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Kernighan and Ritchie say in the Introduction of "The C Programming Language": "C, like any other language, has its blemishes. Some of the operators have the wrong precedence; some parts of the syntax could be better." The C standard did not attempt to correct many of these blemishes, because of the impact of such changes on already existing software.
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Newline indicates the end of a text line; it need not correspond to an actual single character, although for convenience C treats it as one.
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Additional multi-byte encoded characters may be used in string literals, but they are not entirely portable. The latest C standard (C11) allows multi-national Unicode characters to be embedded portably within C source text by using codice_82 or codice_83 encoding (where the codice_84 denotes a hexadecimal character), although this feature is not yet widely implemented.
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The basic C execution character set contains the same characters, along with representations for alert, backspace, and carriage return. Run-time support for extended character sets has increased with each revision of the C standard.
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C89 has 32 reserved words, also known as keywords, which are the words that cannot be used for any purposes other than those for which they are predefined:
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Most of the recently reserved words begin with an underscore followed by a capital letter, because identifiers of that form were previously reserved by the C standard for use only by implementations. Since existing program source code should not have been using these identifiers, it would not be affected when C implementations started supporting these extensions to the programming language. Some standard headers do define more convenient synonyms for underscored identifiers. The language previously included a reserved word called codice_129, but this was seldom implemented, and has now been removed as a reserved word.
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C supports a rich set of operators, which are symbols used within an expression to specify the manipulations to be performed while evaluating that expression. C has operators for:
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C uses the operator codice_135 (used in mathematics to express equality) to indicate assignment, following the precedent of Fortran and PL/I, but unlike ALGOL and its derivatives. C uses the operator codice_145 to test for equality. The similarity between these two operators (assignment and equality) may result in the accidental use of one in place of the other, and in many cases, the mistake does not produce an error message (although some compilers produce warnings). For example, the conditional expression codice_165 might mistakenly be written as codice_166, which will be evaluated as true if codice_75 is not zero after the assignment.
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The C operator precedence is not always intuitive. For example, the operator codice_145 binds more tightly than (is executed prior to) the operators codice_137 (bitwise AND) and codice_138 (bitwise OR) in expressions such as codice_171, which must be written as codice_172 if that is the coder's intent.
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The "hello, world" example, which appeared in the first edition of "K&R", has become the model for an introductory program in most programming textbooks. The program prints "hello, world" to the standard output, which is usually a terminal or screen display.
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The first line of the program contains a preprocessing directive, indicated by codice_15. This causes the compiler to replace that line with the entire text of the codice_174 standard header, which contains declarations for standard input and output functions such as codice_175 and codice_176. The angle brackets surrounding codice_174 indicate that codice_174 can be located using a search strategy that prefers headers provided with the compiler to other headers having the same name, as opposed to double quotes which typically include local or project-specific header files.
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The next line indicates that a function named codice_179 is being defined. The codice_179 function serves a special purpose in C programs; the run-time environment calls the codice_179 function to begin program execution. The type specifier codice_12 indicates that the value that is returned to the invoker (in this case the run-time environment) as a result of evaluating the codice_179 function, is an integer. The keyword codice_9 as a parameter list indicates that this function takes no arguments.
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The next line "calls" (diverts execution to) a function named codice_175, which in this case is supplied from a system library. In this call, the codice_175 function is "passed" (provided with) a single argument, the address of the first character in the string literal codice_188. The string literal is an unnamed array with elements of type codice_13, set up automatically by the compiler with a final 0-valued character to mark the end of the array (codice_175 needs to know this). The codice_191 is an "escape sequence" that C translates to a "newline" character, which on output signifies the end of the current line. The return value of the codice_175 function is of type codice_12, but it is silently discarded since it is not used. (A more careful program might test the return value to determine whether or not the codice_175 function succeeded.) The semicolon codice_195 terminates the statement.
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The closing curly brace indicates the end of the code for the codice_179 function. According to the C99 specification and newer, the codice_179 function, unlike any other function, will implicitly return a value of codice_79 upon reaching the codice_59 that terminates the function. (Formerly an explicit codice_200 statement was required.) This is interpreted by the run-time system as an exit code indicating successful execution.
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The type system in C is static and weakly typed, which makes it similar to the type system of ALGOL descendants such as Pascal. There are built-in types for integers of various sizes, both signed and unsigned, floating-point numbers, and enumerated types (codice_8). Integer type codice_13 is often used for single-byte characters. C99 added a boolean datatype. There are also derived types including arrays, pointers, records (codice_6), and unions (codice_33).
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C is often used in low-level systems programming where escapes from the type system may be necessary. The compiler attempts to ensure type correctness of most expressions, but the programmer can override the checks in various ways, either by using a "type cast" to explicitly convert a value from one type to another, or by using pointers or unions to reinterpret the underlying bits of a data object in some other way.
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Some find C's declaration syntax unintuitive, particularly for function pointers. (Ritchie's idea was to declare identifiers in contexts resembling their use: "declaration reflects use".)
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C's "usual arithmetic conversions" allow for efficient code to be generated, but can sometimes produce unexpected results. For example, a comparison of signed and unsigned integers of equal width requires a conversion of the signed value to unsigned. This can generate unexpected results if the signed value is negative.
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C supports the use of pointers, a type of reference that records the address or location of an object or function in memory. Pointers can be "dereferenced" to access data stored at the address pointed to, or to invoke a pointed-to function. Pointers can be manipulated using assignment or pointer arithmetic. The run-time representation of a pointer value is typically a raw memory address (perhaps augmented by an offset-within-word field), but since a pointer's type includes the type of the thing pointed to, expressions including pointers can be type-checked at compile time. Pointer arithmetic is automatically scaled by the size of the pointed-to data type.
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Pointers are used for many purposes in C. Text strings are commonly manipulated using pointers into arrays of characters. Dynamic memory allocation is performed using pointers; the result of a codice_205 is usually cast to the data type of the data to be stored. Many data types, such as trees, are commonly implemented as dynamically allocated codice_6 objects linked together using pointers. Pointers to other pointers are often used in multi-dimensional arrays and arrays of codice_6 objects. Pointers to functions ("function pointers") are useful for passing functions as arguments to higher-order functions (such as qsort or bsearch), in dispatch tables, or as callbacks to event handlers .
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A "null pointer value" explicitly points to no valid location. Dereferencing a null pointer value is undefined, often resulting in a segmentation fault. Null pointer values are useful for indicating special cases such as no "next" pointer in the final node of a linked list, or as an error indication from functions returning pointers. In appropriate contexts in source code, such as for assigning to a pointer variable, a "null pointer constant" can be written as codice_79, with or without explicit casting to a pointer type, or as the codice_209 macro defined by several standard headers. In conditional contexts, null pointer values evaluate to false, while all other pointer values evaluate to true.
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Void pointers (codice_210) point to objects of unspecified type, and can therefore be used as "generic" data pointers. Since the size and type of the pointed-to object is not known, void pointers cannot be dereferenced, nor is pointer arithmetic on them allowed, although they can easily be (and in many contexts implicitly are) converted to and from any other object pointer type.
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Careless use of pointers is potentially dangerous. Because they are typically unchecked, a pointer variable can be made to point to any arbitrary location, which can cause undesirable effects. Although properly used pointers point to safe places, they can be made to point to unsafe places by using invalid pointer arithmetic; the objects they point to may continue to be used after deallocation (dangling pointers); they may be used without having been initialized (wild pointers); or they may be directly assigned an unsafe value using a cast, union, or through another corrupt pointer. In general, C is permissive in allowing manipulation of and conversion between pointer types, although compilers typically provide options for various levels of checking. Some other programming languages address these problems by using more restrictive reference types.
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Array types in C are traditionally of a fixed, static size specified at compile time. The more recent C99 standard also allows a form of variable-length arrays. However, it is also possible to allocate a block of memory (of arbitrary size) at run-time, using the standard library's codice_205 function, and treat it as an array.
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Since arrays are always accessed (in effect) via pointers, array accesses are typically "not" checked against the underlying array size, although some compilers may provide bounds checking as an option. Array bounds violations are therefore possible and can lead to various repercussions, including illegal memory accesses, corruption of data, buffer overruns, and run-time exceptions.
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C does not have a special provision for declaring multi-dimensional arrays, but rather relies on recursion within the type system to declare arrays of arrays, which effectively accomplishes the same thing. The index values of the resulting "multi-dimensional array" can be thought of as increasing in row-major order. Multi-dimensional arrays are commonly used in numerical algorithms (mainly from applied linear algebra) to store matrices. The structure of the C array is well suited to this particular task. However, in early versions of C the bounds of the array must be known fixed values or else explicitly passed to any subroutine that requires them, and dynamically sized arrays of arrays cannot be accessed using double indexing. (A workaround for this was to allocate the array with an additional "row vector" of pointers to the columns.) C99 introduced "variable-length arrays" which address this issue.
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The following example using modern C (C99 or later) shows allocation of a two-dimensional array on the heap and the use of multi-dimensional array indexing for accesses (which can use bounds-checking on many C compilers):
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The subscript notation codice_212 (where codice_213 designates a pointer) is syntactic sugar for codice_214. Taking advantage of the compiler's knowledge of the pointer type, the address that codice_215 points to is not the base address (pointed to by codice_213) incremented by codice_25 bytes, but rather is defined to be the base address incremented by codice_25 multiplied by the size of an element that codice_213 points to. Thus, codice_212 designates the codice_221th element of the array.
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Furthermore, in most expression contexts (a notable exception is as operand of codice_107), an expression of array type is automatically converted to a pointer to the array's first element. This implies that an array is never copied as a whole when named as an argument to a function, but rather only the address of its first element is passed. Therefore, although function calls in C use pass-by-value semantics, arrays are in effect passed by reference.
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The total size of an array codice_213 can be determined by applying codice_107 to an expression of array type. The size of an element can be determined by applying the operator codice_107 to any dereferenced element of an array codice_77, as in codice_227. Thus, the number of elements in a declared array codice_77 can be determined as codice_229. Note, that if only a pointer to the first element is available as it is often the case in C code because of the automatic conversion described above, the information about the full type of the array and its length are lost.
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One of the most important functions of a programming language is to provide facilities for managing memory and the objects that are stored in memory. C provides three principal ways to allocate memory for objects:
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These three approaches are appropriate in different situations and have various trade-offs. For example, static memory allocation has little allocation overhead, automatic allocation may involve slightly more overhead, and dynamic memory allocation can potentially have a great deal of overhead for both allocation and deallocation. The persistent nature of static objects is useful for maintaining state information across function calls, automatic allocation is easy to use but stack space is typically much more limited and transient than either static memory or heap space, and dynamic memory allocation allows convenient allocation of objects whose size is known only at run-time. Most C programs make extensive use of all three.
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Where possible, automatic or static allocation is usually simplest because the storage is managed by the compiler, freeing the programmer of the potentially error-prone chore of manually allocating and releasing storage. However, many data structures can change in size at runtime, and since static allocations (and automatic allocations before C99) must have a fixed size at compile-time, there are many situations in which dynamic allocation is necessary. Prior to the C99 standard, variable-sized arrays were a common example of this. (See the article on codice_205 for an example of dynamically allocated arrays.) Unlike automatic allocation, which can fail at run time with uncontrolled consequences, the dynamic allocation functions return an indication (in the form of a null pointer value) when the required storage cannot be allocated. (Static allocation that is too large is usually detected by the linker or loader, before the program can even begin execution.)
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Unless otherwise specified, static objects contain zero or null pointer values upon program startup. Automatically and dynamically allocated objects are initialized only if an initial value is explicitly specified; otherwise they initially have indeterminate values (typically, whatever bit pattern happens to be present in the storage, which might not even represent a valid value for that type). If the program attempts to access an uninitialized value, the results are undefined. Many modern compilers try to detect and warn about this problem, but both false positives and false negatives can occur.
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Heap memory allocation has to be synchronized with its actual usage in any program to be reused as much as possible. For example, if the only pointer to a heap memory allocation goes out of scope or has its value overwritten before it is deallocated explicitly, then that memory cannot be recovered for later reuse and is essentially lost to the program, a phenomenon known as a "memory leak." Conversely, it is possible for memory to be freed, but is referenced subsequently, leading to unpredictable results. Typically, the failure symptoms appear in a portion of the program unrelated to the code that causes the error, making it difficult to diagnose the failure. Such issues are ameliorated in languages with automatic garbage collection.
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The C programming language uses libraries as its primary method of extension. In C, a library is a set of functions contained within a single "archive" file. Each library typically has a header file, which contains the prototypes of the functions contained within the library that may be used by a program, and declarations of special data types and macro symbols used with these functions. In order for a program to use a library, it must include the library's header file, and the library must be linked with the program, which in many cases requires compiler flags (e.g., codice_234, shorthand for "link the math library").
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The most common C library is the C standard library, which is specified by the ISO and ANSI C standards and comes with every C implementation (implementations which target limited environments such as embedded systems may provide only a subset of the standard library). This library supports stream input and output, memory allocation, mathematics, character strings, and time values. Several separate standard headers (for example, codice_174) specify the interfaces for these and other standard library facilities.
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Another common set of C library functions are those used by applications specifically targeted for Unix and Unix-like systems, especially functions which provide an interface to the kernel. These functions are detailed in various standards such as POSIX and the Single UNIX Specification.
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Since many programs have been written in C, there are a wide variety of other libraries available. Libraries are often written in C because C compilers generate efficient object code; programmers then create interfaces to the library so that the routines can be used from higher-level languages like Java, Perl, and Python.
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File input and output (I/O) is not part of the C language itself but instead is handled by libraries (such as the C standard library) and their associated header files (e.g. codice_174). File handling is generally implemented through high-level I/O which works through streams. A stream is from this perspective a data flow that is independent of devices, while a file is a concrete device. The high-level I/O is done through the association of a stream to a file. In the C standard library, a buffer (a memory area or queue) is temporarily used to store data before it is sent to the final destination. This reduces the time spent waiting for slower devices, for example a hard drive or solid state drive. Low-level I/O functions are not part of the standard C library but are generally part of "bare metal" programming (programming that's independent of any operating system such as most embedded programming). With few exceptions, implementations include low-level I/O.
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A number of tools have been developed to help C programmers find and fix statements with undefined behavior or possibly erroneous expressions, with greater rigor than that provided by the compiler. The tool lint was the first such, leading to many others.
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Automated source code checking and auditing are beneficial in any language, and for C many such tools exist, such as Lint. A common practice is to use Lint to detect questionable code when a program is first written. Once a program passes Lint, it is then compiled using the C compiler. Also, many compilers can optionally warn about syntactically valid constructs that are likely to actually be errors. MISRA C is a proprietary set of guidelines to avoid such questionable code, developed for embedded systems.
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There are also compilers, libraries, and operating system level mechanisms for performing actions that are not a standard part of C, such as bounds checking for arrays, detection of buffer overflow, serialization, dynamic memory tracking, and automatic garbage collection.
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Tools such as Purify or Valgrind and linking with libraries containing special versions of the memory allocation functions can help uncover runtime errors in memory usage.
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C is widely used for systems programming in implementing operating systems and embedded system applications. This is for several reasons:
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Historically, C was sometimes used for web development using the Common Gateway Interface (CGI) as a "gateway" for information between the web application, the server, and the browser. C may have been chosen over interpreted languages because of its speed, stability, and near-universal availability. It is no longer common practice for web development to be done in C, and many other web development tools exist.
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A consequence of C's wide availability and efficiency is that compilers, libraries and interpreters of other programming languages are often implemented in C. For example, the reference implementations of Python, Perl, Ruby, and PHP are written in C.
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C enables programmers to create efficient implementations of algorithms and data structures, because the layer of abstraction from hardware is thin, and its overhead is low, an important criterion for computationally intensive programs. For example, the GNU Multiple Precision Arithmetic Library, the GNU Scientific Library, Mathematica, and MATLAB are completely or partially written in C. Many languages support calling library functions in C, for example, the Python-based framework NumPy uses C for the high-performance and hardware-interacting aspects.
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C is sometimes used as an intermediate language by implementations of other languages. This approach may be used for portability or convenience; by using C as an intermediate language, additional machine-specific code generators are not necessary. C has some features, such as line-number preprocessor directives and optional superfluous commas at the end of initializer lists, that support compilation of generated code. However, some of C's shortcomings have prompted the development of other C-based languages specifically designed for use as intermediate languages, such as C--. Also, contemporary major compilers GCC and LLVM both feature an intermediate representation that is not C, and those compilers support front ends for many languages including C.
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C has also been widely used to implement end-user applications. However, such applications can also be written in newer, higher-level languages.
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For some purposes, restricted styles of C have been adopted, e.g. MISRA C or CERT C, in an attempt to reduce the opportunity for bugs. Databases such as CWE attempt to count the ways C etc. has vulnerabilities, along with recommendations for mitigation.
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There are tools that can mitigate against some of the drawbacks. Contemporary C compilers include checks which may generate warnings to help identify many potential bugs.
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C has both directly and indirectly influenced many later languages such as C++, C#, D, Go, Java, JavaScript, Perl, PHP, Rust and Unix's C shell. The most pervasive influence has been syntactical; all of the languages mentioned combine the statement and (more or less recognizably) expression syntax of C with type systems, data models, and/or large-scale program structures that differ from those of C, sometimes radically.
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When object-oriented programming languages became popular, C++ and Objective-C were two different extensions of C that provided object-oriented capabilities. Both languages were originally implemented as source-to-source compilers; source code was translated into C, and then compiled with a C compiler.
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The C++ programming language (originally named "C with Classes") was devised by Bjarne Stroustrup as an approach to providing object-oriented functionality with a C-like syntax. C++ adds greater typing strength, scoping, and other tools useful in object-oriented programming, and permits generic programming via templates. Nearly a superset of C, C++ now supports most of C, with a few exceptions.
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Objective-C was originally a very "thin" layer on top of C, and remains a strict superset of C that permits object-oriented programming using a hybrid dynamic/static typing paradigm. Objective-C derives its syntax from both C and Smalltalk: syntax that involves preprocessing, expressions, function declarations, and function calls is inherited from C, while the syntax for object-oriented features was originally taken from Smalltalk.
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Alan Mathison Turing (; 23 June 1912 – 7 June 1954) was an English mathematician, computer scientist, logician, cryptanalyst, philosopher, and theoretical biologist. Turing was highly influential in the development of theoretical computer science, providing a formalisation of the concepts of algorithm and computation with the Turing machine, which can be considered a model of a general-purpose computer. He is widely considered to be the father of theoretical computer science and artificial intelligence.
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Born in Maida Vale, London, Turing was raised in southern England. He graduated at King's College, Cambridge, with a degree in mathematics. Whilst he was a fellow at Cambridge, he published a proof demonstrating that some purely mathematical yes–no questions can never be answered by computation and defined a Turing machine, and went on to prove that the halting problem for Turing machines is undecidable. In 1938, he obtained his PhD from the Department of Mathematics at Princeton University. During the Second World War, Turing worked for the Government Code and Cypher School (GC&CS) at Bletchley Park, Britain's codebreaking centre that produced Ultra intelligence. For a time he led Hut 8, the section that was responsible for German naval cryptanalysis. Here, he devised a number of techniques for speeding the breaking of German ciphers, including improvements to the pre-war Polish bomba method, an electromechanical machine that could find settings for the Enigma machine. Turing played a crucial role in cracking intercepted coded messages that enabled the Allies to defeat the Axis powers in many crucial engagements, including the Battle of the Atlantic.
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After the war, Turing worked at the National Physical Laboratory, where he designed the Automatic Computing Engine (ACE), one of the first designs for a stored-program computer. In 1948, Turing joined Max Newman's Computing Machine Laboratory, at the Victoria University of Manchester, where he helped develop the Manchester computers and became interested in mathematical biology. He wrote a paper on the chemical basis of morphogenesis and predicted oscillating chemical reactions such as the Belousov–Zhabotinsky reaction, first observed in the 1960s. Despite these accomplishments, Turing was never fully recognised in Britain during his lifetime because much of his work was covered by the Official Secrets Act.
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Turing was prosecuted in 1952 for homosexual acts. He accepted hormone treatment with DES, a procedure commonly referred to as chemical castration, as an alternative to prison. Turing died on 7 June 1954, 16 days before his 42nd birthday, from cyanide poisoning. An inquest determined his death as a suicide, but it has been noted that the known evidence is also consistent with accidental poisoning.
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Following a public campaign in 2009, the British Prime Minister Gordon Brown made an official public apology on behalf of the British government for "the appalling way [Turing] was treated". Queen Elizabeth II granted a posthumous pardon in 2013. The term "Alan Turing law" is now used informally to refer to a 2017 law in the United Kingdom that retroactively pardoned men cautioned or convicted under historical legislation that outlawed homosexual acts.
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Turing has an extensive legacy with statues of him and many things named after him, including an annual award for computer science innovations. He appears on the current Bank of England £50 note, which was released on 23 June 2021, to coincide with his birthday. A , as voted by the audience, named him the greatest person of the 20th century.
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Turing was born in Maida Vale, London, while his father, Julius Mathison Turing (1873–1947), was on leave from his position with the Indian Civil Service (ICS) of the British Raj government at Chatrapur, then in the Madras Presidency and presently in Odisha state, in India. Turing's father was the son of a clergyman, the Rev. John Robert Turing, from a Scottish family of merchants that had been based in the Netherlands and included a baronet. Turing's mother, Julius's wife, was Ethel Sara Turing (; 1881–1976), daughter of Edward Waller Stoney, chief engineer of the Madras Railways. The Stoneys were a Protestant Anglo-Irish gentry family from both County Tipperary and County Longford, while Ethel herself had spent much of her childhood in County Clare. Julius and Ethel married on 1 October 1907 at Batholomew's church on Clyde Road, in Dublin.
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Julius's work with the ICS brought the family to British India, where his grandfather had been a general in the Bengal Army. However, both Julius and Ethel wanted their children to be brought up in Britain, so they moved to Maida Vale, London, where Alan Turing was born on 23 June 1912, as recorded by a blue plaque on the outside of the house of his birth, later the Colonnade Hotel. Turing had an elder brother, John (the father of Sir John Dermot Turing, 12th Baronet of the Turing baronets).
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Turing's father's civil service commission was still active and during Turing's childhood years, his parents travelled between Hastings in the United Kingdom and India, leaving their two sons to stay with a retired Army couple. At Hastings, Turing stayed at Baston Lodge, Upper Maze Hill, St Leonards-on-Sea, now marked with a blue plaque. The plaque was unveiled on 23 June 2012, the centenary of Turing's birth.
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Very early in life, Turing showed signs of the genius that he was later to display prominently. His parents purchased a house in Guildford in 1927, and Turing lived there during school holidays. The location is also marked with a blue plaque.
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Turing's parents enrolled him at St Michael's, a primary school at 20 Charles Road, St Leonards-on-Sea, from the age of six to nine. The headmistress recognised his talent, noting that she has "...had clever boys and hardworking boys, but Alan is a genius."
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Between January 1922 and 1926, Turing was educated at Hazelhurst Preparatory School, an independent school in the village of Frant in Sussex (now East Sussex). In 1926, at the age of 13, he went on to Sherborne School, a boarding independent school in the market town of Sherborne in Dorset, where he boarded at Westcott House. The first day of term coincided with the 1926 General Strike, in Britain, but Turing was so determined to attend that he rode his bicycle unaccompanied from Southampton to Sherborne, stopping overnight at an inn.
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Turing's natural inclination towards mathematics and science did not earn him respect from some of the teachers at Sherborne, whose definition of education placed more emphasis on the classics. His headmaster wrote to his parents: "I hope he will not fall between two stools. If he is to stay at public school, he must aim at becoming "educated". If he is to be solely a "Scientific Specialist", he is wasting his time at a public school". Despite this, Turing continued to show remarkable ability in the studies he loved, solving advanced problems in 1927 without having studied even elementary calculus. In 1928, aged 16, Turing encountered Albert Einstein's work; not only did he grasp it, but it is possible that he managed to deduce Einstein's questioning of Newton's laws of motion from a text in which this was never made explicit.
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At Sherborne, Turing formed a significant friendship with fellow pupil Christopher Collan Morcom (13 July 1911 – 13 February 1930), who has been described as Turing's "first love". Their relationship provided inspiration in Turing's future endeavours, but it was cut short by Morcom's death, in February 1930, from complications of bovine tuberculosis, contracted after drinking infected cow's milk some years previously.
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The event caused Turing great sorrow. He coped with his grief by working that much harder on the topics of science and mathematics that he had shared with Morcom. In a letter to Morcom's mother, Frances Isobel Morcom (née Swan), Turing wrote:
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Turing's relationship with Morcom's mother continued long after Morcom's death, with her sending gifts to Turing, and him sending letters, typically on Morcom's birthday. A day before the third anniversary of Morcom's death (13 February 1933), he wrote to Mrs. Morcom:
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1,472 |
Some have speculated that Morcom's death was the cause of Turing's atheism and materialism. Apparently, at this point in his life he still believed in such concepts as a spirit, independent of the body and surviving death. In a later letter, also written to Morcom's mother, Turing wrote:
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1,473 |
After Sherborne, Turing studied as an undergraduate from 1931 to 1934 at King's College, Cambridge, where he was awarded first-class honours in mathematics. In 1935, at the age of 22, he was elected a Fellow of King's College on the strength of a dissertation in which he proved a version of the central limit theorem. Unknown to Turing, this version of the theorem had already been proven, in 1922, by Jarl Waldemar Lindeberg. Despite this, the committee found Turing's methods original and so regarded the work worthy of consideration for the fellowship. Abram Besicovitch's report for the committee went so far as to say that if Turing's work had been published before Lindeberg's, it would have been "an important event in the mathematical literature of that year".
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In 1936, Turing published his paper "On Computable Numbers, with an Application to the Entscheidungsproblem". It was published in the "Proceedings of the London Mathematical Society" journal in two parts, the first on 30 November and the second on 23 December. In this paper, Turing reformulated Kurt Gödel's 1931 results on the limits of proof and computation, replacing Gödel's universal arithmetic-based formal language with the formal and simple hypothetical devices that became known as Turing machines. The "Entscheidungsproblem" (decision problem) was originally posed by German mathematician David Hilbert in 1928. Turing proved that his "universal computing machine" would be capable of performing any conceivable mathematical computation if it were representable as an algorithm. He went on to prove that there was no solution to the "decision problem" by first showing that the halting problem for Turing machines is undecidable: it is not possible to decide algorithmically whether a Turing machine will ever halt. This paper has been called "easily the most influential math paper in history".
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Although Turing's proof was published shortly after Alonzo Church's equivalent proof using his lambda calculus, Turing's approach is considerably more accessible and intuitive than Church's. It also included a notion of a 'Universal Machine' (now known as a universal Turing machine), with the idea that such a machine could perform the tasks of any other computation machine (as indeed could Church's lambda calculus). According to the Church–Turing thesis, Turing machines and the lambda calculus are capable of computing anything that is computable. John von Neumann acknowledged that the central concept of the modern computer was due to Turing's paper. To this day, Turing machines are a central object of study in theory of computation.
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From September 1936 to July 1938, Turing spent most of his time studying under Church at Princeton University, in the second year as a Jane Eliza Procter Visiting Fellow. In addition to his purely mathematical work, he studied cryptology and also built three of four stages of an electro-mechanical binary multiplier. In June 1938, he obtained his PhD from the Department of Mathematics at Princeton; his dissertation, "Systems of Logic Based on Ordinals", introduced the concept of ordinal logic and the notion of relative computing, in which Turing machines are augmented with so-called oracles, allowing the study of problems that cannot be solved by Turing machines. John von Neumann wanted to hire him as his postdoctoral assistant, but he went back to the United Kingdom.
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When Turing returned to Cambridge, he attended lectures given in 1939 by Ludwig Wittgenstein about the foundations of mathematics. The lectures have been reconstructed verbatim, including interjections from Turing and other students, from students' notes. Turing and Wittgenstein argued and disagreed, with Turing defending formalism and Wittgenstein propounding his view that mathematics does not discover any absolute truths, but rather invents them.
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During the Second World War, Turing was a leading participant in the breaking of German ciphers at Bletchley Park. The historian and wartime codebreaker Asa Briggs has said, "You needed exceptional talent, you needed genius at Bletchley and Turing's was that genius."
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From September 1938, Turing worked part-time with the Government Code and Cypher School (GC&CS), the British codebreaking organisation. He concentrated on cryptanalysis of the Enigma cipher machine used by Nazi Germany, together with Dilly Knox, a senior GC&CS codebreaker. Soon after the July 1939 meeting near Warsaw at which the Polish Cipher Bureau gave the British and French details of the wiring of Enigma machine's rotors and their method of decrypting Enigma machine's messages, Turing and Knox developed a broader solution. The Polish method relied on an insecure indicator procedure that the Germans were likely to change, which they in fact did in May 1940. Turing's approach was more general, using crib-based decryption for which he produced the functional specification of the bombe (an improvement on the Polish Bomba).
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On 4 September 1939, the day after the UK declared war on Germany, Turing reported to Bletchley Park, the wartime station of GC&CS. Like all others who came to Bletchley, he was required to sign the Official Secrets Act, in which he agreed not to disclose anything about his work at Bletchley, with severe legal penalties for violating the Act.
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Specifying the bombe was the first of five major cryptanalytical advances that Turing made during the war. The others were: deducing the indicator procedure used by the German navy; developing a statistical procedure dubbed "Banburismus" for making much more efficient use of the bombes; developing a procedure dubbed "Turingery" for working out the cam settings of the wheels of the Lorenz SZ 40/42 ("Tunny") cipher machine and, towards the end of the war, the development of a portable secure voice scrambler at Hanslope Park that was codenamed "Delilah".
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By using statistical techniques to optimise the trial of different possibilities in the code breaking process, Turing made an innovative contribution to the subject. He wrote two papers discussing mathematical approaches, titled "The Applications of Probability to Cryptography" and "Paper on Statistics of Repetitions", which were of such value to GC&CS and its successor GCHQ that they were not released to the UK National Archives until April 2012, shortly before the centenary of his birth. A GCHQ mathematician, "who identified himself only as Richard," said at the time that the fact that the contents had been restricted under the Official Secrets Act for some 70 years demonstrated their importance, and their relevance to post-war cryptanalysis:
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Turing had a reputation for eccentricity at Bletchley Park. He was known to his colleagues as "Prof" and his treatise on Enigma was known as the "Prof's Book". According to historian Ronald Lewin, Jack Good, a cryptanalyst who worked with Turing, said of his colleague:
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Peter Hilton recounted his experience working with Turing in Hut 8 in his "Reminiscences of Bletchley Park" from "A Century of Mathematics in America:"
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While working at Bletchley, Turing, who was a talented long-distance runner, occasionally ran the to London when he was needed for meetings, and he was capable of world-class marathon standards. Turing tried out for the 1948 British Olympic team, but he was hampered by an injury. His tryout time for the marathon was only 11 minutes slower than British silver medallist Thomas Richards' Olympic race time of 2 hours 35 minutes. He was Walton Athletic Club's best runner, a fact discovered when he passed the group while running alone. When asked why he ran so hard in training he replied:
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Due to the problems of counterfactual history, it is hard to estimate the precise effect Ultra intelligence had on the war. However, official war historian Harry Hinsley estimated that this work shortened the war in Europe by more than two years and saved over 14 million lives.
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At the end of the war, a memo was sent to all those who had worked at Bletchley Park, reminding them that the code of silence dictated by the Official Secrets Act did not end with the war but would continue indefinitely. Thus, even though Turing was appointed an Officer of the Order of the British Empire (OBE) in 1946 by King George VI for his wartime services, his work remained secret for many years.
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Within weeks of arriving at Bletchley Park, Turing had specified an electromechanical machine called the bombe, which could break Enigma more effectively than the Polish "bomba kryptologiczna", from which its name was derived. The bombe, with an enhancement suggested by mathematician Gordon Welchman, became one of the primary tools, and the major automated one, used to attack Enigma-enciphered messages.
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The bombe searched for possible correct settings used for an Enigma message (i.e., rotor order, rotor settings and plugboard settings) using a suitable "crib": a fragment of probable plaintext. For each possible setting of the rotors (which had on the order of 10 states, or 10 states for the four-rotor U-boat variant), the bombe performed a chain of logical deductions based on the crib, implemented electromechanically.
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The bombe detected when a contradiction had occurred and ruled out that setting, moving on to the next. Most of the possible settings would cause contradictions and be discarded, leaving only a few to be investigated in detail. A contradiction would occur when an enciphered letter would be turned back into the same plaintext letter, which was impossible with the Enigma. The first bombe was installed on 18 March 1940.
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By late 1941, Turing and his fellow cryptanalysts Gordon Welchman, Hugh Alexander and Stuart Milner-Barry were frustrated. Building on the work of the Poles, they had set up a good working system for decrypting Enigma signals, but their limited staff and bombes meant they could not translate all the signals. In the summer, they had considerable success, and shipping losses had fallen to under 100,000 tons a month; however, they badly needed more resources to keep abreast of German adjustments. They had tried to get more people and fund more bombes through the proper channels, but had failed.
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On 28 October they wrote directly to Winston Churchill explaining their difficulties, with Turing as the first named. They emphasised how small their need was compared with the vast expenditure of men and money by the forces and compared with the level of assistance they could offer to the forces. As Andrew Hodges, biographer of Turing, later wrote, "This letter had an electric effect." Churchill wrote a memo to General Ismay, which read: "ACTION THIS DAY. Make sure they have all they want on extreme priority and report to me that this has been done." On 18 November, the chief of the secret service reported that every possible measure was being taken. The cryptographers at Bletchley Park did not know of the Prime Minister's response, but as Milner-Barry recalled, "All that we did notice was that almost from that day the rough ways began miraculously to be made smooth." More than two hundred bombes were in operation by the end of the war.
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Turing decided to tackle the particularly difficult problem of German naval Enigma "because no one else was doing anything about it and I could have it to myself". In December 1939, Turing solved the essential part of the naval indicator system, which was more complex than the indicator systems used by the other services.
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That same night, he also conceived of the idea of "Banburismus", a sequential statistical technique (what Abraham Wald later called sequential analysis) to assist in breaking the naval Enigma, "though I was not sure that it would work in practice, and was not, in fact, sure until some days had actually broken." For this, he invented a measure of weight of evidence that he called the "ban". "Banburismus" could rule out certain sequences of the Enigma rotors, substantially reducing the time needed to test settings on the bombes. Later this sequential process of accumulating sufficient weight of evidence using decibans (one tenth of a ban) was used in Cryptanalysis of the Lorenz cipher.
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Turing travelled to the United States in November 1942 and worked with US Navy cryptanalysts on the naval Enigma and bombe construction in Washington; he also visited their Computing Machine Laboratory in Dayton, Ohio.
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During this trip, he also assisted at Bell Labs with the development of secure speech devices. He returned to Bletchley Park in March 1943. During his absence, Hugh Alexander had officially assumed the position of head of Hut 8, although Alexander had been "de facto" head for some time (Turing having little interest in the day-to-day running of the section). Turing became a general consultant for cryptanalysis at Bletchley Park.
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In July 1942, Turing devised a technique termed "Turingery" (or jokingly "Turingismus") for use against the Lorenz cipher messages produced by the Germans' new "Geheimschreiber" (secret writer) machine. This was a teleprinter rotor cipher attachment codenamed "Tunny" at Bletchley Park. Turingery was a method of "wheel-breaking", i.e., a procedure for working out the cam settings of Tunny's wheels. He also introduced the Tunny team to Tommy Flowers who, under the guidance of Max Newman, went on to build the Colossus computer, the world's first programmable digital electronic computer, which replaced a simpler prior machine (the Heath Robinson), and whose superior speed allowed the statistical decryption techniques to be applied usefully to the messages. Some have mistakenly said that Turing was a key figure in the design of the Colossus computer. Turingery and the statistical approach of Banburismus undoubtedly fed into the thinking about cryptanalysis of the Lorenz cipher, but he was not directly involved in the Colossus development.
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Following his work at Bell Labs in the US, Turing pursued the idea of electronic enciphering of speech in the telephone system. In the latter part of the war, he moved to work for the Secret Service's Radio Security Service (later HMGCC) at Hanslope Park. At the park, he further developed his knowledge of electronics with the assistance of engineer Donald Bayley. Together they undertook the design and construction of a portable secure voice communications machine codenamed "Delilah". The machine was intended for different applications, but it lacked the capability for use with long-distance radio transmissions. In any case, Delilah was completed too late to be used during the war. Though the system worked fully, with Turing demonstrating it to officials by encrypting and decrypting a recording of a Winston Churchill speech, Delilah was not adopted for use. Turing also consulted with Bell Labs on the development of SIGSALY, a secure voice system that was used in the later years of the war.
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Between 1945 and 1947, Turing lived in Hampton, London, while he worked on the design of the ACE (Automatic Computing Engine) at the National Physical Laboratory (NPL). He presented a paper on 19 February 1946, which was the first detailed design of a stored-program computer. Von Neumann's incomplete "First Draft of a Report on the EDVAC" had predated Turing's paper, but it was much less detailed and, according to John R. Womersley, Superintendent of the NPL Mathematics Division, it "contains a number of ideas which are Dr. Turing's own".
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