LA MACHINE DE TURING COMPILATION REALISATION A L’AIDE DE LEGOS Turing machine This article is about the symbol manipulator. For the de- expressing all tasks accomplishable by computers; nearly ciphering machine, see Bombe. For the test of artificial all programming languages are Turing complete if the intelligence, see Turing test. For the instrumental rock limitations of finite memory are ignored. band, see Turing Machine (band). Classes of automata 1 Overview A Turing machine is an abstract machine[1] that ma- nipulates symbols on a strip of tape according to a ta- A Turing machine is a general example of a CPU that ble of rules; to be more exact, it is a mathematical model controls all data manipulation done by a computer, with of computation that defines such a device.[2] Despite the the canonical machine using sequential memory to store model’s simplicity, given any computer algorithm, a Tur- data. More specifically, it is a machine (automaton) capa- ing machine can be constructed that is capable of simu- ble of enumerating some arbitrary subset of valid strings lating that algorithm’s logic.[3] of an alphabet; these strings are part of a recursively enu- The machine operates on an infinite[4] memory tape di- merable set. vided into cells.[5] The machine positions its head over Assuming a black box, the Turing machine cannot know a cell and “reads” (scans[6]) the symbol there. Then as whether it will eventually enumerate any one specific per the symbol and its present place in a finite table[7] of string of the subset with a given program. This is due user-specified instructions the machine (i) writes a sym- to the fact that the halting problem is unsolvable, which bol (e.g. a digit or a letter from a finite alphabet) in the cell has major implications for the theoretical limits of com- (some models allowing symbol erasure[8] and/or no writ- puting. ing), then (ii) either moves the tape one cell left or right The Turing machine is capable of processing an (some models allow no motion, some models move the unrestricted grammar, which further implies that it is ca- head),[9] then (iii) (as determined by the observed sym- pable of robustly evaluating first-order logic in an infinite bol and the machine’s place in the table) either proceeds number of ways. This is famously demonstrated through to a subsequent instruction or halts[10] the computation. lambda calculus. The Turing machine was invented in 1936 by Alan A Turing machine that is able to simulate any other Tur- Turing,[11][12] who called it an a-machine (automatic ing machine is called a universal Turing machine (UTM, machine).[13] With this model Turing was able to answer or simply a universal machine). A more mathemati- two questions in the negative: (1) Does a machine ex- cally oriented definition with a similar “universal” na- ist that can determine whether any arbitrary machine on ture was introduced by Alonzo Church, whose work on its tape is “circular” (e.g. freezes, or fails to continue lambda calculus intertwined with Turing’s in a formal its computational task); similarly, (2) does a machine ex- theory of computation known as the Church–Turing the- ist that can determine whether any arbitrary machine on sis. The thesis states that Turing machines indeed cap- its tape ever prints a given symbol.[14] Thus by provid- ture the informal notion of effective methods in logic ing a mathematical description of a very simple device and mathematics, and provide a precise definition of capable of arbitrary computations, he was able to prove an algorithm or “mechanical procedure”. Studying their properties of computation in general - and in particular, abstract properties yields many insights into computer the uncomputability of the Hilbert Entscheidungsprob- science and complexity theory. lem (“decision problem”).[15] Thus, Turing machines prove fundamental limitations on the power of mechanical computation.[16] While they can 1.1 Physical description express arbitrary computations, their minimalistic design makes them unsuitable for computation in practice: real- In his 1948 essay, “Intelligent Machinery”, Turing wrote world computers are based on different designs that, un- that his machine consisted of: like Turing machines, use random-access memory. ...an unlimited memory capacity obtained in Turing completeness is the ability for a system of instruc- the form of an infinite tape marked out into tions to simulate a Turing machine. A programming lan- squares, on each of which a symbol could be guage that is Turing complete is theoretically capable of printed. At any moment there is one symbol 1 2 2 INFORMAL DESCRIPTION in the machine; it is called the scanned sym- The alphabet contains a special blank symbol (here bol. The machine can alter the scanned sym- written as '0') and one or more other symbols. The bol, and its behavior is in part determined by tape is assumed to be arbitrarily extendable to the that symbol, but the symbols on the tape else- left and to the right, i.e., the Turing machine is al- where do not affect the behavior of the ma- ways supplied with as much tape as it needs for its chine. However, the tape can be moved back computation. Cells that have not been written be- and forth through the machine, this being one fore are assumed to be filled with the blank symbol. of the elementary operations of the machine. In some models the tape has a left end marked with Any symbol on the tape may therefore eventu- a special symbol; the tape extends or is indefinitely ally have an innings.[17] (Turing 1948, p. 3[18]) extensible to the right. 2. A head that can read and write symbols on the tape 2 Informal description and move the tape left and right one (and only one) cell at a time. In some models the head moves and For visualizations of Turing machines, see Turing the tape is stationary. machine gallery. 3. A state register that stores the state of the Tur- ing machine, one of finitely many. Among these is The Turing machine mathematically models a machine the special start state with which the state register is that mechanically operates on a tape. On this tape are initialized. These states, writes Turing, replace the symbols, which the machine can read and write, one at “state of mind” a person performing computations a time, using a tape head. Operation is fully determined would ordinarily be in. 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 4. A finite table[19] of instructions[20] that, given the is 1, change into state 17; in state 17, if the symbol seen state(qᵢ) the machine is currently in and the sym- is 0, write a 1 and change to state 6;" etc. In the original bol(a) it is reading on the tape (symbol currently article (“On computable numbers, with an application to under the head), tells the machine to do the follow- the Entscheidungsproblem", see also references below), ing in sequence (for the 5-tuple models): Turing imagines not a mechanism, but a person whom he calls the “computer”, who executes these determinis- • tic mechanical rules slavishly (or as Turing puts it, “in a Either erase or write a symbol (replacing a desultory manner”). with a₁), and then • Move the head (which is described by d and q 4 can have values: 'L' for one step left or 'R' for s1 s1 s 3 s1 s0 s1 one step right or 'N' for staying in the same place), and then The head is always over a particular square of the tape; only a • Assume the same or a new state as prescribed finite stretch of squares is shown. The instruction to be performed (go to state qᵢ₁). (q4) is shown over the scanned square. (Drawing after Kleene (1952) p.375.) In the 4-tuple models, erasing or writing a symbol (a₁) and moving the head left or right (d) are spec- ified as separate instructions. Specifically, the table 0 0 0 0 0 1 1 B 0 0 tells the machine to (ia) erase or write a symbol or (ib) move the head left or right, and then (ii) assume q 1 the same or a new state as prescribed, but not both actions (ia) and (ib) in the same instruction. In some Here, the internal state (q1) is shown inside the head, and the il- models, if there is no entry in the table for the cur- lustration describes the tape as being infinite and pre-filled with rent combination of symbol and state then the ma- “0”, the symbol serving as blank. The system’s full state (its com- chine will halt; other models require all entries to be plete configuration) consists of the internal state, any non-blank filled. symbols on the tape (in this illustration “11B”), and the position of the head relative to those symbols including blanks, i.e. “011B”. (Drawing after Minsky (1967) p. 121). Note that every part of the machine (i.e. its state, symbol- collections, and used tape at any given time) and its ac- More precisely, a Turing machine consists of: tions (such as printing, erasing and tape motion) is finite, discrete and distinguishable; it is the unlimited amount of 1. A tape divided into cells, one next to the other. Each tape and runtime that gives it an unbounded amount of cell contains a symbol from some finite alphabet.