Turing machine

The Turing machine is an abstract model of computer execution and storage introduced in 1936 by Alan Turing to give a mathematically precise definition of algorithm or 'mechanical procedure'. As such it is still widely used in theoretical computer science, especially in complexity theory and the theory of computation. The thesis that states that Turing machines indeed capture the informal notion of effective or mechanical method in logic and mathematics is known as the Church-Turing thesis.

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 infinite amount of ordered paper sheets 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', remember the state 17, and go to the following sheet."

Turing machines shouldn't be confused with the Turing test, Turing's attempt to capture the notion of artificial intelligence.

A Turing machine that is able to simulate any other Turing machine is called a universal Turing machine or simply a universal machine as Turing described it in 1947:

It can be shown that a single special machine of that type can be made to do the work of all. It could in fact be made to work as a model of any other machine. The special machine may be called the universal machine.

Definition

Informal description

A Turing machine is a pushdown automaton made more powerful by relaxing the last-in-first-out requirement of its stack. (Interestingly, a 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.)

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 '0') and one or more other symbols. The tape is assumed to be arbitrarily extendible 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.
A head that can read and write symbols on the tape and move left and right.
A state register that stores the state of the Turing machine. The number of different states is always finite and there is one special start state with which the state register is initialized.
An action table (or transition function) that tells the machine what symbol to write, how to move the head ('L' for one step left, and 'R' for one step right) and what its new state will be, given the symbol it has just read on the tape and the state it is currently in. If there is no entry in the table for the current combination of symbol and state then the machine will halt.

Note that every part of the machine is finite, but it is the potentially unlimited amount of tape that gives it an unbounded amount of storage space.

Formal definition

One-tape Turing machine

More formally, a (one-tape) Turing machine is a 7-tuple $M=(Q,\Gamma ,\Sigma ,s,b,F,\delta )$ , where

$Q$ is a finite set of states
$\Gamma$ is a finite set of the tape alphabet
$\Sigma$ is a finite set of the input alphabet ( $\Sigma \subseteq \Gamma$ )
$s\in Q$ is the initial state
$b$ is the blank symbol ( $b\in \Gamma -\Sigma$ )
$F\subseteq Q$ is the set of final or accepting states
$\delta :Q\times \Gamma \rightarrow Q\times \Gamma \times \{L,R\}$ is a partial function called the transition function, where L is left shift, R is right shift.

k-tape Turing machine

A k-tape Turing machine is a 7-tuple $M=(Q,\Gamma ,\Sigma ,s,b,F,\delta )$ , where

$Q$ is a finite set of states
$\Gamma$ is a finite set of the tape alphabet
$\Sigma$ is a finite set of the input alphabet ( $\Sigma \subseteq \Gamma$ )
$s\in Q$ is the initial state
$b$ is the blank symbol ( $b\in \Gamma -\Sigma$ )
$F\subseteq Q$ is the set of final or accepting states
$\delta :Q\times \Gamma ^{k}\rightarrow Q\times (\Gamma \times \{L,R,S\})^{k}$ is a partial function called the transition function, where L is left shift, R is right shift, S is no shift.

Example

The following Turing machine has an alphabet {'0', '1'}, with 0 being the blank symbol. It expects a series of 1's on the tape, with the head initially on the leftmost 1, and doubles the 1's with a 0 in between, i.e., "111" becomes "1110111". The set of states is {s1, s2, s3, s4, s5} and the start state is s1. The action table is as follows.

 Old Read   Wr.      New     Old Read   Wr.      New
 St. Sym.   Sym. Mv. St.     St. Sym.   Sym. Mv. St.
 - - - - - - - - - - - -     - - - - - - - - - - - -
  s1  1  ->  0    R   s2      s4  1  ->  1    L   s4
  s2  1  ->  1    R   s2      s4  0  ->  0    L   s5
  s2  0  ->  0    R   s3      s5  1  ->  1    L   s5
  s3  0  ->  1    L   s4      s5  0  ->  1    R   s1
  s3  1  ->  1    R   s3

A computation of this Turing machine might for example be: (the position of the head is indicated by displaying the cell in bold face)

 Step State Tape   Step State Tape
 - - - - - - - -   - - - - - - - - -
    1  s1   11        9  s2   1001 
    2  s2   01       10  s3   1001
    3  s2   010      11  s3   10010
    4  s3   0100     12  s4   10011
    5  s4   0101     13  s4   10011
    6  s5   0101     14  s5   10011
    7  s5   0101     15  s1   11011
    8  s1   1101       -- halt --

The behavior of this machine can be described as a loop: it starts out in s1, replaces the first 1 with a 0, then uses s2 to move to the right, skipping over 1's and the first 0 encountered. S3 then skips over the next sequence of 1's (initially there are none) and replaces the first 0 it finds with a 1. S4 moves back to the left, skipping over 1's until it finds a 0 and switches to s5. s5 then moves to the left, skipping over 1's until it finds the 0 that was originally written by s1. It replaces that 0 with a 1, moves one position to the right and enters s1 again for another round of the loop. This continues until s1 finds a 0 (this is the 0 right in the middle between the two strings of 1's) at which time the machine halts.

Deterministic and non-deterministic Turing machines

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 non-deterministic Turing machine (NDTM or NTM).

Universal Turing machines

Every Turing machine computes a certain fixed partial function from the input strings over its alphabet. In that sense it behaves like a computer with a fixed program. However, as Alan Turing already described, we can encode the action table of any Turing machine in a string. Thus we might try to construct a Turing machine that expects on its tape a string describing an action table followed by a string describing the input tape, and then computes the tape that the encoded Turing machine would have computed. As Turing showed, such a Turing machine is indeed possible and since it is able to simulate any other Turing machine it is called a universal Turing machine.

With this encoding of action tables as strings, it becomes possible in principle for Turing machines to answer questions about the behavior of other Turing machines. Most of these questions, however, are undecidable, meaning that the function in question cannot be calculated by any Turing machine. For instance, the problem of determining whether a particular Turing machine will halt on a particular input, or on all inputs, known as the Halting problem, was shown to be undecidable in Turing's original paper. Rice's theorem shows that any nontrivial question about the behavior or output of a Turing machine is undecidable.

If we broaden the definition to include any Turing machine that simulates some Turing-complete computational model, not just Turing machines that directly simulate other Turing machines, a universal Turing machine can be fairly simple, using just a few states and a few symbols. For example, only 2 states are needed, since a 2×18 (meaning 2 states, 18 symbols) universal Turing machine is known. A complete list of the smallest known universal Turing machines is: 2×18, 3×10, 4×6, 5×5, 7×4, 10×3, 22×2. These simulate a computational model called tag systems.

A universal Turing machine is Turing-complete. It can calculate any recursive function, decide any recursive language, and accept any recursively enumerable language. According to the Church-Turing thesis, the problems solvable by a universal Turing machine are exactly those problems solvable by an algorithm or an effective method of computation, for any reasonable definition of those terms.

Comparison with real machines

It's 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 and given a finite amount of input is in fact nothing but a deterministic finite automaton, since the machine it runs on can only be in finitely many states. Turing machines would actually only be equivalent to a real machine that is magically given an infinite 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:

Less powerful models of real machines are also much more complex. For example, a Turing machine describing an algorithm may have a few hundred states, while the equivalent DFA on a given machine has quadrillions.
Turing machines can describe algorithms at once over all machines, regardless of how much memory they have; there is a maximum to the amount of memory any machine has now, but this limit can rise arbitrarily in time. Statements about algorithms should be timeless.
Turing machines can describe algorithms without reference to any sort of machine-specific characteristics, such as the memory limit.
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 datatypes available and never have to deal with unexpected conditions such as running out of memory.

On the other hand, there's one way in which Turing machines are often considered a poor model for programs: real programs, such as operating systems and word processors, often receive an unbounded amount of input over time, and never "finish" their task. Turing machines do not model such ongoing computation well, but can still model portions of it, such as individual procedures.

A physical Turing machine

It is not difficult to simulate a Turing machine on a modern computer (except for the limited memory of actually existing computers).

It is also possible to build a Turing machine on a purely mechanical basis. The mathematician Karl Scherer has indeed built such a machine in 1986 using metal and plastic construction sets, and some wood. The 1.5 meter high machine uses the pulling of strings to read, move and write the data (which is represented using ball bearings).

The machine is now exhibited in the entrance of the Department of Computer Science of the University of Heidelberg, Germany.

Also, using a few hundred mirrors, one can build an optical universal Turing machine in one's backyard, using the Horseshoe map. This is based on a work by Stephen Smale.

The concept of a Turing machine was used as an educational tool in the science fiction novel, The Diamond Age (1995), by Neal Stephenson. The main character, Nell, possesses an interactive book which teaches her creative thinking and logic by presenting puzzles in a story as Turing machines which become more and more complex as the story progresses. They begin as a simple chain-fed clockwork device and progresses to abstract economic processes, trades, and finally the interaction of entire fictional kingdoms.

References

Rolf Herken: The Universal Turing Machine - A Half-Century Survey, Springer Verlag, ISBN 3-211-82637-8
Paul Strathern: Turing and the Computer - The big idea, Anchor Books/Doubleday, ISBN 0-385-49243-X
Turing, A., On Computable Numbers, With an Application to the Entscheidungsproblem, Proceedings of the London Mathematical Society, Series 2, Volume 42, 1936; reprinted in M. David (ed.), The Undecidable, Hewlett, NY: Raven Press, 1965;
Boolos, G. and Jeffrey, R., Computability and Logic, 2nd ed., Cambridge: Cambridge University Press, 1980.
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)

External links

Turing Machine on Stanford Encyclopedia of Philosophy
Detailed info on the Church-Turing Hypothesis (Stanford Encyclopedia of Philosophy)
Computing machinery and intelligence - A.M. Turing Original 1950 article by Alan Turing on machine intelligence, where he introduces the famous Turing test
The Turing test and intelligencedocument clarifing the meaning of the Turing test and suggests that meeting the Turing test is already in the process of being achieved
Citations from CiteSeer

Simulators

Visual Turing, a Turing machine interactive simulator/IDE (free software for Windows).
Suzanne Brittons 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 Machine as an educational freeware board game for PCs
Turing Machine II - an educational freeware board game for PCs