Counter-machine model

Although some authors use the name "register machine" synonymously with "counter machine", this article will give details and examples of only of the most primitive speciesthe "counter machine"of the genus "register machine."
Within the species "counter machine" there are a number of varieties: the models of Hermes (1954), Kaphengst (1957), Ershov (1958), Péter (1958), Minsky (1961) and Minsky (1967), Melzak (1961), Lambek (1961), Shepherdson and Sturgis (1963), and Schönhage (1980). These models will be described in more detail in the following.
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:
The only two "arithmetic" instructions are
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:
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:
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:
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 
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.
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. This appears to be a blend of the three base models.
Melzak's physical model is holes { X, Y, Z, etc. } in the ground together with an unlimited supply of pebbles in a special hole S (Sink or Supply or both? Melzak doesn't say).
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:
1961: Lambek "abacus" model: atomizing Melzak's model to X+, X- with test.
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-". He also provides both an informal and "formal" definition of "a program". This form is virtually identical to the Minsky (1961) model, and has been adopted by Boolos-Burgess-Jeffrey (2002).
Abacus model of Boolos-Burgess (1970, etc.), Boolos-Burgess-Jeffrey (2002):
The various editions beginning with 1970 the authors use the Lambek (1961) model of an "infinite abacus". This series of Wikipedia articles is using their symbolism, e.g. " [ r ] +1 → r" "the contents of register identified as number 'r', plus 1, replaces the contents of [is put into] register number 'r' ".
They use Lambek's name "abacus" but follow Melzak's pebble-in-holes model, modified by them to a 'stones-in-boxes' model. Like the original abacus model of Lambek, their model retains the Minsky (1961) use of non-sequential instructionsunlike the "conventional" computer-like default sequential instruction execution, the next instruction Ia is contained within the instruction.
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-":
1963: Shepherdson and Sturgis's model.
On page 218 Shepherdson and Sturgis references Minsky (1961) as it appeared for them in the form of an MIT Lincoln Laboratory report:
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). They "sum up by saying":
Unlimited Register Machine URM: This, their "most flexible machine... consists of a denumerable sequence of registers numbered 1, 2, 3, ..., each of which can store any natural number...Each particular program, however involves only a finite number of these registers" (p. 219). In other words, the number of registers is potentially infinite, and each register's "size" is infinite.
They offer the following instruction set (p. 219), and the following "Notes":
Notes.
Indeed, they show how to reduce this set further, to the following (for an infinite number of registers each of infinite size):
Limited Register Machine LRM: Here they restrict the machine to a finite number of registers N, but they also allow more registers to "be brought in" or removed if empty (cf. p. 228). They show that the remove-register instruction need not require an empty register.
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. This is shown in their Figure 1 as a tape with a read head on the left and a write head on the right, and it can only move the tape right. "A" is their "word" (p. 229):
They also provide a model as "a stack of cards" with the symbols { 0, 1 } (p. 232 and Appendix C p. 248):
1967: Minsky's "Simple Universal Base for a Program Computer".
Ultimately, in Problem 11.7-1 Minsky observes that many bases of computation can be formed from a tiny collection:
The following are definitions of the various instructions he treats:
Minsky (1967) begins with a model that consists of the three operations plus HALT:
He observes that we can dispense with [ 0 ] if we allow for a specific register e.g. w already "empty" (Minsky (1967) p. 206). Later (pages 255ff) he compresses the three { [ 0 ], [ ' ], [ - ] }, into two { [ ' ], [ - ] }.
But he admits the model is easier if he adds some [pseudo]-instructions [ O- ] (combined [ 0 ] and [ - ]) and "go(n)". He builds "go(n)" out of the register w pre-set to 0, so that [O-] (w, (n)) is an unconditional jump.
In his section 11.5 "The equivalence of Program Machines with General-recursive functions" he introduces two new subroutines:
He proceeds to show how to replace the "successor-predecessor" set { [ 0 ], [ ' ], [ - ] } with the "successor-equality" set { [ 0 ], [ ' ], [ ≠ ] }. And then he defines his "REPEAT" [RPT] and shows that we can define any primitive recursive function by the "successor-repeat" set { [ 0 ], [ ' ], [RPT] } (where the range of the [ RPT ] cannot include itself. If it does, we get what is called the mu operator (see also mu recursive functions) (p. 213)):
1980: Schönhage's 0-parameter model RAM0.
Schönhage (1980) developed his computational model in context of a "new" model he called the Storage Machine Modification model (SMM), his variety of pointer machine. His development described a RAM (random-access machine) model with a remarkable instruction set requiring no operands at all, excepting, perhaps, the "conditional jump" (and even that could be achieved without an operand):
The way Schönhage did this is of interest. He (i) atomizes the conventional register "address:datum" into its two parts: "address", and "datum", and (ii) generates the "address" in a specific register n to which the finite-state machine instructions (i.e. the "machine code") would have access, and (iii) provides an "accumulator" register z where all arithmetic operations are to occur.
In his particular RAM0 model has only two "arithmetic operations""Z" for "set contents of register z to zero", and "A" for "add one to contents of register z". The only access to address-register n is via a copy-from-A-to-N instruction called "set address n". To store a "datum" in accumulator z in a given register, the machine uses the contents of n to specify the register's address and register z to supply the datum to be sent to the register.
Peculiarities: A first peculiarity of the Schönhage RAM0 is how it "loads" something into register z: register z first supplies the register-address and then secondly, receives the datum from the registera form of indirect "load". The second peculiarity is the specification of the COMPARE operation. This is a "jump if accumulator-register z="zero" (not, for example, "compare the contents of z to the contents of the register pointed to by n). Apparently if the test fails the machine skips over the next instruction which always must be in the form of "goto λ" where "λ" is the jump-to address. The instruction"compare contents of z to "zero"" is unlike the Schonhage successor-RAM1 model (or any other known successor-models) with the more conventional "compare contents of register z to contents of register a for equality".
Primarily for referencethis is a RAM model, not a counter-machine modelthe following is the Schönhage RAM0 instruction set:
Again, the above instruction set is for a "random-access machine", a "RAM"a counter machine with indirect addressing; instruction "N" allows for indirect storage of the accumulator, and instruction "L" allows for indirect load of the accumulator.
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. 