Turing machine
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A Turing machine is a theoretical device that manipulates symbols according to a table of rules contained on a strip of tape. 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 computer.
The "Turing" machine was described by Alan Turing in 1937,[1] who called it an "a(utomatic)-machine". Turing machines are not intended as a practical computing technology, but rather as a thought experiment representing a computing machine. They help computer scientists understand the limits of mechanical computation.
Turing gave a succinct definition of the experiment in his 1948 essay, "Intelligent Machinery". Referring to his 1936 publication, Turing wrote that the Turing machine, here called a Logical Computing Machine, consisted of:
...an infinite memory capacity obtained in the form of an infinite tape marked out into squares, on each of which a symbol could be printed. At any moment there is one symbol in the machine; it is called the scanned symbol. The machine can alter the scanned symbol and its behavior is in part determined by that symbol, but the symbols on the tape elsewhere do not affect the behavior of the machine. However, the tape can be moved back and forth through the machine, this being one of the elementary operations of the machine. Any symbol on the tape may therefore eventually have an innings.[2] (Turing 1948, p. 61)
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'.
Studying their abstract properties yields many insights into computer science and complexity theory.[3]
Informal description
- For visualizations of Turing machines, see Turing machine gallery.
The Turing machine mathematically models a machine that mechanically operates on a tape on which symbols are written which it can read and write one at a time using a tape head. Operation is fully determined by a finite set of elementary instructions such as "in state 42, if the symbol seen is 0, write a 1; if the symbol seen is 1, shift to the right, and change into state 17; in state 17, if the symbol seen is 0, write a 1 and change to state 6;" etc. In the original article ("On computable numbers, with an application to the Entscheidungsproblem", see also references below), Turing imagines not a mechanism, but a person whom he calls the "computer", who executes these deterministic mechanical rules slavishly (or as Turing puts it, "in a desultory manner").
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 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.
- 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 finite TABLE ("action table", or transition function) of instructions (usually quintuples [5-tuples] : qiaj→qi1aj1dk, but sometimes 4-tuples) that, given the state(qi) the machine is currently in and the symbol(aj) it is reading on the tape (symbol currently under HEAD) tells the machine to do the following in sequence (for the 5-tuple models):
- Either erase or write a symbol (instead of aj written aj1), and then
- Move the head (which is described by dk and can have values: 'L' for one step left or 'R' for one step right or 'N' for staying in the same place), and then
- Assume the same or a new state as prescribed (go to state qi1).
In the 4-tuple models, erase or write a symbol (aj1) and move the head left or right (dk) are specified as separate instructions. Specifically, the TABLE tells the machine to (ia) erase or 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, one of finitely many. There is one special start state with which the state register is initialized. These states, writes Turing, replace the "state of mind" a person performing computations would ordinarily be in.
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
Hopcroft and Ullman (1979, p. 148) formally define a (one-tape) Turing machine as a 7-tuple 
where
- Q is a finite, non-empty set of states
- Γ is a finite, non-empty set of the tape alphabet/symbols
is the blank symbol (the only symbol allowed to occur on the tape infinitely often at any step during the computation)
is the set of input symbols
is the initial state
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. (A relatively uncommon variant allows "no shift", say N, as a third element of the latter set.)
Anything that operates according to these specifications is a Turing machine.
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 = { 1 }
- δ = see state-table below
- q0 = A = initial state
- F = the one element set of final states {HALT}
Initially all tape cells are marked with 0.
| 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 | 1 | R | B | 1 | L | A | 1 | L | B |
| 1 | 1 | L | C | 1 | R | B | 1 | R | HALT |
Additional details required to visualize or implement Turing machines
In the words of van Emde Boas (1990), p. 6: "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."
For instance,
- There will need to be some decision on what the symbols actually look like, and a failproof way of reading and writing symbols indefinitely.
- The shift left and shift right operations may shift the tape head across the tape, but when actually building a Turing machine it is more practical to make the tape slide back and forth under the head instead.
- The tape can be finite, and automatically extended with blanks as needed (which is closest to the mathematical definition), but it is more common to think of it as stretching infinitely at both ends and being pre-filled with blanks except on the explicitly given finite fragment the tape head is on. (This is, of course, not implementable in practice.) The tape cannot be fixed in length, since that would not correspond to the given definition and would seriously limit the range of computations the machine can perform to those of a linear bounded automaton.
Alternative definitions
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): (qi, Sj, Sk/E/N, L/R/N, qm)
- ( current state qi , symbol scanned Sj , print symbol Sk/erase E/none N , move_tape_one_square left L/right R/none N , new state qm )
Other authors (Minsky (1967) p. 119, Hopcroft and Ullman (1979) p. 158, Stone (1972) p. 9) adopt a different convention, with new state qm listed immediately after the scanned symbol Sj:
- (definition 2): (qi, Sj, qm, Sk/E/N, L/R/N)
- ( current state qi , symbol scanned Sj , new state qm , print symbol Sk/erase E/none N , move_tape_one_square left L/right R/none N )
For the remainder of this article "definition 1" (the Turing/Davis convention) will be used.
| 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 Sk" 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 types of 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 | qi | Sj | Print(Sk) | Left L | qm | (qi, Sj, Sk, L, qm) | "blank" = S0, 1=S1, etc. | |
| N2 | qi | Sj | Print(Sk) | Right R | qm | (qi, Sj, Sk, R, qm) | "blank" = S0, 1=S1, etc. | |
| N3 | qi | Sj | Print(Sk) | None N | qm | (qi, Sj, Sk, N, qm) | "blank" = S0, 1=S1, etc. | (qi, Sj, Sk, qm) |
| 4 | qi | Sj | None N | Left L | qm | (qi, Sj, N, L, qm) | (qi, Sj, L, qm) | |
| 5 | qi | Sj | None N | Right R | qm | (qi, Sj, N, R, qm) | (qi, Sj, R, qm) | |
| 6 | qi | Sj | None N | None N | qm | (qi, Sj, N, N, qm) | Direct "jump" | (qi, Sj, N, qm) |
| 7 | qi | Sj | Erase | Left L | qm | (qi, Sj, E, L, qm) | ||
| 8 | qi | Sj | Erase | Right R | qm | (qi, Sj, E, R, qm) | ||
| 9 | qi | Sj | Erase | None N | qm | (qi, Sj, E, N, qm) | (qi, Sj, E, qm) |
Any Turing table (list of instructions) can be constructed from the above nine 5-tuples. For technical reasons, the three non-printing or "N" instructions (4, 5, 6) can usually be dispensed with. For examples see Turing machine examples.
Less frequently the use of 4-tuples are encountered: 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, as it can mean two things. 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", (its internal state) and the machine's (or person's) "state of progress" through the computation - the current state of the total system. 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.
A variant of this is seen 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.
"State" in the context of Turing machines should be clarified as to which is being described: (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.
Turing's biographer Andrew Hodges (1983: 107) has noted and discussed this confusion.
Turing machine "state" diagrams
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Notes
- ^ The idea came to him in mid-1935 (perhaps, see more in the History section) after a question posed by M. H. A. Newman in his lectures -- "Was there a definite method, or as Newman put it, a mechanical process which could be applied to a mathematical statement, and which would come up with the answer as to whether it was provable" (Hodges 1983:93). Turing submitted his paper on 31 May 1936 to the London Mathematical Society for its Proceedings (cf Hodges 1983:112), but it was published in early 1937 -- offprints available February 1937 (cf Hodges 1983:129).
- ^ See the definition of "innings" on Wiktionary
- ^ Vitanyi, Paul M.B. (2009). "Turing machine". Scholarpedia 4 (3). ISSN 1941-6016. http://www.scholarpedia.org/article/Turing_machine#Importance_of_the_Turing_machine. Retrieved 23 April 2010. "In the last three-quarter of a century the Turing machine model has proven to be of priceless value for the development of the science of dataprocessing. All theory development reaches back to this format. The model has become so dominant that new other models that are not polynomial-time reducible to Turing machines are viewed as not realistic (the so-called polynomial-time Computability thesis).".
References
Primary literature, reprints, and compilations
- 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 (ed.) (1965), The Undecidable, Raven Press, Hewlett, NY.
- 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. In particular see the footnotes 11 with corrections to the universal computing machine coding and footnote 14 with comments on Turing's first and second proofs.
- Turing, A.M. (1936). "On Computable Numbers, with an Application to the Entscheidungsproblem". Proceedings of the London Mathematical Society. 2 42: 230–65. 1937. doi:10.1112/plms/s2-42.1.230. (and Turing, A.M. (1938). "On Computable Numbers, with an Application to the Entscheidungsproblem: A correction". Proceedings of the London Mathematical Society. 2 43: 544–6. 1937. doi:10.1112/plms/s2-43.6.544.). Reprinted in many collections, e.g. in The Undecidable pp. 115–154; available on the web in many places, e.g. at Scribd.
- Alan Turing, 1948, "Intelligent Machinery." Reprinted in "Cybernetics: Key Papers." Ed. C.R. Evans and A.D.J. Robertson. Baltimore: University Park Press, 1968. p. 31.
- F. C. Hennie and R. E. Stearns. Two-tape simulation of multitape Turing machines. JACM, 13(4):533–546, 1966.
Computability theory
- Boolos, George; Richard Jeffrey (1989, 1999). Computability and Logic (3rd ed.). Cambridge UK: Cambridge University Press. ISBN 0-521-20402-X.
- Boolos, George; John Burgess, Richard Jeffrey, (2002). Computability and Logic (4th ed.). Cambridge UK: Cambridge University Press. ISBN 0-521-00758-5 (pb.). 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.
- Martin Davis (1958). Computability and Unsolvability. McGraw-Hill Book Company, Inc, New York.. 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; Ron Sigal, Elaine J. Weyuker (1994). Computability, Complexity, and Languages and Logic: Fundamentals of Theoretical Computer Science (2nd ed.). San Diego: Academic Press, Harcourt, Brace & Company. ISBN 0122063821.
- Hennie, Fredrick (1977). Introduction to Computability. Addison–Wesley, Reading, Mass.. On pages 90–103 Hennie discusses the UTM with examples and flow-charts, but no actual 'code'.
- John Hopcroft and Jeffrey Ullman, (1979). Introduction to Automata Theory, Languages and Computation (1st ed.). Addison–Wesley, Reading Mass. ISBN 0-201-02988-X.. A difficult book. Centered around the issues of machine-interpretation of "languages", NP-completeness, etc.
- Hopcroft, John E.; Rajeev Motwani, Jeffrey D. Ullman (2001). Introduction to Automata Theory, Languages, and Computation (2nd ed.). Reading Mass: Addison–Wesley. ISBN 0201441241. 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.). Reading, Mass.: Addison–Wesley Publishing Company.. 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.
- Zohar Manna, 1974, Mathematical Theory of Computation. Reprinted, Dover, 2003. ISBN 9780486432380
- 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 ed.). Addison Wesley. ISBN 0-201-53082-1. Chapter 2: Turing machines, pp. 19–56.
- 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.). New York: McGraw–Hill Book Company. ISBN 0-07-061726-0.
- 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.
Church's thesis
- Nachum Dershowitz; Yuri Gurevich (September 2008). "A natural axiomatization of computability and proof of Church's Thesis". Bulletin of Symbolic Logic 14 (3). http://research.microsoft.com/en-us/um/people/gurevich/Opera/188.pdf. Retrieved 2008-10-15.
- Roger Penrose (1989, 1990). The Emperor's New Mind (2nd ed.). Oxford University Press, New York. ISBN 0-19-851973-7.
Small Turing machines
- Rogozhin, Yurii, 1998, "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)
- Stephen Wolfram, 2002, A New Kind of Science, Wolfram Media, ISBN 1-57955-008-8
- Brunfiel, Geoff, Student snags maths prize, Nature, October 24. 2007.
- Jim Giles (2007), Simplest 'universal computer' wins student $25,000, New Scientist, October 24, 2007.
- Alex Smith, Universality of Wolfram’s 2, 3 Turing Machine, Submission for the Wolfram 2, 3 Turing Machine Research Prize.
- Vaughan Pratt, 2007, "Simple Turing machines, Universality, Encodings, etc.", FOM email list. October 29, 2007.
- Martin Davis, 2007, "Smallest universal machine", and Definition of universal Turing machine FOM email list. October 26–27, 2007.
- Alasdair Urquhart, 2007 "Smallest universal machine", FOM email list. October 26, 2007.
- Hector Zenil (Wolfram Research), 2007 "smallest universal machine", FOM email list. October 29, 2007.
- Todd Rowland, 2007, "Confusion on FOM", Wolfram Science message board, October 30, 2007.
Other
- Martin Davis (2000). Engines of Logic: Mathematicians and the origin of the Computer (1st ed.). W. W. Norton & Company, New York. ISBN 0-393-32229-7 pbk..
- Robin Gandy, "The Confluence of Ideas in 1936", pp. 51–102 in Rolf Herken, see below.
- Stephen Hawking (editor), 2005, God Created the Integers: The Mathematical Breakthroughs that Changed History, Running Press, Philadelphia, ISBN 978-0-7624-1922-7. Includes Turing's 1936–1937 paper, with brief commentary and biography of Turing as written by Hawking.
- Rolf Herken (1995). 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.
- Ivars Peterson (1988). The Mathematical Tourist: Snapshots of Modern Mathematics (1st ed.). W. H. Freeman and Company, New York. ISBN 0-7167-2064-7 (pbk.).
- Paul Strathern (1997). Turing and the Computer—The Big Idea. Anchor Books/Doubleday. ISBN 0-385-49243-X.
- Hao Wang, "A variant to Turing's theory of computing machines", Journal of the Association for Computing Machinery (JACM) 4, 63–92 (1957).
- Charles Petzold, Petzold, Charles, The Annotated Turing, John Wiley & Sons, Inc., ISBN 0-47022-905-5
- Arora, Sanjeev; Barak, Boaz, "Complexity Theory: A Modern Approach", Cambridge University Press, 2009, ISBN 978-0-521-42426-4, section 1.4, "Machines as strings and the universal Turing machine" and 1.7, "Proof of theorem 1.9"
- Isaiah Pinchas Kantorovitz, "A note on turing machine computability of rule driven systems", ACM SIGACT News December 2005.
External links
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Wikimedia Commons has media related to: Turing machine |
- 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 DNA-tape processor.
- The Turing machine—Summary about the Turing machine, its functionality and historical facts
- 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. The contest has ended, with the proof affirming the machine's universality.
- Turing Machine in Conway's Game of Life by Paul Rendell
- "Turing Machine Causal Networks" by Enrique Zeleny, Wolfram Demonstrations Project.
- Turing Machines at the Open Directory Project
- n bands Turing Machine Simulator
- Implementation of a Turing Machine
- Video of a physical Turing Machine running
- Turing machine calculations at wolframalpha.com
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