Turing machine

Template:Two other uses

Turing machines are extremely basic abstract symbol-manipulating devices which, despite their simplicity, can be adapted to simulate the logic of any computer that could possibly be constructed. They were described in 1936 by Alan Turing. Though they were intended to be technically feasible, Turing machines were not meant to be a practical computing technology, but a thought experiment about the limits of mechanical computation; thus they were not actually constructed. Studying their abstract properties yields many insights into computer science and complexity theory.

A Turing machine that is able to simulate any other Turing machine is called a Universal Turing machine (UTM, or simply a universal machine). A more mathematically-oriented definition with a similar "universal" nature was introduced by Alonzo Church, whose work on lambda calculus intertwined with Turing's in a formal theory of computation known as the Church-Turing thesis. The thesis states that Turing machines indeed capture the informal notion of effective method in logic and mathematics, and provide a precise definition of an algorithm or 'mechanical procedure'.

Informal description

For visualizations of Turing machines, see Turing machine gallery.

The concept of the Turing machine is based on the idea of a person executing a well-defined procedure by changing the contents of an unlimited paper tape, which is divided into squares that can contain one of a finite set of symbols. The person needs to remember one of a finite set of states and the procedure is formulated in very basic steps in the form of "If your state is 42 and the symbol you see is a '0' then replace this with a '1', move one symbol to the right, and assume state 17 as your new state."

More precisely, a Turing machine consists of:

A TAPE which is divided into cells, one next to the other. Each cell contains a symbol from some finite alphabet. The alphabet contains a special blank symbol (here written as 'B') and one or more other symbols. The tape is assumed to be arbitrarily extendable to the left and to the right, i.e., the Turing machine is always supplied with as much tape as it needs for its computation. Cells that have not been written to before are assumed to be filled with the blank symbol. In some models the tape has a left end marked with a special symbol; the tape extends or is indefinitely extensible to the right. The symbols are sometimes referred to as colors.
A HEAD that can read and write symbols on the tape and move the tape left and right one (and only one) cell at a time. In some models the head moves and the tape is stationary.
A TABLE ("action table", or transition function) of instructions (usually quintuples or 5-tuples but sometimes 4-tuples) that, given the state the machine is currently in and the symbol it is reading on the tape tells the machine to do the following in sequence (for the 5-tuple models): (i) either erase or write a symbol, and then (ii) move the head ('L' for one step left or 'R' for one step right), and then (iii) assume the same or a new state as prescribed. In the 4-tuple models the TABLE tells the machine to (ia) erase or to write a symbol or (ib) move the head left or right, and then (ii) assume the same or a new state as prescribed, but not both actions (ia) and (ib) in the same instruction. In some models, if there is no entry in the table for the current combination of symbol and state then the machine will halt; other models require all entries to be filled.
A state register that stores the state of the Turing table. The number of different states is always finite and there is one special start state with which the state register is initialized. Turing defined this as a "note of instructions" to preserve the computation of the "computer" (a person) who is working in a "desultory manner":
"This note is the counterpart of the 'state of mind'." (Undecidable, p. 139)

Note that every part of the machine—its state- and symbol-collections—and its actions—printing, erasing and tape motion—is finite, discrete and distinguishable; it is the potentially unlimited amount of tape that gives it an unbounded amount of storage space.

Examples of Turing machines

To see examples of the following models, see Turing machine examples:

Turing's very first machine
Copy routine
3-state busy beaver

Formal definition of single-tape Turing machine

A nutshell formal description of a "Turing machine":

"A Turing machine is a finite-state machine associated with an external storage or memory medium." (Minsky (1967), p. 117)

"A Turing machine is essentially a finite-state sequential machine that has the ability to communicate with an external store of information." (Booth (1967), p. 354)

The finite state machine is represented by the state-table together with its state register. The "external storage medium" is the tape. The input to the state machine is the scanned symbol on the tape. The output of the state machine is a symbol to print or the erase command and tape motion-command left or right.

Hopcroft and Ullman (1979, p. 148) formally define a (one-tape) Turing machine as a 7-tuple $M=\langle Q,\Gamma ,b,\Sigma ,\delta ,q_{0},F\rangle$ where

$Q$ is a finite set of states
$\Gamma$ is a finite set of the tape alphabet/symbols
$b\in \Gamma$ is the blank symbol (the only symbol allowed to occur on the tape infinitely often at any step during the computation)
$\Sigma$ , a subset of $\Gamma$ not including b is the set of input symbols
$\delta :Q\times \Gamma \rightarrow Q\times \Gamma \times \{L,R\}$ is a partial function called the transition function, where L is left shift, R is right shift.
$q_{0}\in Q$ is the initial state
$F\subseteq Q$ is the set of final or accepting states

The 7-tuple for the 3-state busy beaver looks like this (see more about this busy beaver at Turing machine examples):

Q = { A, B, C, HALT }

Γ = { 0, 1 }

b = 0 = "blank"

Σ = { }, empty set

δ = see state-table below

q₀ = A = initial state

F = the one element set of final states {HALT}

State table for 3 state, 2 symbol busy beaver
Tape symbol	Current state A			Current state B			Current state C
Tape symbol	Write symbol	Move tape	Next state	Write symbol	Move tape	Next state	Write symbol	Move tape	Next state
0	1	R	B	1	L	A	1	L	B
1	1	L	C	1	R	B	1	N	HALT

This specification is insufficient, however. van Emde Boas (1990) observes that:

"The set-theoretical object [his formal seven-tuple description similar to the above] provides only partial information on how the machine will behave and what its computations will look like" (p. 6).

Examples readily demonstrate that, as well as the behavior of the components of the informal description, the form of the input and output "parameters", and the position of the head at the start must be specified as well. For example, Turing (1936) places his "figures" "1" and "0" on alternate squares. Other models form the input as tight-packed unary 1s with blanks between the various strings, yet others place strings of 0's, each spaced by 1, on truly blank tape, etc.; the output may or may not appear in a similar manner. Thus to fully "describe" a "computation" of a Turing machine Stone (1972) states it is necessary to state the following:

The tape alphabet
The form in which the parameters are presented on the tape
The initial state of the Turing machine
The form in which answers will be represented on the tape when the Turing machine halts
The machine program" (Stone, p. 10)

Turing instructions—quintuples (5-tuples)

Definitions in literature sometimes differ slightly, to make arguments or proofs easier or clearer, but this is always done in such a way that the resulting machine has the same computational power. For example, changing the set $\{L,R\}$ to $\{L,R,N\}$ , where N ("None" or "No-operation") would allow the machine to stay on the same tape cell instead of moving left or right, does not increase the machine's computational power.

The most common convention represents each "Turing instruction" in a "Turing table" by one of nine 5-tuples, per the convention of Turing/Davis (Turing (1936 in Undecidable, p. 126-127 and Davis (2000) p. 152):

(definition 1): (q_i, S_j, S_k/E/N, L/R/N, q_m)

( current state q_i , symbol scanned S_j , print symbol S_k PS_k/erase E/none N , move_tape_one_ square left L/right R/None N , new state q_m )

Other authors (Minsky (1967) p. 119, Hopcroft and Ullman (1979) p. 158, Stone (1972) p. 9) adopt a different convention, with new state q_m listed immediately after the scanned symbol S_j:

(definition 2): (q_i, S_j, q_m, S_k/E/N, L/R/N)

( current state q_i, symbol scanned S_j , new state q_m , print symbol S_k PS_k/erase E/none N , move_tape_one_square left L/none N/right R )

For remainder of this article we will use "definition 1" i.e. the Turing/Davis convention.

Example: state table for the 3-state 2-symbol busy beaver reduced to 5-tuples
Current state	Scanned symbol	Print symbol	Move tape	Final (i.e. next) state	5-tuples
A	0	1	R	B	(A, 0, 1, R, B)
A	1	1	L	C	(A, 1, 1, L, C)
B	0	1	L	A	(B, 0, 1, L, A)
B	1	1	R	B	(B, 1, 1, R, B)
C	0	1	L	B	(C, 0, 1, L, B)
C	1	1	N	H	(C, 1, 1, N, H)

In the following table, Turing's original model allowed only the first three lines that he called N1, N2, N3 (cf Turing in Undecidable, p. 126). He allowed for erasure of the "scanned square" by naming a 0th symbol S₀ = "erase" or "blank", etc. However, he did not allow for non-printing, so every instruction-line includes "print symbol S_k" or "erase" (cf footnote 12 in Post (1947), Undecidable p. 300). The abbreviations are Turing's (Undecidable p.119). Subsequent to Turing's original paper in 1936–1937, machine-models have allowed all nine possible five-tuples:

	Current m-configuration (Turing state)	Tape symbol	Print-operation	Tape-motion	Final m-configuration (Turing state)	5-tuple	5-tuple comments	4-tuple
N1	q_i	S_j	Print(S_k)	Left L	q_m	(q_i, S_j, S_k, L, q_m)	"blank" = S₀, 1=S₁, etc.
N2	q_i	S_j	Print(S_k)	Right R	q_m	(q_i, S_j, S_k, R, q_m)	"blank" = S₀, 1=S₁, etc.
N3	q_i	S_j	Print(S_k)	None N	q_m	(q_i, S_j, S_k, N, q_m)	"blank" = S₀, 1=S₁, etc.	(q_i, S_j, S_k, q_m)

4	q_i	S_j	None N	Left L	q_m	(q_i, S_j, N, L, q_m)		(q_i, S_j, L, q_m)
5	q_i	S_j	None N	Right R	q_m	(q_i, S_j, N, R, q_m)		(q_i, S_j, R, q_m)
6	q_i	S_j	None N	None N	q_m	(q_i, S_j, N, N, q_m)	Direct "jump"	(q_i, S_j, N, q_m)
7	q_i	S_j	Erase	Left L	q_m	(q_i, S_j, E, L, q_m)
8	q_i	S_j	Erase	Right R	q_m	(q_i, S_j, E, R, q_m)
9	q_i	S_j	Erase	None N	q_m	(q_i, S_j, E, N, q_m)		(q_i, S_j, E, q_m)

We can construct any Turing table (list of instructions) from the above nine 5-tuples. For technical reasons usually we can dispense with the three non-printing or "N" instructions (4, 5, 6). For examples see Turing machine examples.

Less frequently we encounter the use of 4-tuples—these represent a further atomization of the Turing instructions (cf Post (1947), Boolos & Jeffrey (1974, 1999), Davis-Sigal-Weyuker (1994)); also see more at Post-Turing machine.

The "state"

The word "state" used in context of Turing machines can be a source of confusion. Most commentators after Turing have used "state" to mean the name/designator of the current instruction to be performed—i.e. the contents of the state register. But Turing (1936) made a strong distinction between a record of what he called the machine's "m-configuration", and the machine's (or person's) "state of progress" through the computation. What Turing called "the state formula" includes both the current instruction and all the symbols on the tape:

"Thus the state of progress of the computation at any stage is completely determined by the note of instructions and the symbols on the tape. That is, the state of the system may be described by a single expression (sequence of symbols) consisting of the symbols on the tape followed by Δ (which we suppose not to appear elsewhere) and then by the note of instructions. This expression is called the 'state formula' (Undecidable, p.139–140, emphasis added)

Earlier in his paper Turing carried this even further: he gives an example where he places a symbol of the current "m-configuration"—the instruction's label—beneath the scanned square, together with all the symbols on the tape (Undecidable, p.121); this he calls "the complete configuration" (Undecidable, p. 118). To print the "complete configuration" on one line he places the state-label/m-configuration to the left of the scanned symbol.

We see a variant of this in Kleene (1952) where Kleene shows how to write the Gödel number of a machine's "situation": he places the "m-configuration" symbol q₄ over the scanned square in roughly the center of the 6 non-blank squares on the tape (see the Turing-tape figure in this article) and puts it to the right of the scanned square. But Kleene refers to "q₄" itself as "the machine state" (Kleene, p. 374-375). Hopcroft and Ullman call this composite the "instantaneous description" and follow the Turing convention of putting the "current state" (instruction-label, m-configuration) to the left of the scanned symbol (p.149).

Example: total state of 3-state 2-symbol busy beaver after 3 "moves" (taken from example "run" in the figure below):

1A1

This means: after three moves the tape has … 000110000 … on it, the head is scanning the right-most 1, and the state is A. Blanks (in this case represented by "0"s) can be part of the total state as shown here: B01 ; the tape has a single 1 on it, but the head is scanning the 0 ("blank") to its left and the state is B.

Thus when we speak of "state" in the context of Turing machines we should clarify whether we are describing: (i) the current instruction, or (ii) the list of symbols on the tape together with the current instruction, or (iii) the list of symbols on the tape together with the current instruction placed to the left of the scanned symbol or to the right of the scanned symbol.

Hodges—Turing's biographer—has observed this confusion, and his commentary on it appears in a footnote (cf Hodges (1983) p. 107).

Turing machine "state" diagrams

The table for the 3-state busy beaver ("P" = print/write a "1")
Tape symbol	Current state A			Current state B			Current state C
	Write symbol	Move tape	Next state	Write symbol	Move tape	Next state	Write symbol	Move tape	Next state
0	P	R	B	P	L	A	P	L	B
1	P	L	C	P	R	B	P	N	HALT

File:State diagram 3 state busy beaver 2B.gif

The "3-state busy beaver" Turing Machine in a finite state representation. Each circle represents a "state" of the TABLE—an "m-configuration" or "instruction". "Direction" of a state transition is shown by an arrow. The label (e.g.. 0/P,R) near the outgoing state (at the "tail" of the arrow) specifies the scanned symbol that causes a particular transition (e.g. 0) followed by a slash /, followed by the subsequent "behaviors" of the machine, e.g. "P Print" then move tape "R Right". No general accepted format exists. The convention shown is after McClusky (1965), Booth (1965), Hill and Peterson (1974).

To the right: the above TABLE as expressed as a "state transition" diagram".

Usually large TABLES are better left as tables (Booth, p. 74). They are more readily simulated by computer in tabular form (Booth, p. 74). However, certain concepts—e.g. machines with "reset" states and machines with repeating patterns (cf Hill and Peterson p. 244ff)—can be more readily seen when viewed as a drawing.

Whether a drawing represents an improvement on its TABLE must be decided by the reader for the particular context. See Finite state machine for more.

File:State diagram 3 state busy beaver 4 .JPG

The evolution of the busy-beaver's computation starts at the top and proceeds to the bottom.

The reader should again be cautioned that such diagrams represent a snapshot of their TABLE frozen in time, not the course ("trajectory") of a computation through time and/or space. While every time the busy beaver machine "runs" it will always follow the same state-trajectory, this is not true for the "copy" machine that can be provided with variable input "parameters".

The diagram "Progress of the computation" shows the 3-state busy beaver's "state" (instruction) progress through its computation from start to finish. On the far right is the Turing "complete configuration" (Kleene "situation", Hopcroft-Ullman "instantaneous description") at each step. If we were to stop the machine and clear to blank both the "state register" and entire tape, these "configurations" could be used to rekindle a computation anywhere in its progress (cf Turing (1936) Undecidable pp. 139–140).

Models equivalent to the Turing machine model

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 (1967)). 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 Church-Turing thesis hypothesizes this to be true: that anything that can be “computed” can be computed by some Turing machine.)

A Turing machine is equivalent to a pushdown automaton made more powerful by relaxing the last-in-first-out requirement of its stack. (Interestingly, this seemingly minor relaxation enables the Turing machine to perform such a wide variety of computations that it can serve as a model for the computational capabilities of all modern computer software.)

At the other extreme, some very simple models turn out to be Turing-equivalent, i.e. to have the same computational power as the Turing machine model.

Common equivalent models are the multi-tape Turing machine, machines with input and output, and the non-deterministic Turing machine (NDTM) as opposed to the deterministic Turing machine (DTM) for which the action table has at most one entry for each combination of symbol and state.

For more on this topic see Turing machine equivalents, in particular Register machine and Post-Turing machine.

Read only right moving Turing Machines are equivalent to NDFA's (as well as DFA's by conversion using the NDFA to DFA conversion algorithm).

For practical and didactical intentions the equivalent register machine can be used as a usual assembly programming language.

Choice c-machines, Oracle o-machines

Early in his paper (1936) Turing makes a distinction between an "automatic machine"—its "motion … completely determined by the configuration" and a "choice machine":

"…whose motion is only partially determined by the configuration … When such a machine reaches one of these ambiguous configurations, it cannot go on until some arbitrary choice has been made by an external operator. This would be the case if we were using machines to deal with axiomatic systems" (p. 118 Undecidable)

Turing (1936) does not elaborate further excepting a footnote in which he describes how to use an a-machine to "find all the provable formulae of the [Hilbert] calculus" rather than use a choice machine. He "suppose[s] that the choices are always between two possibilities 0 and 1. Each proof will then be determined by a sequence of choices i₁, i₂, …, i_n (i₁ = 0 or 1, i₂ = 0 or 1, …, i_n = 0 or 1), and hence the number 2ⁿ + i₁2^n-1 + i₂2^n-2 + … +in completely determines the proof. The automatic machine carries out successively proof 1, proof 2, proof 3, …"(Footnote ‡ The Undecidable:138)

This is indeed the technique by which a deterministic (i.e. a-) Turing machine can be used to mimic the action of a nondeterministic Turing machine; Turing solved the matter in a footnote and appears to dismiss it from further consideration.

An oracle machine or o-machine is a Turing a-machine that pauses its computation at state "o" while, to complete its calculation, it "awaits the decision" of "the oracle"—an unspecified entity "apart from saying that it cannot be a machine" (Turing (1939), Undecidable p. 166–168). The concept is now actively used by mathematicians.

Universal Turing machines

As Turing wrote in Undecidable, p. 128:

"It is possible to invent a single machine which can be used to compute any computable sequence. If this machine U is supplied with the tape on the beginning of which is written the string of quintuples separated by semicolons of some computing machine M, then U will compute the same sequence as M." (italics added)

We now take this remarkable finding for granted. But at the time (1936) it was astonishing. The model of computation that Turing called his "universal machine"—"U" for short—is considered by some (cf Davis (2000)) to have been the fundamental theoretical breakthrough that led to the notion of the stored program computer.

"Turing's paper … contains, in essence, the invention of the modern computer and some of the programming techniques that accompanied it" (Minsky (1967), p. 104).

As part of his New Kind of Science, Stephen Wolfram announced his discovery of a 2-state 5-symbol universal Turing machine. Wolfram's example was the smallest universal Turing machine then known since it has the smallest product (2,5)=10 of any known universal Turing machine.

On May 14th, 2007, Wolfram announced a US$25,000 prize for the proof or disproof of the conjecture that an even simpler, 2-state 3-symbol Turing machine is universal [1]. A 2-state 3-symbol universal Turing machine was proven by Alex Smith, an undergraduate studying Electronic and Computer Engineering at the University of Birmingham, UK. The prize was won on October 24th, 2007. [2][3][4].

Comparison with real machines

It is often said that Turing machines, unlike simpler automata, are as powerful as real machines, and are able to execute any operation that a real program can. What is missed in this statement is that almost any particular program running on a particular machine is in fact nothing but a deterministic finite automaton, since the machine it runs on can only be in finitely many configurations. Turing machines would actually only be equivalent to a machine that had an unlimited amount of storage space. We might ask, then, why Turing machines are useful models of real computers. There are a number of ways to answer this:

Anything a real computer can compute, a Turing machine can also compute. For example: "A Turing machine can simulate any type of subroutine found in programming languages, including recursive procedures and any of the known parameter-passing mechanisms" (Hopcroft and Ullman p. 157). Thus, a statement about the limitations of Turing machines will also apply to real computers.
The difference lies only with the ability of a Turing machine to manipulate an unbounded amount of data. However, given a finite amount of time, a Turing machine (like a real machine) can only manipulate a finite amount of data.
Like a Turing machine, a real machine can have its storage space enlarged as needed, by acquiring more disks or other storage media. If the supply of these runs short, the Turing machine may become less useful as a model. But the fact is that neither Turing machines nor real machines need astronomical amounts of storage space in order to perform useful computation. The processing time required is usually much more of a problem.
Real machines are much more complex than a Turing machine. For example, a Turing machine describing an algorithm may have a few hundred states, while the equivalent deterministic finite automaton on a given real machine has quadrillions. This makes the DFA representation infeasible to analyze.
Turing machines describe algorithms independent of how much memory they utilize. There is a limit to the memory possessed by any current machine, but this limit can rise arbitrarily in time. Turing machines allow us to make statements about algorithms which will (theoretically) hold forever, regardless of advances in conventional computing machine architecture.
Turing machines simplify the statement of algorithms. Algorithms running on Turing-equivalent abstract machines are usually more general than their counterparts running on real machines, because they have arbitrary-precision data types available and never have to deal with unexpected conditions (including, but not limited to, running out of memory).

One way in which Turing machines are a poor model for programs is that many real programs, such as operating systems and word processors, are written to receive unbounded input over time, and therefore do not halt. Turing machines do not model such ongoing computation well (but can still model portions of it, such as individual procedures).

Limitations of Turing machines in computational complexity theory

A limitation of Turing Machines is that they do not model the strengths of a particular arrangement well. For instance, modern stored-program computers are actually instances of a more specific form of abstract machine known as the random access stored program machine or RASP machine model. Like the Universal Turing machine the RASP stores its "program" in "memory" external to its finite-state machine's "instructions". Unlike the Universal Turing Machine, the RASP has an infinite number of distinguishable, numbered but unbounded "registers"—memory "cells" that can contain any integer (cf Elgot and Robinson (1964), Hartmanis (1971), and in particular Cook-Rechow (1973); references at random access machine). The RASP's finite-state machine is equipped with the capability for indirect addressing (e.g. the contents of one register can "point to" the address of any other, arbitrary register); thus the RASP's "program" can address any register in the register-sequence. The upshot of this distinction is that there are computational optimizations that can be performed based on the memory indices, which are not possible in a general Turing Machine; thus when Turing Machines are used as the basis for bounding running times, a 'false lower bound' can be proven on certain algorithms' running times (due to the false simplifying assumption of a Turing Machine). An example of this is binary search, which violates the Turing machine's linear Ω(n) lower bound on searching an ordered list.

History

Historical background: computational machinery

Robin Gandy—a student of Alan Turing (1912–1954) and his life-long friend—traces the lineage of the notion of "calculating machine" back to Babbage (circa 1834) and actually proposes "Babbage's Thesis":

"That the whole of development and operations of analysis are now capable of being executed by machinery" (italics in Babbage as cited by Gandy, p. 54)

Gandy's analysis of Babbage's Analytical Engine describes the following five operations (cf p. 52–53):

The arithmetic functions +, -, x where - indicates "proper" subtraction x - y = 0 if y >= x
Any sequence of operations is an operation
Iteration of an operation (repeating n times an operation P)
Conditional iteration (repeating n times an operation P conditional on the "success" of test T)
Conditional transfer (i.e. conditional "goto")

Gandy states that "the functions which can be calculated by (1), (2), and (4) are precisely those which are Turing computable." (p. 53). He cites other proposals for "universal calculating machines" included those of Percy Ludgate (1909), Leonardo Torres y Quevedo (1914), M. d'Ocagne (1922), Louis Couffignal (1933), Vannevar Bush (1936), Howard Aiken (1937). However:

"... the emphasis is on programming a fixed iterable sequence of arithmetical operations. The fundamental importance of conditional iteration and conditional transfer for a general theory of calculating machines is not recognized ..."(Gandy p. 55)

The Entscheidungsproblem (the "decision problem"): Hilbert's tenth question of 1900

With regards to Hilbert's problems posed by the famous mathematician David Hilbert in 1900, an aspect of problem #10 had been floating about for almost 30 years before it was framed precisely. Hilbert's original expression for #10 is as follows:

"10. Determination of the solvability of a Diophantine equation. Given a Diophantine equation with any number of unknown quantities and with rational integral coefficients: To devise a process according to which it can be determined in a finite number of operations whether the equation is solvable in rational integers.

"The Entscheidungsproblem [decision problem for first-order logic] is solved when we know a procedure that allows for any given logical expression to decide by finitely many operations its validity or satisfiability ... The Entscheidungsproblem must be considered the main problem of mathematical logic. ..." (quoted, with this translation and the original German, in Nachum Dershowitz and Yuri Gurevich, "A Natural Axiomatization of Church's Thesis", 2007:1 [5])

By 1922, this notion of "Entscheidungsproblem" had developed a bit, and H. Behmann stated that

"... most general form of the Entscheidungsproblem [is] as follows:

"A quite definite generally applicable prescription is required which will allow one to decide in a finite number of steps the truth or falsity of a given purely logical assertion ..." (Gandy p. 57 quoting Behmann)

"Behmann remarks that ... the general problem is equivalent to the problem of deciding which mathematical propositions are true." (Gandy p. 57)

If one were able to solve the Entscheidungsproblem then one would have a "procedure for solving many (or even all) mathematical problems" (Gandy p. 92).

By the 1928 international congress of mathematicians Hilbert "made his questions quite precise. First, was mathematics complete ... Second, was mathematics consistent ... And thirdly, was mathematics decidable?" (Hodges p. 91, Hawking p. 1121). The first two questions were answered in 1930 by Kurt Gödel at the very same meeting where Hilbert delivered his retirement speech (much to the chagrin of Hilbert); the third — the Entscheidungsproblem — had to wait until the mid-1930's.

The problem was: An answer first required a precise definition of "definite general applicable prescription", what Alonzo Church would come to call "effective calculability", and in 1928 no such definition existed. But over the next 6–7 years Emil Post developed his definition of a worker moving from room to room writing and erasing marks per a list of instructions (Post 1936), as did Princeton professor Church and his two students Stephen Kleene and J. B. Rosser by use of Church's λ-calculus and Gödel's recursion theory (1934). Church's paper (published 15 April 1936) showed that the Entscheidungsproblem was indeed "undecidable" and beat Turing to the punch by almost a year (Turing's paper submitted 28 May 1936, published January 1937). In the meantime, Emil Post submittted a brief paper in the fall of 1936 so Turing at least had priority over Post. While Church refereed Turing's paper, Turing had time to study Church's paper and add an Appendix where he sketched a proof that Church's λ-calculus and his machines would compute the same functions.

"But what Church had done was something rather different, and in a certain sense weaker . . . the Turing construction was more direct, and provided an argument from first principles, closing the gap in Church's demonstration." (Hodges p. 112).

And Post had only proposed a definition of calculability and criticized Church's "definition", but had proved nothing.

Alan Turing's a- (automatic-)machine

In the spring of 1935 Alan M. Turing, the young Master's student at King's College Cambridge UK took on the challenge; he had been stimulated by the lectures of the logician M. H. A. Newman "and learned from them of Gödel's work and the Entscheidungsproblem ... Newman used the word 'mechanical' ... In his obituary of Turing 1955 Newman writes:

"To the question 'what is a "mechanical" process?' Turing returned the characteristic answer 'Something that can be done by a machine' and he embarked on the highly congenial task of analysing the general notion of a computing machine." (Gandy p. 74)

Gandy states that:

"I suppose, but do not know, that Turing, right from the start of his work, had as his goal a proof of the undecidability of the Entscheidungsproblem. He told me that the 'main idea' of the paper came to him when he was lying in Grantchester meadows in the summer of 1935. The 'main idea' might have either been his analysis of computation or his realization that there was a universal machine, and so a diagonal argument to prove unsolvability." (p. 76)

While Gandy believed that Newman's statement above is "misleading", this opinion is not shared by all. Turing had had a life-long interest in machines: "Alan had dreamt of inventing typewriters as a boy; [his mother] Mrs. Turing had a typewriter; and he could well have begun by asking himself what was meant by calling a typewriter 'mechanical'" (Hodges p. 96). While at Princeton pursuing his PhD, Turing built a Boolean-logic multiplier (see below). His PhD thesis, titled "Systems of Logic Based on Ordinals", contains the following definition of "a computable function":

"It was stated above that 'a function is effectively calculable if its values can be found by some purely mechanical process'. We may take this statement literally, understanding by a purely mechanical process one which could be carried out by a machine. It is possible to give a mathematical description, in a certain normal form, of the structures of these machines. The development of these ideas leads to the author's definition of a computable function, and to an identification of computability with effective calculabiity. It is not difficult, though somewhat laborious, to prove that these three definitions [the 3rd is the λ-calculus] are equivalent ..." (Turing (1939) in The Undecidable p. 160)

When Turing returned to the UK he ultimately became jointly responsible for breaking the German secret codes created by encryption machines called "The Enigma"; he also became involved in the design of the ACE (Automatic Computing Machine), "[Turing's] ACE proposal was effectively self-contained, and its roots lay not in the EDVAC [the USA's initiative], but in his own universal machine" (Hodges p. 318). Arguments still continue concerning the origin and nature of what has been named by Kleene (1952) Turing's Thesis. But what Turing did prove with his computational-machine model appears in his paper On Computable Numbers, With an Application to the Entscheidungsproblem (1937):

"[that] the Hilbert Entscheidungsproblem can have no solution ... I propose, therefore to show that there can be no general process for determining whether a given formula U of the functional calculus K is provable, i.e. that there can be no machine which, supplied with any one U of these formulae, will eventually say whether U is provable." (from Turing's paper reprinted in The Undecidable, p. 145)

Turing's example (his second proof): If one is to ask for a general procedure to tell us: "Does this machine ever print 0", the question is "undecidable."

1937–1970: The "digital computer", the birth of "Computer science"

In 1937, while at Princeton working on his PhD thesis, Turing built a digital (Boolean-logic) multiplier from scratch, making his own electromechanical relays (Hodges p. 138). "Alan's task was to embody the logical design of a Turing machine in a network of relay-operated switches . . ." (Hodges p. 138). While Turing might have been just curious and experimenting, quite-earnest work in the same direction was going in Germany ( Konrad Zuse (1938)), and in the United States (Howard Aiken) and George Stibitz (1937); the fruits of their labors were used by the Axis and Allied military in World War II (cf Hodges p. 298–299). In the early- to mid-1950's Hao Wang and Marvin Minsky reduced the Turing machine to a simpler form (a precursor to the Post-Turing machine of Martin Davis ); simultaneously European researchers were reducing the new-fangled electronic computer to a computer-like theoretical object equivalent to what was now being called a "Turing machine" (Turing by this time had killed himself by eating a poisoned apple). In the late 1950's and early 1960's, the coincidentally-parallel developments of Melzak and Lambek (1961), Minsky (1961), and Shepherdson and Sturgis (1961) carried the European work further and reduced the Turing machine to a more friendly, computer-like abstract model called the counter machine; Elgot and Robinson (1964), Hartmanis (1971), Cook and Reckhow (1973) carried this work even further with the register machine and random access machine models —but basically all are just multi-tape Turing machines with an arithmetic-like instruction set.

1970–present: The Turing machine as a model of computation

Today the counter, register and random-access machines and their sire the Turing machine continue to be the models of choice for theorists investigating questions in the theory of computation. In particular, computational complexity theory makes use of the Turing machine:

"Depending on the objects one likes to manipulate in the computations (numbers like nonnegative integers or aphanumeric strings), two models have obtained a domiant position in machine-based complexity theory:

"the off-line multitape Turing machine. . . , which represents the standard model for string-oriented computation, and

"the random access machine (RAM) as introduced by Cook and Reckhow . . . , which models the idealized Von Neumann style computer." (van Emde Boas 1990:4)

"Only in the related area of analysis of algorithms this role is taken over by the RAM model"(van Emde Boas 1990:16)

References

Brunfiel, Geoff, Student snags maths prize, Nature, October 24. 2007.
Boolos, George (1989, 1999). Computability and Logic (3rd ed. ed.). Cambridge UK: Cambridge University Press. ISBN 0-521-20402-X. {{cite book}}: |edition= has extra text (help); Check date values in: |year= (help); Unknown parameter |coauthors= ignored (|author= suggested) (help)CS1 maint: year (link)
Boolos, George (2002). Computability and Logic (4th ed. ed.). Cambridge UK: Cambridge University Press. ISBN 0-521-00758-5 (pb.). {{cite book}}: |edition= has extra text (help); Unknown parameter |coauthors= ignored (|author= suggested) (help)CS1 maint: extra punctuation (link) Some parts have been significantly rewritten by Burgess. Presentation of Turing machines in context of Lambek "abacus machines" (cf Register machine) and recursive functions, showing their equivalence.
Taylor L. Booth (1967), Sequential Machines and Automata Theory, John Wiley and Sons, Inc., New York. Graduate level engineering text; ranges over a wide variety of topics, Chapter IX Turing Machines includes some recursion theory.
B. Jack Copeland ed. (2004), The Essential Turing: Seminal Writings in Computing, Logic, Philosophy, Artificial Intelligence, and Artificial Life plus The Secrets of Enigma, Clarendon Press (Oxford University Press), Oxford UK, ISBN 0-19-825079-7. Contains the Turing papers plus a draft letter to Emil Post re his criticism of "Turing's convention", and Donald W. Davies' Corrections to Turing's Universal Computing Machine
Martin Davis (1958). Computability and Unsolvability. McGraw-Hill Book Company, Inc, New York. {{cite book}}: Cite has empty unknown parameter: |1= (help). On pages 12-20 he gives examples of 5-tuple tables for Addition, The Successor Function, Subtraction (x > = y), Proper Subtraction (0 if x < y), The Identity Function and various identity functions, and Multiplication.
Davis, Martin (1994). Second Edition: Computability, Complexity, and Languages and Logic: Fundamentals of Theoretical Computer Science (2nd ed. ed.). San Diego: Academic Press, Harcourt, Brace & Company. {{cite book}}: |edition= has extra text (help); Text "ISBN 0-12-206382-1" ignored (help)
Martin Davis (ed.) (1965), The Undecidable, Raven Press, Hewlett, NY.
Martin Davis (2000). Engines of Logic: Mathematicians and the origin of the Computer (1st edition ed.). W. W. Norton & Company, New York. ISBN 0-393-32229-7 pbk. {{cite book}}: |edition= has extra text (help)
Jim Giles, Simplest 'universal computer' wins student $25,000, New Scientist, October 24, 2007.
Robin Gandy, "The Confluence of Ideas in 1936", pp.51–102 in Rolf Herken, see below.
Hennie, Fredrick (1977). Introduction to Computability. Addison-Wesley, Reading, Mass. {{cite book}}: Text "id: QA248.5H4 1977" ignored (help). On pages 90–103 Hennie discusses the UTM with examples and flow-charts, but no actual 'code'.
Stephen Hawking (editor), 2005, God Created the Integers: The Mathematical Breakthroughs that Changed History", Running Press, Philadelphia, ISBN-13: 978-0-7624-1922-7. Includes Turing's 1936–1937 paper, with brief commentary and biography of Turing as written by Hawking.
Rolf Herken. The Universal Turing Machine—A Half-Century Survey. Springer Verlag. ISBN 3-211-82637-8.
Andrew Hodges, Alan Turing: The Enigma, Simon and Schuster, New York. Cf Chapter "The Spirit of Truth" for a history leading to, and a discussion of, his proof.
John Hopcroft and Jeffrey Ullman, (1979). Introduction to Automata Theory, Languages and Computation (1st edition ed.). Addison-Wesley, Reading Mass. ISBN 0-201-02988-X. {{cite book}}: |edition= has extra text (help)CS1 maint: extra punctuation (link) A difficult book. Centered around the issues of machine-interpretation of "languages", NP-Completeness, etc.
Hopcroft, John E. (2001). Introduction to Automata Theory, Languages, and Computation (2nd ed. ed.). Reading Mass: Addison-Wesley. {{cite book}}: |edition= has extra text (help); Text "ISBN 0-201-44124-1" ignored (help) Distinctly different and less intimidating than the first edition.
Stephen Kleene (1952), Introduction to Metamathematics, North-Holland Publishing Company, Amsterdam Netherlands, 10th impression (with corrections of 6th reprint 1971). Graduate level text; most of Chapter XIII Computable functions is on Turing machine proofs of computability of recursive functions, etc.
Knuth, Donald E. (1973). Volume 1/Fundamental Algorithms: The Art of computer Programming (2nd ed. ed.). Reading, Mass.: Addison-Wesley Publishing Company. {{cite book}}: |edition= has extra text (help). With reference to the role of Turing machines in the development of computation (both hardware and software) see 1.4.5 History and Bibliography pp. 225ff and 2.6 History and Bibliographypp. 456ff.
Marvin Minsky, Computation: Finite and Infinite Machines, Prentice-Hall, Inc., N.J., 1967. See Chapter 8, Section 8.2 "Unsolvability of the Halting Problem." Excellent, i.e. relatively readable, sometimes funny.
Christos Papadimitriou (1993). Computational Complexity (1st edition ed.). Addison Wesley. ISBN 0-201-53082-1. {{cite book}}: |edition= has extra text (help) Chapter 2: Turing machines, pp.19–56.
Emil Post (1936), "Finite Combinatory Processes—Formulation 1", Journal of Symbolic Logic, 1, 103–105, 1936. Reprinted in The Undecidable pp.289ff.
Emil Post (1947), "Recursive Unsolvability of a Problem of Thue", Journal of Symbolic Logic, vol. 12, pp. 1–11. Reprinted in The Undecidable pp.293ff. In the Appendix of this paper Post comments on and gives corrections to Turing's paper of 1936–1937.
Roger Penrose (1989, 1990). The Emperor's New Mind (2nd edition ed.). Oxford University Press, New York. ISBN 0-19-851973-7. {{cite book}}: |edition= has extra text (help); Check date values in: |year= (help)CS1 maint: year (link)
Ivars Peterson (1988). The Mathematical Tourist: Snapshots of Modern Mathematics (1st edition ed.). W. H. Freeman and Company, New York. ISBN 0-7167-2064-7 (pbk.). {{cite book}}: |edition= has extra text (help)
Rogozhin, Yurii, "A Universal Turing Machine with 22 States and 2 Symbols", Romanian Journal Of Information Science and Technology, 1(3), 259–265, 1998. (surveys known results about small universal Turing machines)
Michael Sipser (1997). Introduction to the Theory of Computation. PWS Publishing. ISBN 0-534-94728-X. Chapter 3: The Church-Turing Thesis, pp.125–149.
Stone, Harold S. (1972). Introduction to Computer Organization and Data Structures (1st ed. ed.). New York: McGraw-Hill Book Company. ISBN 0-07-061726-0. {{cite book}}: |edition= has extra text (help)
Paul Strathern. Turing and the Computer—The Big Idea. Anchor Books/Doubleday. ISBN 0-385-49243-X.
Alan Turing (1936), "On Computable Numbers, With an Application to the Entscheidungsproblem", Proceedings of the London Mathematical Society, Series 2, Volume 42 (1936). Eprint. Reprinted in The Undecidable pp.115–154.
Peter van Emde Boas 1990, Machine Models and Simulations, pp. 3–66, in Jan van Leeuwen, ed., Handbook of Theoretical Computer Science, Volume A: Algorithms and Complexity, The MIT Press/Elsevier, [place?], ISBN 0-444-88071-2 (Volume A). QA76.H279 1990. Valuable survey, with 141 references.
Hao Wang, "A variant to Turing's theory of computing machines", Journal of the Association for Computing Machinery (JACM) 4, 63–92 (1957).
Stephen Wolfram, Wolfram, Stephen, A New Kind of Science, Wolfram Media, ISBN 1-57955-008-8

External links

online Turing Machine emulator Online emulator
Turing Machine on Stanford Encyclopedia of Philosophy
Detailed info on the Church-Turing Hypothesis (Stanford Encyclopedia of Philosophy)
Turing Machine-Like Models in Molecular Biology, to understand life mechanisms with a the DNA-tape processor.
The Turing machine—Summary about the Turing machine, its functionality and historical facts
Can the cosmos be modeled by a Turing machine? Private web site concludes that the cosmos cannot be represented as a Turing computation.
The Turing Machine and Universal Computation—The paper discusses the concept of the Turing Machine in a historical context and why it provides an important connection between mathematics and computing.
The Wolfram 2,3 Turing Machine Research Prize—Stephen Wolfram's $25,000 prize for the proof or disproof of the universality of the potentially smallest universal Turing Machine.
Turing Machine in Conway's Game of Life by Paul Rendell

Simulators

Visual Turing Machine, a visual designer and simulator (Free software, cross-platform)
Visual Turing, a Turing machine interactive simulator/IDE (freeware, Windows)
Simple Java demonstration of a Turing Machine that adds and multiplies small numbers.
JFLAP, tool for simulating finite automata, pushdown automata and Turing machines (multi-tape and building blocks) (Free software)
Turing Machine Simulator, basic, easy-to-use Turing Machine simulator written in C# (Free software, .NET)
Visual Automata Simulator, a tool for simulating, visualizing and transforming finite state automata and Turing Machines (Free software, cross-platform)
Suzanne Britton's Turing Machine Simulator (Java applet)
C++ Simulator of a Nondeterministic and Deterministic Multitape Turing Machine (Free software)
C++ Simulator of a Universal Turing Machine (which accepts Multitape Turing Machine) (Free software)
Turing Train Terminal—A working Turing machine built out of model trains
TMML—Describing a Turing Machine with XML
TuringMachine function in Mathematica—Documentation for the Mathematica function that generates the evolution of Turing machines.