 Turing machine equivalents

Turing machine(s) Machina  Universal Turing machine
 Alternating Turing machine
 Quantum Turing machine
 Readonly Turing machine
 Readonly right moving Turing Machines
 Probabilistic Turing machine
 Multitrack Turing machine
 Turing machine equivalents
 Turing machine examples
 Wolfram's 2state 3symbol..
Science This box: view · Turing machine. Many of the machines described here have articles that offer much more information. Contents
 1 Machines equivalent to the Turing machine model
 2 Tapebased Turing machines
 3 Register machine models
 4 Machines with input and output
 5 Other equivalent machines and methods
 6 References
Machines equivalent to the Turing machine model
Turing equivalence:
Many machines that might be thought to have more computational capability than a simple universal Turing machine can be shown to have no more power (Hopcroft and Ullman p. 159, cf Minsky). They might compute faster, perhaps, or use less memory, or their instruction set might be smaller, but they cannot compute more powerfully (i.e. more mathematical functions). (Recall that the ChurchTuring thesis hypothesizes this to be true: that anything that can be “computed” can be computed by some Turing machine.)
While none of the following models have been shown to have more power than the singletape, oneway infinite, multisymbol Turingmachine model, their authors defined and used them to investigate questions and solve problems more easily than they could have if they had stayed with Turing's amachine model.
The sequentialmachine models:
All of the following are called "sequential machine models" to distinguish them from "parallel machine models" (van Emde Boas (1990) p. 18).
Tapebased Turing machines
 For more return to the article Turing machine
Turing's amachine model: Turing's (1936) amachine (his name) was leftended, rightendinfinite. He provided symbols əə to mark the left end. Any of finite number of tape symbols were permitted. The instructions (if a universal machine), and the "input" and "out" were written on only on "Fsquares", and markers were to appear on "Esquares". In essence he divided his machine into two tapes that always moved together. The instructions appeared in a tabular form called "5tuples" and were not executed sequentially.
Singletape machines with restricted symbols and/or restricted instructions
The following models are single tape Turing machines but restricted with (i) restricted tape symbols { mark, blank }, and/or (ii) sequential, computerlike instructions, and/or (iii) machineactions fully atomized.
Post's "Formulation 1" model of computation
 For more see the article PostTuring machine
Emil Post (1936) in an independent description of a computational process, reduced the symbols allowed to the equivalent binary set of marks on the tape { "mark", "blank"=not_mark }. He changed the notion of "tape" from 1way infinite to the right to an infinite set of rooms each with a sheet of paper in both directions. He atomized the Turing 5tuples into 4tuples—motion instructions separate from print/erase instructions. Although his (1936) model is ambiguous about this, Post's (1947) model did not require sequential instruction execution.
His extremely simple model can emulate any Turing machine, and although his 1936 Formulation 1 does not use the word "program" or "machine", it is effectively a formulation of a very primitive programmable computer and associated programming language, with the boxes acting as an unbounded bitstring memory, and the set of instructions constituting a program.
Wang machines
For more details on this topic, see Wang Bmachine.In an influential paper, Hao Wang (1954, 1957) reduced Post's "formulation 1" to machines that still use a twoway infinite binary tape, but whose instructions are simpler — being the "atomic" components of Post's instructions — and are by default executed sequentially (like a "computer program"). His stated principal purpose was to offer, as an alternative to Turing's theory, one that "is more economical in the basic operations". His results were "program formulations" of a variety of such machines, including the 5instruction Wang Wmachine with the instructionset
 { SHIFTLEFT, SHIFTRIGHT, MARKSQUARE, ERASESQUARE, JUMPIFSQUAREMARKEDto xxx }
and his mostseverely reduced 4instruction Wang Bmachine ("B" for "basic") with the instructionset
 { SHIFTLEFT, SHIFTRIGHT, MARKSQUARE, JUMPIFSQUAREMARKEDto xxx }
which has not even an ERASESQUARE instruction.
Many authors later introduced variants of the machines discussed by Wang:
Minsky (1961) evolved Wang's notion with his version of the (multitape) "counter machine" model that allowed SHIFTLEFT and SHIFTRIGHT motion of the separate heads but no printing at all. In this case the tapes would be leftended, each end marked with a single "mark" to indicate the end. He was able to reduce this to a single tape, but at the expense of introducing multitapesquare motion equivalent to multiplication and division rather than the much simpler { SHIFTLEFT = DECREMENT, SHIFTRIGHT = INCREMENT }.
Davis, adding an explicit HALT instruction to one of the machines discussed by Wang, used a model with the instructionset
 { SHIFTLEFT, SHIFTRIGHT, ERASE, MARK, JUMPIFSQUAREMARKEDto xxx, JUMPto xxx, HALT }
and also considered versions with tapealphabets of size larger than 2.
Böhm's theoretical machine language P"
 For details see the article P"
In keeping with Wang's project to seek a Turingequivalent theory "economical in the basic operations", and wishing to avoid unconditional jumps, a notable theoretical language is the 4instruction language P" introduced by Corrado Böhm in 1964 — the first "GOTOless" imperative "structured programming" language to be proved Turingcomplete.
Multitape Turing machines
In practical analysis, various types of multitape Turing machines are often used. Multitape machines are similar to singletape machines, but there is some constant k number of independent tapes.
The TABLE has full independent control over all the heads, any of all of which move and print/erase their own tapes (cf AhoHopcroftUllman 1974 p. 26). Most models have tapes with left ends, right ends unbounded.
This model intuitively seems much more powerful than the singletape model, but any multitape machine, no matter how large the k, can be simulated by a singletape machine using only quadratically more computation time (Papadimitriou 1994, Thrm 2.1). Thus, multitape machines cannot calculate any more functions than singletape machines, and none of the robust complexity classes (such as polynomial time) are affected by a change between singletape and multitape machines.
Twostack Turing machine
Twostack Turing machines have a readonly input and two storage tapes. If a head moves left on either tape a blank is printed on that tape, but one symbol from a “library” can be printed.
Formal definition: multitape Turing machine
A ktape Turing machine can be described as a 6tuple where
 Q is a finite set of states
 Γ is a finite set of the tape alphabet
 is the initial state
 is the blank symbol
 is the set of final or accepting states
 is a partial function called the transition function, where L is left shift, R is right shift, S is no shift.
Deterministic and nondeterministic Turing machines
 For more see the article nondeterministic Turing machine.
If the action table has at most one entry for each combination of symbol and state then the machine is a "deterministic Turing machine" (DTM). If the action table contains multiple entries for a combination of symbol and state then the machine is a "nondeterministic Turing machine" (NDTM). The two are computationally equivalent, that is, it is possible to turn any NDTM into a DTM (and vice versa).
Oblivious Turing machines
An oblivious Turing machine is a Turing machine where movement of the various heads are fixed functions of time, independent of the input. In other words, there is a predetermined sequence in which the various tapes are scanned, advanced, and written to. Pippenger and Fischer (1979) showed that any computation that can be performed by a multitape Turing machine in n steps can be performed by an oblivious twotape Turing machine in O(n log n) steps.
Register machine models

 For more see the article Register machine.
van Emde Boas (1990) includes all machines of this type in one category (group, class, collection)  "the register machine". However, historically the literature has also called the most primitive member of this group i.e. "the counter machine"  "the register machine". And the most primitive embodiment of a "counter machine" is sometimes called the "Minsky machine".
The "counter machine", also called a "register machine" model
 For more see the article Counter machine.
The primitive model register machine is, in effect, a multitape 2symbol PostTuring machine with its behavior restricted so its tapes act like simple "counters".
By the time of Melzak, Lambek, and Minsky (all 1961) the notion of a "computer program" produced a different type of simple machine with many leftended tapes cut from a PostTuring tape. In all cases the models permit only two tape symbols { mark, blank }.
Some versions represent the positive integers as only a strings/stack of marks allowed in a "register" (i.e. leftended tape), and a blank tape represented by the count "0". Minsky (1961) eliminated the PRINT instruction at the expense of providing his model with a mandatory single mark at the leftend of each tape.
In this model the singleended tapesasregisters are thought of as "counters", their instructions restricted to only two (or three if the TEST/DECREMENT instruction is atomized). Two common instruction sets are the following:
 (1): { INC ( r ), DEC ( r ), JZ ( r,z ) }, i.e.
 { INCrement contents of register #r; DECrement contents of register #r; IF contents of #r=Zero THEN Jumpto Instruction #z}
 (2): { CLR ( r ); INC ( r ); JE ( r_{i}, r_{j}, z ) }, i.e.
 { CLeaR contents of register r; INCrement contents of r; compare contents of r_{i} to r_{j} and if Equal then Jump to instruction z}
Although his model is more complicated than this simple description, the Melzak "pebble" model extended this notion of "counter" to permit multi pebble adds and subtracts.
The Random Access Machine (RAM) model
 For more see the article Random access machine.
Melzak (1961) recognized a couple serious defects in his register/countermachine model: (i) Without a form of indirect addressing he would not be able to "easily" show the model is Turing equivalent, (ii) The program and registers were in different "spaces", so selfmodifying programs would not be easy. When Melzak added indirect addressing to his model he created a random access machine model.
(However, with Gödel numbering of the instructions Minsky (1961) offered a proof that with such numbering the general recursive functions were indeed possible; in Minsky (1967) he offers proof that μ recursion is indeed possible).
Unlike the RASP model, the RAM model does not allow the machine's actions to modify its instructions. Sometimes the model works only registertoregister with no accumulator, but most models seem to include an accumulator.
van Emde Boas (1990) divides the various RAM models into a number of subtypes:
 SRAM, the "successor RAM" with only one arithmetic instruction, the successor (INCREMENT h). The others include "CLEAR h", and an IF equalitybetweenregister THEN jumpto xxx.
 RAM: the standard model with addition and subtraction
 MRAM: the RAM agumented with multiplication and division
 BRAM, MBRAM: Bitwise Boolean versions of the RAM and MRAM
 N****: Nondeterministic versions of any of the above with an N before the name
The Random Access Stored Program (RASP) machine model
 For more see the article Random access stored program machine.
The RASP is a RAM with the instructions stored together with their data in the same 'space'  i.e. sequence of registers. The notion of a RASP was described at least as early as Kiphengst (1959). His model had a "mill"  an accumulator, but now the instructions were in the registers with the data—the socalled von Neumann architecture. When the RASP has alternating even and odd registers—the even holding the "operation code" (instruction) and the odd holding its "operand" (parameter), then indirect addressing is achieved by simply modifying an instruction's operand (cf Cook and Reckhow 1973).
The original RASP model of Elgot and Robinson (1964) had only three instructions in the fashion of the registermachine model, but they placed them in the register space together with their data. (Here COPY takes the place of CLEAR when one register e.g. "z" or "0" starts with and always contains 0. This trick is not unusual. The unit 1 in register "unit" or "1" is also useful.)
 { INC ( r ), COPY ( r_{i}, r_{j} ), JE ( r_{i}, r_{i}, z ) }
The RASP models allow indirect as well as directaddressing; some allow "immediate" instructions too, e.g. "Load accumulator with the constant 3". The instructions may be of a highly restricted set such as the following 16 instructions of Hartmanis (1971). This model uses an accumulator A. The mnemonics are those that the authors used (their CLA is "load accumulator" with constant or from register; STO is "store accumulator"). Their syntax is the following, excepting the jumps: "n, <n>, <<n>>" for "immediate", "direct" and "indirect"). Jumps are via two "Transfer instructions" TRA—unconditional jump by directly "n" or indirectly "< n >" jamming contents of register n into the instruction counter, TRZ (conditional jump if Accumulator is zero in the same manner as TRA):
 { ADD n , ADD < n >, ADD << n >>, SUB n, SUB < n >, SUB << n >>, CLA n, CLA < n >, CLA << n >>, STO < n >, STO << n >>, TRA n, TRA < n >, TRZ n, TRA < n >, HALT }
The Pointer machine model
 For more see the article Pointer machine.
A relative latecomer is Schönhage's Storage Modification Machine (1970) or pointer machine. Another version is the KolmogorovUspensii machine, and the Knuth "linking automaton" proposal. (For references see pointer machine). Like a statemachine diagram, a node emits at least two labelled "edges" (arrows) that point to another node or nodes which in turn point to other nodes, etc. The outside world points at the center node.
Machines with input and output
Any of the above tapebased machines can be equipped with input and output tapes; any of the above registerbased machines can be equipped with dedicated input and output registers. For example, the Schönhage pointermachine model has two instructions called "input λ_{0},λ_{1}" and "output β" (Schönhage 1990 p. 493)
It is difficult to study sublinear space complexity on multitape machines with the traditional model, because an input of size n already takes up space n. Thus, to study small DSPACE classes, we must use a different model. In some sense, if we never "write to" the input tape, we don't want to charge ourself for this space. And if we never "read from" our output tape, we don't want to charge ourself for this space.
We solve this problem by introducing a kstring Turing machine with input and output. This is the same as an ordinary kstring Turing machine, except that the transition function δ is restricted so that the input tape can never be changed, and so that the output head can never move left. This model allows us to define deterministic space classes smaller than linear. Turing machines with inputandoutput also have the same time complexity as other Turing machines; in the words of Papaditriou 1994 Prop 2.2:
 For any kstring Turing machine M operating within time bound f(n)) there is a (k+2)string Turing machine M’ with input and output, which operates within time bound O(f(n)).
kstring Turing machines with input and output are used in the formal definition of the complexity resource DSPACE in, for example, Papadimitriou 1994 (Def. 2.6).
Other equivalent machines and methods
 Multidimensional Turing machine: For example, a model by Schönhage (1990) uses the four headmovement commands { North, South, East, West }.
 Singletape, multihead Turing machine: In an undecidability proof of the "problem of tag", Minsky 1961 and Shepherdson and Sturgis (1963) described machines with a single tape that could write along the tape with one head and read further along the tape with another.
 Markov's (1954) Normal Algorithm is another remarkably simple computational model equivalent to the Turing machines.
References
 For more references see the following, or return to the article Turing machine:
 Peter van Emde Boas, Machine Models and Simulations, pp. 366, located in:
 Jan van Leeuwen, ed. Handbook of Theoretical Computer Science. Volume A: Algorithms and Complexity, The MIT Press/Elsevier, 1990. ISBN 0262720140 (volume A). QA76.H279 1990.
 A thorough and helpful survey with respect to machine models and complexity theory, with definitions of e.g. MLOGSPACE, etc., and a categorization of "sequential machine" models. With 141 references (!)
 Jan van Leeuwen, ed. Handbook of Theoretical Computer Science. Volume A: Algorithms and Complexity, The MIT Press/Elsevier, 1990. ISBN 0262720140 (volume A). QA76.H279 1990.
 A. A. Markov (1954) Theory of algorithms. [Translated by Jacques J. SchorrKon and PST staff] Imprint Moscow, Academy of Sciences of the USSR, 1954 [i.e. Jerusalem, Israel Program for Scientific Translations, 1961; available from the Office of Technical Services, U.S. Dept. of Commerce, Washington] Description 444 p. 28 cm. Added t.p. in Russian Translation of Works of the Mathematical Institute, Academy of Sciences of the USSR, v. 42. Original title: Teoriya algerifmov. [QA248.M2943 Dartmouth College library. U.S. Dept. of Commerce, Office of Technical Services, number OTS 6051085.]
 Pippenger, Nicholas; Fischer, Michael J. (1979), "Relations Among Complexity Measures", JACM 26 (3): 361–381, doi:10.1145/322123.322138
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Turing machine equivalents
 Turing machine equivalents

Turing machine(s) Machina  Universal Turing machine
 Alternating Turing machine
 Quantum Turing machine
 Readonly Turing machine
 Readonly right moving Turing Machines
 Probabilistic Turing machine
 Multitrack Turing machine
 Turing machine equivalents
 Turing machine examples
 Wolfram's 2state 3symbol..
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