Switch statement

From Wikipedia, the free encyclopedia
https://en.wikipedia.org/wiki/Switch_statement

In computer programming languages, a switch statement is a type of selection control mechanism used to allow the value of a variable or expression to change the control flow of program execution via search and map.

Switch statements function somewhat similarly to the if statement used in programming languages like C/C++, C#, Visual Basic .NET, Java and exist in most high-level imperative programming languages such as Pascal, Ada, C/C++, C#, Visual Basic .NET, Java, and in many other types of language, using such keywords as switch, case, select, or inspect.

Switch statements come in two main variants: a structured switch, as in Pascal, which takes exactly one branch, and an unstructured switch, as in C, which functions as a type of goto. The main reasons for using a switch include improving clarity, by reducing otherwise repetitive coding, and (if the heuristics permit) also offering the potential for faster execution through easier compiler optimization in many cases.

An example of a switch statement in C

switch (age) { case 1: printf("You're one."); break; case 2: printf("You're two."); break; case 3: printf("You're three."); case 4: printf("You're three or four."); break; default: printf("You're not 1, 2, 3 or 4!"); }

History

In his 1952 text Introduction to Metamathematics, Stephen Kleene formally proved that the CASE function (the IF-THEN-ELSE function being its simplest form) is a primitive recursive function, where he defines the notion "definition by cases" in the following manner:

"#F. The function φ defined thus

φ(x₁ , ... , x_n ) =

• φ₁(x₁ , ... , x_n ) if Q₁(x₁ , ... , x_n ),
• . . . . . . . . . . . .
• φ_m(x₁ , ... , x_n ) if Q_m(x₁ , ... , x_n ),
• φ_m+1(x₁ , ... , x_n ) otherwise,

where Q₁ , ... , Q_m are mutually exclusive predicates (or φ(x₁ , ... , x_n) shall have the value given by the first clause which applies) is primitive recursive in φ₁, ..., φ_m+1, Q₁, ..., Q_m+1.

— Stephen Kleene,

Kleene provides a proof of this in terms of the Boolean-like recursive functions "sign-of" sg( ) and "not sign of" ~sg( ) (Kleene 1952:222-223); the first returns 1 if its input is positive and −1 if its input is negative.

Boolos-Burgess-Jeffrey make the additional observation that "definition by cases" must be both mutually exclusive and collectively exhaustive. They too offer a proof of the primitive recursiveness of this function (Boolos-Burgess-Jeffrey 2002:74-75).

The IF-THEN-ELSE is the basis of the McCarthy formalism: its usage replaces both primitive recursion and the mu-operator.

The earliest Fortran compilers supported the computed GOTO statement for multi-way branching. Early ALGOL compilers supported a SWITCH data type which contains a list of "designational expressions". A GOTO statement could reference a switch variable and, by providing an index, branch to the desired destination. With experience it was realized that a more formal multi-way construct, with single point of entrance and exit, was needed. Languages such as BCPL, ALGOL-W, and ALGOL-68 introduced forms of this construct which have survived through modern languages.

Typical syntax

In most languages, programmers write a switch statement across many individual lines using one or two keywords. A typical syntax involves:

• the first select, followed by an expression which is often referred to as the control expression or control variable of the switch statement
• subsequent lines defining the actual cases (the values), with corresponding sequences of statements for execution when a match occurs
• In languages with fallthrough behavior, a break statement typically follows a case statement to end said statement. [Wells]
• In some languages, e.g., PL/I, the control expression is optional; if there is no control expression then each alternative begins with a WHEN clause containing a Boolean expression and a match occurs for the first case for which that expression evaluates to true. This usage is similar to the if/then/elseif/else structures in some other languages, e.g., Perl.
• In some languages, e.g., Rexx, no control expression is allowed and each alternative begins with a WHEN clause containing a Boolean expression and a match occurs for the first case for which that expression evaluates to true.

Each alternative begins with the particular value, or list of values (see below), that the control variable may match and which will cause the control to goto the corresponding sequence of statements. The value (or list/range of values) is usually separated from the corresponding statement sequence by a colon or by an implication arrow. In many languages, every case must also be preceded by a keyword such as case or when.

An optional default case is typically also allowed, specified by a default, otherwise, or else keyword. This executes when none of the other cases match the control expression. In some languages, such as C, if no case matches and the default is omitted the switch statement simply does nothing. In others, like PL/I, an error is raised.

Semantics

Semantically, there are two main forms of switch statements.

The first form are structured switches, as in Pascal, where exactly one branch is taken, and the cases are treated as separate, exclusive blocks. This functions as a generalized if–then–else conditional, here with any number of branches, not just two.

The second form are unstructured switches, as in C, where the cases are treated as labels within a single block, and the switch functions as a generalized goto. This distinction is referred to as the treatment of fallthrough, which is elaborated below.

Fallthrough

In many languages, only the matching block is executed, and then execution continues at the end of the switch statement. These include the Pascal family (Object Pascal, Modula, Oberon, Ada, etc.) as well as PL/I, modern forms of Fortran and BASIC dialects influenced by Pascal, most functional languages, and many others. To allow multiple values to execute the same code (and avoid needing to duplicate code), Pascal-type languages permit any number of values per case, given as a comma-separated list, as a range, or as a combination.

Languages derived from C language, and more generally those influenced by Fortran's computed GOTO, instead feature fallthrough, where control moves to the matching case, and then execution continues ("falls through") to the statements associated with the next case in the source text. This also allows multiple values to match the same point without any special syntax: they are just listed with empty bodies. Values can be special conditioned with code in the case body. In practice, fallthrough is usually prevented with a break keyword at the end of the matching body, which exits execution of the switch block, but this can cause bugs due to unintentional fallthrough if the programmer forgets to insert the break statement. This is thus seen by many as a language wart, and warned against in some lint tools. Syntactically, the cases are interpreted as labels, not blocks, and the switch and break statements explicitly change control flow. Some languages influenced by C, such as JavaScript, retain default fallthrough, while others remove fallthrough, or only allow it in special circumstances. Notable variations on this in the C-family include C#, in which all blocks must be terminated with a break or return unless the block is empty (i.e. fallthrough is used as a way to specify multiple values).

In some cases languages provide optional fallthrough. For example, Perl does not fall through by default, but a case may explicitly do so using a continue keyword. This prevents unintentional fallthrough but allows it when desired. Similarly, Bash defaults to not falling through when terminated with ;;, but allows fallthrough with ;& or ;;& instead.

An example of a switch statement that relies on fallthrough is Duff's device.

Compilation

Optimizing compilers such as GCC or Clang may compile a switch statement into either a branch table or a binary search through the values in the cases. A branch table allows the switch statement to determine with a small, constant number of instructions which branch to execute without having to go through a list of comparisons, while a binary search takes only a logarithmic number of comparisons, measured in the number of cases in the switch statement.

Normally, the only method of finding out if this optimization has occurred is by actually looking at the resultant assembly or machine code output that has been generated by the compiler.

Advantages and disadvantages

In some languages and programming environments, the use of a case or switch statement is considered superior to an equivalent series of if else if statements because it is:

• Easier to debug (e.g. setting breakpoints on code vs. a call table, if the debugger has no conditional breakpoint capability)
• Easier for a person to read
• Easier to understand, and consequently easier to maintain
• Fixed depth: a sequence of "if else if" statements may yield deep nesting, making compilation more difficult (especially in automatically generated code)
• Easier to verify that all values are handled. Compilers can issue a warning if some enum values are not handled.

Additionally, an optimized implementation may execute much faster than the alternative, because it is often implemented by using an indexed branch table. For example, deciding program flow based on a single character's value, if correctly implemented, is vastly more efficient than the alternative, reducing instruction path lengths considerably. When implemented as such, a switch statement essentially becomes a perfect hash.

In terms of the control-flow graph, a switch statement consists of two nodes (entrance and exit), plus one edge between them for each option. By contrast, a sequence of "if...else if...else if" statements has an additional node for every case other than the first and last, together with a corresponding edge. The resulting control-flow graph for the sequences of "if"s thus has many more nodes and almost twice as many edges, with these not adding any useful information. However, the simple branches in the if statements are individually conceptually easier than the complex branch of a switch statement. In terms of cyclomatic complexity, both of these options increase it by k−1 if given k cases.

Switch expressions

Switch expressions are introduced in Java SE 12, 19 March 2019, as a preview feature. Here a whole switch expression can be used to return a value. There is also a new form of case label, case L-> where the right-hand-side is a single expression. This also prevents fall through and requires that cases are exhaustive. In Java SE 13 the yield statement is introduced, and in Java SE 14 switch expressions become a standard language feature. For example:

int ndays = switch (month) {
    case JAN, MAR, MAY, JUL, AUG, OCT, DEC -> 31;
    case APR, JUN, SEP, NOV -> 30;
    case FEB -> {
        if (year % 400 == 0) yield 29;
        else if (year % 100 == 0) yield 28;
        else if (year % 4 == 0) yield 29;
        else yield 28; }
};

Alternative uses

Many languages evaluate expressions inside switch blocks at runtime, allowing a number of less obvious uses for the construction. This prohibits certain compiler optimizations, so is more common in dynamic and scripting languages where the enhanced flexibility is more important than the performance overhead.

PHP

For example, in PHP, a constant can be used as the "variable" to check against, and the first case statement which evaluates to that constant will be executed:

switch (true) {
    case ($x == 'hello'):
        foo();
        break;
    case ($z == 'howdy'): break;
}
switch (5) {
    case $x: break;
    case $y: break;
}

This feature is also useful for checking multiple variables against one value rather than one variable against many values. COBOL also supports this form (and other forms) in the EVALUATE statement. PL/I has an alternative form of the SELECT statement where the control expression is omitted altogether and the first WHEN that evaluates to true is executed.

Ruby

In Ruby, due to its handling of === equality, the statement can be used to test for variable’s class:

case input
when Array then puts 'input is an Array!'
when Hash then puts 'input is a Hash!'
end

Ruby also returns a value that can be assigned to a variable, and doesn’t actually require the case to have any parameters (acting a bit like an else if statement):

catfood =
  case
  when cat.age <= 1
    junior
  when cat.age > 10
    senior
  else
    normal
  end

Assembler

A switch statement in assembly language:

switch:
  cmp ah, 00h
  je a
  cmp ah, 01h
  je b
  jmp swtend   ; No cases match or "default" code here
a:
  push ah
  mov al, 'a'
  mov ah, 0Eh
  mov bh, 00h
  int 10h
  pop ah
  jmp swtend   ; Equivalent to "break"
b:
  push ah
  mov al, 'b'
  mov ah, 0Eh
  mov bh, 00h
  int 10h
  pop ah
  jmp swtend   ; Equivalent to "break"
  ...
swtend:

Python

For Python 3.10.6, PEPs 634-636 were accepted, which added match and case keywords. Unlike other languages, Python notably doesn't exhibit fallthrough behavior.

letter = input("Put in a single letter: ").strip()[0].casefold()  # First non-whitespace character of the input, lowercase
match letter:
    case "a" | "e" | "i" | "o" | "u":  # Unlike conditions in if statements, the `or` keyword cannot be used here to differentiate between cases
        print(f"Letter {letter} is a vowel!")
    case "y":
        print(f"Letter {letter} may be a vowel.")
    case _: # `case _` is equivalent to `default` from C and others
        print(f"Letter {letter} is not a vowel!")

Exception handling

A number of languages implement a form of switch statement in exception handling, where if an exception is raised in a block, a separate branch is chosen, depending on the exception. In some cases a default branch, if no exception is raised, is also present. An early example is Modula-3, which use the TRY...EXCEPT syntax, where each EXCEPT defines a case. This is also found in Delphi, Scala, and Visual Basic .NET.

Alternatives

Some alternatives to switch statements can be:

• A series of if-else conditionals that examine the target one value at a time. Fallthrough behavior can be achieved with a sequence of if conditionals each without the else clause.
• A lookup table, which contains, as keys, the case values and, as values, the part under the case statement.

(In some languages, only actual data types are allowed as values in the lookup table. In other languages, it is also possible to assign functions as lookup table values, gaining the same flexibility as a real switch statement. See Control table article for more detail on this).

Lua does not support case/switch statements. This lookup technique is one way to implement switch statements in the Lua language, which has no built-in switch. In some cases, lookup tables are more efficient than non-optimized switch statements since many languages can optimize table lookups, whereas switch statements are not optimized unless the range of values is small with few gaps. A non-optimized, non-binary search lookup, however, will almost certainly be slower than either a non-optimized switch or the equivalent multiple if-else statements.

• A control table (that may be implemented as a simple lookup table) can also be customized to accommodate multiple conditions on multiple inputs if required and usually exhibits greater 'visual compactness' than an equivalent switch (that can occupy many statements).
• Pattern matching, which is used to implement switch-like functionality in many functional languages.

Turing machine

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Turing_machine

A physical Turing machine model constructed by Mike Davey. A true Turing machine would have a tape of infinite length; however, physical models can only have a finite amount of tape.

A Turing machine is a mathematical model of computation describing an abstract machine that manipulates symbols on a strip of tape according to a table of rules. Despite the model's simplicity, it is capable of implementing any computer algorithm.

The machine operates on an infinite memory tape divided into discrete cells, each of which can hold a single symbol drawn from a finite set of symbols called the alphabet of the machine. It has a "head" that, at any point in the machine's operation, is positioned over one of these cells, and a "state" selected from a finite set of states. At each step of its operation, the head reads the symbol in its cell. Then, based on the symbol and the machine's own present state, the machine writes a symbol into the same cell, and moves the head one step to the left or the right, or halts the computation. The choice of which replacement symbol to write, which direction to move the head, and whether to halt is based on a finite table that specifies what to do for each combination of the current state and the symbol that is read. As with a real computer program, it is possible for a Turing machine to go into an infinite loop which will never halt.

The Turing machine was invented in 1936 by Alan Turing, who called it an "a-machine" (automatic machine). It was Turing's doctoral advisor, Alonzo Church, who later coined the term "Turing machine" in a review. With this model, Turing was able to answer two questions in the negative:

• Does a machine exist that can determine whether any arbitrary machine on its tape is "circular" (e.g., freezes, or fails to continue its computational task)?
• Does a machine exist that can determine whether any arbitrary machine on its tape ever prints a given symbol?

Thus by providing a mathematical description of a very simple device capable of arbitrary computations, he was able to prove properties of computation in general—and in particular, the uncomputability of the Entscheidungsproblem, or 'decision problem' (whether every mathematical statement is provable or disprovable).

Turing machines proved the existence of fundamental limitations on the power of mechanical computation.

While they can express arbitrary computations, their minimalist design makes them too slow for computation in practice: real-world computers are based on different designs that, unlike Turing machines, use random-access memory.

Turing completeness is the ability for a computational model or a system of instructions to simulate a Turing machine. A programming language that is Turing complete is theoretically capable of expressing all tasks accomplishable by computers; nearly all programming languages are Turing complete if the limitations of finite memory are ignored.

Overview

A Turing machine is an idealised model of a central processing unit (CPU) that controls all data manipulation done by a computer, with the canonical machine using sequential memory to store data. Typically, the sequential memory is represented as a tape of infinite length on which the machine can perform read and write operations.

In the context of formal language theory, a Turing machine (automaton) is capable of enumerating some arbitrary subset of valid strings of an alphabet. A set of strings which can be enumerated in this manner is called a recursively enumerable language. The Turing machine can equivalently be defined as a model that recognises valid input strings, rather than enumerating output strings.

Given a Turing machine M and an arbitrary string s, it is generally not possible to decide whether M will eventually produce s. This is due to the fact that the halting problem is unsolvable, which has major implications for the theoretical limits of computing.

The Turing machine is capable of processing an unrestricted grammar, which further implies that it is capable of robustly evaluating first-order logic in an infinite number of ways. This is famously demonstrated through lambda calculus.

A Turing machine that is able to simulate any other Turing machine is called a universal Turing machine (UTM, or simply a universal machine). Another mathematical formalism, lambda calculus, with a similar "universal" nature was introduced by Alonzo Church. Church's work intertwined with Turing's to form the basis for the Church–Turing thesis. This thesis states that Turing machines, lambda calculus, and other similar formalisms of computation do indeed capture the informal notion of effective methods in logic and mathematics and thus provide a model through which one can reason about an algorithm or "mechanical procedure" in a mathematically precise way without being tied to any particular formalism. Studying the abstract properties of Turing machines has yielded many insights into computer science, computability theory, and complexity theory.

Physical description

In his 1948 essay, "Intelligent Machinery", Turing wrote that his machine consists of:

...an unlimited 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.

— Turing 1948, p. 3

Description

For examples of Turing machines, see Turing machine examples.

The Turing machine mathematically models a machine that mechanically operates on a tape. On this tape are symbols, which the machine 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, 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").

The head is always over a particular square of the tape; only a finite stretch of squares is shown. The state of the machine (q₄) is shown over the square to be processed. (Drawing after Kleene (1952) p. 375.)

Here, the internal state (q₁) is shown inside the head, and the illustration describes the tape as being infinite and pre-filled with "0", the symbol serving as blank. The system's full state (its "complete configuration") consists of the internal state, any non-blank 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.)

More explicitly, a Turing machine consists of:

• A tape 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, so that the Turing machine is always supplied with as much tape as it needs for its computation. Cells that have not been written 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 state register that stores the state of the Turing machine, one of finitely many. Among these is the special start state with which the state register is initialised. These states, writes Turing, replace the "state of mind" a person performing computations would ordinarily be in.
• A finite table of instructions that, given the state(q_i) the machine is currently in and the symbol(a_j) it is reading on the tape (the symbol currently under the head), tells the machine to do the following in sequence (for the 5-tuple models):

1. Either erase or write a symbol (replacing a_j with a_j1).
2. Move the head (which is described by d_k and can have values: 'L' for one step left or 'R' for one step right or 'N' for staying in the same place).
3. Assume the same or a new state as prescribed (go to state q_i1).

In the 4-tuple models, erasing or writing a symbol (a_j1) and moving the head left or right (d_k) are specified as separate instructions. 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.

Every part of the machine (i.e. its state, symbol-collections, and used tape at any given time) and its actions (such as printing, erasing and tape motion) is finite, discrete and distinguishable; it is the unlimited amount of tape and runtime that gives it an unbounded amount of storage space.

Formal definition

Following Hopcroft & Ullman (1979, p. 148), a (one-tape) Turing machine can be formally defined as a 7-tuple ${\displaystyle M=⟨Q,\mathrm{\Gamma },b,\mathrm{\Sigma },\delta ,q_0,F⟩}$ where

• ${\displaystyle \mathrm{\Gamma }}$ is a finite, non-empty set of tape alphabet symbols;
• ${\displaystyle b\in \mathrm{\Gamma }}$ is the blank symbol (the only symbol allowed to occur on the tape infinitely often at any step during the computation);
• ${\displaystyle \mathrm{\Sigma }\subseteq \mathrm{\Gamma }\setminus \{b\}}$ is the set of input symbols, that is, the set of symbols allowed to appear in the initial tape contents;
• ${\displaystyle Q}$ is a finite, non-empty set of states;
• ${\displaystyle q_0\in Q}$ is the initial state;
• ${\displaystyle F\subseteq Q}$ is the set of final states or accepting states. The initial tape contents is said to be accepted by ${\displaystyle M}$ if it eventually halts in a state from ${\displaystyle F}$.
• ${\displaystyle \delta :(Q\setminus F)\times \mathrm{\Gamma }\rightharpoonup Q\times \mathrm{\Gamma }\times \{L,R\}}$ is a partial function called the transition function, where L is left shift, R is right shift. If ${\displaystyle \delta }$ is not defined on the current state and the current tape symbol, then the machine halts; intuitively, the transition function specifies the next state transited from the current state, which symbol to overwrite the current symbol pointed by the head, and the next head movement.

3-state Busy Beaver. Black icons represent location and state of head; square colors represent 1s (orange) and 0s (white); time progresses vertically from the top until the HALT state at the bottom.

A variant allows "no shift", say N, as a third element of the set of directions ${\displaystyle \{L,R\}}$.

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

• ${\displaystyle Q=\{\text{A},\text{B},\text{C},\text{HALT}\}}$ (states);
• ${\displaystyle \mathrm{\Gamma }=\{0,1\}}$ (tape alphabet symbols);
• ${\displaystyle b=0}$ (blank symbol);
• ${\displaystyle \mathrm{\Sigma }=\{1\}}$ (input symbols);
• ${\displaystyle q_0=\text{A}}$ (initial state);
• ${\displaystyle F=\{\text{HALT}\}}$ (final states);
• ${\displaystyle \delta =}$ see state-table below (transition function).

Initially all tape cells are marked with ${\displaystyle 0}$.

State table for 3-state, 2-symbol busy beaver

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 visualise 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 many decisions 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 one or 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 if the tape was proportional to the input size, or finite-state machine if it was strictly fixed-length.

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, the set could be changed from ${\displaystyle \{L,R\}}$ to ${\displaystyle \{L,R,N\}}$, where N ("None" or "No-operation") would allow the machine to stay on the same tape cell instead of moving left or right. This would 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 The 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/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/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.

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 The 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), The Undecidable, p. 300). The abbreviations are Turing's (The 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", 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 is supposed to not to appear elsewhere) and then by the note of instructions. This expression is called the "state formula".

— The Undecidable, pp. 139–140, emphasis added

Earlier in his paper Turing carried this even further: he gives an example where he placed a symbol of the current "m-configuration"—the instruction's label—beneath the scanned square, together with all the symbols on the tape (The Undecidable, p. 121); this he calls "the complete configuration" (The 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), that is, the instantaneous description is the composite of non-blank symbols to the left, state of the machine, the current symbol scanned by the head, and the non-blank symbols to the right.

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: the current instruction, or the list of symbols on the tape together with the current instruction, or 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.

"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	R	HALT

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 (1967), 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.

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 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 three-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 the machine were to be stopped and cleared 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) The Undecidable, pp. 139–140).

Equivalent models

See also: Turing machine equivalents, Register machine, and Post–Turing machine

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). (The Church–Turing thesis hypothesises this to be true for any kind of machine: that anything that can be "computed" can be computed by some Turing machine.)

A Turing machine is equivalent to a single-stack pushdown automaton (PDA) that has been made more flexible and concise by relaxing the last-in-first-out (LIFO) requirement of its stack. In addition, a Turing machine is also equivalent to a two-stack PDA with standard LIFO semantics, by using one stack to model the tape left of the head and the other stack for the tape to the right.

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, multi-track 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.

Read-only, right-moving Turing machines are equivalent to DFAs (as well as NFAs by conversion using the NFA to DFA conversion algorithm).

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

A relevant question is whether or not the computation model represented by concrete programming languages is Turing equivalent. While the computation of a real computer is based on finite states and thus not capable to simulate a Turing machine, programming languages themselves do not necessarily have this limitation. Kirner et al., 2009 have shown that among the general-purpose programming languages some are Turing complete while others are not. For example, ANSI C is not Turing complete, as all instantiations of ANSI C (different instantiations are possible as the standard deliberately leaves certain behaviour undefined for legacy reasons) imply a finite-space memory. This is because the size of memory reference data types, called pointers, is accessible inside the language. However, other programming languages like Pascal do not have this feature, which allows them to be Turing complete in principle. It is just Turing complete in principle, as memory allocation in a programming language is allowed to fail, which means the programming language can be Turing complete when ignoring failed memory allocations, but the compiled programs executable on a real computer cannot.

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.

— The Undecidable, p. 118

Turing (1936) does not elaborate further except in 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 + ... +i_n completely determines the proof. The automatic machine carries out successively proof 1, proof 2, proof 3, ..." (Footnote ‡, The Undecidable, p. 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 entity unspecified by Turing "apart from saying that it cannot be a machine" (Turing (1939), The Undecidable, p. 166–168).

Universal Turing machines

Main article: Universal Turing machine

An implementation of a Turing machine

As Turing wrote in The Undecidable, p. 128 (italics added):

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.

This finding is now taken for granted, but at the time (1936) it was considered astonishing.^{[citation needed]} 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

In terms of computational complexity, a multi-tape universal Turing machine need only be slower by logarithmic factor compared to the machines it simulates. This result was obtained in 1966 by F. C. Hennie and R. E. Stearns. (Arora and Barak, 2009, theorem 1.9)

Comparison with real machines

A Turing machine realization using Lego pieces

Turing machines are more powerful than some other kinds of automata, such as finite-state machines and pushdown automata. According to the Church–Turing thesis, they are as powerful as real machines, and are able to execute any operation that a real program can. What is neglected in this statement is that, because a real machine can only have a finite number of configurations, it is nothing but a finite-state machine, whereas a Turing machine has an unlimited amount of storage space available for its computations.

There are a number of ways to explain why Turing machines are useful models of real computers:

• 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). A large enough FSA can also model any real computer, disregarding IO. 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.
• Descriptions of real machine programs using simpler abstract models are often much more complex than descriptions using Turing machines. For example, a Turing machine describing an algorithm may have a few hundred states, while the equivalent deterministic finite automaton (DFA) 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 use. 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.
• Algorithms running on Turing-equivalent abstract machines can have arbitrary-precision data types available and never have to deal with unexpected conditions (including, but not limited to, running out of memory).

Another Turing machine realization

Limitations

Computational complexity theory

Further information: 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 be used as an address to specify another 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, an algorithm that can be shown to perform more quickly when using the RASP model of computation rather than the Turing machine model.

Interaction

In the early days of computing, computer use was typically limited to batch processing, i.e., non-interactive tasks, each producing output data from given input data. Computability theory, which studies computability of functions from inputs to outputs, and for which Turing machines were invented, reflects this practice.

Since the 1970s, interactive use of computers became much more common. In principle, it is possible to model this by having an external agent read from the tape and write to it at the same time as a Turing machine, but this rarely matches how interaction actually happens; therefore, when describing interactivity, alternatives such as I/O automata are usually preferred.

Comparison with the arithmetic model of computation

The arithmetic model of computation differs from the Turing model in two aspects:

• In the arithmetic model, every real number requires a single memory cell, whereas in the Turing model the storage size of a real number depends on the number of bits required to represent it.
• In the arithmetic model, every basic arithmetic operation on real numbers (addition, subtraction, multiplication and division) can be done in a single step, whereas in the Turing model the run-time of each arithmetic operation depends on the length of the operands.

Some algorithms run in polynomial time in one model but not in the other one. For example:

• The Euclidean algorithm runs in polynomial time in the Turing model, but not in the arithmetic model.
• The algorithm that reads n numbers and then computes ${\displaystyle 2^{2^n}}$ by repeated squaring runs in polynomial time in the Arithmetic model, but not in the Turing model. This is because the number of bits required to represent the outcome is exponential in the input size.

However, if an algorithm runs in polynomial time in the arithmetic model, and in addition, the binary length of all involved numbers is polynomial in the length of the input, then it is always polynomial-time in the Turing model. Such an algorithm is said to run in strongly polynomial time.

History

See also: Algorithm and Church–Turing thesis

Historical background: computational machinery

Robin Gandy (1919–1995)—a student of Alan Turing (1912–1954), and his lifelong friend—traces the lineage of the notion of "calculating machine" back to Charles 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):

1. The arithmetic functions +, −, ×, where − indicates "proper" subtraction: x − y = 0 if y ≥ x.
2. Any sequence of operations is an operation.
3. Iteration of an operation (repeating n times an operation P).
4. Conditional iteration (repeating n times an operation P conditional on the "success" of test T).
5. 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" including those of Percy Ludgate (1909), Leonardo Torres Quevedo (1914), Maurice 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 regard 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 No. 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 Dershowitz and Gurevich, 2008

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.

— ibid.

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

— ibid., 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-1930s.

The problem was that an answer first required a precise definition of "definite general applicable prescription", which Princeton professor 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 Church and his two students Stephen Kleene and J. B. Rosser by use of Church's lambda-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 submitted 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 lambda-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 criticised Church's "definition", but had proved nothing.

Alan Turing's a-machine

In the spring of 1935, Turing as a young Master's student at King's College, Cambridge, 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.

— ibid., p. 76

While Gandy believed that Newman's statement above is "misleading", this opinion is not shared by all. Turing had a lifelong 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 calculability. 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

Alan Turing invented the "a-machine" (automatic machine) in 1936.^[7] 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 and offprints were available in February 1937 (cf. Hodges 1983:129) It was Turing's doctoral advisor, Alonzo Church, who later coined the term "Turing machine" in a review. With this model, Turing was able to answer two questions in the negative:

• Does a machine exist that can determine whether any arbitrary machine on its tape is "circular" (e.g., freezes, or fails to continue its computational task)?
• Does a machine exist that can determine whether any arbitrary machine on its tape ever prints a given symbol?

Thus by providing a mathematical description of a very simple device capable of arbitrary computations, he was able to prove properties of computation in general—and in particular, the uncomputability of the Entscheidungsproblem ('decision problem').

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 Engine), "[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 as 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 initially 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 both the Axis and Allied militaries in World War II (cf. Hodges p. 298–299). In the early to mid-1950s 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". In the late 1950s and early 1960s, 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: 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 alphanumeric strings), two models have obtained a dominant 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 idealised 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