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mathematical logic Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of forma ...
and
theoretical computer science computer science (TCS) is a subset of general computer science and mathematics that focuses on mathematical aspects of computer science such as the theory of computation, lambda calculus, and type theory. It is difficult to circumscribe the ...
a register machine is a generic class of
abstract machine An abstract machine is a computer science theoretical model that allows for a detailed and precise analysis of how a computer system functions. It is analogous to a mathematical function in that it receives inputs and produces outputs based on pr ...
s used in a manner similar to a
Turing machine A Turing machine is a mathematical model of computation describing an abstract machine that manipulates symbols on a strip of tape according to a table of rules. Despite the model's simplicity, it is capable of implementing any computer alg ...
. All the models are Turing equivalent.


Overview

The register machine gets its name from its use of one or more " registers". In contrast to the tape and head used by a
Turing machine A Turing machine is a mathematical model of computation describing an abstract machine that manipulates symbols on a strip of tape according to a table of rules. Despite the model's simplicity, it is capable of implementing any computer alg ...
, the model uses multiple, uniquely addressed registers, each of which holds a single positive
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
. There are at least four sub-classes found in literature, here listed from most primitive to the most like a
computer A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations ( computation) automatically. Modern digital electronic computers can perform generic sets of operations known as programs. These prog ...
: *
Counter machine A counter machine is an abstract machine used in a formal logic and theoretical computer science to model of computation, model computation. It is the most primitive of the four types of register machines. A counter machine comprises a set of one o ...
– the most primitive and reduced theoretical model of a computer hardware. Lacks indirect addressing. Instructions are in the finite state machine in the manner of the
Harvard architecture The Harvard architecture is a computer architecture with separate storage and signal pathways for instructions and data. It contrasts with the von Neumann architecture, where program instructions and data share the same memory and pathway ...
. *
Pointer machine In theoretical computer science a pointer machine is an "atomistic" ''abstract computational machine'' model akin to the random-access machine. A pointer algorithm is an algorithm restricted to the pointer machine model. Depending on the type, a ...
– a blend of counter machine and RAM models. Less common and more abstract than either model. Instructions are in the finite state machine in the manner of the Harvard architecture. *
Random-access machine In computer science, random-access machine (RAM) is an abstract machine in the general class of register machines. The RAM is very similar to the counter machine but with the added capability of 'indirect addressing' of its registers. Like the c ...
(RAM) – a counter machine with indirect addressing and, usually, an augmented instruction set. Instructions are in the finite state machine in the manner of the Harvard architecture. *
Random-access stored-program machine In theoretical computer science the random-access stored-program (RASP) machine model is an abstract machine used for the purposes of algorithm development and algorithm complexity theory. The RASP is a random-access machine (RAM) model that, un ...
model (RASP) – a RAM with instructions in its registers analogous to the
Universal Turing machine In computer science, a universal Turing machine (UTM) is a Turing machine that can simulate an arbitrary Turing machine on arbitrary input. The universal machine essentially achieves this by reading both the description of the machine to be simu ...
; thus it is an example of the
von Neumann architecture The von Neumann architecture — also known as the von Neumann model or Princeton architecture — is a computer architecture based on a 1945 description by John von Neumann, and by others, in the '' First Draft of a Report on the EDVAC''. T ...
. But unlike a computer, the model is ''idealized'' with effectively infinite registers (and if used, effectively infinite special registers such as an accumulator). Compared to a computer, the instruction set is much reduced in number and complexity. Any properly defined register machine model is Turing equivalent. Computational speed is very dependent on the model specifics. In practical computer science, a similar concept known as a
virtual machine In computing, a virtual machine (VM) is the virtualization/ emulation of a computer system. Virtual machines are based on computer architectures and provide functionality of a physical computer. Their implementations may involve specialized h ...
is sometimes used to minimise dependencies on underlying machine architectures. Such machines are also used for teaching. The term "register machine" is sometimes used to refer to a virtual machine in textbooks.


Formal definition

A register machine consists of: #An unbounded number of labeled, discrete, unbounded registers unbounded in extent (capacity): a finite (or infinite in some models) set of registers r_0 \ldots r_n each considered to be of infinite extent and each of which holds a single non-negative integer (0, 1, 2, ...). The registers may do their own arithmetic, or there may be one or more special registers that do the arithmetic e.g. an "accumulator" and/or "address register". ''See also
Random-access machine In computer science, random-access machine (RAM) is an abstract machine in the general class of register machines. The RAM is very similar to the counter machine but with the added capability of 'indirect addressing' of its registers. Like the c ...
.'' #Tally counters or marks: discrete, indistinguishable objects or marks of only one sort suitable for the model. In the most-reduced
counter machine A counter machine is an abstract machine used in a formal logic and theoretical computer science to model of computation, model computation. It is the most primitive of the four types of register machines. A counter machine comprises a set of one o ...
model, per each arithmetic operation only one object/mark is either added to or removed from its location/tape. In some counter machine models (e.g. Melzak (1961), Minsky (1961)) and most RAM and RASP models more than one object/mark can be added or removed in one operation with "addition" and usually "subtraction"; sometimes with "multiplication" and/or "division". Some models have control operations such as "copy" (variously: "move", "load", "store") that move "clumps" of objects/marks from register to register in one action. #A (very) limited set of instructions: the instructions tend to divide into two classes: arithmetic and control. The instructions are drawn from the two classes to form "instruction-sets", such that an instruction set must allow the model to be Turing equivalent (it must be able to compute any partial recursive function). ##Arithmetic: arithmetic instructions may operate on all registers or on just a special register (e.g. accumulator). They are ''usually'' chosen from the following sets (but exceptions abound): ##*Counter machine: ##*Reduced RAM, RASP: ##*Augmented RAM, RASP: All of the reduced instructions plus: ##Control: ##*Counter machine models: optional ##*RAM and RASP models: most have , or ##*All models: at least one ''conditional "jump"'' (branch, goto) following test of a register e.g. ##*All models optional: ##Register-addressing method: ##*Counter machine: no indirect addressing, immediate operands possible in highly atomized models ##*RAM and RASP: indirect addressing available, immediate operands typical ##Input-output: optional in all models #State register: A special Instruction Register "IR", finite and separate from the registers above, stores the current instruction to be executed and its address in the TABLE of instructions; this register and its TABLE is located in the finite state machine. #*The IR is off-limits to all models. In the case of the RAM and RASP, for purposes of determining the "address" of a register, the model can select either (i) in the case of direct addressing—the address specified by the TABLE and temporarily located in the IR or (ii) in the case of indirect addressing—the contents of the register specified by the IR's instruction. #*The IR is ''not'' the "program counter" (PC) of the RASP (or conventional computer). The PC is just another register similar to an accumulator, but dedicated to holding the number of the RASP's current register-based instruction. Thus a RASP has ''two'' "instruction/program" registers—(i) the IR (finite state machine's Instruction Register), and (ii) a PC (Program Counter) for the program located in the registers. (As well as a register dedicated to "the PC", a RASP may dedicate another register to "the Program-Instruction Register" (going by any number of names such as "PIR, "IR", "PR", etc.) #List of labeled instructions, usually in sequential order: A finite list of instructions I_1 \ldots I_m. In the case of the counter machine, random-access machine (RAM) and pointer machine the instruction store is in the "TABLE" of the finite state machine; thus these models are example of the Harvard architecture. In the case of the RASP the program store is in the registers; thus this is an example of the von Neumann architecture. ''See also Random-access machine and Random-access stored-program machine.''
Usually, like
computer program A computer program is a sequence or set of instructions in a programming language for a computer to Execution (computing), execute. Computer programs are one component of software, which also includes software documentation, documentation and oth ...
s, the instructions are listed in sequential order; unless a jump is successful the default sequence continues in numerical order. An exception to this is the abacus (Lambek (1961), Minsky (1961)) counter machine models—every instruction has at least one "next" instruction identifier "z", and the conditional branch has two. #*Observe also that the abacus model combines two instructions, JZ then DEC: e.g. .
See McCarthy Formalism for more about the ''conditional expression'' "IF r=0 THEN ztrue ELSE zfalse" (cf McCarthy (1960)).


Historical development of the register machine model

Two trends appeared in the early 1950s—the first to characterize the computer as a Turing machine, the second to define computer-like models—models with sequential instruction sequences and conditional jumps—with the power of a Turing machine, i.e. a so-called Turing equivalence. Need for this work was carried out in context of two "hard" problems: the unsolvable word problem posed by
Emil Post Emil Leon Post (; February 11, 1897 – April 21, 1954) was an American mathematician and logician. He is best known for his work in the field that eventually became known as computability theory. Life Post was born in Augustów, Suwałki Gove ...
—his problem of "tag"—and the very "hard" problem of
Hilbert's problems Hilbert's problems are 23 problems in mathematics published by German mathematician David Hilbert in 1900. They were all unsolved at the time, and several proved to be very influential for 20th-century mathematics. Hilbert presented ten of the pro ...
—the 10th question around
Diophantine equation In mathematics, a Diophantine equation is an equation, typically a polynomial equation in two or more unknowns with integer coefficients, such that the only solutions of interest are the integer ones. A linear Diophantine equation equates to a ...
s. Researchers were questing for Turing-equivalent models that were less "logical" in nature and more "arithmetic" (cf Melzak (1961) p. 281, Shepherdson–Sturgis (1963) p. 218). The first trend—toward characterizing computers—seems to have originatedSee the "Note" in Shepherdson and Sturgis 1963:219. In their Appendix A the authors follow up with a listing and discussions of Kaphengst's, Ershov's and Péter's instruction sets (cf p. 245ff). with Hans Hermes (1954),
Rózsa Péter Rózsa Péter, born Rózsa Politzer, (17 February 1905 – 16 February 1977) was a Hungarian mathematician and logician. She is best known as the "founding mother of recursion theory". Early life and education Péter was born in Budapest, ...
(1958), and Heinz Kaphengst (1959), the second trend with Hao Wang (1954, 1957) and, as noted above, furthered along by Zdzislaw Alexander Melzak (1961),
Joachim Lambek Joachim "Jim" Lambek (5 December 1922 – 23 June 2014) was a German-born Canadian mathematician. He was Peter Redpath Emeritus Professor of Pure Mathematics at McGill University, where he earned his PhD degree in 1950 with Hans Zassenhaus ...
(1961) and
Marvin Minsky Marvin Lee Minsky (August 9, 1927 – January 24, 2016) was an American cognitive and computer scientist concerned largely with research of artificial intelligence (AI), co-founder of the Massachusetts Institute of Technology's AI laboratory ...
(1961, 1967). The last five names are listed explicitly in that order by Yuri Matiyasevich. He follows up with: :"Register machines ome 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 http://logic.pdmi.ras.ru/yumat/H10Pbook/commch_5htm. ) It appears that Lambek, Melzak, Minsky and Shepherdson and Sturgis independently anticipated the same idea at the same time. See Note On Precedence below. The history begins with Wang's model.


Wang's (1954, 1957) model: Post–Turing machine

Wang's work followed from Emil Post's (1936) paper and led Wang to his definition of his
Wang B-machine As presented by Hao Wang (1954, 1957), his basic machine B is an extremely simple computational model equivalent to the Turing machine. It is "the first formulation of a Turing-machine theory in terms of computer-like models" (Minsky, 1967: 200). ...
—a two-symbol
Post–Turing machine A Post–Turing machineRajendra Kumar, ''Theory of Automata'', Tata McGraw-Hill Education, 2010, p. 343. is a "program formulation" of a type of Turing machine, comprising a variant of Emil Post's Turing-equivalent model of computation. Post's mode ...
computation model with only four atomic instructions: : To these four both Wang (1954, 1957) and then C.Y. Lee (1961) added another instruction from the Post set , and then a Post's unconditional jump (or to make things easier, the conditional jump JUMP_IF_blank_to_instruction_z, or both. Lee named this a "W-machine" model: : 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. Difficulties with the Wang/Post–Turing model: 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). Indeed, as examples at Turing machine examples, Post–Turing machine and
partial function In mathematics, a partial function from a set to a set is a function from a subset of (possibly itself) to . The subset , that is, the domain of viewed as a function, is called the domain of definition of . If equals , that is, if is de ...
show, the work can be "complicated".


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. We just have to be careful to write our instructions so that a test-for-zero and jump occurs ''before'' we decrement otherwise our machine will "fall off the end" or "bump against the end"—we will have an instance of a
partial function In mathematics, a partial function from a set to a set is a function from a subset of (possibly itself) to . The subset , that is, the domain of viewed as a function, is called the domain of definition of . If equals , that is, if is de ...
. Before a decrement our machine must always ask the question: "Is the tape/counter empty? If so then I can't decrement, otherwise I can." Minsky (1961) and Shepherdson–Sturgis (1963) prove that only a few tapes—as few as one—still allow the machine to be Turing equivalent ''IF'' the data on the tape is represented as a Gödel number (or some other uniquely encodable-decodable number); this number will evolve as the computation proceeds. In the one tape version with Gödel number encoding the counter machine must be able to (i) multiply the Gödel number by a constant (numbers "2" or "3"), and (ii) divide by a constant (numbers "2" or "3") and jump if the remainder is zero. Minsky (1967) shows that the need for this bizarre instruction set can be relaxed to and the convenience instructions if two tapes are available. A simple Gödelization is still required, however. A similar result appears in Elgot–Robinson (1964) with respect to their RASP model.


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

A RASP or random-access stored-program machine begins as a counter machine with its "program of instruction" placed in its "registers". Analogous to, but independent of, the finite state machine's "Instruction Register", at least one of the registers (nicknamed the "program counter" (PC)) and one or more "temporary" registers maintain a record of, and operate on, the current instruction's number. The finite state machine's TABLE of instructions is responsible for (i) fetching the current ''program'' instruction from the proper register, (ii) parsing the ''program'' instruction, (iii) fetching operands specified by the ''program '' instruction, and (iv) executing the ''program'' instruction. Except there is a problem: If based on the ''counter machine'' chassis this computer-like,
von Neumann Von Neumann may refer to: * John von Neumann (1903–1957), a Hungarian American mathematician * Von Neumann family * Von Neumann (surname), a German surname * Von Neumann (crater), a lunar impact crater See also * Von Neumann algebra * Von Ne ...
machine will not be Turing equivalent. It cannot compute everything that is computable. Intrinsically the model is bounded by the size of its (very-) ''finite'' state machine's instructions. The counter machine based RASP can compute any
primitive recursive function In computability theory, a primitive recursive function is roughly speaking a function that can be computed by a computer program whose loops are all "for" loops (that is, an upper bound of the number of iterations of every loop can be determined ...
(e.g. multiplication) but not all
mu recursive function In mathematical logic and computer science, a general recursive function, partial recursive function, or μ-recursive function is a partial function from natural numbers to natural numbers that is "computable" in an intuitive sense – as well as i ...
s (e.g. the
Ackermann function In computability theory, the Ackermann function, named after Wilhelm Ackermann, is one of the simplest and earliest-discovered examples of a total computable function that is not primitive recursive. All primitive recursive functions are total ...
). 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. Computed goto: A RASP ''program'' of instructions that modifies the "goto address" in a conditional- or unconditional-jump ''program'' instruction. But this does not solve the problem (unless one resorts to Gödel numbers). What is necessary is a method to fetch the address of a program instruction that lies (far) "beyond/above" the upper bound of the ''finite'' state machine's instruction register and TABLE. :Example: A counter machine equipped with only four unbounded registers can e.g. multiply any two numbers ( m, n ) together to yield p—and thus be a primitive recursive function—no matter how large the numbers m and n; moreover, less than 20 instructions are required to do this! e.g. :However, with only 4 registers, this machine has not nearly big enough to build a RASP that can execute the multiply algorithm as a ''program''. No matter how big we build our finite state machine there will always be a ''program'' (including its parameters) which is larger. So by definition the bounded program machine that does not use unbounded encoding tricks such as Gödel numbers cannot be ''universal''. Minsky (1967) hints at the issue in his investigation of a counter machine (he calls them "program computer models") equipped with the instructions . He doesn't tell us how to fix the problem, but he does observe that: : "... 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) But Elgot and Robinson solve the problem: They augment their P0 RASP with an indexed set of instructions—a somewhat more complicated (but more flexible) form of indirect addressing. Their P'0 model addresses the registers by adding the contents of the "base" register (specified in the instruction) to the "index" specified explicitly in the instruction (or vice versa, swapping "base" and "index"). Thus the indexing P'0 instructions have one more parameter than the non-indexing P0 instructions: : Example: INC ( rbase, index ) ; effective address will be base+ index, where the natural number "index" is derived from the finite-state machine instruction itself.


Hartmanis (1971)

By 1971 Hartmanis has simplified the indexing to
indirection In computer programming, indirection (also called dereferencing) is the ability to reference something using a name, reference, or container instead of the value itself. The most common form of indirection is the act of manipulating a value throug ...
for use in his RASP model. Indirect addressing: A pointer-register supplies the finite state machine with the address of the target register required for the instruction. Said another way: The ''contents'' of the pointer-register is the ''address'' of the "target" register to be used by the instruction. If the pointer-register is unbounded, the RAM, and a suitable RASP built on its chassis, will be Turing equivalent. The target register can serve either as a source or destination register, as specified by the instruction. 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) describe the RAM

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.


Precedence

Minsky was working at the
MIT Lincoln Laboratory The MIT Lincoln Laboratory, located in Lexington, Massachusetts, is a United States Department of Defense federally funded research and development center chartered to apply advanced technology to problems of national security. Research and dev ...
and published his work there; his paper was received for publishing in the ''Annals of Mathematics'' on August 15, 1960, but not published until November 1961. While receipt occurred a full year before the work of Melzak and Lambek was received and published (received, respectively, May and June 15, 1961, and published side-by-side September 1961). That (i) both were Canadians and published in the
Canadian Mathematical Bulletin The ''Canadian Mathematical Bulletin'' (french: Bulletin Canadien de Mathématiques) is a mathematics journal, established in 1958 and published quarterly by the Canadian Mathematical Society. The current editors-in-chief of the journal are Anton ...
, (ii) neither would have had reference to Minsky's work because it was not yet published in a peer-reviewed journal, but (iii) Melzak references Wang, and Lambek references Melzak, leads one to hypothesize that their work occurred simultaneously and independently. 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: : produce 0 i.e. 0 → n : increment a number i.e. n+1 → n ::"i.e. of performing the operations which generate the natural numbers" (p. 246) : copy a number i.e. n → m : to "change the course of a computation", either comparing two numbers or decrementing until 0 Indeed, Shepherson and Sturgis conclude ::"The various minimal systems are very similar"( p. 246) By order of ''publishing'' date the work of Kaphengst (1959), Ershov (1958), Peter (1958) were first.


See also

*
Counter machine A counter machine is an abstract machine used in a formal logic and theoretical computer science to model of computation, model computation. It is the most primitive of the four types of register machines. A counter machine comprises a set of one o ...
**
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 "co ...
*
Pointer machine In theoretical computer science a pointer machine is an "atomistic" ''abstract computational machine'' model akin to the random-access machine. A pointer algorithm is an algorithm restricted to the pointer machine model. Depending on the type, a ...
*
Random-access machine In computer science, random-access machine (RAM) is an abstract machine in the general class of register machines. The RAM is very similar to the counter machine but with the added capability of 'indirect addressing' of its registers. Like the c ...
*
Random-access stored-program machine In theoretical computer science the random-access stored-program (RASP) machine model is an abstract machine used for the purposes of algorithm development and algorithm complexity theory. The RASP is a random-access machine (RAM) model that, un ...
*
Turing machine A Turing machine is a mathematical model of computation describing an abstract machine that manipulates symbols on a strip of tape according to a table of rules. Despite the model's simplicity, it is capable of implementing any computer alg ...
**
Universal Turing machine In computer science, a universal Turing machine (UTM) is a Turing machine that can simulate an arbitrary Turing machine on arbitrary input. The universal machine essentially achieves this by reading both the description of the machine to be simu ...
** Turing machine gallery ** Turing machine examples *
Wang B-machine As presented by Hao Wang (1954, 1957), his basic machine B is an extremely simple computational model equivalent to the Turing machine. It is "the first formulation of a Turing-machine theory in terms of computer-like models" (Minsky, 1967: 200). ...
*
Post–Turing machine A Post–Turing machineRajendra Kumar, ''Theory of Automata'', Tata McGraw-Hill Education, 2010, p. 343. is a "program formulation" of a type of Turing machine, comprising a variant of Emil Post's Turing-equivalent model of computation. Post's mode ...
- description plus examples *
Algorithm In mathematics and computer science, an algorithm () is a finite sequence of rigorous instructions, typically used to solve a class of specific problems or to perform a computation. Algorithms are used as specifications for performing ...
** Algorithm characterizations *
Halting problem In computability theory, the halting problem is the problem of determining, from a description of an arbitrary computer program and an input, whether the program will finish running, or continue to run forever. Alan Turing proved in 1936 that a ...
* Busy beaver *
Stack machine In computer science, computer engineering and programming language implementations, a stack machine is a computer processor or a virtual machine in which the primary interaction is moving short-lived temporary values to and from a push down ...
* WDR paper computer


Bibliography

Background texts: The following bibliography of source papers includes a number of texts to be used as background. The mathematics that led to the flurry of papers about abstract machines in the 1950s and 1960s can be found in van Heijenoort (1967)—an assemblage of original papers spanning the 50 years from Frege (1879) to Gödel (1931). Davis (ed.) ''The Undecidable'' (1965) carries the torch onward beginning with Gödel (1931) through Gödel's (1964) postscriptum (p. 71); the original papers of
Alan Turing Alan Mathison Turing (; 23 June 1912 – 7 June 1954) was an English mathematician, computer scientist, logician, cryptanalyst, philosopher, and theoretical biologist. Turing was highly influential in the development of theoretical ...
(1936-7) and Emil Post (1936) are included in ''The Undecidable''. The mathematics of Church, Rosser and Kleene that appear as reprints of original papers in ''The Undecidable'' is carried further in Kleene (1952), a mandatory text for anyone pursuing a deeper understanding of the mathematics behind the machines. Both Kleene (1952) and Davis (1958) are referenced by a number of the papers. 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. The papers: The papers begin with Wang (1957) and his dramatic simplification of the Turing machine. Turing (1936), Kleene (1952), Davis (1958) and in particular Post (1936) are cited in Wang (1957); in turn, Wang is referenced by Melzak (1961), Minsky (1961) and Shepherdson–Sturgis (1961-3) as they independently reduce the Turing tapes to "counters". Melzak (1961) provides his pebble-in-holes counter machine model with indirection but doesn't carry the treatment further. The work of Elgot–Robinson (1964) define the RASP—the computer-like random-access stored-program machines—and appear to be the first to investigate the failure of the bounded
counter machine A counter machine is an abstract machine used in a formal logic and theoretical computer science to model of computation, model computation. It is the most primitive of the four types of register machines. A counter machine comprises a set of one o ...
to calculate the mu-recursive functions. This failure—except with the draconian use of Gödel numbers in the manner of Minsky (1961))—leads to their definition of "indexed" instructions (i.e. indirect addressing) for their RASP model. Elgot–Robinson (1964) and more so Hartmanis (1971) investigate RASPs with self-modifying programs. Hartmanis (1971) specifies an instruction set with indirection, citing lecture notes of Cook (1970). For use in investigations of computational complexity Cook and his graduate student Reckhow (1973) provide the definition of a RAM (their model and mnemonic convention are similar to Melzak's, but offer him no reference in the paper). The pointer machines are an offshoot of Knuth (1968, 1973) and independently Schönhage (1980). For the most part the papers contain mathematics beyond the undergraduate level—in particular the
primitive recursive function In computability theory, a primitive recursive function is roughly speaking a function that can be computed by a computer program whose loops are all "for" loops (that is, an upper bound of the number of iterations of every loop can be determined ...
s and
mu recursive function In mathematical logic and computer science, a general recursive function, partial recursive function, or μ-recursive function is a partial function from natural numbers to natural numbers that is "computable" in an intuitive sense – as well as i ...
s presented elegantly in Kleene (1952) and less in depth, but useful nonetheless, in Boolos-Burgess-Jeffrey (2002). 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.


References

Notes Sources *
George Boolos George Stephen Boolos (; 4 September 1940 – 27 May 1996) was an American philosopher and a mathematical logician who taught at the Massachusetts Institute of Technology. Life Boolos is of Greek- Jewish descent. He graduated with an A.B. ...
,
John P. Burgess John Patton Burgess (born 5 June 1948) is an American philosopher. He is John N. Woodhull Professor of Philosophy at Princeton University where he specializes in logic and philosophy of mathematics. Education and career Burgess received his Ph ...
,
Richard Jeffrey Richard Carl Jeffrey (August 5, 1926 – November 9, 2002) was an American philosopher, logician, and probability theorist. He is best known for developing and championing the philosophy of radical probabilism and the associated heuristic of ...
(2002), ''Computability and Logic: Fourth Edition'', Cambridge University Press, Cambridge, England. The original Boolos-Jeffrey 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 4-tuple form) and recursion the other two. *
Arthur Burks Arthur Walter Burks (October 13, 1915 – May 14, 2008) was an American mathematician who worked in the 1940s as a senior engineer on the project that contributed to the design of the ENIAC, the first general-purpose electronic digital computer. ...
,
Herman Goldstine Herman Heine Goldstine (September 13, 1913 – June 16, 2004) was a mathematician and computer scientist, who worked as the director of the IAS machine at Princeton University's Institute for Advanced Study and helped to develop ENIAC, th ...
,
John von Neumann John von Neumann (; hu, Neumann János Lajos, ; December 28, 1903 – February 8, 1957) was a Hungarian-American mathematician, physicist, computer scientist, engineer and polymath. He was regarded as having perhaps the widest c ...
(1946), "Preliminary discussion of the logical design of an electronic computing instrument", reprinted pp. 92ff in
Gordon Bell Chester Gordon Bell (born August 19, 1934) is an American electrical engineer and manager. An early employee of Digital Equipment Corporation (DEC) 1960–1966, Bell designed several of their PDP machines and later became Vice President of Engi ...
and
Allen Newell Allen Newell (March 19, 1927 – July 19, 1992) was a researcher in computer science and cognitive psychology at the RAND Corporation and at Carnegie Mellon University’s School of Computer Science, Tepper School of Business, and Depart ...
(1971), ''Computer Structures: Readings and Examples'', McGraw-Hill Book Company, New York. . * Stephen A. Cook and Robert A. Reckhow (1972), ''Time-bounded random access machines'', Journal of Computer Systems Science 7 (1973), 354–375. * Martin Davis (1958), ''Computability & Unsolvability'', McGraw-Hill Book Company, Inc. New York. * Calvin Elgot and
Abraham Robinson Abraham Robinson (born Robinsohn; October 6, 1918 – April 11, 1974) was a mathematician who is most widely known for development of nonstandard analysis, a mathematically rigorous system whereby infinitesimal and infinite numbers were reincorp ...
(1964), "Random-Access Stored-Program Machines, an Approach to Programming Languages", ''Journal of the Association for Computing Machinery'', Vol. 11, No. 4 (October, 1964), pp. 365–399. *
J. Hartmanis ''J. The Jewish News of Northern California'', formerly known as ''Jweekly'', is a weekly print newspaper in Northern California, with its online edition updated daily. It is owned and operated by San Francisco Jewish Community Publications In ...
(1971), "Computational Complexity of Random Access Stored Program Machines," ''Mathematical Systems Theory'' 5, 3 (1971) pp. 232–245. *
John Hopcroft John Edward Hopcroft (born October 7, 1939) is an American theoretical computer scientist. His textbooks on theory of computation (also known as the Cinderella book) and data structures are regarded as standards in their fields. He is the IBM P ...
,
Jeffrey Ullman Jeffrey David Ullman (born November 22, 1942) is an American computer scientist and the Stanford W. Ascherman Professor of Engineering, Emeritus, at Stanford University. His textbooks on compilers (various editions are popularly known as the d ...
(1979). ''Introduction to Automata Theory, Languages and Computation'', 1st ed., Reading Mass: Addison-Wesley. . A difficult book centered around the issues of machine-interpretation of "languages", NP-Completeness, etc. *
Stephen Kleene Stephen Cole Kleene ( ; January 5, 1909 – January 25, 1994) was an American mathematician. One of the students of Alonzo Church, Kleene, along with Rózsa Péter, Alan Turing, Emil Post, and others, is best known as a founder of the branch of ...
(1952), ''Introduction to Metamathematics'', North-Holland Publishing Company, Amsterdam, Netherlands. . *
Donald Knuth Donald Ervin Knuth ( ; born January 10, 1938) is an American computer scientist, mathematician, and professor emeritus at Stanford University. He is the 1974 recipient of the ACM Turing Award, informally considered the Nobel Prize of computer sc ...
(1968), ''The Art of Computer Programming'', Second Edition 1973, Addison-Wesley, Reading, Massachusetts. Cf pages 462-463 where he defines "a new kind of abstract machine or 'automaton' which deals with linked structures." * The manuscript was received by the journal on June 15, 1961. In his Appendix II, Lambek proposes a "formal definition of 'program'. He references Melzak (1961) and Kleene (1952) ''Introduction to Metamathematics''. * The manuscript was received by the journal on May 15, 1961. 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." * * 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 John is a common English name and surname: * John (given name) * John (surname) John may also refer to: New Testament Works * Gospel of John, a title often shortened to John * First Epistle of John, often shortened to 1 John * Seco ...
and
H. E. Sturgis H is the eighth letter of the Latin alphabet. H may also refer to: Musical symbols * H number, Harry Halbreich reference mechanism for music by Honegger and Martinů * H, B (musical note) * H, B major People * H. (noble) (died after 127 ...
(1961) received December 1961 "Computability of Recursive Functions", ''Journal of the Association of Computing Machinery'' (JACM) 10:217-255, 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), 366-379. * Ershov, A. P. "On operator algorithms", (Russian) ''Dok. Akad. Nauk'' 122 (1958), 967-970. English translation, Automat. Express 1 (1959), 20-23. * Péter, Rózsa "Graphschemata und rekursive Funktionen", ''Dialectica'' 12 (1958), 373. *Hermes, Hans "Die Universalität programmgesteuerter Rechenmaschinen". ''Math.-Phys. Semesterberichte'' (Göttingen) 4 (1954), 42-53. *
Arnold Schönhage Arnold Schönhage (born 1 December 1934 in Lockhausen, now Bad Salzuflen) is a German mathematician and computer scientist. Schönhage was professor at the Rheinische Friedrich-Wilhelms-Universität, Bonn, and also in Tübingen and Konstanz. ...
(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. resp. ''Storage Modification Machines'', in ''Theoretical Computer Science'' (1979), pp. 36–37 *
Peter van Emde Boas Peter van Emde Boas (born 3 April 1945, Amsterdam) is a Dutch computer scientist and professor at the University of Amsterdam The University of Amsterdam (abbreviated as UvA, nl, Universiteit van Amsterdam) is a public research university l ...
, "Machine Models and Simulations" pp. 3–66, in:
Jan van Leeuwen Jan van Leeuwen (born December 17, 1946, in Waddinxveen) is a Dutch computer scientist and Emeritus professor of computer science at the Department of Information and Computing Sciences at Utrecht University.
, ed. ''Handbook of Theoretical Computer Science. Volume A: Algorithms and Complexity'', The MIT PRESS/Elsevier, 1990. (volume A). QA 76.H279 1990. van Emde Boas' treatment of SMMs appears on pp. 32–35. 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; 63–92. Presented at the meeting of the Association, June 23–25, 1954. *


External links

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Igblan - Minsky Register Machines
{{DEFAULTSORT:Register Machine Models of computation