 Counter machine

A counter machine is an abstract machine used in formal logic and theoretical computer science to model computation. It is the most primitive of the four types of register machines. A counter machine comprises a set of one or more unbounded registers, each of which can hold a single nonnegative integer, and a list of (usually sequential) arithmetic and control instructions for the machine to follow.
Basic features
For a given counter machine model the instruction set is tiny—from just one to six or seven instructions. Most models contain a few arithmetic operations and at least one conditional operation (if condition is true, then jump). Three base models, each using three instructions, are drawn from the following collection. (The abbreviations are arbitrary.)

 CLR (r): CLeaR register r. (Set r to zero.)
 INC (r): INCrement the contents of register r.
 DEC (r): DECrement the contents of register r.
 CPY (r_{j}, r_{k}): CoPY the contents of register r_{j} to register r_{k} leaving the contents of r_{j} intact.
 JZ (r, z): IF register r contains Zero THEN Jump to instruction z ELSE continue in sequence.
 JE (r_{j}, r_{k}, z): IF the contents of register r_{j} Equals the contents of register r_{k} THEN Jump to instruction z ELSE continue in sequence.
In addition, a machine usually has a HALT instruction, which stops the machine (normally after the result has been computed).
Using the instructions mentioned above, various authors have discussed certain counter machines:
 set 1: { INC (r), DEC (r), JZ (r, z) }, (Minsky (1961, 1967), Lambek (1961))
 set 2: { CLR (r), INC (r), JE (r_{j}, r_{k}, z) }, (Ershov (1958), Peter (1958) as interpreted by ShepherdsonSturgis (1964); Minsky (1967); Schönhage (1980))
 set 3: { INC (r), CPY (r_{j}, r_{k}), JE (r_{j}, r_{k}, z) }, (ElgotRobinson (1964), Minsky (1967))
The three counter machine base models have the same computational power since the instructions of one model can be derived from those of another. All are equivalent to the computational power of Turing machines (but only if Gödel numbers are used to encode data in the register or registers; otherwise their power is equivalent to the primitive recursive functions). Due to their unary processing style, counter machines are typically exponentially slower than comparable Turing machines.
Alternative names, alternative models
Main article: Counter machine modelsThe counter machine models go by a number of different names that may help to distinguish them by their peculiarities. In the following the instruction "JZDEC ( r )" is a compound instruction that tests to see if a register r is empty; if so then jump to instruction I_{z}, else if not then DECrement the contents of r:

 ShepherdsonSturgis machine, because these authors formally stated their model in an easily accessible exposition (1963). Uses instruction set (1) augmented with additional convenience instructions ( JNZ is "Jump if Not Zero, used in place of JZ ):

 { INC ( r ), DEC ( r ), CLR ( r ), CPY ( r_{j}, r_{k} ), JNZ ( r, z ), J ( z ) }
 Minsky machine, because Marvin Minsky (1961) formalized the model. Usually uses instruction set (1), but instructionexecution is not defaultsequential so the additional parameter 'z' appears to specify the next instruction after INC and as the alternative in JZDEC:

 { INC ( r, z ), JZDEC ( r, z_{true}, z_{false}) }
 Program machine, Program computer, the names Minsky (1967) gave the model because, like a computer its instructions proceed sequentially unless a conditional jump is successful. Uses (usually) instruction set (1) but may be augmented similar to the ShephersonSturgis model. JZDEC is often split apart:

 { INC ( r ), DEC ( r ), JZ ( r, z_{true} )}
 Abacus machine, the name Lambek (1961) gave to his simplification of the Melzak (1961) model, and what BoolosBurgessJeffrey (2002) calls it. Uses instruction set (1) but with an additional parameter z to specify the next instruction in the manner of the Minsky (1961) model;

 { INC ( r, z ), JZDEC (r, z_{true}, z_{false} ) }
 Lambek machine, an alternative name BoolosBurgessJeffrey (2002) gave to the abacus machine. Uses instruction set (1reduced) with an additional parameter to specify the next instruction:

 { INC ( r, z ), JZDEC ( r, z_{true}, z_{false} ) }
 Successor machine, because it uses the 'successor operation' of, and closely resembles, the Peano axioms. Used as a base for the successor RAM model. Uses instruction set (2) by e.g. Schönhage as a base for his RAM0 and RAM1 models that lead to his pointer machine SMM model, also discussed briefly in van Emde Boas (1990):


 { CLR ( r ), INC ( r ), JE ( r_{j}, r_{k}, z ) }

 ElgotRobinson model, used to define their RASP model (1964). This model requires one empty register at the start ( e.g. [r0] =0 ). (They augmented the same model with indirect addressing by use of an additional register to be used as an "index" register.)

 { INC (r), CPY ( r_{s}, r_{d} ), JE ( r_{j}, r_{k}, z ) }
 Other counter machines: Minsky (1967) demonstrates how to build the three base models (program/Minsky/Lambekabacus, successor, and ElgotRobinson) from the superset of available instructions described in the lead paragraph of this article. The Melzak (1961) model is quite different from the above because it includes 'add' and 'subtract' rather than 'increment' and 'decrement'. The proofs of Minsky (1961, 1967) that a single register will suffice for Turing equivalence requires the two instructions { MULtiply k, and DIV k } to encode and decode the Gödel number in the register that represents the computation. Minsky shows that if two or more registers are available then the simpler INC, DEC etc. are adequate (but the Gödel number is still required to demonstrate Turing equivalence; also demonstrated in ElgotRobinson 1964 ).
Formal definition
A counter machine consists of:
 Labeled unbounded integervalued registers: a finite (or infinite in some models) set of registers r_{0} ... r_{n} each of which can hold any single nonnegative integer (0, 1, 2, ...  i.e. unbounded). The registers do their own arithmetic; there may or may not be one or more special registers e.g. "accumulator" (See Random access machine for more on this).
 A state register that stores/identifies the current instruction to be executed. This register is finite and separate from the registers above; thus the counter machine model is an example of the Harvard architecture
 List of labelled, sequential instructions: A finite list of instructions I_{0} ... I_{m}. The program store (instructions of the finite state machine) is not in the same physical "space" as the registers. Usually, but not always, like computer programs the instructions are listed in sequential order; unless a jump is successful, the default sequence continues in numerical order. Each of the instructions in the list is from a (very) small set, but this set does not include indirection. The historically most models drew their instructions from this set:

 { Increment (r), Decrement (r), Clear (r); Copy (r_{j},r_{k}), conditional Jump if contents of r=0, conditional Jump if r_{j}=r_{k}, unconditional Jump, HALT }
 Some models have either further atomized some of the above into noparameter instructions, or combined them into a single instruction such as "Decrement" preceded by conditional jumpifzero "JZ ( r, z )" . The atomization of instructions or the inclusion of convenience instructions causes no change in conceptual power, as any program in one variant can be straightforwardly translated to the other.

 Alternative instructionsets are discussed in the supplement Register machine models.
Example: COPY the count from register #1 to #2
This example shows how to create three more useful instructions: clear, unconditional jump, and copy.
 CLR ( j ): Clear the contents of register r_{j} to zero.
 J ( z ): Unconditionally jump to instruction I_{z}.
 CPY ( s, d ): Copy the contents of source register r_{s} to destination register r_{d}. Afterward r_{s} will contain its original count (unlike MOVE which empties the source register, i.e., clears it to zero).
The basic set (1) is used as defined here:
Instruction Effect on register "j" Effect on Instruction Counter Register ICR Summary INC ( j ) [j] +1 → j [IC] +1 → IC Increment contents of register j; next instruction DEC ( j ) [j] 1 → j [IC] +1 → IC Decrement contents of register j; next instruction JZ ( j, z) IF [j] = 0 THEN I_{z} → IC ELSE [IC] +1 → IC If contents of register j=0 then instruction z else next instruction HALT Initial conditions:
Initially, register #2 contains "2". Registers #0, #1 and #3 are empty (contain "0"). Register #0 remains unchanged throughout calculations because it is used for the unconditional jump. Register #1 is a scratch pad. The program begins with instruction 1.
Final conditions:
The program HALTs with the contents of register #2 at its original count and the contents of register #3 equal to the original contents of register #2, i.e.,
 [2] = [3].
Program Highlevel Description:
The program COPY ( #2, #3) has two parts. In the first part the program moves the contents of source register #2 to both scratchpad #1 and to destination register #3; thus #1 and #3 will be copies of one another and of the original count in #2, but #2 is cleared in the process of decrementing it to zero. Unconditional jumps J (z) are done by tests of register #0, which always contains the number 0:
 [#2] →#3; [#2] →#1; 0 →#2
In the second the part the program moves (returns, restores) the contents of scratchpad #1 back to #2, clearing the scratchpad #1 in the process:
 [#1] →#2; 0 →#1
Program: The program, highlighted in yellow, is shown written lefttoright in the upper right.
A "run" of the program is shown below. Time runs down the page. The instructions are in yellow, the registers in blue. The program is flipped 90 degrees, with the instruction numbers (addresses) along the top, the instruction mnemonics under the addresses, and the instruction parameters under the mnemonics (one per cell):
1 2 3 4 5 6 7 8 9 10 ← Instruction number (address) JZ DEC INC INC JZ JZ DEC INC JZ H ← Instruction 2 2 3 1 0 1 1 2 0 ← Register number 6 1 10 6 ← Jumpto instruction number step IC Inst # reg # Jaddr reg0 reg1 reg2 reg3 reg4 IC start 0 0 2 0 0 1 move [#2] to #1 and #3: 1 1 JZ 2 6 0 0 2 0 0 1→2 JZ Jump fails: register #2 contains 2 2 2 DEC 2 0 0 0 2→1 0 0 2→3 DEC Decrement register #2 from 2 to 1 3 3 INC 3 0 0 0 1 0→1 0 3→4 INC Increment register #3 from 0 to 1 4 4 INC 1 0 0 0→1 1 1 0 4→5 INC Increment register #1 from 0 to 1 5 5 JZ 0 1 0 1 1 1 0 5→1 JZ UJump: register #0 is empty 6 1 JZ 2 6 0 1 1 1 0 1→2 JZ Jump fails: register #2 contains 1 7 2 DEC 2 0 0 1 1→0 1 0 2→3 DEC Decrement register #2 from 1 to 0 8 3 INC 3 0 0 1 0 1→2 0 3→4 INC Increment register #3 from 1 to 2 9 4 INC 1 0 0 1→2 0 2 0 4→5 INC Increment register #1 from 1 to 2 10 5 JZ 0 1 0 2 0 2 0 5→1 JZ UJump: register #0 is empty 11 1 JZ 2 6 0 2 0 2 0 1→6 JZ Jump !: register #2 is empty move [1] to 2: 12 6 JZ 1 10 0 2 0 2 0 6→7 JZ Jump fails: register #1 contains 2 13 7 DEC 1 0 0 2→1 0 2 0 7→8 DEC Decrement register #1 from 2 to 1 14 8 INC 2 0 0 1 0→1 2 0 8→9 INC Increment register #2 from 0 to 1 15 9 JZ 0 6 0 1 1 2 0 9→6 JZ UJump: register #0 is empty 16 6 JZ 1 10 0 1 1 2 0 6→7 JZ Jump fails: register #1 contains 1 17 7 DEC 1 0 0 1→0 1 2 0 7→8 DEC Decrement register #1 from 1 to 0 18 8 INC 2 0 0 0 1→2 2 0 8→9 INC Increment register #2 from 1 to 2 19 9 JZ 0 6 0 0 2 2 0 9→6 JZ UJump: register #0 is empty 20 6 JZ 1 10 0 0 2 2 0 6→10 JZ Jump !: register #1 is empty 21 10 H 0 0 0 0 2 2 0 10→10 H HALT The partial recursive functions: building "convenience instructions" using recursion
The example above demonstrates how the first basic instructions { INC, DEC, JZ } can create three more instructions  unconditional jump J, CLR, CPY. In a sense CPY used both CLR and J plus the base set. If register #3 had had contents initially, the sum of contents of #2 and #3 would have ended up in #3. So to be fully accurate CPY program should have preceded its moves with CLR (1) and CLR (2).
However, we do see that ADD would have been possible, easily. And in fact the following is summary of how the primitive recursive functions such as ADD, MULtiply and EXPonent can come about (see BoolosBurgessJeffrey (2002) p. 4551).
 Beginning instruction set: { DEC, INC, JZ, H }
 Define unconditional "Jump J (z)" in terms of JZ ( r0, z ) given that r0 contains 0.
 { J, DEC, INC, JZ, H }
 Define "CLeaR ( r ) in terms of the above:
 { CLR, J, DEC, INC, JZ, H }
 Define "CoPY ( r_{j}, r_{k} )" while preserving contents of r_{j} in terms of the above:
 { CPY, CLR, J, DEC, INC, JZ, H }
 The above is the instruction set of ShepherdsonSturgis (1963).
 Define "ADD ( r_{j}, r_{k}, r_{i} )", (perhaps preserving the contents of r_{j}, and r_{k} ), by use of the above:
 { ADD, CPY, CLR, J, DEC, INC, JZ, H }
 Define " MULtiply ( r_{j}, r_{k}, r_{i} )" (MUL) (perhaps preserving the contents of r1 r2), in terms of the above:
 { MUL, ADD, CPY, CLR, J, DEC, INC, JZ, H }
 Define "EXPonential ( r_{j}, r_{k}, r_{i} )" (EXP) (perhaps preserving the contents of r_{j}, r_{k} ) in terms of the above,
 { EXP, MUL, ADD, CPY, CLR, J, DEC, INC, JZ, H }
In general, we can build any partial or total primitive recursive function that we wish, by using the same methods. Indeed Minsky (1967), ShepherdsonSturgis (1963) and BoolosBurgessJeffrey (2002) give demonstrations of how to form the five primitive recursive function "operators" (15 below) from the base set (1).
But what about full Turing equivalence? We need to add the sixth operator  the μ operator  to obtain the full equivalence, capable of creating the total and partial recursive functions:
 Zero function (or constant function)
 Successor function
 Identity function
 Composition function
 Primitive recursion (induction)
 μ operator (unbounded search operator)
The authors show that this is done easily within any of the available base sets (1, 2, or 3) (an example can be found at μ operator ). However, the reader needs to be cautioned that, even though the μ operator is easily created by the base instruction set doesn't mean that an arbitrary partial recursive function can be easily created with a base model  Turing equivalence and partial recursive functions imply an unbounded μ operator, one that can scurry to the ends of the register chain ad infinitum searching for its goal. The problem is: registers must be called out explicily by number/name e.g. INC (47,528) and DEC (39,347) and this will exhaust the finite state machine's TABLE of instructions. No matter how "big" the finite state machine is, we can find a function that uses a "bigger" number of registers.
Problems with the counter machine model
 The problems are discussed in detail in the article Random access machine. The problems fall into two major classes and a third "inconvenience" class:
(1) Unbounded capacities of registers versus bounded capacities of statemachine instructions: How will the machine create constants larger than the capacity of its finite state machine?
(2) Unbounded numbers of registers versus bounded numbers of statemachine instructions: How will the machine access registers with addressnumbers beyond the reach/capability of its finite state machine?
(3) The fully reduced models are cumbersome:
Shepherdson and Sturgis (1963) are unapologetic about their 6instruction set. They have made their choice based on "ease of programming... rather than economy" (p. 219 footnote 1).
Shepherdson and Sturgis' instructions ( [r] indicates "contents of register r"):

 INCREMENT ( r ) ; [r] +1 → r
 DECREMENT ( r ) ; [r] 1 → r
 CLEAR ( r ) ; 0 → r
 COPY ( r_{s} to r_{d} ) ; [r_{s}] → r_{d}
 JUMPUNCONDITIONAL to instruction I_{z}
 JUMP IF [r] =0 to instruction I_{z}
Minsky (1967) expanded his 2instruction set { INC (z), JZDEC (r, I_{z}) } to { CLR (r), INC (r), JZDEC (r, I_{z}), J (I_{z}) } before his proof that a "Universal Program Machine" can be built with only two registers (p. 255ff).
Twocounter machines are Turing equivalent (with a caveat)
For every Turing machine, there is a 2CM that simulates it, given that the 2CM's input and output are properly encoded. This is proved in Minsky's book (Computation, 1967, p.255258), and an alternative proof is sketched below in three steps. First, a Turing machine can be simulated by a finitestate machine (FSM) equipped with two stacks. Then, two stacks can be simulated by four counters. Finally, four counters can be simulated by two counters.
Step 1: A Turing machine can be simulated by two stacks.
A Turing machine consists of an FSM and an infinite tape, initially filled with zeros, upon which the machine can write ones and zeros. At any time, the read/write head of the machine points to one cell on the tape. This tape can be conceptually cut in half at that point. Each half of the tape can be treated as a stack, where the top is the cell nearest the read/write head, and the bottom is some distance away from the head, with all zeros on the tape beyond the bottom. Accordingly, a Turing machine can be simulated by an FSM plus two stacks. Moving the head left or right is equivalent to popping a bit from one stack and pushing it onto the other. Writing is equivalent to changing the bit before pushing it.
Step 2: A stack can be simulated by two counters.
A stack containing zeros and ones can be simulated by two counters, when the bits on the stack are thought of as representing a binary number, with the top being the least significant bit. Pushing a zero onto the stack is equivalent to doubling the number. Pushing a one is equivalent to doubling and adding 1. Popping is equivalent to dividing by 2, where the remainder is the bit that was popped. Two counters can simulate this stack, in which one of the counters holds a number whose binary representation represents the bits on the stack, and the other counter is used as a scratchpad. To double the number in the first counter, the FSM can initialize the second counter to zero, then repeatedly decrement the first counter once and increment the second counter twice. This continues until the first counter reaches zero. At that point, the second counter will hold the doubled number. Halving is performed by decrementing one counter twice and increment the other once, and repeating until the first counter reaches zero. The remainder can be determined by whether it reached zero after an even or an odd number of tries.
Step 3: Four counters can be simulated by two counters.
As before, one of the counters is used as scratchpad. The other, real counter holds an integer whose prime factorization is 2^{a}3^{b}5^{c}7^{d}. The exponents a, b, c, and d can be thought of as four virtual counters that are being simulated. If the real counter is set to zero then incremented once, that is equivalent to setting all the virtual counters to zero. If the real counter is doubled, that is equivalent to incrementing a, and if it's halved, that's equivalent to decrementing a. By a similar procedure, it can be multiplied or divided by 3, which is equivalent to incrementing or decrementing b. Similarly, c and d can be incremented or decremented. To check if a virtual counter such as c is equal to zero, just divide the real counter by 5, see what the remainder is, then multiply by 5 and add back the remainder. That leaves the real counter unchanged. The remainder will have been nonzero if and only if c was zero.
As a result, an FSM with two counters can simulate four counters, which are in turn simulating two stacks, which are simulating a Turing machine. Therefore, an FSM plus two counters is at least as powerful as a Turing machine. A Turing machine can easily simulate an FSM with two counters, therefore the two machines have equivalent power.
The caveat: *If* its counters are initialised to N and 0, then a 2CM cannot calculate 2^{N}
This result, together with a list of other functions of N that are not calculable by a twocounter machine — when initialised with N in one counter and 0 in the other — such as N^{2}, sqrt(N), log_{2}(N), etc., appears in a paper by Schroeppel (1972). The result is not surprising, because the twocounter machine model was proved (by Minsky) to be universal only when the argument N is appropriately encoded (by Gödelization) to simulate a Turing machine whose initial tape contains N encoded in unary; moreover, the output of the twocounter machine will be similarly encoded. This phenomenon is typical of very small bases of computation whose universality is proved only by simulation (e.g., many Turing tarpits, the smallestknown universal Turing machines, etc.).
The proof is preceded by some interesting theorems:
 "Theorem: A threecounter machine can simulate a Turing machine" (p. 2, also cf Minsky 1967:170174)
 "Theorem: A 3CM [threecounter machine] can compute any partial recursive function of one variable. It starts with the argument [i.e. N] in a counter, and (if it halts) leaves the answer [i.e. F(N)] in a counter." (p. 3)
 "Theorem: A counter machine can be simulated by a 2CM [twocounter machine], provided an obscure coding is accepted for the input and output" [p. 3; the "obscure coding" is: 2^{W}3^{X}5^{Y}7^{Z} where the simulated counters are W, X, Y, Z]
 "Theorem: Any counter machine can be simulated by a 2CM, provided an obscure coding is accepted for the input and output." (p. 3)

 "Corollary: the Halting Problem for 2CM's is unsolvable.
 "Corollary: A 2CM can compute any partial recursive function of one argument, provided the input is coded as 2^{N} and the output (if the machine halts) is coded as 2^{answer}." (p. 3)
 "Theorem: There is no two counter machine that calculates 2^{N} [if one counter is initialised to N]." (p. 11)
With regard to the second theorem that "A 3CM can compute any partial recursive function" the author challenges the reader with a "Hard Problem: Multiply two numbers using ony three counters" (p. 2); he's not kidding, it is a hard problem, and he asks for a better solution. The main proof is clever and difficult and involves the notion that twocounter machines cannot compute arithmetic sequences with nonlinear growth rates (p. 15) i.e. "the function 2^{X} grows more rapidly than any arithmetic progression." (p. 11).
See also
 Counter machine:Reference model
 Halting problem
 Pointer machine
 PostTuring machine
 Random access machine
 Register machine
 Turing machine
 Wang Bmachine
References
 George Boolos, John P. Burgess, Richard Jeffrey (2002), Computability and Logic: Fourth Edition, Cambridge University Press, Cambridge, England. The original BoolosJeffrey text has been extensively revised by Burgess: more advanced than an introductory textbook. "Abacus machine" model is extensively developed in Chapter 5 Abacus Computability; it is one of three models extensively treated and compared  the Turing machine (still in Boolos' original 4tuple form) and recursion the other two.
 Arthur Burks, Herman Goldstine, John von Neumann (1946), Preliminary discussion of the logical design of an electronic computing instrument, reprinted pp. 92ff in Gordon Bell and Allen Newell (1971), Computer Structures: Readings and Examples, mcGrawHill Book Company, New York. ISBN 0070043574 .
 Stephen A. Cook and Robert A. Reckhow (1972), Timebounded random access machines, Journal of Computer Systems Science 7 (1973), 354375.
 Martin Davis (1958), Computability & Unsolvability, McGrawHill Book Company, Inc. New York.
 Calvin Elgot and Abraham Robinson (1964), RandomAccess StoredProgram Machines, an Approach to Programming Languages, Journal of the Association for Computing Machinery, Vol. 11, No. 4 (October, 1964), pp. 365399.
 Fischer, Patrick C.; Meyer, A. R.; Rosenberg, Arnold L. (1968), "Counter machines and counter languages", Mathematical Systems Theory 2: 265–283, MR0235932. Develops time hierarchy and space hierarchy theorems for counter machines, analogous to the hierarchies for Turing machines.
 J. Hartmanis (1971), "Computational Complexity of Random Access Stored Program Machines," Mathematical Systems Theory 5, 3 (1971) pp. 232245.
 Hopcroft, John; Jeffrey Ullman (1979). Introduction to Automata Theory, Languages and Computation (1st ed.). Reading Mass: AddisonWesley. ISBN 020102988X. A difficult book centered around the issues of machineinterpretation of "languages", NPCompleteness, etc.
 Stephen Kleene (1952), Introduction to Metamathematics, NorthHolland Publishing Company, Amsterdam, Netherlands. ISBN 0720421039.
 Donald Knuth (1968), The Art of Computer Programming, Second Edition 1973, AddisonWesley, Reading, Massachusetts. Cf pages 462463 where he defines "a new kind of abstract machine or 'automaton' which deals with linked structures."
 Joachim Lambek (1961, received 15 June 1961), How to Program an Infinite Abacus, Mathematical Bulletin, vol. 4, no. 3. September 1961 pages 295302. In his Appendix II, Lambek proposes a "formal definition of 'program'. He references Melzak (1961) and Kleene (1952) Introduction to Metamathematics.
 Z. A. Melzak (1961, received 15 May 1961), An informal Arthmetical Approach to Computability and Computation, Canadian Mathematical Bulletin, vol. 4, no. 3. September 1961 pages 279293. Melzak offers no references but acknowledges "the benefit of conversations with Drs. R. Hamming, D. McIlroy and V. Vyssots of the Bell telephone Laborators and with Dr. H. Wang of Oxford University."
 Marvin Minsky (1961, received August 15, 1960). "Recursive Unsolvability of Post's Problem of 'Tag' and Other Topics in Theory of Turing Machines". Annals of Math (Annals of Mathematics) 74 (3): 437–455. doi:10.2307/1970290. JSTOR 1970290.
 Marvin Minsky (1967). Computation: Finite and Infinite Machines (1st ed.). Englewood Cliffs, N. J.: PrenticeHall, Inc.. In particular see chapter 11: Models Similar to Digital Computers and chapter 14: Very Simple Bases for Computability. In the former chapter he defines "Program machines" and in the later chapter he discusses "Universal Program machines with Two Registers" and "...with one register", etc.
 John C. Shepherdson and H. E. Sturgis (1961) received December 1961 Computability of Recursive Functions, Journal of the Association of Computing Machinery (JACM) 10:217255, 1963. An extremely valuable reference paper. In their Appendix A the authors cite 4 others with reference to "Minimality of Instructions Used in 4.1: Comparison with Similar Systems".

 Kaphengst, Heinz, Eine Abstrakte programmgesteuerte Rechenmaschine', Zeitschrift fur mathematische Logik und Grundlagen der Mathematik:5 (1959), 366379.
 Ershov, A. P. On operator algorithms, (Russian) Dok. Akad. Nauk 122 (1958), 967970. English translation, Automat. Express 1 (1959), 2023.
 Péter, Rózsa Graphschemata und rekursive Funktionen, Dialectica 12 (1958), 373.
 Hermes, Hans Die Universalität programmgesteuerter Rechenmaschinen. Math.Phys. Semesterberichte (Göttingen) 4 (1954), 4253.
 A. Schōnhage (1980), Storage Modification Machines, Society for Industrial and Applied Mathematics, SIAM J. Comput. Vol. 9, No. 3, August 1980. Wherein Schōnhage shows the equivalence of his SMM with the "successor RAM" (Random Access Machine), etc.
 Rich Schroeppel, May 1972, "A Two counter Machine Cannot Calculate 2^{N}", Massachusetts Institute of Technology, A. I. Laboratory, Artificial Intelligence Memo #257. The author references Minsky 1967 and notes that "Frances Yao independently proved the noncomputability using a similar method in April 1971."
 Peter van Emde Boas, Machine Models and Simulations pp.366, appearing in:

 Jan van Leeuwen, ed. "Handbbook of Theoretical Computer Science. Volumne A: Algorithms and Complexity, The MIT PRESS/Elsevier, 1990. ISBN 0444880712 (volume A). QA 76.H279 1990.
 van Emde Boas' treatment of SMMs appears on pp. 3235. This treatment clarifies Schōnhage 1980  it closely follows but expands slightly the Schōnhage treatment. Both references may be needed for effective understanding.
 Hao Wang (1957), A Variant to Turing's Theory of Computing Machines, JACM (Journal of the Association for Computing Machinery) 4; 6392. Presented at the meeting of the Association, June 2325, 1954.
External links
Categories: Register machines

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