Register machine
Encyclopedia
In mathematical logic
and theoretical computer science
a register machine is a generic class of abstract machine
s used in a manner similar to a Turing machine
. All the models are Turing equivalent
.
—in place of a Turing machine's tape and head (or tapes and heads) the model uses multiple, uniquely-addressed registers, each of which holds a single positive integer
.
There are at least 4 sub-classes found in the 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.
In practical computer science, a similar concept known as a virtual machine
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.
A register machine consists of:
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—his problem of "tag"—and the very "hard" problem of Hilbert's problems
—the 10th question around Diophantine equation
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 originated with Hans Hermes (1954) 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
(1961), Marvin Minsky
(1961, 1967), and John Shepherdson and Howard E. Sturgis (1963).
The last five names are listed explicitly in that order by Yuri Matiyasevich
. He follows up with:
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.
—a two-symbol Post-Turing machine
computation model with only four atomic instructions:
To these four both Wang (1954, 1957) and then C.Y. Lee (1961) added another other instruction from the Post set { ERASE }, and then a Post's unconditional jump { JUMP_to_ instruction_z } (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:
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):
Indeed as examples at Turing machine examples
, Post-Turing machine and partial function
show, the work can be "complicated".
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
. 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 { INC (r), JZDEC (r, z) } and the convenience instructions { CLR (r), J (r) } 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.
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.
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.
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, (ii) 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 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
(e.g. multiplication) but not all mu recursive functions (e.g. the Ackermann function
).
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 number
s). 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.
Minsky (1967) hints at the issue in his investigation of a counter machine (he calls them "program computer models") equipped with the instructions { CLR (r), INC (r), and RPT ("a" times the instructions m to n) }. He doesn't tell us how to fix the problem, but he does observe that:
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:
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).
( 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.
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, (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:
Indeed, Shepherson and Sturgis conclude
By order of publishing date the work of Kaphengst (1959), Ershov (1958), Peter (1958) were first. Does context matter? An answer would require close examination of the papers. Conclusions and opinions about this will be left to the reader.
Mathematical logic
Mathematical logic is a subfield of mathematics with close connections to foundations of mathematics, theoretical computer science and philosophical logic. The field includes both the mathematical study of logic and the applications of formal logic to other areas of mathematics...
and theoretical computer science
Theoretical computer science
Theoretical computer science is a division or subset of general computer science and mathematics which focuses on more abstract or mathematical aspects of computing....
a register machine is a generic class of abstract machine
Abstract machine
An abstract machine, also called an abstract computer, is a theoretical model of a computer hardware or software system used in automata theory...
s used in a manner similar to a Turing machine
Turing machine
A Turing machine is a theoretical device that manipulates symbols on a strip of tape according to a table of rules. Despite its simplicity, a Turing machine can be adapted to simulate the logic of any computer algorithm, and is particularly useful in explaining the functions of a CPU inside a...
. All the models are Turing equivalent
Turing completeness
In computability theory, a system of data-manipulation rules is said to be Turing complete or computationally universal if and only if it can be used to simulate any single-taped Turing machine and thus in principle any computer. A classic example is the lambda calculus...
.
Overview
The register machine gets its name from its one or more "registers"Processor register
In computer architecture, a processor register is a small amount of storage available as part of a CPU or other digital processor. Such registers are addressed by mechanisms other than main memory and can be accessed more quickly...
—in place of a Turing machine's tape and head (or tapes and heads) the model uses multiple, uniquely-addressed registers, each of which holds a single positive integer
Integer
The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...
.
There are at least 4 sub-classes found in the literature, here listed from most primitive to the most like a computer
Computer
A computer is a programmable machine designed to sequentially and automatically carry out a sequence of arithmetic or logical operations. The particular sequence of operations can be changed readily, allowing the computer to solve more than one kind of problem...
:
- counter machineCounter machineA 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...
– the most primitive and reduced model. Lacks indirect addressing. Instructions are in the finite state machine in the manner of the Harvard architectureHarvard architectureThe Harvard architecture is a computer architecture with physically separate storage and signal pathways for instructions and data. The term originated from the Harvard Mark I relay-based computer, which stored instructions on punched tape and data in electro-mechanical counters...
. - Pointer machinePointer machineIn theoretical computer science a pointer machine is an "atomistic" abstract computational machine model akin to the Random access machine....
– 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 machineRandom access machineIn computer science, random access machine 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...
(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 machineRandom access stored program machineIn theoretical computer science the Random Access Stored Program machine model is an abstract machine used for the purposes of algorithm development and algorithm complexity theory....
model (RASP) – a RAM with instructions in its registers analogous to the Universal Turing machineUniversal Turing machineIn computer science, a universal Turing machine 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 simulated as well as the input thereof from its own tape. Alan...
; thus it is an example of the von Neumann architectureVon Neumann architectureThe term Von Neumann architecture, aka the Von Neumann model, derives from a computer architecture proposal by the mathematician and early computer scientist John von Neumann and others, dated June 30, 1945, entitled First Draft of a Report on the EDVAC...
. But unlike a computer the model is idealized with effectively-infinite registers (and if used, effectively-infinite special registers such as an accumulator). Unlike a computer or even a RISC, the instruction set is much reduced in the number of instructions.
Any properly-defined register machine model is Turing equivalent
Turing completeness
In computability theory, a system of data-manipulation rules is said to be Turing complete or computationally universal if and only if it can be used to simulate any single-taped Turing machine and thus in principle any computer. A classic example is the lambda calculus...
. Computational speed is very dependent on the model specifics.
In practical computer science, a similar concept known as a virtual machine
Virtual machine
A virtual machine is a "completely isolated guest operating system installation within a normal host operating system". Modern virtual machines are implemented with either software emulation or hardware virtualization or both together.-VM Definitions:A virtual machine is a software...
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
- No standard terminology exists; each author is responsible for defining in prose the meanings of their mnemonics or symbols. Many authors use a "register-transfer"-like symbolism to explain the actions of their models, but again they are responsible for defining its syntax.
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
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 machineRandom access machineIn computer science, random access machine 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...
. - Tally counters or marks: discrete, indistinguishable objects or marks of only one sort suitable for the model. In the most-reduced counter machineCounter machineA 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...
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 equivalentTuring completenessIn computability theory, a system of data-manipulation rules is said to be Turing complete or computationally universal if and only if it can be used to simulate any single-taped Turing machine and thus in principle any computer. A classic example is the lambda calculus...
(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: { Increment (r), Decrement (r), Clear-to-zero (r) }
- Reduced RAM, RASP: { Increment (r), Decrement (r), Clear-to-zero (r), Load-immediate-constant k, Add (r1,r2), proper-Subtract (r1,r2), Increment accumulator, Decrement accumulator, Clear accumulator, Add to accumulator contents of register r, proper-Subtract from accumulator contents of register r, }
- Augmented RAM, RASP: All of the reduced instructions plus: { Multiply, Divide, various Boolean bit-wise (left-shift, bit test, etc.)}
- Control:
- Counter machine models: optional { Copy (r1,r2) }
- RAM and RASP models: most have { Copy (r1,r2) }, or { Load Accumulator from r, Store accumulator into r, Load Accumulator with immediate constant }
- All models: at least one conditional "jump" (branch, goto) following test of a register e.g. { Jump-if-zero, Jump-if-not-zero (i.e. Jump-if-positive), Jump-if-equal, Jump-if-not equal }
- All models optional: { unconditional program jump (goto) }
- 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
- 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):
- 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 computerComputerA computer is a programmable machine designed to sequentially and automatically carry out a sequence of arithmetic or logical operations. The particular sequence of operations can be changed readily, allowing the computer to solve more than one kind of problem...
). 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
. 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 architectureHarvard architectureThe Harvard architecture is a computer architecture with physically separate storage and signal pathways for instructions and data. The term originated from the Harvard Mark I relay-based computer, which stored instructions on punched tape and data in electro-mechanical counters...
. In the case of the RASP the program store is in the registers; thus this is an example of the von Neumann architectureVon Neumann architectureThe term Von Neumann architecture, aka the Von Neumann model, derives from a computer architecture proposal by the mathematician and early computer scientist John von Neumann and others, dated June 30, 1945, entitled First Draft of a Report on the EDVAC...
. See also Random access machine and Random access stored program machineRandom access stored program machineIn theoretical computer science the Random Access Stored Program machine model is an abstract machine used for the purposes of algorithm development and algorithm complexity theory....
.
Usually, like computer programComputer programA computer program is a sequence of instructions written to perform a specified task with a computer. A computer requires programs to function, typically executing the program's instructions in a central processor. The program has an executable form that the computer can use directly to execute...
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. { INC ( r, z ), JZDEC ( r, ztrue, zfalse ) }.
See McCarthy FormalismMcCarthy FormalismIn computer science and recursion theory the McCarthy Formalism of computer scientist John McCarthy clarifies the notion of recursive functions by use of the IF-THEN-ELSE construction common to computer science, together with the four of the operators of primitive recursive functions: zero,...
for more about the conditional expression "IF r=0 THEN ztrue ELSE zfalse" (cf McCarthy (1960)).
- Observe also that the abacus model combines two instructions, JZ then DEC: e.g. { INC ( r, z ), JZDEC ( r, ztrue, zfalse ) }.
Historical development of the register machine model
Two trends appeared in the early 1950s—the first to characterize the computerComputer
A computer is a programmable machine designed to sequentially and automatically carry out a sequence of arithmetic or logical operations. The particular sequence of operations can be changed readily, allowing the computer to solve more than one kind of problem...
as a Turing machine
Turing machine
A Turing machine is a theoretical device that manipulates symbols on a strip of tape according to a table of rules. Despite its simplicity, a Turing machine can be adapted to simulate the logic of any computer algorithm, and is particularly useful in explaining the functions of a CPU inside a...
, 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
Turing completeness
In computability theory, a system of data-manipulation rules is said to be Turing complete or computationally universal if and only if it can be used to simulate any single-taped Turing machine and thus in principle any computer. A classic example is the lambda calculus...
. Need for this work was carried out in context of two "hard" problems: the unsolvable word problem posed by Emil Post—his problem of "tag"—and the very "hard" problem of Hilbert's problems
Hilbert's problems
Hilbert's problems form a list of twenty-three problems in mathematics published by German mathematician David Hilbert in 1900. The problems were all unsolved at the time, and several of them were very influential for 20th century mathematics...
—the 10th question around Diophantine equation
Diophantine equation
In mathematics, a Diophantine equation is an indeterminate polynomial equation that allows the variables to be integers only. Diophantine problems have fewer equations than unknown variables and involve finding integers that work correctly for all equations...
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 originated with Hans Hermes (1954) 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 Lambek
Joachim Lambek is Peter Redpath Emeritus Professor of Pure Mathematics at McGill University, where he earned his Ph.D. degree in 1950 with Hans Julius Zassenhaus as advisor. He is called Jim by his friends.- Scholarly work :...
(1961), Marvin Minsky
Marvin Minsky
Marvin Lee Minsky is an American cognitive scientist in the field of artificial intelligence , co-founder of Massachusetts Institute of Technology's AI laboratory, and author of several texts on AI and philosophy.-Biography:...
(1961, 1967), and John Shepherdson and Howard E. Sturgis (1963).
The last five names are listed explicitly in that order by Yuri Matiyasevich
Yuri Matiyasevich
Yuri Vladimirovich Matiyasevich, is a Russian mathematician and computer scientist. He is best known for his negative solution of Hilbert's tenth problem, presented in his doctoral thesis, at LOMI .- Biography :* In 1962-1963 studied at Saint Petersburg Lyceum 239...
. 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 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.
(1954, 1957) Wang's model: Post-Turing machine
Wang's work followed from Emil Post's (1936) paper and led Wang to his definition of his Wang B-machineWang B-machine
As presented by Hao Wang , 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" . With only 4 sequential instructions it is very similar to, but even simpler than,...
—a two-symbol Post-Turing machine
Post-Turing machine
A Post–Turing machine is a "program formulation" of an especially simple type of Turing machine, comprising a variant of Emil Post's Turing-equivalent model of computation described below. A Post–Turing machine is a "program formulation" of an especially simple type of Turing machine, comprising a...
computation model with only four atomic instructions:
- { LEFT, RIGHT, PRINT, JUMP_if_marked_to_instruction_z }
To these four both Wang (1954, 1957) and then C.Y. Lee (1961) added another other instruction from the Post set { ERASE }, and then a Post's unconditional jump { JUMP_to_ instruction_z } (or to make things easier, the conditional jump JUMP_IF_blank_to_instruction_z, or both. Lee named this a "W-machine" model:
- { LEFT, RIGHT, PRINT, ERASE, JUMP_if_marked, [maybe JUMP or JUMP_IF_blank] }
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
Martin Davis
Martin David Davis, is an American mathematician, known for his work on Hilbert's tenth problem . He received his Ph.D. from Princeton University in 1950, where his adviser was Alonzo Church . He is Professor Emeritus at New York University. He is the co-inventor of the Davis-Putnam and the DPLL...
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 under one wishing to work on and keep it separate from other numbers" (Shepherdson and Sturgis (1963) p. 218).
Indeed as examples at Turing machine examples
Turing machine examples
-Turing's very first example:The following table is Turing's very first example :With regard to what actions the machine actually does, Turing states the following:...
, Post-Turing machine and partial function
Partial function
In mathematics, a partial function from X to Y is a function ƒ: X' → Y, where X' is a subset of X. It generalizes the concept of a function by not forcing f to map every element of X to an element of Y . If X' = X, then ƒ is called a total function and is equivalent to a function...
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 (ie. 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
Partial function
In mathematics, a partial function from X to Y is a function ƒ: X' → Y, where X' is a subset of X. It generalizes the concept of a function by not forcing f to map every element of X to an element of Y . If X' = X, then ƒ is called a total function and is equivalent to a function...
. 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."
- For example of the addition algorithm written for a counter machine see Algorithm examplesAlgorithm examplesThis article Algorithm examples supplements Algorithm and Algorithm characterizations.- An example: Algorithm specification of addition m+n :Choice of machine model:...
, and for an example of (im-) proper subtraction see Partial functionPartial functionIn mathematics, a partial function from X to Y is a function ƒ: X' → Y, where X' is a subset of X. It generalizes the concept of a function by not forcing f to map every element of X to an element of Y . If X' = X, then ƒ is called a total function and is equivalent to a function...
.
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
Gödel number
In mathematical logic, a Gödel numbering is a function that assigns to each symbol and well-formed formula of some formal language a unique natural number, called its Gödel number. The concept was famously used by Kurt Gödel for the proof of his incompleteness theorems...
(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 { INC (r), JZDEC (r, z) } and the convenience instructions { CLR (r), J (r) } 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.
(1961) Melzak's 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
Turing completeness
In computability theory, a system of data-manipulation rules is said to be Turing complete or computationally universal if and only if it can be used to simulate any single-taped Turing machine and thus in principle any computer. A classic example is the lambda calculus...
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 machineRandom access stored program machine
In theoretical computer science the Random Access Stored Program machine model is an abstract machine used for the purposes of algorithm development and algorithm complexity theory....
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, (ii) 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 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
Primitive recursive function
The primitive recursive functions are defined using primitive recursion and composition as central operations and are a strict subset of the total µ-recursive functions...
(e.g. multiplication) but not all mu recursive functions (e.g. the Ackermann function
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...
).
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 number
Gödel number
In mathematical logic, a Gödel numbering is a function that assigns to each symbol and well-formed formula of some formal language a unique natural number, called its Gödel number. The concept was famously used by Kurt Gödel for the proof of his incompleteness theorems...
s). 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. { 1: CLR ( p ), 2: JZ ( m, done ), 3 outer_loop: JZ ( n, done ), 4: CPY ( m, temp ), 5: inner_loop: JZ ( m, outer_loop ), 6: DEC ( m ), 7: INC ( p ), 8: J ( inner_loop ), 9: outer_loop: DEC ( n ), 10 J ( outer_loop ), HALT }
- 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 not universal.
Minsky (1967) hints at the issue in his investigation of a counter machine (he calls them "program computer models") equipped with the instructions { CLR (r), INC (r), and RPT ("a" times the instructions m to n) }. 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 [rbase] + 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 indirectionIndirection
In computer programming, indirection 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 through its memory address. For example, accessing a variable through the use of a...
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 machineRandom access machine
In computer science, random access machine 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...
( RAM—i.e. a machine with indirection and the Harvard architecture
Harvard architecture
The Harvard architecture is a computer architecture with physically separate storage and signal pathways for instructions and data. The term originated from the Harvard Mark I relay-based computer, which stored instructions on punched tape and data in electro-mechanical counters...
). In a sense we are back to Melzak (1961) but with a much simpler model than Melzak's.
Precedence
Minsky was working at the M.I.T. Lincoln LabsLincoln Laboratory
MIT Lincoln Laboratory, located in Lexington, Massachusetts, is a United States Department of Defense research and development center chartered to apply advanced technology to problems of national security. Research and development activities focus on long-term technology development as well as...
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, (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. Does context matter? An answer would require close examination of the papers. Conclusions and opinions about this will be left to the reader.