Register machine

There are at least four sub-classes found in literature, here listed from most primitive to the most like a computer:

Any properly defined register machine model is Turing equivalent. Computational speed is very dependent on the model specifics.

The last five names are listed explicitly in that order by Yuri Matiyasevich. He follows up with:

"Register machines [some authors use "register machine" synonymous with "counter-machine"] are particularly suitable for constructing Diophantine equations. Like Turing machines, they have very primitive instructions and, in addition, they deal with numbers" (Yuri Matiyasevich (1993), Hilbert's Tenth Problem, commentary to Chapter 5 of the book, at . )

It appears that Lambek, Melzak, Minsky and Shepherdson and Sturgis independently anticipated the same idea at the same time. See Note On Precedence below.

Wang expressed hope that his model would be "a rapprochement" (p. 63) between the theory of Turing machines and the practical world of the computer.

Wang's work was highly influential. We find him referenced by Minsky (1961) and (1967), Melzak (1961), Shepherdson and Sturgis (1963). Indeed, Shepherdson and Sturgis (1963) remark that:

"...we have tried to carry a step further the 'rapprochement' between the practical and theoretical aspects of computation suggested by Wang" (p. 218)

Martin Davis eventually evolved this model into the (2-symbol) Post–Turing machine.

Except there was a problem: the Wang model (the six instructions of the 7-instruction Post–Turing machine) was still a single-tape Turing-like device, however nice its sequential program instruction-flow might be. Both Melzak (1961) and Shepherdson and Sturgis (1963) observed this (in the context of certain proofs and investigations):

"...a Turing machine has a certain opacity... a Turing machine is slow in (hypothetical) operation and, usually, complicated. This makes it rather hard to design it, and even harder to investigate such matters as time or storage optimization or a comparison between efficiency of two algorithms. (Melzak (1961) p. 281)
"...although not difficult ... proofs are complicated and tedious to follow for two reasons: (1) A Turing machine has only head so that one is obliged to break down the computation into very small steps of operations on a single digit. (2) It has only one tape so that one has to go to some trouble to find the number one wishes to work on and keep it separate from other numbers" (Shepherdson and Sturgis (1963) p. 218).
Minsky, Melzak-Lambek and Shepherdson–Sturgis models "cut the tape" into many

So why not 'cut the tape' so each is infinitely long (to accommodate any size integer) but left-ended, and call these three tapes "Post–Turing (i.e. Wang-like) tapes"? The individual heads will move left (for decrement) and right (for increment). In one sense the heads indicate "the tops of the stack" of concatenated marks. Or in Minsky (1961) and Hopcroft and Ullman (1979, p. 171ff) the tape is always blank except for a mark at the left end—at no time does a head ever print or erase.

Melzak's (1961) model is different: clumps of pebbles go into and out of holes

Melzak's (1961) model is significantly different. He took his own model, flipped the tapes vertically, called them "holes in the ground" to be filled with "pebble counters". Unlike Minsky's "increment" and "decrement", Melzak allowed for proper subtraction of any count of pebbles and "adds" of any count of pebbles.

He defines indirect addressing for his model (p. 288) and provides two examples of its use (p. 89); his "proof" (p. 290-292) that his model is Turing equivalent is so sketchy that the reader cannot tell whether or not he intended the indirect addressing to be a requirement for the proof.

Legacy of Melzak's model is Lambek's simplification and the reappearance of his mnemonic conventions in Cook and Reckhow 1973.

Lambek (1961) atomizes Melzak's model into the Minsky (1961) model: INC and DEC-with-test

Lambek (1961) took Melzak's ternary model and atomized it down to the two unary instructions—X+, X- if possible else jump—exactly the same two that Minsky (1961) had come up with.

However, like the Minsky (1961) model, the Lambek model does execute its instructions in a default-sequential manner—both X+ and X- carry the identifier of the next instruction, and X- also carries the jump-to instruction if the zero-test is successful.

Elgot–Robinson (1964) and the problem of the RASP without indirect addressing

Elgot–Robinson investigate the possibility of allowing their RASP model to "self modify" its program instructions. The idea was an old one, proposed by Burks-Goldstine-von Neumann (1946-7), and sometimes called "the computed goto." Melzak (1961) specifically mentions the "computed goto" by name but instead provides his model with indirect addressing.

"... the program computer has to have some way to keep track of how many RPT's remain to be done, and this might exhaust any particular amount of storage allowed in the finite part of the computer. RPT operations require infinite registers of their own, in general, and they must be treated differently from the other kinds of operations we have considered." (p. 214)

By 1971 Hartmanis has simplified the indexing to indirection for use in his RASP model.

Note that the finite state machine does not have to explicitly specify this target register's address. It just says to the rest of the machine: Get me the contents of the register pointed to by my pointer-register and then do xyz with it. It must specify explicitly by name, via its instruction, this pointer-register (e.g. "N", or "72" or "PC", etc.) but it doesn't have to know what number the pointer-register actually contains (perhaps 279,431).

Cook and Reckhow (1973) cite Hartmanis (1971) and simplify his model to what they call a random-access machine (RAM—i.e. a machine with indirection and the Harvard architecture). In a sense we are back to Melzak (1961) but with a much simpler model than Melzak's.

Almost exactly the same thing happened to Shepherdson and Sturgis. Their paper was received in December 1961—just a few months after Melzak and Lambek's work was received. Again, they had little (at most 1 month) or no benefit of reviewing the work of Minsky. They were careful to observe in footnotes that papers by Ershov, Kaphengst and Peter had "recently appeared" (p. 219). These were published much earlier but appeared in the German language in German journals so issues of accessibility present themselves.

The final paper of Shepherdson and Sturgis did not appear in a peer-reviewed journal until 1963. And as they fairly and honestly note in their Appendix A, the 'systems' of Kaphengst (1959), Ershov (1958), Peter (1958) are all so similar to what results were obtained later as to be indistinguishable to a set of the following:

"i.e. of performing the operations which generate the natural numbers" (p. 246)
to "change the course of a computation", either comparing two numbers or decrementing until 0

By order of publishing date the work of Kaphengst (1959), Ershov (1958), Peter (1958) were first.

For a good treatment of the counter machine see Minsky (1967) Chapter 11 "Models similar to Digital Computers"—he calls the counter machine a "program computer". A recent overview is found at van Emde Boas (1990). A recent treatment of the Minsky (1961)/Lambek (1961) model can be found Boolos-Burgess-Jeffrey (2002); they reincarnate Lambek's "abacus model" to demonstrate equivalence of Turing machines and partial recursive functions, and they provide a graduate-level introduction to both abstract machine models (counter- and Turing-) and the mathematics of recursion theory. Beginning with the first edition Boolos-Burgess (1970) this model appeared with virtually the same treatment.

All texts and papers excepting the four starred have been witnessed. These four are written in German and appear as references in Shepherdson–Sturgis (1963) and Elgot–Robinson (1964); Shepherdson–Sturgis (1963) offer a brief discussion of their results in Shepherdson–Sturgis' Appendix A. The terminology of at least one paper (Kaphengst (1959) seems to hark back to the Burke-Goldstine-von Neumann (1946-7) analysis of computer architecture.