Minsky, Lambek, Shepherdson and Sturgis, and Schönhage. These are explained below.
The models in more detail
1954: Hermes' model
Shepherdson and Sturgis 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" (Shepherdson and Sturgis, p. 219).
Shepherdson and Sturgis observe that: "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" (Shepherdson and Sturgis, p. 219)
The only two arithmetic instructions are
Successor operation
Testing two numbers for equality
The rest of the operations are transfers from register-to-accumulator or accumulator-to-register or test-jumps.
Kaphengst's paper is written in German; Sheperdson and Sturgis's translation uses terms such as "mill" and "orders".
The machine contains "a mill" (accumulator). Kaphengst designates his mill/accumulator with the "infinity" symbol but we will use "A" in the following description. It also contains an "order register" ("order" as in "instruction", not as in "sequence"). (This usage came from the Burks–Goldstine–von Neumann (1946) report's description of "...an Electronic Computing Instrument".) The order/instruction register is register "0". 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".
The instructions are stored in the registers: "...so the machine, like an actual computer, is capable of doing arithmetic operations on its own program" (p. 244).
Thus this model is actually a random-access machine. In the following, "[ r ]" indicates "contents of" register r, etc.
Shepherdson and Sturgis remove the mill/accumulator A and reduce the Kaphengst instructions to register-to-register "copy", arithmetic operation "increment", and "register-to-register compare". Observe that there is no decrement. This model, almost verbatim, is to be found in Minsky (1967); see more in the section below.
1958: Ershov's class of operator algorithms
Shepherdson and Sturgis (1963) observe that Ersov's model allows for storage of the program in the registers. They assert that Ersov's model is as follows:
1958: Péter's "treatment" Shepherdson and Sturgis (1963) observe that Péter's "treatment" (they are not too specific here) has an equivalence to the instructions shown in the following table. They comment specifically about these instructions, that: "from the point of view of proving as quickly as possible the computability of all partial recursive functions Péter's is perhaps the best; for proving their computability by Turing machines a further analysis of the copying operation is necessary along the lines we have taken above." (Shepherdson and Sturgis (1963) p. 246)
1961: Minsky's model of a partial recursive function reduced to a "program" of only two instructions In his inquiry into problems of Emil Post (the tag system) and Hilbert's 10th problem (Hilbert's problems, Diophantine equation) led Minsky to the following definition of: "an interesting basis for recursive function theory involving programs of only the simplest arithmetic operations" (Minsky (1961) p. 437).
His "Theorem Ia" asserts that any partial recursive function is represented by "a program operating on two integers S1 and S2 using instructions Ij of the forms (cf. Minsky (1961) p. 449):
The first theorem is the context of a second "Theorem IIa" that
"...represents any partial recursive function by a program operating on one integer S [contained in a single register r1] using instructions Ij of the forms":
In this second form the machine uses Gödel numbers to process "the integer S". He asserts that the first machine/model does not need to do this if it has 4 registers available to it.
1961: Melzak model: a single ternary instruction with addition and proper subtraction "It is our object to describe a primitive device, to be called a Q-machine, which arrives at effective computability via arithmetic rather than via logic. Its three operations are keeping tally, comparing non-negative integers, and transferring" (Melzak (1961) p. 281)