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A prototypical example of an algorithm is the Euclidean algorithm, which is used to determine the maximum common divisor of two integers; an example (there are others) is described by the flowchart above and as an example in a later section. offer an informal meaning of the word "algorithm" in the following quotation:
No human being can write fast enough, or long enough, or small enough† ( †"smaller and smaller without limit ... you'd be trying to write on molecules, on atoms, on electrons") to list all members of an enumerably infinite set by writing out their names, one after another, in some notation. But humans can do something equally useful, in the case of certain enumerably infinite sets: They can give explicit instructions for determining the nth member of the set, for arbitrary finite n. Such instructions are to be given quite explicitly, in a form in which they could be followed by a computing machine, or by a human who is capable of carrying out only very elementary operations on symbols. An "enumerably infinite set" is one whose elements can be put into one-to-one correspondence with the integers. Thus Boolos and Jeffrey are saying that an algorithm implies instructions for a process that "creates" output integers from an arbitrary "input" integer or integers that, in theory, can be arbitrarily large. | [
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For example, an algorithm can be an algebraic equation such as y = m + n (i.e., two arbitrary "input variables" m and n that produce an output y), but various authors' attempts to define the notion indicate that the word implies much more than this, something on the order of (for the addition example):
Precise instructions (in a language understood by "the computer") for a fast, efficient, "good" process that specifies the "moves" of "the computer" (machine or human, equipped with the necessary internally contained information and capabilities) to find, decode, and then process arbitrary input integers/symbols m and n, symbols + and = ... and "effectively" produce, in a "reasonable" time, output-integer y at a specified place and in a specified format. The concept of algorithm is also used to define the notion of decidability—a notion that is central for explaining how formal systems come into being starting from a small set of axioms and rules. In logic, the time that an algorithm requires to complete cannot be measured, as it is not apparently related to the customary physical dimension. From such uncertainties, that characterize ongoing work, stems the unavailability of a definition of algorithm that suits both concrete (in some sense) and abstract usage of the term. Most algorithms are intended to be implemented as computer programs. | [
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However, algorithms are also implemented by other means, such as in a biological neural network (for example, the human brain implementing arithmetic or an insect looking for food), in an electrical circuit, or in a mechanical device. Formalization
Algorithms are essential to the way computers process data. Many computer programs contain algorithms that detail the specific instructions a computer should perform—in a specific order—to carry out a specified task, such as calculating employees' paychecks or printing students' report cards. Thus, an algorithm can be considered to be any sequence of operations that can be simulated by a Turing-complete system. Authors who assert this thesis include Minsky (1967), Savage (1987) and Gurevich (2000):
Minsky: "But we will also maintain, with Turing ... that any procedure which could "naturally" be called effective, can, in fact, be realized by a (simple) machine. Although this may seem extreme, the arguments ... in its favor are hard to refute". Gurevich: "… Turing's informal argument in favor of his thesis justifies a stronger thesis: every algorithm can be simulated by a Turing machine … according to Savage [1987], an algorithm is a computational process defined by a Turing machine".Turing machines can define computational processes that do not terminate. The informal definitions of algorithms generally require that the algorithm always terminates. This requirement renders the task of deciding whether a formal procedure is an algorithm impossible in the general case—due to a major theorem of computability theory known as the halting problem. | [
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Typically, when an algorithm is associated with processing information, data can be read from an input source, written to an output device and stored for further processing. Stored data are regarded as part of the internal state of the entity performing the algorithm. In practice, the state is stored in one or more data structures. For some of these computational processes, the algorithm must be rigorously defined: specified in the way it applies in all possible circumstances that could arise. This means that any conditional steps must be systematically dealt with, case-by-case; the criteria for each case must be clear (and computable). Because an algorithm is a precise list of precise steps, the order of computation is always crucial to the functioning of the algorithm. Instructions are usually assumed to be listed explicitly, and are described as starting "from the top" and going "down to the bottom"—an idea that is described more formally by flow of control. So far, the discussion on the formalization of an algorithm has assumed the premises of imperative programming. This is the most common conception—one which attempts to describe a task in discrete, "mechanical" means. Unique to this conception of formalized algorithms is the assignment operation, which sets the value of a variable. It derives from the intuition of "memory" as a scratchpad. An example of such an assignment can be found below. For some alternate conceptions of what constitutes an algorithm, see functional programming and logic programming. | [
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] |
Expressing algorithms
Algorithms can be expressed in many kinds of notation, including natural languages, pseudocode, flowcharts, drakon-charts, programming languages or control tables (processed by interpreters). Natural language expressions of algorithms tend to be verbose and ambiguous, and are rarely used for complex or technical algorithms. Pseudocode, flowcharts, drakon-charts and control tables are structured ways to express algorithms that avoid many of the ambiguities common in the statements based on natural language. Programming languages are primarily intended for expressing algorithms in a form that can be executed by a computer, but are also often used as a way to define or document algorithms. There is a wide variety of representations possible and one can express a given Turing machine program as a sequence of machine tables (see finite-state machine, state transition table and control table for more), as flowcharts and drakon-charts (see state diagram for more), or as a form of rudimentary machine code or assembly code called "sets of quadruples" (see Turing machine for more). Representations of algorithms can be classed into three accepted levels of Turing machine description, as follows:
1 High-level description
"...prose to describe an algorithm, ignoring the implementation details. At this level, we do not need to mention how the machine manages its tape or head." 2 Implementation description
"...prose used to define the way the Turing machine uses its head and the way that it stores data on its tape. At this level, we do not give details of states or transition function." | [
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3 Formal description
Most detailed, "lowest level", gives the Turing machine's "state table". For an example of the simple algorithm "Add m+n" described in all three levels, see Examples. Design
Algorithm design refers to a method or a mathematical process for problem-solving and engineering algorithms. The design of algorithms is part of many solution theories of operation research, such as dynamic programming and divide-and-conquer. Techniques for designing and implementing algorithm designs are also called algorithm design patterns, with examples including the template method pattern and the decorator pattern. One of the most important aspects of algorithm design is resource (run-time, memory usage) efficiency; the big O notation is used to describe e.g. an algorithm's run-time growth as the size of its input increases. Typical steps in the development of algorithms:
Problem definition
Development of a model
Specification of the algorithm
Designing an algorithm
Checking the correctness of the algorithm
Analysis of algorithm
Implementation of algorithm
Program testing
Documentation preparation
Computer algorithms
"Elegant" (compact) programs, "good" (fast) programs : The notion of "simplicity and elegance" appears informally in Knuth and precisely in Chaitin:
Knuth: " ... we want good algorithms in some loosely defined aesthetic sense. One criterion ... is the length of time taken to perform the algorithm .... Other criteria are adaptability of the algorithm to computers, its simplicity and elegance, etc." | [
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Chaitin: " ... a program is 'elegant,' by which I mean that it's the smallest possible program for producing the output that it does"
Chaitin prefaces his definition with: "I'll show you can't prove that a program is 'elegant—such a proof would solve the Halting problem (ibid). Algorithm versus function computable by an algorithm: For a given function multiple algorithms may exist. This is true, even without expanding the available instruction set available to the programmer. Rogers observes that "It is ... important to distinguish between the notion of algorithm, i.e. procedure and the notion of function computable by algorithm, i.e. mapping yielded by procedure. The same function may have several different algorithms". Unfortunately, there may be a tradeoff between goodness (speed) and elegance (compactness)—an elegant program may take more steps to complete a computation than one less elegant. An example that uses Euclid's algorithm appears below. Computers (and computors), models of computation: A computer (or human "computor") is a restricted type of machine, a "discrete deterministic mechanical device" that blindly follows its instructions. Melzak's and Lambek's primitive models reduced this notion to four elements: (i) discrete, distinguishable locations, (ii) discrete, indistinguishable counters (iii) an agent, and (iv) a list of instructions that are effective relative to the capability of the agent. Minsky describes a more congenial variation of Lambek's "abacus" model in his "Very Simple Bases for Computability". | [
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Minsky's machine proceeds sequentially through its five (or six, depending on how one counts) instructions unless either a conditional IF-THEN GOTO or an unconditional GOTO changes program flow out of sequence. Besides HALT, Minsky's machine includes three assignment (replacement, substitution) operations: ZERO (e.g. the contents of location replaced by 0: L ← 0), SUCCESSOR (e.g. L ← L+1), and DECREMENT (e.g. L ← L − 1). Rarely must a programmer write "code" with such a limited instruction set. But Minsky shows (as do Melzak and Lambek) that his machine is Turing complete with only four general types of instructions: conditional GOTO, unconditional GOTO, assignment/replacement/substitution, and HALT. However, a few different assignment instructions (e.g. DECREMENT, INCREMENT, and ZERO/CLEAR/EMPTY for a Minsky machine) are also required for Turing-completeness; their exact specification is somewhat up to the designer. The unconditional GOTO is a convenience; it can be constructed by initializing a dedicated location to zero e.g. the instruction " Z ← 0 "; thereafter the instruction IF Z=0 THEN GOTO xxx is unconditional. Simulation of an algorithm: computer (computor) language: Knuth advises the reader that "the best way to learn an algorithm is to try it . . . immediately take pen and paper and work through an example". But what about a simulation or execution of the real thing? The programmer must translate the algorithm into a language that the simulator/computer/computor can effectively execute. | [
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Stone gives an example of this: when computing the roots of a quadratic equation the computor must know how to take a square root. If they don't, then the algorithm, to be effective, must provide a set of rules for extracting a square root. This means that the programmer must know a "language" that is effective relative to the target computing agent (computer/computor). But what model should be used for the simulation? Van Emde Boas observes "even if we base complexity theory on abstract instead of concrete machines, arbitrariness of the choice of a model remains. It is at this point that the notion of simulation enters". When speed is being measured, the instruction set matters. For example, the subprogram in Euclid's algorithm to compute the remainder would execute much faster if the programmer had a "modulus" instruction available rather than just subtraction (or worse: just Minsky's "decrement"). Structured programming, canonical structures: Per the Church–Turing thesis, any algorithm can be computed by a model known to be Turing complete, and per Minsky's demonstrations, Turing completeness requires only four instruction types—conditional GOTO, unconditional GOTO, assignment, HALT. Kemeny and Kurtz observe that, while "undisciplined" use of unconditional GOTOs and conditional IF-THEN GOTOs can result in "spaghetti code", a programmer can write structured programs using only these instructions; on the other hand "it is also possible, and not too hard, to write badly structured programs in a structured language". | [
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Tausworthe augments the three Böhm-Jacopini canonical structures: SEQUENCE, IF-THEN-ELSE, and WHILE-DO, with two more: DO-WHILE and CASE. An additional benefit of a structured program is that it lends itself to proofs of correctness using mathematical induction. Canonical flowchart symbols: The graphical aide called a flowchart, offers a way to describe and document an algorithm (and a computer program of one). Like the program flow of a Minsky machine, a flowchart always starts at the top of a page and proceeds down. Its primary symbols are only four: the directed arrow showing program flow, the rectangle (SEQUENCE, GOTO), the diamond (IF-THEN-ELSE), and the dot (OR-tie). The Böhm–Jacopini canonical structures are made of these primitive shapes. Sub-structures can "nest" in rectangles, but only if a single exit occurs from the superstructure. The symbols, and their use to build the canonical structures are shown in the diagram. Examples
Algorithm example
One of the simplest algorithms is to find the largest number in a list of numbers of random order. Finding the solution requires looking at every number in the list. From this follows a simple algorithm, which can be stated in a high-level description in English prose, as:
High-level description:
If there are no numbers in the set then there is no highest number. Assume the first number in the set is the largest number in the set. For each remaining number in the set: if this number is larger than the current largest number, consider this number to be the largest number in the set. | [
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When there are no numbers left in the set to iterate over, consider the current largest number to be the largest number of the set. (Quasi-)formal description:
Written in prose but much closer to the high-level language of a computer program, the following is the more formal coding of the algorithm in pseudocode or pidgin code:
Input: A list of numbers L.
Output: The largest number in the list L.
if L.size = 0 return null
largest ← L[0]
for each item in L, do
if item > largest, then
largest ← item
return largest
Euclid's algorithm
In mathematics, the Euclidean algorithm, or Euclid's algorithm, is an efficient method for computing the greatest common divisor (GCD) of two integers (numbers), the largest number that divides them both without a remainder. It is named after the ancient Greek mathematician Euclid, who first described it in his Elements (c. 300 BC). It is one of the oldest algorithms in common use. It can be used to reduce fractions to their simplest form, and is a part of many other number-theoretic and cryptographic calculations. Euclid poses the problem thus: "Given two numbers not prime to one another, to find their greatest common measure". He defines "A number [to be] a multitude composed of units": a counting number, a positive integer not including zero. | [
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To "measure" is to place a shorter measuring length s successively (q times) along longer length l until the remaining portion r is less than the shorter length s. In modern words, remainder r = l − q×s, q being the quotient, or remainder r is the "modulus", the integer-fractional part left over after the division. For Euclid's method to succeed, the starting lengths must satisfy two requirements: (i) the lengths must not be zero, AND (ii) the subtraction must be "proper"; i.e., a test must guarantee that the smaller of the two numbers is subtracted from the larger (or the two can be equal so their subtraction yields zero). Euclid's original proof adds a third requirement: the two lengths must not be prime to one another. Euclid stipulated this so that he could construct a reductio ad absurdum proof that the two numbers' common measure is in fact the greatest. While Nicomachus' algorithm is the same as Euclid's, when the numbers are prime to one another, it yields the number "1" for their common measure. So, to be precise, the following is really Nicomachus' algorithm. Computer language for Euclid's algorithm
Only a few instruction types are required to execute Euclid's algorithm—some logical tests (conditional GOTO), unconditional GOTO, assignment (replacement), and subtraction. A location is symbolized by upper case letter(s), e.g. S, A, etc. | [
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The varying quantity (number) in a location is written in lower case letter(s) and (usually) associated with the location's name. For example, location L at the start might contain the number l = 3009. An inelegant program for Euclid's algorithm
The following algorithm is framed as Knuth's four-step version of Euclid's and Nicomachus', but, rather than using division to find the remainder, it uses successive subtractions of the shorter length s from the remaining length r until r is less than s. The high-level description, shown in boldface, is adapted from Knuth 1973:2–4:
INPUT:
[Into two locations L and S put the numbers l and s that represent the two lengths]:
INPUT L, S
[Initialize R: make the remaining length r equal to the starting/initial/input length l]:
R ← L
E0: [Ensure r ≥ s.]
[Ensure the smaller of the two numbers is in S and the larger in R]:
IF R > S THEN
the contents of L is the larger number so skip over the exchange-steps 4, 5 and 6:
GOTO step 7
ELSE
swap the contents of R and S.
L ← R (this first step is redundant, but is useful for later discussion). | [
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] |
R ← S
S ← L
E1: [Find remainder]: Until the remaining length r in R is less than the shorter length s in S, repeatedly subtract the measuring number s in S from the remaining length r in R.
IF S > R THEN
done measuring so
GOTO 10
ELSE
measure again,
R ← R − S
[Remainder-loop]:
GOTO 7. E2: [Is the remainder zero? ]: EITHER (i) the last measure was exact, the remainder in R is zero, and the program can halt, OR (ii) the algorithm must continue: the last measure left a remainder in R less than measuring number in S.
IF R = 0 THEN
done so
GOTO step 15
ELSE
CONTINUE TO step 11,
E3: [Interchange s and r]: The nut of Euclid's algorithm. Use remainder r to measure what was previously smaller number s; L serves as a temporary location. L ← R
R ← S
S ← L
[Repeat the measuring process]:
GOTO 7
OUTPUT:
[Done. S contains the greatest common divisor]:
PRINT S
DONE:
HALT, END, STOP. An elegant program for Euclid's algorithm
The flowchart of "Elegant" can be found at the top of this article. In the (unstructured) Basic language, the steps are numbered, and the instruction LET [] = [] is the assignment instruction symbolized by ←. | [
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5 REM Euclid's algorithm for greatest common divisor
6 PRINT "Type two integers greater than 0"
10 INPUT A,B
20 IF B=0 THEN GOTO 80
30 IF A > B THEN GOTO 60
40 LET B=B-A
50 GOTO 20
60 LET A=A-B
70 GOTO 20
80 PRINT A
90 END
How "Elegant" works: In place of an outer "Euclid loop", "Elegant" shifts back and forth between two "co-loops", an A > B loop that computes A ← A − B, and a B ≤ A loop that computes B ← B − A. This works because, when at last the minuend M is less than or equal to the subtrahend S (Difference = Minuend − Subtrahend), the minuend can become s (the new measuring length) and the subtrahend can become the new r (the length to be measured); in other words the "sense" of the subtraction reverses. The following version can be used with programming languages from the C-family:
// Euclid's algorithm for greatest common divisor
int euclidAlgorithm (int A, int B){
A=abs(A);
B=abs(B);
while (B!=0){
while (A>B) A=A-B;
B=B-A;
}
return A;
}
Testing the Euclid algorithms
Does an algorithm do what its author wants it to do? A few test cases usually give some confidence in the core functionality. But tests are not enough. For test cases, one source uses 3009 and 884. Knuth suggested 40902, 24140. | [
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Another interesting case is the two relatively prime numbers 14157 and 5950. But "exceptional cases" must be identified and tested. Will "Inelegant" perform properly when R > S, S > R, R = S? Ditto for "Elegant": B > A, A > B, A = B? (Yes to all). What happens when one number is zero, both numbers are zero? ("Inelegant" computes forever in all cases; "Elegant" computes forever when A = 0.) What happens if negative numbers are entered? Fractional numbers? If the input numbers, i.e. the domain of the function computed by the algorithm/program, is to include only positive integers including zero, then the failures at zero indicate that the algorithm (and the program that instantiates it) is a partial function rather than a total function. A notable failure due to exceptions is the Ariane 5 Flight 501 rocket failure (June 4, 1996). Proof of program correctness by use of mathematical induction: Knuth demonstrates the application of mathematical induction to an "extended" version of Euclid's algorithm, and he proposes "a general method applicable to proving the validity of any algorithm". Tausworthe proposes that a measure of the complexity of a program be the length of its correctness proof. Measuring and improving the Euclid algorithms
Elegance (compactness) versus goodness (speed): With only six core instructions, "Elegant" is the clear winner, compared to "Inelegant" at thirteen instructions. | [
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However, "Inelegant" is faster (it arrives at HALT in fewer steps). Algorithm analysis indicates why this is the case: "Elegant" does two conditional tests in every subtraction loop, whereas "Inelegant" only does one. As the algorithm (usually) requires many loop-throughs, on average much time is wasted doing a "B = 0?" test that is needed only after the remainder is computed. Can the algorithms be improved? : Once the programmer judges a program "fit" and "effective"—that is, it computes the function intended by its author—then the question becomes, can it be improved? The compactness of "Inelegant" can be improved by the elimination of five steps. But Chaitin proved that compacting an algorithm cannot be automated by a generalized algorithm; rather, it can only be done heuristically; i.e., by exhaustive search (examples to be found at Busy beaver), trial and error, cleverness, insight, application of inductive reasoning, etc. Observe that steps 4, 5 and 6 are repeated in steps 11, 12 and 13. Comparison with "Elegant" provides a hint that these steps, together with steps 2 and 3, can be eliminated. This reduces the number of core instructions from thirteen to eight, which makes it "more elegant" than "Elegant", at nine steps. The speed of "Elegant" can be improved by moving the "B=0?" test outside of the two subtraction loops. | [
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This change calls for the addition of three instructions (B = 0?, A = 0?, GOTO). Now "Elegant" computes the example-numbers faster; whether this is always the case for any given A, B, and R, S would require a detailed analysis. Algorithmic analysis
It is frequently important to know how much of a particular resource (such as time or storage) is theoretically required for a given algorithm. Methods have been developed for the analysis of algorithms to obtain such quantitative answers (estimates); for example, an algorithm which adds up the elements of a list of n numbers would have a time requirement of O(n), using big O notation. At all times the algorithm only needs to remember two values: the sum of all the elements so far, and its current position in the input list. Therefore, it is said to have a space requirement of O(1), if the space required to store the input numbers is not counted, or O(n) if it is counted. Different algorithms may complete the same task with a different set of instructions in less or more time, space, or 'effort' than others. For example, a binary search algorithm (with cost O(log n)) outperforms a sequential search (cost O(n) ) when used for table lookups on sorted lists or arrays. | [
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Formal versus empirical
The analysis, and study of algorithms is a discipline of computer science, and is often practiced abstractly without the use of a specific programming language or implementation. In this sense, algorithm analysis resembles other mathematical disciplines in that it focuses on the underlying properties of the algorithm and not on the specifics of any particular implementation. Usually pseudocode is used for analysis as it is the simplest and most general representation. However, ultimately, most algorithms are usually implemented on particular hardware/software platforms and their algorithmic efficiency is eventually put to the test using real code. For the solution of a "one off" problem, the efficiency of a particular algorithm may not have significant consequences (unless n is extremely large) but for algorithms designed for fast interactive, commercial or long life scientific usage it may be critical. Scaling from small n to large n frequently exposes inefficient algorithms that are otherwise benign. Empirical testing is useful because it may uncover unexpected interactions that affect performance. Benchmarks may be used to compare before/after potential improvements to an algorithm after program optimization. Empirical tests cannot replace formal analysis, though, and are not trivial to perform in a fair manner. Execution efficiency
To illustrate the potential improvements possible even in well-established algorithms, a recent significant innovation, relating to FFT algorithms (used heavily in the field of image processing), can decrease processing time up to 1,000 times for applications like medical imaging. In general, speed improvements depend on special properties of the problem, which are very common in practical applications. | [
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Speedups of this magnitude enable computing devices that make extensive use of image processing (like digital cameras and medical equipment) to consume less power. Classification
There are various ways to classify algorithms, each with its own merits. By implementation
One way to classify algorithms is by implementation means. Recursion
A recursive algorithm is one that invokes (makes reference to) itself repeatedly until a certain condition (also known as termination condition) matches, which is a method common to functional programming. Iterative algorithms use repetitive constructs like loops and sometimes additional data structures like stacks to solve the given problems. Some problems are naturally suited for one implementation or the other. For example, towers of Hanoi is well understood using recursive implementation. Every recursive version has an equivalent (but possibly more or less complex) iterative version, and vice versa. Logical
An algorithm may be viewed as controlled logical deduction. This notion may be expressed as: Algorithm = logic + control. The logic component expresses the axioms that may be used in the computation and the control component determines the way in which deduction is applied to the axioms. This is the basis for the logic programming paradigm. In pure logic programming languages, the control component is fixed and algorithms are specified by supplying only the logic component. The appeal of this approach is the elegant semantics: a change in the axioms produces a well-defined change in the algorithm. Serial, parallel or distributed
Algorithms are usually discussed with the assumption that computers execute one instruction of an algorithm at a time. Those computers are sometimes called serial computers. | [
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An algorithm designed for such an environment is called a serial algorithm, as opposed to parallel algorithms or distributed algorithms. Parallel algorithms take advantage of computer architectures where several processors can work on a problem at the same time, whereas distributed algorithms utilize multiple machines connected with a computer network. Parallel or distributed algorithms divide the problem into more symmetrical or asymmetrical subproblems and collect the results back together. The resource consumption in such algorithms is not only processor cycles on each processor but also the communication overhead between the processors. Some sorting algorithms can be parallelized efficiently, but their communication overhead is expensive. Iterative algorithms are generally parallelizable. Some problems have no parallel algorithms and are called inherently serial problems. Deterministic or non-deterministic
Deterministic algorithms solve the problem with exact decision at every step of the algorithm whereas non-deterministic algorithms solve problems via guessing although typical guesses are made more accurate through the use of heuristics. Exact or approximate
While many algorithms reach an exact solution, approximation algorithms seek an approximation that is closer to the true solution. The approximation can be reached by either using a deterministic or a random strategy. Such algorithms have practical value for many hard problems. One of the examples of an approximate algorithm is the Knapsack problem, where there is a set of given items. Its goal is to pack the knapsack to get the maximum total value. Each item has some weight and some value. Total weight that can be carried is no more than some fixed number X. So, the solution must consider weights of items as well as their value. | [
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Quantum algorithm
They run on a realistic model of quantum computation. The term is usually used for those algorithms which seem inherently quantum, or use some essential feature of Quantum computing such as quantum superposition or quantum entanglement. By design paradigm
Another way of classifying algorithms is by their design methodology or paradigm. There is a certain number of paradigms, each different from the other. Furthermore, each of these categories includes many different types of algorithms. Some common paradigms are:
Brute-force or exhaustive search
This is the naive method of trying every possible solution to see which is best. Divide and conquer
A divide and conquer algorithm repeatedly reduces an instance of a problem to one or more smaller instances of the same problem (usually recursively) until the instances are small enough to solve easily. One such example of divide and conquer is merge sorting. Sorting can be done on each segment of data after dividing data into segments and sorting of entire data can be obtained in the conquer phase by merging the segments. A simpler variant of divide and conquer is called a decrease and conquer algorithm, which solves an identical subproblem and uses the solution of this subproblem to solve the bigger problem. Divide and conquer divides the problem into multiple subproblems and so the conquer stage is more complex than decrease and conquer algorithms. An example of a decrease and conquer algorithm is the binary search algorithm. Search and enumeration
Many problems (such as playing chess) can be modeled as problems on graphs. A graph exploration algorithm specifies rules for moving around a graph and is useful for such problems. | [
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This category also includes search algorithms, branch and bound enumeration and backtracking. Randomized algorithm
Such algorithms make some choices randomly (or pseudo-randomly). They can be very useful in finding approximate solutions for problems where finding exact solutions can be impractical (see heuristic method below). For some of these problems, it is known that the fastest approximations must involve some randomness. Whether randomized algorithms with polynomial time complexity can be the fastest algorithms for some problems is an open question known as the P versus NP problem. There are two large classes of such algorithms:
Monte Carlo algorithms return a correct answer with high-probability. E.g. RP is the subclass of these that run in polynomial time. Las Vegas algorithms always return the correct answer, but their running time is only probabilistically bound, e.g. ZPP. Reduction of complexity
This technique involves solving a difficult problem by transforming it into a better-known problem for which we have (hopefully) asymptotically optimal algorithms. The goal is to find a reducing algorithm whose complexity is not dominated by the resulting reduced algorithm's. For example, one selection algorithm for finding the median in an unsorted list involves first sorting the list (the expensive portion) and then pulling out the middle element in the sorted list (the cheap portion). This technique is also known as transform and conquer. Back tracking
In this approach, multiple solutions are built incrementally and abandoned when it is determined that they cannot lead to a valid full solution. | [
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Optimization problems
For optimization problems there is a more specific classification of algorithms; an algorithm for such problems may fall into one or more of the general categories described above as well as into one of the following:
Linear programming
When searching for optimal solutions to a linear function bound to linear equality and inequality constraints, the constraints of the problem can be used directly in producing the optimal solutions. There are algorithms that can solve any problem in this category, such as the popular simplex algorithm. Problems that can be solved with linear programming include the maximum flow problem for directed graphs. If a problem additionally requires that one or more of the unknowns must be an integer then it is classified in integer programming. A linear programming algorithm can solve such a problem if it can be proved that all restrictions for integer values are superficial, i.e., the solutions satisfy these restrictions anyway. In the general case, a specialized algorithm or an algorithm that finds approximate solutions is used, depending on the difficulty of the problem. Dynamic programming
When a problem shows optimal substructures—meaning the optimal solution to a problem can be constructed from optimal solutions to subproblems—and overlapping subproblems, meaning the same subproblems are used to solve many different problem instances, a quicker approach called dynamic programming avoids recomputing solutions that have already been computed. For example, Floyd–Warshall algorithm, the shortest path to a goal from a vertex in a weighted graph can be found by using the shortest path to the goal from all adjacent vertices. Dynamic programming and memoization go together. | [
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The main difference between dynamic programming and divide and conquer is that subproblems are more or less independent in divide and conquer, whereas subproblems overlap in dynamic programming. The difference between dynamic programming and straightforward recursion is in caching or memoization of recursive calls. When subproblems are independent and there is no repetition, memoization does not help; hence dynamic programming is not a solution for all complex problems. By using memoization or maintaining a table of subproblems already solved, dynamic programming reduces the exponential nature of many problems to polynomial complexity. The greedy method
A greedy algorithm is similar to a dynamic programming algorithm in that it works by examining substructures, in this case not of the problem but of a given solution. Such algorithms start with some solution, which may be given or have been constructed in some way, and improve it by making small modifications. For some problems they can find the optimal solution while for others they stop at local optima, that is, at solutions that cannot be improved by the algorithm but are not optimum. The most popular use of greedy algorithms is for finding the minimal spanning tree where finding the optimal solution is possible with this method. Huffman Tree, Kruskal, Prim, Sollin are greedy algorithms that can solve this optimization problem. The heuristic method
In optimization problems, heuristic algorithms can be used to find a solution close to the optimal solution in cases where finding the optimal solution is impractical. These algorithms work by getting closer and closer to the optimal solution as they progress. | [
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In principle, if run for an infinite amount of time, they will find the optimal solution. Their merit is that they can find a solution very close to the optimal solution in a relatively short time. Such algorithms include local search, tabu search, simulated annealing, and genetic algorithms. Some of them, like simulated annealing, are non-deterministic algorithms while others, like tabu search, are deterministic. When a bound on the error of the non-optimal solution is known, the algorithm is further categorized as an approximation algorithm. By field of study
Every field of science has its own problems and needs efficient algorithms. Related problems in one field are often studied together. Some example classes are search algorithms, sorting algorithms, merge algorithms, numerical algorithms, graph algorithms, string algorithms, computational geometric algorithms, combinatorial algorithms, medical algorithms, machine learning, cryptography, data compression algorithms and parsing techniques. Fields tend to overlap with each other, and algorithm advances in one field may improve those of other, sometimes completely unrelated, fields. For example, dynamic programming was invented for optimization of resource consumption in industry but is now used in solving a broad range of problems in many fields. By complexity
Algorithms can be classified by the amount of time they need to complete compared to their input size:
Constant time: if the time needed by the algorithm is the same, regardless of the input size. E.g. an access to an array element. Logarithmic time: if the time is a logarithmic function of the input size. E.g. binary search algorithm. | [
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Linear time: if the time is proportional to the input size. E.g. the traverse of a list. Polynomial time: if the time is a power of the input size. E.g. the bubble sort algorithm has quadratic time complexity. Exponential time: if the time is an exponential function of the input size. E.g. Brute-force search. Some problems may have multiple algorithms of differing complexity, while other problems might have no algorithms or no known efficient algorithms. There are also mappings from some problems to other problems. Owing to this, it was found to be more suitable to classify the problems themselves instead of the algorithms into equivalence classes based on the complexity of the best possible algorithms for them. Continuous algorithms
The adjective "continuous" when applied to the word "algorithm" can mean:
An algorithm operating on data that represents continuous quantities, even though this data is represented by discrete approximations—such algorithms are studied in numerical analysis; or
An algorithm in the form of a differential equation that operates continuously on the data, running on an analog computer. Legal issues
Algorithms, by themselves, are not usually patentable. In the United States, a claim consisting solely of simple manipulations of abstract concepts, numbers, or signals does not constitute "processes" (USPTO 2006), and hence algorithms are not patentable (as in Gottschalk v. Benson). However practical applications of algorithms are sometimes patentable. For example, in Diamond v. Diehr, the application of a simple feedback algorithm to aid in the curing of synthetic rubber was deemed patentable. | [
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The patenting of software is highly controversial, and there are highly criticized patents involving algorithms, especially data compression algorithms, such as Unisys' LZW patent. Additionally, some cryptographic algorithms have export restrictions (see export of cryptography). History: Development of the notion of "algorithm"
Ancient Near East
The earliest evidence of algorithms is found in the Babylonian mathematics of ancient Mesopotamia (modern Iraq). A Sumerian clay tablet found in Shuruppak near Baghdad and dated to circa 2500 BC described the earliest division algorithm. During the Hammurabi dynasty circa 1800-1600 BC, Babylonian clay tablets described algorithms for computing formulas. Algorithms were also used in Babylonian astronomy. Babylonian clay tablets describe and employ algorithmic procedures to compute the time and place of significant astronomical events. Algorithms for arithmetic are also found in ancient Egyptian mathematics, dating back to the Rhind Mathematical Papyrus circa 1550 BC. Algorithms were later used in ancient Hellenistic mathematics. Two examples are the Sieve of Eratosthenes, which was described in the Introduction to Arithmetic by Nicomachus, and the Euclidean algorithm, which was first described in Euclid's Elements (c. 300 BC). Discrete and distinguishable symbols
Tally-marks: To keep track of their flocks, their sacks of grain and their money the ancients used tallying: accumulating stones or marks scratched on sticks or making discrete symbols in clay. Through the Babylonian and Egyptian use of marks and symbols, eventually Roman numerals and the abacus evolved (Dilson, p. 16–41). Tally marks appear prominently in unary numeral system arithmetic used in Turing machine and Post–Turing machine computations. | [
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] |
Manipulation of symbols as "place holders" for numbers: algebra
Muhammad ibn Mūsā al-Khwārizmī, a Persian mathematician, wrote the Al-jabr in the 9th century. The terms "algorism" and "algorithm" are derived from the name al-Khwārizmī, while the term "algebra" is derived from the book Al-jabr. In Europe, the word "algorithm" was originally used to refer to the sets of rules and techniques used by Al-Khwarizmi to solve algebraic equations, before later being generalized to refer to any set of rules or techniques. This eventually culminated in Leibniz's notion of the calculus ratiocinator (ca 1680):
Cryptographic algorithms
The first cryptographic algorithm for deciphering encrypted code was developed by Al-Kindi, a 9th-century Arab mathematician, in A Manuscript On Deciphering Cryptographic Messages. He gave the first description of cryptanalysis by frequency analysis, the earliest codebreaking algorithm. Mechanical contrivances with discrete states
The clock: Bolter credits the invention of the weight-driven clock as "The key invention [of Europe in the Middle Ages]", in particular, the verge escapement that provides us with the tick and tock of a mechanical clock. "The accurate automatic machine" led immediately to "mechanical automata" beginning in the 13th century and finally to "computational machines"—the difference engine and analytical engines of Charles Babbage and Countess Ada Lovelace, mid-19th century. | [
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Lovelace is credited with the first creation of an algorithm intended for processing on a computer—Babbage's analytical engine, the first device considered a real Turing-complete computer instead of just a calculator—and is sometimes called "history's first programmer" as a result, though a full implementation of Babbage's second device would not be realized until decades after her lifetime. Logical machines 1870 – Stanley Jevons' "logical abacus" and "logical machine": The technical problem was to reduce Boolean equations when presented in a form similar to what is now known as Karnaugh maps. Jevons (1880) describes first a simple "abacus" of "slips of wood furnished with pins, contrived so that any part or class of the [logical] combinations can be picked out mechanically ... More recently, however, I have reduced the system to a completely mechanical form, and have thus embodied the whole of the indirect process of inference in what may be called a Logical Machine" His machine came equipped with "certain moveable wooden rods" and "at the foot are 21 keys like those of a piano [etc.] ...". With this machine he could analyze a "syllogism or any other simple logical argument". This machine he displayed in 1870 before the Fellows of the Royal Society. | [
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Another logician John Venn, however, in his 1881 Symbolic Logic, turned a jaundiced eye to this effort: "I have no high estimate myself of the interest or importance of what are sometimes called logical machines ... it does not seem to me that any contrivances at present known or likely to be discovered really deserve the name of logical machines"; see more at Algorithm characterizations. But not to be outdone he too presented "a plan somewhat analogous, I apprehend, to Prof. Jevon's abacus ... [And] [a]gain, corresponding to Prof. Jevons's logical machine, the following contrivance may be described. I prefer to call it merely a logical-diagram machine ... but I suppose that it could do very completely all that can be rationally expected of any logical machine". Jacquard loom, Hollerith punch cards, telegraphy and telephony – the electromechanical relay: Bell and Newell (1971) indicate that the Jacquard loom (1801), precursor to Hollerith cards (punch cards, 1887), and "telephone switching technologies" were the roots of a tree leading to the development of the first computers. By the mid-19th century the telegraph, the precursor of the telephone, was in use throughout the world, its discrete and distinguishable encoding of letters as "dots and dashes" a common sound. By the late 19th century the ticker tape (ca 1870s) was in use, as was the use of Hollerith cards in the 1890 U.S. census. Then came the teleprinter (ca. 1910) with its punched-paper use of Baudot code on tape. | [
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Telephone-switching networks of electromechanical relays (invented 1835) was behind the work of George Stibitz (1937), the inventor of the digital adding device. As he worked in Bell Laboratories, he observed the "burdensome' use of mechanical calculators with gears. "He went home one evening in 1937 intending to test his idea... When the tinkering was over, Stibitz had constructed a binary adding device". Davis (2000) observes the particular importance of the electromechanical relay (with its two "binary states" open and closed):
It was only with the development, beginning in the 1930s, of electromechanical calculators using electrical relays, that machines were built having the scope Babbage had envisioned." Mathematics during the 19th century up to the mid-20th century
Symbols and rules: In rapid succession, the mathematics of George Boole (1847, 1854), Gottlob Frege (1879), and Giuseppe Peano (1888–1889) reduced arithmetic to a sequence of symbols manipulated by rules. Peano's The principles of arithmetic, presented by a new method (1888) was "the first attempt at an axiomatization of mathematics in a symbolic language". But Heijenoort gives Frege (1879) this kudos: Frege's is "perhaps the most important single work ever written in logic. ... in which we see a " 'formula language', that is a lingua characterica, a language written with special symbols, "for pure thought", that is, free from rhetorical embellishments ... constructed from specific symbols that are manipulated according to definite rules". | [
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The work of Frege was further simplified and amplified by Alfred North Whitehead and Bertrand Russell in their Principia Mathematica (1910–1913). The paradoxes: At the same time a number of disturbing paradoxes appeared in the literature, in particular, the Burali-Forti paradox (1897), the Russell paradox (1902–03), and the Richard Paradox. The resultant considerations led to Kurt Gödel's paper (1931)—he specifically cites the paradox of the liar—that completely reduces rules of recursion to numbers. Effective calculability: In an effort to solve the Entscheidungsproblem defined precisely by Hilbert in 1928, mathematicians first set about to define what was meant by an "effective method" or "effective calculation" or "effective calculability" (i.e., a calculation that would succeed). In rapid succession the following appeared: Alonzo Church, Stephen Kleene and J.B. Rosser's λ-calculus a finely honed definition of "general recursion" from the work of Gödel acting on suggestions of Jacques Herbrand (cf. Gödel's Princeton lectures of 1934) and subsequent simplifications by Kleene. Church's proof that the Entscheidungsproblem was unsolvable, Emil Post's definition of effective calculability as a worker mindlessly following a list of instructions to move left or right through a sequence of rooms and while there either mark or erase a paper or observe the paper and make a yes-no decision about the next instruction. | [
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Alan Turing's proof of that the Entscheidungsproblem was unsolvable by use of his "a- [automatic-] machine"—in effect almost identical to Post's "formulation", J. Barkley Rosser's definition of "effective method" in terms of "a machine". Kleene's proposal of a precursor to "Church thesis" that he called "Thesis I", and a few years later Kleene's renaming his Thesis "Church's Thesis" and proposing "Turing's Thesis". Emil Post (1936) and Alan Turing (1936–37, 1939)
Emil Post (1936) described the actions of a "computer" (human being) as follows:
"...two concepts are involved: that of a symbol space in which the work leading from problem to answer is to be carried out, and a fixed unalterable set of directions. His symbol space would be
"a two-way infinite sequence of spaces or boxes... The problem solver or worker is to move and work in this symbol space, being capable of being in, and operating in but one box at a time.... a box is to admit of but two possible conditions, i.e., being empty or unmarked, and having a single mark in it, say a vertical stroke. "One box is to be singled out and called the starting point. ...a specific problem is to be given in symbolic form by a finite number of boxes [i.e., INPUT] being marked with a stroke. | [
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Likewise, the answer [i.e., OUTPUT] is to be given in symbolic form by such a configuration of marked boxes...
"A set of directions applicable to a general problem sets up a deterministic process when applied to each specific problem. This process terminates only when it comes to the direction of type (C ) [i.e., STOP]". See more at Post–Turing machine
Alan Turing's work preceded that of Stibitz (1937); it is unknown whether Stibitz knew of the work of Turing. Turing's biographer believed that Turing's use of a typewriter-like model derived from a youthful interest: "Alan had dreamt of inventing typewriters as a boy; Mrs. Turing had a typewriter, and he could well have begun by asking himself what was meant by calling a typewriter 'mechanical'". Given the prevalence of Morse code and telegraphy, ticker tape machines, and teletypewriters we might conjecture that all were influences. Turing—his model of computation is now called a Turing machine—begins, as did Post, with an analysis of a human computer that he whittles down to a simple set of basic motions and "states of mind". But he continues a step further and creates a machine as a model of computation of numbers. "Computing is normally done by writing certain symbols on paper. We may suppose this paper is divided into squares like a child's arithmetic book...I assume then that the computation is carried out on one-dimensional paper, i.e., on a tape divided into squares. | [
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I shall also suppose that the number of symbols which may be printed is finite...
"The behavior of the computer at any moment is determined by the symbols which he is observing, and his "state of mind" at that moment. We may suppose that there is a bound B to the number of symbols or squares which the computer can observe at one moment. If he wishes to observe more, he must use successive observations. We will also suppose that the number of states of mind which need be taken into account is finite...
"Let us imagine that the operations performed by the computer to be split up into 'simple operations' which are so elementary that it is not easy to imagine them further divided." Turing's reduction yields the following:
"The simple operations must therefore include:
"(a) Changes of the symbol on one of the observed squares
"(b) Changes of one of the squares observed to another square within L squares of one of the previously observed squares. "It may be that some of these change necessarily invoke a change of state of mind. The most general single operation must, therefore, be taken to be one of the following:
"(A) A possible change (a) of symbol together with a possible change of state of mind. "(B) A possible change (b) of observed squares, together with a possible change of state of mind"
"We may now construct a machine to do the work of this computer." | [
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] |
A few years later, Turing expanded his analysis (thesis, definition) with this forceful expression of it:
"A function is said to be "effectively calculable" if its values can be found by some purely mechanical process. Though it is fairly easy to get an intuitive grasp of this idea, it is nevertheless desirable to have some more definite, mathematical expressible definition ... [he discusses the history of the definition pretty much as presented above with respect to Gödel, Herbrand, Kleene, Church, Turing, and Post] ... 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 ... . "† We shall use the expression "computable function" to mean a function calculable by a machine, and we let "effectively calculable" refer to the intuitive idea without particular identification with any one of these definitions". J.B. Rosser (1939) and S.C. Kleene (1943)
J. Barkley Rosser defined an 'effective [mathematical] method' in the following manner (italicization added):
"'Effective method' is used here in the rather special sense of a method each step of which is precisely determined and which is certain to produce the answer in a finite number of steps. | [
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With this special meaning, three different precise definitions have been given to date. [his footnote #5; see discussion immediately below]. The simplest of these to state (due to Post and Turing) says essentially that an effective method of solving certain sets of problems exists if one can build a machine which will then solve any problem of the set with no human intervention beyond inserting the question and (later) reading the answer. All three definitions are equivalent, so it doesn't matter which one is used. Moreover, the fact that all three are equivalent is a very strong argument for the correctness of any one." (Rosser 1939:225–226)
Rosser's footnote No. 5 references the work of (1) Church and Kleene and their definition of λ-definability, in particular, Church's use of it in his An Unsolvable Problem of Elementary Number Theory (1936); (2) Herbrand and Gödel and their use of recursion, in particular, Gödel's use in his famous paper On Formally Undecidable Propositions of Principia Mathematica and Related Systems I (1931); and (3) Post (1936) and Turing (1936–37) in their mechanism-models of computation. Stephen C. Kleene defined as his now-famous "Thesis I" known as the Church–Turing thesis. But he did this in the following context (boldface in original):
"12. Algorithmic theories... | [
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In setting up a complete algorithmic theory, what we do is to describe a procedure, performable for each set of values of the independent variables, which procedure necessarily terminates and in such manner that from the outcome we can read a definite answer, "yes" or "no," to the question, "is the predicate value true?"" (Kleene 1943:273)
History after 1950
A number of efforts have been directed toward further refinement of the definition of "algorithm", and activity is on-going because of issues surrounding, in particular, foundations of mathematics (especially the Church–Turing thesis) and philosophy of mind (especially arguments about artificial intelligence). For more, see Algorithm characterizations. See also
Abstract machine
Algorithm engineering
Algorithm characterizations
Algorithmic bias
Algorithmic composition
Algorithmic entities
Algorithmic synthesis
Algorithmic technique
Algorithmic topology
Garbage in, garbage out
Introduction to Algorithms (textbook)
List of algorithms
List of algorithm general topics
List of important publications in theoretical computer science – Algorithms
Regulation of algorithms
Theory of computation
Computability theory
Computational complexity theory
Notes
Bibliography
Bell, C. Gordon and Newell, Allen (1971), Computer Structures: Readings and Examples, McGraw–Hill Book Company, New York. . Includes an excellent bibliography of 56 references. ,
: cf. Chapter 3 Turing machines where they discuss "certain enumerable sets not effectively (mechanically) enumerable". Campagnolo, M.L., Moore, C., and Costa, J.F. (2000) An analog characterization of the subrecursive functions. In Proc. of the 4th Conference on Real Numbers and Computers, Odense University, pp. | [
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91–109
Reprinted in The Undecidable, p. 89ff. The first expression of "Church's Thesis". See in particular page 100 (The Undecidable) where he defines the notion of "effective calculability" in terms of "an algorithm", and he uses the word "terminates", etc. Reprinted in The Undecidable, p. 110ff. Church shows that the Entscheidungsproblem is unsolvable in about 3 pages of text and 3 pages of footnotes. Davis gives commentary before each article. Papers of Gödel, Alonzo Church, Turing, Rosser, Kleene, and Emil Post are included; those cited in the article are listed here by author's name. Davis offers concise biographies of Leibniz, Boole, Frege, Cantor, Hilbert, Gödel and Turing with von Neumann as the show-stealing villain. Very brief bios of Joseph-Marie Jacquard, Babbage, Ada Lovelace, Claude Shannon, Howard Aiken, etc. ,
Yuri Gurevich, Sequential Abstract State Machines Capture Sequential Algorithms, ACM Transactions on Computational Logic, Vol 1, no 1 (July 2000), pp. 77–111. Includes bibliography of 33 sources. , 3rd edition 1976[? ], (pbk.) , . Cf. Chapter "The Spirit of Truth" for a history leading to, and a discussion of, his proof. Presented to the American Mathematical Society, September 1935. Reprinted in The Undecidable, p. 237ff. | [
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Kleene's definition of "general recursion" (known now as mu-recursion) was used by Church in his 1935 paper An Unsolvable Problem of Elementary Number Theory that proved the "decision problem" to be "undecidable" (i.e., a negative result). Reprinted in The Undecidable, p. 255ff. Kleene refined his definition of "general recursion" and proceeded in his chapter "12. Algorithmic theories" to posit "Thesis I" (p. 274); he would later repeat this thesis (in Kleene 1952:300) and name it "Church's Thesis"(Kleene 1952:317) (i.e., the Church thesis). Kosovsky, N.K. Elements of Mathematical Logic and its Application to the theory of Subrecursive Algorithms, LSU Publ., Leningrad, 1981
A.A. Markov (1954) Theory of algorithms. [Translated by Jacques J. Schorr-Kon and PST staff] Imprint Moscow, Academy of Sciences of the USSR, 1954 [i.e., Jerusalem, Israel Program for Scientific Translations, 1961; available from the Office of Technical Services, U.S. Dept. of Commerce, Washington] Description 444 p. 28 cm. Added t.p. in Russian Translation of Works of the Mathematical Institute, Academy of Sciences of the USSR, v. 42. Original title: Teoriya algerifmov. [QA248.M2943 Dartmouth College library. U.S. Dept. of Commerce, Office of Technical Services, number OTS .] Minsky expands his "...idea of an algorithm – an effective procedure..." in chapter 5.1 Computability, Effective Procedures and Algorithms. Infinite machines. Reprinted in The Undecidable, pp. 289ff. | [
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Post defines a simple algorithmic-like process of a man writing marks or erasing marks and going from box to box and eventually halting, as he follows a list of simple instructions. This is cited by Kleene as one source of his "Thesis I", the so-called Church–Turing thesis. Reprinted in The Undecidable, p. 223ff. Herein is Rosser's famous definition of "effective method": "...a method each step of which is precisely predetermined and which is certain to produce the answer in a finite number of steps... a machine which will then solve any problem of the set with no human intervention beyond inserting the question and (later) reading the answer" (p. 225–226, The Undecidable)
Cf. in particular the first chapter titled: Algorithms, Turing Machines, and Programs. His succinct informal definition: "...any sequence of instructions that can be obeyed by a robot, is called an algorithm" (p. 4). . Corrections, ibid, vol. 43(1937) pp. 544–546. Reprinted in The Undecidable, p. 116ff. Turing's famous paper completed as a Master's dissertation while at King's College Cambridge UK. Reprinted in The Undecidable, pp. 155ff. Turing's paper that defined "the oracle" was his PhD thesis while at Princeton. United States Patent and Trademark Office (2006), 2106.02 **>Mathematical Algorithms: 2100 Patentability, Manual of Patent Examining Procedure (MPEP). Latest revision August 2006
Further reading
Knuth, Donald E. (2000). Selected Papers on Analysis of Algorithms. | [
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] |
Stanford, California: Center for the Study of Language and Information. Knuth, Donald E. (2010). Selected Papers on Design of Algorithms. Stanford, California: Center for the Study of Language and Information. External links
Dictionary of Algorithms and Data Structures – National Institute of Standards and Technology
Algorithm repositories
The Stony Brook Algorithm Repository – State University of New York at Stony Brook
Collected Algorithms of the ACM – Association for Computing Machinery
The Stanford GraphBase – Stanford University
Articles with example pseudocode
Mathematical logic
Theoretical computer science | [
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Arabic (, or , or ) is a Semitic language that first emerged in the 1st to 4th centuries CE. It is the lingua franca of the Arab world and the liturgical language of Islam. It is named after the Arabs, a term initially used to describe peoples living in the Arabian Peninsula bounded by eastern Egypt in the west, Mesopotamia in the east, and the Anti-Lebanon mountains and northern Syria in the north, as perceived by ancient Greek geographers. The ISO assigns language codes to 32 varieties of Arabic, including its standard form, Modern Standard Arabic, also referred to as Literary Arabic, which is modernized Classical Arabic. This distinction exists primarily among Western linguists; Arabic speakers themselves generally do not distinguish between Modern Standard Arabic and Classical Arabic, but rather refer to both as ( "the eloquent Arabic") or simply (). Arabic is widely taught in schools and universities around the world and is used to varying degrees in workplaces, governments and the media. Arabic, in its Modern Standard Arabic form, is an official language of 26 states and 1 disputed territory, the third most after English and French;
it is also the liturgical language of the religion of Islam, since the Quran and the Hadiths were written in Classical Arabic. During the early Middle Ages, Arabic was a major vehicle of culture in the Mediterranean region, especially in science, mathematics and philosophy. As a result, many European languages have also borrowed many words from it. | [
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Arabic influence, mainly in vocabulary, is seen in European languages—mainly Spanish and to a lesser extent Portuguese, Catalan, and Sicilian—owing to both the proximity of Christian European and Muslim Arabized civilizations and the long-lasting Muslim culture and Arabic language presence, mainly in Southern Iberia, during the Al-Andalus era. The Maltese language is a Semitic language developed from a dialect of Arabic and written in the Latin alphabet. The Balkan languages, including Greek and Bulgarian, have also acquired a significant number of words of Arabic origin through contact with Ottoman Turkish. Arabic has influenced many other languages around the globe throughout its history especially languages of Muslim cultures and countries that were conquered by Muslims. Some of the most influenced languages are Persian, Turkish, Hindustani (Hindi and Urdu), Kashmiri, Kurdish, Bosnian, Kazakh, Bengali, Malay (Indonesian and Malaysian), Maldivian, Pashto, Punjabi, Albanian, Armenian, Azerbaijani, Sicilian, Spanish, Greek, Bulgarian, Tagalog, Sindhi, Odia Hebrew and Hausa and some languages in parts of Africa. Conversely, Arabic has borrowed words from other languages, including Aramaic as well as Hebrew, Latin, Greek, Persian and to a lesser extent Turkish (due to the Ottoman Empire), English and French (due to their colonization of the Levant) and other Semitic languages such as Abyssinian. Arabic is the liturgical language of 1.9 billion Muslims, and Arabic is one of six official languages of the United Nations. | [
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All varieties of Arabic combined are spoken by perhaps as many as 422 million speakers (native and non-native) in the Arab world, making it the fifth most spoken language in the world, and the fourth most used language on the internet in terms of users. In 2011, Bloomberg Businessweek ranked Arabic the fourth most useful language for business, after English, Standard Mandarin Chinese, and French. Arabic is written with the Arabic alphabet, which is an abjad script and is written from right to left, although the spoken varieties are sometimes written in ASCII Latin from left to right with no standardized orthography. Classification
Arabic is usually, but not universally, classified as a Central Semitic language. It is related to languages in other subgroups of the Semitic language group (Northwest Semitic, South Semitic, East Semitic, West Semitic), such as Aramaic, Syriac, Hebrew, Ugaritic, Phoenician, Canaanite, Amorite, Ammonite, Eblaite, epigraphic Ancient North Arabian, epigraphic Ancient South Arabian, Ethiopic, Modern South Arabian, and numerous other dead and modern languages. Linguists still differ as to the best classification of Semitic language sub-groups. The Semitic languages changed a great deal between Proto-Semitic and the emergence of the Central Semitic languages, particularly in grammar. Innovations of the Central Semitic languages—all maintained in Arabic—include:
The conversion of the suffix-conjugated stative formation (jalas-) into a past tense. The conversion of the prefix-conjugated preterite-tense formation (yajlis-) into a present tense. | [
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The elimination of other prefix-conjugated mood/aspect forms (e.g., a present tense formed by doubling the middle root, a perfect formed by infixing a after the first root consonant, probably a jussive formed by a stress shift) in favor of new moods formed by endings attached to the prefix-conjugation forms (e.g., -u for indicative, -a for subjunctive, no ending for jussive, -an or -anna for energetic). The development of an internal passive. There are several features which Classical Arabic, the modern Arabic varieties, as well as the Safaitic and Hismaic inscriptions share which are unattested in any other Central Semitic language variety, including the Dadanitic and Taymanitic languages of the northern Hejaz. These features are evidence of common descent from a hypothetical ancestor, Proto-Arabic. The following features can be reconstructed with confidence for Proto-Arabic:
negative particles * ; * to Classical Arabic
G-passive participle
prepositions and adverbs , , , ,
a subjunctive in -
-demonstratives
leveling of the - allomorph of the feminine ending
complementizer and subordinator
the use of - to introduce modal clauses
independent object pronoun in
vestiges of nunation
History
Old Arabic
Arabia boasted a wide variety of Semitic languages in antiquity. In the southwest, various Central Semitic languages both belonging to and outside of the Ancient South Arabian family (e.g. Southern Thamudic) were spoken. It is also believed that the ancestors of the Modern South Arabian languages (non-Central Semitic languages) were also spoken in southern Arabia at this time. To the north, in the oases of northern Hejaz, Dadanitic and Taymanitic held some prestige as inscriptional languages. | [
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In Najd and parts of western Arabia, a language known to scholars as Thamudic C is attested. In eastern Arabia, inscriptions in a script derived from ASA attest to a language known as Hasaitic. Finally, on the northwestern frontier of Arabia, various languages known to scholars as Thamudic B, Thamudic D, Safaitic, and Hismaic are attested. The last two share important isoglosses with later forms of Arabic, leading scholars to theorize that Safaitic and Hismaic are in fact early forms of Arabic and that they should be considered Old Arabic. Linguists generally believe that "Old Arabic" (a collection of related dialects that constitute the precursor of Arabic) first emerged around the 1st century CE. Previously, the earliest attestation of Old Arabic was thought to be a single 1st century CE inscription in Sabaic script at Qaryat Al-Faw, in southern present-day Saudi Arabia. However, this inscription does not participate in several of the key innovations of the Arabic language group, such as the conversion of Semitic mimation to nunation in the singular. It is best reassessed as a separate language on the Central Semitic dialect continuum. It was also thought that Old Arabic coexisted alongside—and then gradually displaced--epigraphic Ancient North Arabian (ANA), which was theorized to have been the regional tongue for many centuries. ANA, despite its name, was considered a very distinct language, and mutually unintelligible, from "Arabic". Scholars named its variant dialects after the towns where the inscriptions were discovered (Dadanitic, Taymanitic, Hismaic, Safaitic). | [
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However, most arguments for a single ANA language or language family were based on the shape of the definite article, a prefixed h-. It has been argued that the h- is an archaism and not a shared innovation, and thus unsuitable for language classification, rendering the hypothesis of an ANA language family untenable. Safaitic and Hismaic, previously considered ANA, should be considered Old Arabic due to the fact that they participate in the innovations common to all forms of Arabic.The earliest attestation of continuous Arabic text in an ancestor of the modern Arabic script are three lines of poetry by a man named Garm(')allāhe found in En Avdat, Israel, and dated to around 125 CE. This is followed by the Namara inscription, an epitaph of the Lakhmid king Imru' al-Qays bar 'Amro, dating to 328 CE, found at Namaraa, Syria. From the 4th to the 6th centuries, the Nabataean script evolves into the Arabic script recognizable from the early Islamic era. There are inscriptions in an undotted, 17-letter Arabic script dating to the 6th century CE, found at four locations in Syria (Zabad, Jabal 'Usays, Harran, Umm al-Jimaal). The oldest surviving papyrus in Arabic dates to 643 CE, and it uses dots to produce the modern 28-letter Arabic alphabet. The language of that papyrus and of the Qur'an are referred to by linguists as "Quranic Arabic", as distinct from its codification soon thereafter into "Classical Arabic". | [
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Old Hejazi and Classical Arabic
In late pre-Islamic times, a transdialectal and transcommunal variety of Arabic emerged in the Hejaz which continued living its parallel life after literary Arabic had been institutionally standardized in the 2nd and 3rd century of the Hijra, most strongly in Judeo-Christian texts, keeping alive ancient features eliminated from the "learned" tradition (Classical Arabic). This variety and both its classicizing and "lay" iterations have been termed Middle Arabic in the past, but they are thought to continue an Old Higazi register. It is clear that the orthography of the Qur'an was not developed for the standardized form of Classical Arabic; rather, it shows the attempt on the part of writers to record an archaic form of Old Higazi. In the late 6th century AD, a relatively uniform intertribal "poetic koine" distinct from the spoken vernaculars developed based on the Bedouin dialects of Najd, probably in connection with the court of al-Ḥīra. During the first Islamic century, the majority of Arabic poets and Arabic-writing persons spoke Arabic as their mother tongue. Their texts, although mainly preserved in far later manuscripts, contain traces of non-standardized Classical Arabic elements in morphology and syntax. Standardization
Abu al-Aswad al-Du'ali (c. 603–689) is credited with standardizing Arabic grammar, or an-naḥw ( "the way"), and pioneering a system of diacritics to differentiate consonants ( nuqat l-i'jām "pointing for non-Arabs") and indicate vocalization ( at-tashkil). | [
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Al-Khalil ibn Ahmad al-Farahidi (718 – 786) compiled the first Arabic dictionary, Kitāb al-'Ayn ( "The Book of the Letter ع"), and is credited with establishing the rules of Arabic prosody. Al-Jahiz (776-868) proposed to Al-Akhfash al-Akbar an overhaul of the grammar of Arabic, but it would not come to pass two centuries. The standardization of Arabic reached completion around the end of the 8th century. The first comprehensive description of the ʿarabiyya "Arabic", Sībawayhi's al-Kitāb, is based first of all upon a corpus of poetic texts, in addition to Qur'an usage and Bedouin informants whom he considered to be reliable speakers of the ʿarabiyya. Spread
Arabic spread with the spread of Islam. Following the early Muslim conquests, Arabic gained vocabulary from Middle Persian and Turkish. In the early Abbasid period, many Classical Greek terms entered Arabic through translations carried out at Baghdad's House of Wisdom. By the 8th century, knowledge of Classical Arabic had become an essential prerequisite for rising into the higher classes throughout the Islamic world, both for Muslims and non-Muslims. For example, Maimonides, the Andalusi Jewish philosopher, authored works in Judeo-Arabic—Arabic written in Hebrew script—including his famous The Guide for the Perplexed ( Dalālat al-ḥāʾirīn). Development
Ibn Jinni of Mosul, a pioneer in phonology, wrote prolifically in the 10th century on Arabic morphology and phonology in works such as Kitāb Al-Munṣif, Kitāb Al-Muḥtasab, and . Ibn Mada' of Cordoba (1116–1196) realized the overhaul of Arabic grammar first proposed by Al-Jahiz 200 years prior. | [
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The Maghrebi lexicographer Ibn Manzur compiled (لسان العرب, "Tongue of Arabs"), a major reference dictionary of Arabic, in 1290. Neo-Arabic
Charles Ferguson's koine theory (Ferguson 1959) claims that the modern Arabic dialects collectively descend from a single military koine that sprang up during the Islamic conquests; this view has been challenged in recent times. Ahmad al-Jallad proposes that there were at least two considerably distinct types of Arabic on the eve of the conquests: Northern and Central (Al-Jallad 2009). The modern dialects emerged from a new contact situation produced following the conquests. Instead of the emergence of a single or multiple koines, the dialects contain several sedimentary layers of borrowed and areal features, which they absorbed at different points in their linguistic histories. According to Veersteegh and Bickerton, colloquial Arabic dialects arose from pidginized Arabic formed from contact between Arabs and conquered peoples. Pidginization and subsequent creolization among Arabs and arabized peoples could explain relative morphological and phonological simplicity of vernacular Arabic compared to Classical and MSA. In around the 11th and 12th centuries in al-Andalus, the zajal and muwashah poetry forms developed in the dialectical Arabic of Cordoba and the Maghreb. Nahda
The Nahda was a cultural and especially literary renaissance of the 19th century in which writers sought "to fuse Arabic and European forms of expression." According to James L. Gelvin, "Nahda writers attempted to simplify the Arabic language and script so that it might be accessible to a wider audience." | [
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In the wake of the industrial revolution and European hegemony and colonialism, pioneering Arabic presses, such as the Amiri Press established by Muhammad Ali (1819), dramatically changed the diffusion and consumption of Arabic literature and publications. Rifa'a al-Tahtawi proposed the establishment of in 1836 and led a translation campaign that highlighted the need for a lexical injection in Arabic, to suit concepts of the industrial and post-industrial age. In response, a number of Arabic academies modeled after the Académie française were established with the aim of developing standardized additions to the Arabic lexicon to suit these transformations, first in Damascus (1919), then in Cairo (1932), Baghdad (1948), Rabat (1960), Amman (1977), (1993), and Tunis (1993). In 1997, a bureau of Arabization standardization was added to the Educational, Cultural, and Scientific Organization of the Arab League. These academies and organizations have worked toward the Arabization of the sciences, creating terms in Arabic to describe new concepts, toward the standardization of these new terms throughout the Arabic-speaking world, and toward the development of Arabic as a world language. This gave rise to what Western scholars call Modern Standard Arabic. From the 1950s, Arabization became a postcolonial nationalist policy in countries such as Tunisia, Algeria, Morocco, and Sudan. Classical, Modern Standard and spoken Arabic
Arabic usually refers to Standard Arabic, which Western linguists divide into Classical Arabic and Modern Standard Arabic. It could also refer to any of a variety of regional vernacular Arabic dialects, which are not necessarily mutually intelligible. | [
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Classical Arabic is the language found in the Quran, used from the period of Pre-Islamic Arabia to that of the Abbasid Caliphate. Classical Arabic is prescriptive, according to the syntactic and grammatical norms laid down by classical grammarians (such as Sibawayh) and the vocabulary defined in classical dictionaries (such as the Lisān al-ʻArab). Modern Standard Arabic (MSA) largely follows the grammatical standards of Classical Arabic and uses much of the same vocabulary. However, it has discarded some grammatical constructions and vocabulary that no longer have any counterpart in the spoken varieties and has adopted certain new constructions and vocabulary from the spoken varieties. Much of the new vocabulary is used to denote concepts that have arisen in the industrial and post-industrial era, especially in modern times. Due to its grounding in Classical Arabic, Modern Standard Arabic is removed over a millennium from everyday speech, which is construed as a multitude of dialects of this language. These dialects and Modern Standard Arabic are described by some scholars as not mutually comprehensible. The former are usually acquired in families, while the latter is taught in formal education settings. However, there have been studies reporting some degree of comprehension of stories told in the standard variety among preschool-aged children. The relation between Modern Standard Arabic and these dialects is sometimes compared to that of Classical Latin and Vulgar Latin vernaculars (which became Romance languages) in medieval and early modern Europe. This view though does not take into account the widespread use of Modern Standard Arabic as a medium of audiovisual communication in today's mass media—a function Latin has never performed. | [
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] |
MSA is the variety used in most current, printed Arabic publications, spoken by some of the Arabic media across North Africa and the Middle East, and understood by most educated Arabic speakers. "Literary Arabic" and "Standard Arabic" ( ) are less strictly defined terms that may refer to Modern Standard Arabic or Classical Arabic. Some of the differences between Classical Arabic (CA) and Modern Standard Arabic (MSA) are as follows:
Certain grammatical constructions of CA that have no counterpart in any modern vernacular dialect (e.g., the energetic mood) are almost never used in Modern Standard Arabic. Case distinctions are very rare in Arabic vernaculars. As a result, MSA is generally composed without case distinctions in mind, and the proper cases are added after the fact, when necessary. Because most case endings are noted using final short vowels, which are normally left unwritten in the Arabic script, it is unnecessary to determine the proper case of most words. The practical result of this is that MSA, like English and Standard Chinese, is written in a strongly determined word order and alternative orders that were used in CA for emphasis are rare. In addition, because of the lack of case marking in the spoken varieties, most speakers cannot consistently use the correct endings in extemporaneous speech. As a result, spoken MSA tends to drop or regularize the endings except when reading from a prepared text. The numeral system in CA is complex and heavily tied in with the case system. | [
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This system is never used in MSA, even in the most formal of circumstances; instead, a significantly simplified system is used, approximating the system of the conservative spoken varieties. MSA uses much Classical vocabulary (e.g., 'to go') that is not present in the spoken varieties, but deletes Classical words that sound obsolete in MSA. In addition, MSA has borrowed or coined many terms for concepts that did not exist in Quranic times, and MSA continues to evolve. Some words have been borrowed from other languages—notice that transliteration mainly indicates spelling and not real pronunciation (e.g., 'film' or 'democracy'). However, the current preference is to avoid direct borrowings, preferring to either use loan translations (e.g., 'branch', also used for the branch of a company or organization; 'wing', is also used for the wing of an airplane, building, air force, etc. ), or to coin new words using forms within existing roots ( 'apoptosis', using the root m/w/t 'death' put into the Xth form, or 'university', based on 'to gather, unite'; 'republic', based on 'multitude'). An earlier tendency was to redefine an older word although this has fallen into disuse (e.g., 'telephone' < 'invisible caller (in Sufism)'; 'newspaper' < 'palm-leaf stalk'). Colloquial or dialectal Arabic refers to the many national or regional varieties which constitute the everyday spoken language and evolved from Classical Arabic. | [
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Colloquial Arabic has many regional variants; geographically distant varieties usually differ enough to be mutually unintelligible, and some linguists consider them distinct languages. However, research indicates a high degree of mutual intelligibility between closely related Arabic variants for native speakers listening to words, sentences, and texts; and between more distantly related dialects in interactional situations. The varieties are typically unwritten. They are often used in informal spoken media, such as soap operas and talk shows, as well as occasionally in certain forms of written media such as poetry and printed advertising. The only variety of modern Arabic to have acquired official language status is Maltese, which is spoken in (predominantly Catholic) Malta and written with the Latin script. It is descended from Classical Arabic through Siculo-Arabic, but is not mutually intelligible with any other variety of Arabic. Most linguists list it as a separate language rather than as a dialect of Arabic. Even during Muhammad's lifetime, there were dialects of spoken Arabic. Muhammad spoke in the dialect of Mecca, in the western Arabian peninsula, and it was in this dialect that the Quran was written down. However, the dialects of the eastern Arabian peninsula were considered the most prestigious at the time, so the language of the Quran was ultimately converted to follow the eastern phonology. It is this phonology that underlies the modern pronunciation of Classical Arabic. | [
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The phonological differences between these two dialects account for some of the complexities of Arabic writing, most notably the writing of the glottal stop or hamzah (which was preserved in the eastern dialects but lost in western speech) and the use of (representing a sound preserved in the western dialects but merged with in eastern speech). Language and dialect
The sociolinguistic situation of Arabic in modern times provides a prime example of the linguistic phenomenon of diglossia, which is the normal use of two separate varieties of the same language, usually in different social situations. Tawleed is the process of giving a new shade of meaning to an old classical word. For example, al-hatif lexicographically, means the one whose sound is heard but whose person remains unseen. Now the term al-hatif is used for a telephone. Therefore, the process of tawleed can express the needs of modern civilization in a manner that would appear to be originally Arabic. In the case of Arabic, educated Arabs of any nationality can be assumed to speak both their school-taught Standard Arabic as well as their native dialects, which depending on the region may be mutually unintelligible. Some of these dialects can be considered to constitute separate languages which may have “sub-dialects” of their own. When educated Arabs of different dialects engage in conversation (for example, a Moroccan speaking with a Lebanese), many speakers code-switch back and forth between the dialectal and standard varieties of the language, sometimes even within the same sentence. Arabic speakers often improve their familiarity with other dialects via music or film. | [
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The issue of whether Arabic is one language or many languages is politically charged, in the same way it is for the varieties of Chinese, Hindi and Urdu, Serbian and Croatian, Scots and English, etc. In contrast to speakers of Hindi and Urdu who claim they cannot understand each other even when they can, speakers of the varieties of Arabic will claim they can all understand each other even when they cannot. While there is a minimum level of comprehension between all Arabic dialects, this level can increase or decrease based on geographic proximity: for example, Levantine and Gulf speakers understand each other much better than they do speakers from the Maghreb. The issue of diglossia between spoken and written language is a significant complicating factor: A single written form, significantly different from any of the spoken varieties learned natively, unites a number of sometimes divergent spoken forms. For political reasons, Arabs mostly assert that they all speak a single language, despite significant issues of mutual incomprehensibility among differing spoken versions. From a linguistic standpoint, it is often said that the various spoken varieties of Arabic differ among each other collectively about as much as the Romance languages. This is an apt comparison in a number of ways. The period of divergence from a single spoken form is similar—perhaps 1500 years for Arabic, 2000 years for the Romance languages. | [
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Also, while it is comprehensible to people from the Maghreb, a linguistically innovative variety such as Moroccan Arabic is essentially incomprehensible to Arabs from the Mashriq, much as French is incomprehensible to Spanish or Italian speakers but relatively easily learned by them. This suggests that the spoken varieties may linguistically be considered separate languages. Influence of Arabic on other languages
The influence of Arabic has been most important in Islamic countries, because it is the language of the Islamic sacred book, the Quran. Arabic is also an important source of vocabulary for languages such as Amharic, Azerbaijani, Baluchi, Bengali, Berber, Bosnian, Chaldean, Chechen, Chittagonian, Croatian, Dagestani, Dhivehi, English, German, Gujarati, Hausa, Hindi, Kazakh, Kurdish, Kutchi, Kyrgyz, Malay (Malaysian and Indonesian), Pashto, Persian, Punjabi, Rohingya, Romance languages (French, Catalan, Italian, Portuguese, Sicilian, Spanish, etc.) Saraiki, Sindhi, Somali, Sylheti, Swahili, Tagalog, Tigrinya, Turkish, Turkmen, Urdu, Uyghur, Uzbek, Visayan and Wolof, as well as other languages in countries where these languages are spoken.Modern Hebrew has been also influenced by Arabic especially during the process of revival, as MSA was used as a source for modern Hebrew vocabulary and roots, as well as much of Modern Hebrew's slang. The Education Minister of France Jean-Michel Blanquer has emphasized the learning and usage of Arabic in French schools. In addition, English has many Arabic loanwords, some directly, but most via other Mediterranean languages. | [
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Examples of such words include admiral, adobe, alchemy, alcohol, algebra, algorithm, alkaline, almanac, amber, arsenal, assassin, candy, carat, cipher, coffee, cotton, ghoul, hazard, jar, kismet, lemon, loofah, magazine, mattress, sherbet, sofa, sumac, tariff, and zenith. Other languages such as Maltese and Kinubi derive ultimately from Arabic, rather than merely borrowing vocabulary or grammatical rules. Terms borrowed range from religious terminology (like Berber taẓallit, "prayer", from salat ( )), academic terms (like Uyghur mentiq, "logic"), and economic items (like English coffee) to placeholders (like Spanish fulano, "so-and-so"), everyday terms (like Hindustani lekin, "but", or Spanish taza and French tasse, meaning "cup"), and expressions (like Catalan a betzef, "galore, in quantity"). Most Berber varieties (such as Kabyle), along with Swahili, borrow some numbers from Arabic. Most Islamic religious terms are direct borrowings from Arabic, such as (salat), "prayer", and (imam), "prayer leader." In languages not directly in contact with the Arab world, Arabic loanwords are often transferred indirectly via other languages rather than being transferred directly from Arabic. For example, most Arabic loanwords in Hindustani and Turkish entered through Persian. Older Arabic loanwords in Hausa were borrowed from Kanuri. Most Arabic loanwords in Yoruba entered through Hausa. | [
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Arabic words also made their way into several West African languages as Islam spread across the Sahara. Variants of Arabic words such as kitāb ("book") have spread to the languages of African groups who had no direct contact with Arab traders. Since, throughout the Islamic world, Arabic occupied a position similar to that of Latin in Europe, many of the Arabic concepts in the fields of science, philosophy, commerce, etc. were coined from Arabic roots by non-native Arabic speakers, notably by Aramaic and Persian translators, and then found their way into other languages. This process of using Arabic roots, especially in Kurdish and Persian, to translate foreign concepts continued through to the 18th and 19th centuries, when swaths of Arab-inhabited lands were under Ottoman rule. Influence of other languages on Arabic
The most important sources of borrowings into (pre-Islamic) Arabic are from the related (Semitic) languages Aramaic, which used to be the principal, international language of communication throughout the ancient Near and Middle East, and Ethiopic. In addition, many cultural, religious and political terms have entered Arabic from Iranian languages, notably Middle Persian, Parthian, and (Classical) Persian, and Hellenistic Greek (kīmiyāʼ has as origin the Greek khymia, meaning in that language the melting of metals; see Roger Dachez, Histoire de la Médecine de l'Antiquité au XXe siècle, Tallandier, 2008, p. 251), alembic (distiller) from ambix (cup), almanac (climate) from almenichiakon (calendar). | [
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] |
(For the origin of the last three borrowed words, see Alfred-Louis de Prémare, Foundations of Islam, Seuil, L'Univers Historique, 2002.) Some Arabic borrowings from Semitic or Persian languages are, as presented in De Prémare's above-cited book:
madīnah/medina (مدينة, city or city square), a word of Aramaic origin “madenta” (in which it means "a state"). jazīrah (جزيرة), as in the well-known form الجزيرة "Al-Jazeera," means "island" and has its origin in the Syriac ܓܙܝܪܗ gazarta. lāzaward (لازورد) is taken from Persian لاژورد lājvard, the name of a blue stone, lapis lazuli. This word was borrowed in several European languages to mean (light) blue – azure in English, azur in French and azul in Portuguese and Spanish. A comprehensive overview of the influence of other languages on Arabic is found in Lucas & Manfredi (2020). Arabic alphabet and nationalism
There have been many instances of national movements to convert Arabic script into Latin script or to Romanize the language. Currently, the only language derived from Classical Arabic to use Latin script is Maltese. Lebanon
The Beirut newspaper La Syrie pushed for the change from Arabic script to Latin letters in 1922. The major head of this movement was Louis Massignon, a French Orientalist, who brought his concern before the Arabic Language Academy in Damascus in 1928. Massignon's attempt at Romanization failed as the Academy and population viewed the proposal as an attempt from the Western world to take over their country. | [
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Sa'id Afghani, a member of the Academy, mentioned that the movement to Romanize the script was a Zionist plan to dominate Lebanon. Said Akl created a Latin-based alphabet for Lebanese and used it in a newspaper he founded, Lebnaan, as well as in some books he wrote. Egypt
After the period of colonialism in Egypt, Egyptians were looking for a way to reclaim and re-emphasize Egyptian culture. As a result, some Egyptians pushed for an Egyptianization of the Arabic language in which the formal Arabic and the colloquial Arabic would be combined into one language and the Latin alphabet would be used. There was also the idea of finding a way to use Hieroglyphics instead of the Latin alphabet, but this was seen as too complicated to use. A scholar, Salama Musa agreed with the idea of applying a Latin alphabet to Arabic, as he believed that would allow Egypt to have a closer relationship with the West. He also believed that Latin script was key to the success of Egypt as it would allow for more advances in science and technology. This change in alphabet, he believed, would solve the problems inherent with Arabic, such as a lack of written vowels and difficulties writing foreign words that made it difficult for non-native speakers to learn. Ahmad Lutfi As Sayid and Muhammad Azmi, two Egyptian intellectuals, agreed with Musa and supported the push for Romanization. The idea that Romanization was necessary for modernization and growth in Egypt continued with Abd Al-Aziz Fahmi in 1944. He was the chairman for the Writing and Grammar Committee for the Arabic Language Academy of Cairo. | [
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However, this effort failed as the Egyptian people felt a strong cultural tie to the Arabic alphabet. In particular, the older Egyptian generations believed that the Arabic alphabet had strong connections to Arab values and history, due to the long history of the Arabic alphabet (Shrivtiel, 189) in Muslim societies. The language of the Quran and its influence on poetry
The Quran introduced a new way of writing to the world. People began studying and applying the unique styles they learned from the Quran to not only their own writing, but also their culture. Writers studied the unique structure and format of the Quran in order to identify and apply the figurative devices and their impact on the reader. Quran's figurative devices
The Quran inspired musicality in poetry through the internal rhythm of the verses. The arrangement of words, how certain sounds create harmony, and the agreement of rhymes create the sense of rhythm within each verse. At times, the chapters of the Quran only have the rhythm in common. The repetition in the Quran introduced the true power and impact repetition can have in poetry. The repetition of certain words and phrases made them appear more firm and explicit in the Quran. The Quran uses constant metaphors of blindness and deafness to imply unbelief. Metaphors were not a new concept to poetry, however the strength of extended metaphors was. The explicit imagery in the Quran inspired many poets to include and focus on the feature in their own work. The poet ibn al-Mu'tazz wrote a book regarding the figures of speech inspired by his study of the Quran. | [
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Poet Badr Shakir al-Sayyab expresses his political opinion in his work through imagery inspired by the forms of more harsher imagery used in the Quran. The Quran uses figurative devices in order to express the meaning in the most beautiful form possible. The study of the pauses in the Quran as well as other rhetoric allow it to be approached in a multiple ways. Structure
Although the Quran is known for its fluency and harmony, the structure can be best described as not always being inherently chronological, but can also flow thematically instead (the chapters in the Quran have segments that flow in chronological order, however segments can transition into other segments not related in chronology, but could be related in topic). The suras, also known as chapters of the Quran, are not placed in chronological order. The only constant in their structure is that the longest are placed first and shorter ones follow. The topics discussed in the chapters can also have no direct relation to each other (as seen in many suras) and can share in their sense of rhyme. The Quran introduces to poetry the idea of abandoning order and scattering narratives throughout the text. Harmony is also present in the sound of the Quran. The elongations and accents present in the Quran create a harmonious flow within the writing. Unique sound of the Quran recited, due to the accents, create a deeper level of understanding through a deeper emotional connection. The Quran is written in a language that is simple and understandable by people. The simplicity of the writing inspired later poets to write in a more clear and clear-cut style. | [
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The words of the Quran, although unchanged, are to this day understandable and frequently used in both formal and informal Arabic. The simplicity of the language makes memorizing and reciting the Quran a slightly easier task. Culture and the Quran
The writer al-Khattabi explains how culture is a required element to create a sense of art in work as well as understand it. He believes that the fluency and harmony which the Quran possess are not the only elements that make it beautiful and create a bond between the reader and the text. While a lot of poetry was deemed comparable to the Quran in that it is equal to or better than the composition of the Quran, a debate rose that such statements are not possible because humans are incapable of composing work comparable to the Quran. Because the structure of the Quran made it difficult for a clear timeline to be seen, Hadith were the main source of chronological order. The Hadith were passed down from generation to generation and this tradition became a large resource for understanding the context. Poetry after the Quran began possessing this element of tradition by including ambiguity and background information to be required to understand the meaning. After the Quran came down to the people, the tradition of memorizing the verses became present. It is believed that the greater the amount of the Quran memorized, the greater the faith. As technology improved over time, hearing recitations of the Quran became more available as well as more tools to help memorize the verses. | [
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The tradition of Love Poetry served as a symbolic representation of a Muslim's desire for a closer contact with their Lord. While the influence of the Quran on Arabic poetry is explained and defended by numerous writers, some writers such as Al-Baqillani believe that poetry and the Quran are in no conceivable way related due to the uniqueness of the Quran. Poetry's imperfections prove his points that they cannot be compared with the fluency the Quran holds. Arabic and Islam
Classical Arabic is the language of poetry and literature (including news); it is also mainly the language of the Quran. Classical Arabic is closely associated with the religion of Islam because the Quran was written in it. Most of the world's Muslims do not speak Classical Arabic as their native language, but many can read the Quranic script and recite the Quran. Among non-Arab Muslims, translations of the Quran are most often accompanied by the original text. At present, Modern Standard Arabic (MSA) is also used in modernized versions of literary forms of the Quran. Some Muslims present a monogenesis of languages and claim that the Arabic language was the language revealed by God for the benefit of mankind and the original language as a prototype system of symbolic communication, based upon its system of triconsonantal roots, spoken by man from which all other languages were derived, having first been corrupted. Judaism has a similar account with the Tower of Babel. Dialects and descendants
Colloquial Arabic is a collective term for the spoken dialects of Arabic used throughout the Arab world, which differ radically from the literary language. | [
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The main dialectal division is between the varieties within and outside of the Arabian peninsula, followed by that between sedentary varieties and the much more conservative Bedouin varieties. All the varieties outside of the Arabian peninsula (which include the large majority of speakers) have many features in common with each other that are not found in Classical Arabic. This has led researchers to postulate the existence of a prestige koine dialect in the one or two centuries immediately following the Arab conquest, whose features eventually spread to all newly conquered areas. These features are present to varying degrees inside the Arabian peninsula. Generally, the Arabian peninsula varieties have much more diversity than the non-peninsula varieties, but these have been understudied. Within the non-peninsula varieties, the largest difference is between the non-Egyptian North African dialects (especially Moroccan Arabic) and the others. Moroccan Arabic in particular is hardly comprehensible to Arabic speakers east of Libya (although the converse is not true, in part due to the popularity of Egyptian films and other media). One factor in the differentiation of the dialects is influence from the languages previously spoken in the areas, which have typically provided a significant number of new words and have sometimes also influenced pronunciation or word order; however, a much more significant factor for most dialects is, as among Romance languages, retention (or change of meaning) of different classical forms. Thus Iraqi aku, Levantine fīh and North African kayən all mean 'there is', and all come from Classical Arabic forms (yakūn, fīhi, kā'in respectively), but now sound very different. | [
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Examples
Transcription is a broad IPA transcription, so minor differences were ignored for easier comparison. Also, the pronunciation of Modern Standard Arabic differs significantly from region to region. Koiné
According to Charles A. Ferguson, the following are some of the characteristic features of the koiné that underlies all the modern dialects outside the Arabian peninsula. Although many other features are common to most or all of these varieties, Ferguson believes that these features in particular are unlikely to have evolved independently more than once or twice and together suggest the existence of the koine:
Loss of the dual number except on nouns, with consistent plural agreement (cf. feminine singular agreement in plural inanimates). Change of a to i in many affixes (e.g., non-past-tense prefixes ti- yi- ni-; wi- 'and'; il- 'the'; feminine -it in the construct state). Loss of third-weak verbs ending in w (which merge with verbs ending in y). Reformation of geminate verbs, e.g., 'I untied' → . Conversion of separate words lī 'to me', laka 'to you', etc. into indirect-object clitic suffixes. Certain changes in the cardinal number system, e.g., 'five days' → , where certain words have a special plural with prefixed t.
Loss of the feminine elative (comparative). Adjective plurals of the form 'big' → . Change of nisba suffix > . Certain lexical items, e.g., 'bring' < 'come with'; 'see'; 'what' (or similar) < 'which thing'; (relative pronoun). Merger of and . | [
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Dialect groups
Egyptian Arabic is spoken by around 53 million people in Egypt (55 million worldwide). It is one of the most understood varieties of Arabic, due in large part to the widespread distribution of Egyptian films and television shows throughout the Arabic-speaking world
Levantine Arabic includes North Levantine Arabic, South Levantine Arabic and Cypriot Arabic. It is spoken by about 21 million people in Lebanon, Syria, Jordan, Palestine, Israel, Cyprus and Turkey. Lebanese Arabic is a variety of Levantine Arabic spoken primarily in Lebanon. Jordanian Arabic is a continuum of mutually intelligible varieties of Levantine Arabic spoken by the population of the Kingdom of Jordan. Palestinian Arabic is a name of several dialects of the subgroup of Levantine Arabic spoken by the Palestinians in Palestine, by Arab citizens of Israel and in most Palestinian populations around the world. Samaritan Arabic, spoken by only several hundred in the Nablus region
Cypriot Maronite Arabic, spoken in Cyprus
Maghrebi Arabic, also called "Darija" spoken by about 70 million people in Morocco, Algeria, Tunisia and Libya. It also forms the basis of Maltese via the extinct Sicilian Arabic dialect. Maghrebi Arabic is very hard to understand for Arabic speakers from the Mashriq or Mesopotamia, the most comprehensible being Libyan Arabic and the most difficult Moroccan Arabic. The others such as Algerian Arabic can be considered in between the two in terms of difficulty. Libyan Arabic spoken in Libya and neighboring countries. | [
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Tunisian Arabic spoken in Tunisia and North-eastern Algeria
Algerian Arabic spoken in Algeria
Judeo-Algerian Arabic was spoken by Jews in Algeria until 1962
Moroccan Arabic spoken in Morocco
Hassaniya Arabic (3 million speakers), spoken in Mauritania, Western Sahara, some parts of the Azawad in northern Mali, southern Morocco and south-western Algeria. Andalusian Arabic, spoken in Spain until the 16th century. Siculo-Arabic (Sicilian Arabic), was spoken in Sicily and Malta between the end of the 9th century and the end of the 12th century and eventually evolved into the Maltese language. Maltese, spoken on the island of Malta, is the only fully separate standardized language to have originated from an Arabic dialect (the extinct Siculo-Arabic dialect), with independent literary norms. Maltese has evolved independently of Modern Standard Arabic and its varieties into a standardized language over the past 800 years in a gradual process of Latinisation. Maltese is therefore considered an exceptional descendant of Arabic that has no diglossic relationship with Standard Arabic or Classical Arabic. Maltese is also different from Arabic and other Semitic languages since its morphology has been deeply influenced by Romance languages, Italian and Sicilian. It is also the only Semitic language written in the Latin script. In terms of basic everyday language, speakers of Maltese are reported to be able to understand less than a third of what is said to them in Tunisian Arabic, which is related to Siculo-Arabic, whereas speakers of Tunisian are able to understand about 40% of what is said to them in Maltese. This asymmetric intelligibility is considerably lower than the mutual intelligibility found between Maghrebi Arabic dialects. | [
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Maltese has its own dialects, with urban varieties of Maltese being closer to Standard Maltese than rural varieties. Mesopotamian Arabic, spoken by about 41.2 million people in Iraq (where it is called "Aamiyah"), eastern Syria and southwestern Iran (Khuzestan) and in the southeastern of Turkey (in the eastern Mediterranean, Southeastern Anatolia Region)
North Mesopotamian Arabic is a spoken north of the Hamrin Mountains in Iraq, in western Iran, northern Syria, and in southeastern Turkey (in the eastern Mediterranean Region, Southeastern Anatolia Region, and southern Eastern Anatolia Region). Judeo-Mesopotamian Arabic, also known as Iraqi Judeo Arabic and Yahudic, is a variety of Arabic spoken by Iraqi Jews of Mosul. Baghdad Arabic is the Arabic dialect spoken in Baghdad, and the surrounding cities and it is a subvariety of Mesopotamian Arabic. Baghdad Jewish Arabic is the dialect spoken by the Iraqi Jews of Baghdad. South Mesopotamian Arabic (Basrawi dialect) is the dialect spoken in southern Iraq, such as Basra, Dhi Qar and Najaf. Khuzestani Arabic is the dialect spoken in the Iranian province of Khuzestan. This dialect is a mix of Southen Mesopotamian Arabic and Gulf Arabic. Khorasani Arabic spoken in the Iranian province of Khorasan. Kuwaiti Arabic is a Gulf Arabic dialect spoken in Kuwait. Sudanese Arabic is spoken by 17 million people in Sudan and some parts of southern Egypt. Sudanese Arabic is quite distinct from the dialect of its neighbor to the north; rather, the Sudanese have a dialect similar to the Hejazi dialect. | [
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Juba Arabic spoken in South Sudan and southern Sudan
Gulf Arabic, spoken by around four million people, predominantly in Kuwait, Bahrain, some parts of Oman, eastern Saudi Arabia coastal areas and some parts of UAE and Qatar. Also spoken in Iran's Bushehr and Hormozgan provinces. Although Gulf Arabic is spoken in Qatar, most Qatari citizens speak Najdi Arabic (Bedawi). Omani Arabic, distinct from the Gulf Arabic of Eastern Arabia and Bahrain, spoken in Central Oman. With recent oil wealth and mobility has spread over other parts of the Sultanate. Hadhrami Arabic, spoken by around 8 million people, predominantly in Hadhramaut, and in parts of the Arabian Peninsula, South and Southeast Asia, and East Africa by Hadhrami descendants. Yemeni Arabic spoken in Yemen, and southern Saudi Arabia by 15 million people. Similar to Gulf Arabic. Najdi Arabic, spoken by around 10 million people, mainly spoken in Najd, central and northern Saudi Arabia. Most Qatari citizens speak Najdi Arabic (Bedawi). Hejazi Arabic (6 million speakers), spoken in Hejaz, western Saudi Arabia
Saharan Arabic spoken in some parts of Algeria, Niger and Mali
Baharna Arabic (600,000 speakers), spoken by Bahrani Shiʻah in Bahrain and Qatif, the dialect exhibits many big differences from Gulf Arabic. It is also spoken to a lesser extent in Oman. Judeo-Arabic dialects – these are the dialects spoken by the Jews that had lived or continue to live in the Arab World. As Jewish migration to Israel took hold, the language did not thrive and is now considered endangered. So-called Qəltu Arabic. | [
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Chadian Arabic, spoken in Chad, Sudan, some parts of South Sudan, Central African Republic, Niger, Nigeria, Cameroon
Central Asian Arabic, spoken in Uzbekistan, Tajikistan and Afghanistan, is highly endangered
Shirvani Arabic, spoken in Azerbaijan and Dagestan until the 1930s, now extinct. Phonology
History
Of the 29 Proto-Semitic consonants, only one has been lost: , which merged with , while became (see Semitic languages). Various other consonants have changed their sound too, but have remained distinct. An original lenited to , and – consistently attested in pre-Islamic Greek transcription of Arabic languages – became palatalized to or by the time of the Quran and , , or after early Muslim conquests and in MSA (see Arabic phonology#Local variations for more detail). An original voiceless alveolar lateral fricative became . Its emphatic counterpart was considered by Arabs to be the most unusual sound in Arabic (Hence the Classical Arabic's appellation or "language of the "); for most modern dialects, it has become an emphatic stop with loss of the laterality or with complete loss of any pharyngealization or velarization, . (The classical pronunciation of pharyngealization still occurs in the Mehri language, and the similar sound without velarization, , exists in other Modern South Arabian languages.) Other changes may also have happened. Classical Arabic pronunciation is not thoroughly recorded and different reconstructions of the sound system of Proto-Semitic propose different phonetic values. One example is the emphatic consonants, which are pharyngealized in modern pronunciations but may have been velarized in the eighth century and glottalized in Proto-Semitic. | [
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Reduction of and between vowels occurs in a number of circumstances and is responsible for much of the complexity of third-weak ("defective") verbs. Early Akkadian transcriptions of Arabic names shows that this reduction had not yet occurred as of the early part of the 1st millennium BC. The Classical Arabic language as recorded was a poetic koine that reflected a consciously archaizing dialect, chosen based on the tribes of the western part of the Arabian Peninsula, who spoke the most conservative variants of Arabic. Even at the time of Muhammed and before, other dialects existed with many more changes, including the loss of most glottal stops, the loss of case endings, the reduction of the diphthongs and into monophthongs , etc. Most of these changes are present in most or all modern varieties of Arabic. An interesting feature of the writing system of the Quran (and hence of Classical Arabic) is that it contains certain features of Muhammad's native dialect of Mecca, corrected through diacritics into the forms of standard Classical Arabic. Among these features visible under the corrections are the loss of the glottal stop and a differing development of the reduction of certain final sequences containing : Evidently, final became as in the Classical language, but final became a different sound, possibly (rather than again in the Classical language). This is the apparent source of the alif maqṣūrah 'restricted alif' where a final is reconstructed: a letter that would normally indicate or some similar high-vowel sound, but is taken in this context to be a logical variant of alif and represent the sound . | [
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Although Classical Arabic was a unitary language and is now used in Quran, its pronunciation varies somewhat from country to country and from region to region within a country. It is influenced by colloquial dialects. Literary Arabic
The "colloquial" spoken dialects of Arabic are learned at home and constitute the native languages of Arabic speakers. "Formal" Modern Standard Arabic is learned at school; although many speakers have a native-like command of the language, it is technically not the native language of any speakers. Both varieties can be both written and spoken, although the colloquial varieties are rarely written down and the formal variety is spoken mostly in formal circumstances, e.g., in radio and TV broadcasts, formal lectures, parliamentary discussions and to some extent between speakers of different colloquial dialects. Even when the literary language is spoken, however, it is normally only spoken in its pure form when reading a prepared text out loud and communication between speakers of different colloquial dialects. When speaking extemporaneously (i.e. making up the language on the spot, as in a normal discussion among people), speakers tend to deviate somewhat from the strict literary language in the direction of the colloquial varieties. | [
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In fact, there is a continuous range of "in-between" spoken varieties: from nearly pure Modern Standard Arabic (MSA), to a form that still uses MSA grammar and vocabulary but with significant colloquial influence, to a form of the colloquial language that imports a number of words and grammatical constructions in MSA, to a form that is close to pure colloquial but with the "rough edges" (the most noticeably "vulgar" or non-Classical aspects) smoothed out, to pure colloquial. The particular variant (or register) used depends on the social class and education level of the speakers involved and the level of formality of the speech situation. Often it will vary within a single encounter, e.g., moving from nearly pure MSA to a more mixed language in the process of a radio interview, as the interviewee becomes more comfortable with the interviewer. This type of variation is characteristic of the diglossia that exists throughout the Arabic-speaking world. Although Modern Standard Arabic (MSA) is a unitary language, its pronunciation varies somewhat from country to country and from region to region within a country. The variation in individual "accents" of MSA speakers tends to mirror corresponding variations in the colloquial speech of the speakers in question, but with the distinguishing characteristics moderated somewhat. It is important in descriptions of "Arabic" phonology to distinguish between pronunciation of a given colloquial (spoken) dialect and the pronunciation of MSA by these same speakers. Although they are related, they are not the same. | [
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For example, the phoneme that derives from Classical Arabic has many different pronunciations in the modern spoken varieties, e.g., including the proposed original . Speakers whose native variety has either or will use the same pronunciation when speaking MSA. Even speakers from Cairo, whose native Egyptian Arabic has , normally use when speaking MSA. The of Persian Gulf speakers is the only variant pronunciation which isn't found in MSA; is used instead, but may use [j] in MSA for comfortable pronunciation. Another reason of different pronunciations is influence of colloquial dialects. The differentiation of pronunciation of colloquial dialects is the influence from other languages previously spoken and some still presently spoken in the regions, such as Coptic in Egypt, Berber, Punic, or Phoenician in North Africa, Himyaritic, Modern South Arabian, and Old South Arabian in Yemen and Oman, and Aramaic and Canaanite languages (including Phoenician) in the Levant and Mesopotamia. Another example: Many colloquial varieties are known for a type of vowel harmony in which the presence of an "emphatic consonant" triggers backed allophones of nearby vowels (especially of the low vowels , which are backed to in these circumstances and very often fronted to in all other circumstances). In many spoken varieties, the backed or "emphatic" vowel allophones spread a fair distance in both directions from the triggering consonant; in some varieties (most notably Egyptian Arabic), the "emphatic" allophones spread throughout the entire word, usually including prefixes and suffixes, even at a distance of several syllables from the triggering consonant. | [
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Speakers of colloquial varieties with this vowel harmony tend to introduce it into their MSA pronunciation as well, but usually with a lesser degree of spreading than in the colloquial varieties. (For example, speakers of colloquial varieties with extremely long-distance harmony may allow a moderate, but not extreme, amount of spreading of the harmonic allophones in their MSA speech, while speakers of colloquial varieties with moderate-distance harmony may only harmonize immediately adjacent vowels in MSA.) Vowels
Modern Standard Arabic has six pure vowels (while most modern dialects have eight pure vowels which includes the long vowels ), with short and corresponding long vowels . There are also two diphthongs: and . The pronunciation of the vowels differs from speaker to speaker, in a way that tends to reflect the pronunciation of the corresponding colloquial variety. Nonetheless, there are some common trends. Most noticeable is the differing pronunciation of and , which tend towards fronted , or in most situations, but a back in the neighborhood of emphatic consonants. Some accents and dialects, such as those of the Hejaz region, have an open or a central in all situations. The vowel varies towards too. Listen to the final vowel in the recording of at the beginning of this article, for example. The point is, Arabic has only three short vowel phonemes, so those phonemes can have a very wide range of allophones. The vowels and are often affected somewhat in emphatic neighborhoods as well, with generally more back or centralized allophones, but the differences are less great than for the low vowels. | [
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The pronunciation of short and tends towards and , respectively, in many dialects. The definition of both "emphatic" and "neighborhood" vary in ways that reflect (to some extent) corresponding variations in the spoken dialects. Generally, the consonants triggering "emphatic" allophones are the pharyngealized consonants ; ; and , if not followed immediately by . Frequently, the fricatives also trigger emphatic allophones; occasionally also the pharyngeal consonants (the former more than the latter). Many dialects have multiple emphatic allophones of each vowel, depending on the particular nearby consonants. In most MSA accents, emphatic coloring of vowels is limited to vowels immediately adjacent to a triggering consonant, although in some it spreads a bit farther: e.g., 'time'; 'homeland'; 'downtown' (sometimes or similar). In a non-emphatic environment, the vowel in the diphthong is pronounced or : hence 'sword' but 'summer'. However, in accents with no emphatic allophones of (e.g., in the Hejaz), the pronunciation or occurs in all situations. Consonants
The phoneme is represented by the Arabic letter () and has many standard pronunciations. is characteristic of north Algeria, Iraq, and most of the Arabian peninsula but with an allophonic in some positions; occurs in most of the Levant and most of North Africa; and is used in most of Egypt and some regions in Yemen and Oman. Generally this corresponds with the pronunciation in the colloquial dialects. | [
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In some regions in Sudan and Yemen, as well as in some Sudanese and Yemeni dialects, it may be either or , representing the original pronunciation of Classical Arabic. Foreign words containing may be transcribed with , , , , , or , mainly depending on the regional spoken variety of Arabic or the commonly diacriticized Arabic letter. In northern Egypt, where the Arabic letter () is normally pronounced , a separate phoneme , which may be transcribed with , occurs in a small number of mostly non-Arabic loanwords, e.g., 'jacket'. () can be pronounced as . In some places of Maghreb it can be also pronounced as . and () are velar, post-velar, or uvular. In many varieties, () are epiglottal in Western Asia. is pronounced as velarized in الله , the name of God, q.e. Allah, when the word follows a, ā, u or ū (after i or ī it is unvelarized: bismi l–lāh ). Some speakers velarize other occurrences of in MSA, in imitation of their spoken dialects. The emphatic consonant was actually pronounced , or possibly —either way, a highly unusual sound. The medieval Arabs actually termed their language 'the language of the Ḍād' (the name of the letter used for this sound), since they thought the sound was unique to their language. (In fact, it also exists in a few other minority Semitic languages, e.g., Mehri.) | [
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Arabic has consonants traditionally termed "emphatic" (), which exhibit simultaneous pharyngealization as well as varying degrees of velarization (depending on the region), so they may be written with the "Velarized or pharyngealized" diacritic () as: . This simultaneous articulation is described as "Retracted Tongue Root" by phonologists. In some transcription systems, emphasis is shown by capitalizing the letter, for example, is written ; in others the letter is underlined or has a dot below it, for example, . Vowels and consonants can be phonologically short or long. Long (geminate) consonants are normally written doubled in Latin transcription (i.e. bb, dd, etc. ), reflecting the presence of the Arabic diacritic mark , which indicates doubled consonants. In actual pronunciation, doubled consonants are held twice as long as short consonants. This consonant lengthening is phonemically contrastive: 'he accepted' vs. 'he kissed'. Syllable structure
Arabic has two kinds of syllables: open syllables (CV) and (CVV)—and closed syllables (CVC), (CVVC) and (CVCC). The syllable types with two morae (units of time), i.e. CVC and CVV, are termed heavy syllables, while those with three morae, i.e. CVVC and CVCC, are superheavy syllables. | [
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Superheavy syllables in Classical Arabic occur in only two places: at the end of the sentence (due to pausal pronunciation) and in words such as 'hot', 'stuff, substance', 'they disputed with each other', where a long occurs before two identical consonants (a former short vowel between the consonants has been lost). (In less formal pronunciations of Modern Standard Arabic, superheavy syllables are common at the end of words or before clitic suffixes such as 'us, our', due to the deletion of final short vowels.) In surface pronunciation, every vowel must be preceded by a consonant (which may include the glottal stop ). There are no cases of hiatus within a word (where two vowels occur next to each other, without an intervening consonant). Some words do have an underlying vowel at the beginning, such as the definite article al- or words such as 'he bought', 'meeting'. When actually pronounced, one of three things happens:
If the word occurs after another word ending in a consonant, there is a smooth transition from final consonant to initial vowel, e.g., 'meeting' . If the word occurs after another word ending in a vowel, the initial vowel of the word is elided, e.g., 'house of the director' . If the word occurs at the beginning of an utterance, a glottal stop is added onto the beginning, e.g., 'The house is ...' . Stress
Word stress is not phonemically contrastive in Standard Arabic. It bears a strong relationship to vowel length. | [
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The basic rules for Modern Standard Arabic are:
A final vowel, long or short, may not be stressed. Only one of the last three syllables may be stressed. Given this restriction, the last heavy syllable (containing a long vowel or ending in a consonant) is stressed, if it is not the final syllable. If the final syllable is super heavy and closed (of the form CVVC or CVCC) it receives stress. If no syllable is heavy or super heavy, the first possible syllable (i.e. third from end) is stressed. As a special exception, in Form VII and VIII verb forms stress may not be on the first syllable, despite the above rules: Hence 'he subscribed' (whether or not the final short vowel is pronounced), 'he subscribes' (whether or not the final short vowel is pronounced), 'he should subscribe (juss.)'. Likewise Form VIII 'he bought', 'he buys'. Examples: 'book', 'writer', 'desk', 'desks', 'library' (but 'library' in short pronunciation), (Modern Standard Arabic) 'they wrote' = (dialect), (Modern Standard Arabic) 'they wrote it' = (dialect), (Modern Standard Arabic) 'they (dual, fem) wrote', (Modern Standard Arabic) 'I wrote' = (short form or dialect). Doubled consonants count as two consonants: 'magazine', "place". | [
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These rules may result in differently stressed syllables when final case endings are pronounced, vs. the normal situation where they are not pronounced, as in the above example of 'library' in full pronunciation, but 'library' in short pronunciation. The restriction on final long vowels does not apply to the spoken dialects, where original final long vowels have been shortened and secondary final long vowels have arisen from loss of original final -hu/hi. Some dialects have different stress rules. In the Cairo (Egyptian Arabic) dialect a heavy syllable may not carry stress more than two syllables from the end of a word, hence 'school', 'Cairo'. This also affects the way that Modern Standard Arabic is pronounced in Egypt. In the Arabic of Sanaa, stress is often retracted: 'two houses', 'their table', 'desks', 'sometimes', 'their school'. (In this dialect, only syllables with long vowels or diphthongs are considered heavy; in a two-syllable word, the final syllable can be stressed only if the preceding syllable is light; and in longer words, the final syllable cannot be stressed.) Levels of pronunciation
The final short vowels (e.g., the case endings -a -i -u and mood endings -u -a) are often not pronounced in this language, despite forming part of the formal paradigm of nouns and verbs. The following levels of pronunciation exist:
Full pronunciation with pausa
This is the most formal level actually used in speech. | [
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All endings are pronounced as written, except at the end of an utterance, where the following changes occur:
Final short vowels are not pronounced. (But possibly an exception is made for feminine plural -na and shortened vowels in the jussive/imperative of defective verbs, e.g., irmi! 'throw!'".) The entire indefinite noun endings -in and -un (with nunation) are left off. The ending -an is left off of nouns preceded by a tāʾ marbūṭah ة (i.e. the -t in the ending -at- that typically marks feminine nouns), but pronounced as -ā in other nouns (hence its writing in this fashion in the Arabic script). The tāʼ marbūṭah itself (typically of feminine nouns) is pronounced as h. (At least, this is the case in extremely formal pronunciation, e.g., some Quranic recitations. In practice, this h is usually omitted.) Formal short pronunciation
This is a formal level of pronunciation sometimes seen. It is somewhat like pronouncing all words as if they were in pausal position (with influence from the colloquial varieties). The following changes occur:
Most final short vowels are not pronounced. However, the following short vowels are pronounced:
feminine plural -na
shortened vowels in the jussive/imperative of defective verbs, e.g., irmi! 'throw!' second-person singular feminine past-tense -ti and likewise anti 'you (fem. sg.)' sometimes, first-person singular past-tense -tu
sometimes, second-person masculine past-tense -ta and likewise anta 'you (masc. sg.)' | [
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final -a in certain short words, e.g., laysa 'is not', sawfa (future-tense marker)
The nunation endings -an -in -un are not pronounced. However, they are pronounced in adverbial accusative formations, e.g., تَقْرِيبًا 'almost, approximately', عَادَةً 'usually'. The tāʾ marbūṭah ending ة is unpronounced, except in construct state nouns, where it sounds as t (and in adverbial accusative constructions, e.g., عَادَةً 'usually', where the entire -tan is pronounced). The masculine singular nisbah ending is actually pronounced and is unstressed (but plural and feminine singular forms, i.e. when followed by a suffix, still sound as ). Full endings (including case endings) occur when a clitic object or possessive suffix is added (e.g., 'us/our'). Informal short pronunciation
This is the pronunciation used by speakers of Modern Standard Arabic in extemporaneous speech, i.e. when producing new sentences rather than simply reading a prepared text. It is similar to formal short pronunciation except that the rules for dropping final vowels apply even when a clitic suffix is added. Basically, short-vowel case and mood endings are never pronounced and certain other changes occur that echo the corresponding colloquial pronunciations. Specifically:
All the rules for formal short pronunciation apply, except as follows. The past tense singular endings written formally as -tu -ta -ti are pronounced -t -t -ti. But masculine is pronounced in full. Unlike in formal short pronunciation, the rules for dropping or modifying final endings are also applied when a clitic object or possessive suffix is added (e.g., 'us/our'). | [
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If this produces a sequence of three consonants, then one of the following happens, depending on the speaker's native colloquial variety:
A short vowel (e.g., -i- or -ǝ-) is consistently added, either between the second and third or the first and second consonants. Or, a short vowel is added only if an otherwise unpronounceable sequence occurs, typically due to a violation of the sonority hierarchy (e.g., -rtn- is pronounced as a three-consonant cluster, but -trn- needs to be broken up). Or, a short vowel is never added, but consonants like r l m n occurring between two other consonants will be pronounced as a syllabic consonant (as in the English words "butter bottle bottom button"). When a doubled consonant occurs before another consonant (or finally), it is often shortened to a single consonant rather than a vowel added. (However, Moroccan Arabic never shortens doubled consonants or inserts short vowels to break up clusters, instead tolerating arbitrary-length series of arbitrary consonants and hence Moroccan Arabic speakers are likely to follow the same rules in their pronunciation of Modern Standard Arabic.) The clitic suffixes themselves tend also to be changed, in a way that avoids many possible occurrences of three-consonant clusters. In particular, -ka -ki -hu generally sound as -ak -ik -uh. Final long vowels are often shortened, merging with any short vowels that remain. Depending on the level of formality, the speaker's education level, etc., various grammatical changes may occur in ways that echo the colloquial variants:
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masculine plural nominative -ūn vs. oblique -īn) will be leveled, with the oblique form used everywhere. (However, in words like 'father' and 'brother' with special long-vowel case endings in the construct state, the nominative is used everywhere, hence 'father of', 'brother of'.) Feminine plural endings in verbs and clitic suffixes will often drop out, with the masculine plural endings used instead. If the speaker's native variety has feminine plural endings, they may be preserved, but will often be modified in the direction of the forms used in the speaker's native variety, e.g. -an instead of -na. Dual endings will often drop out except on nouns and then used only for emphasis (similar to their use in the colloquial varieties); elsewhere, the plural endings are used (or feminine singular, if appropriate). Colloquial varieties
Vowels
As mentioned above, many spoken dialects have a process of emphasis spreading, where the "emphasis" (pharyngealization) of emphatic consonants spreads forward and back through adjacent syllables, pharyngealizing all nearby consonants and triggering the back allophone in all nearby low vowels. The extent of emphasis spreading varies. For example, in Moroccan Arabic, it spreads as far as the first full vowel (i.e. sound derived from a long vowel or diphthong) on either side; in many Levantine dialects, it spreads indefinitely, but is blocked by any or ; while in Egyptian Arabic, it usually spreads throughout the entire word, including prefixes and suffixes. In Moroccan Arabic, also have emphatic allophones and , respectively. | [
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Unstressed short vowels, especially , are deleted in many contexts. Many sporadic examples of short vowel change have occurred (especially → and interchange ↔). Most Levantine dialects merge short /i u/ into in most contexts (all except directly before a single final consonant). In Moroccan Arabic, on the other hand, short triggers labialization of nearby consonants (especially velar consonants and uvular consonants), and then short /a i u/ all merge into , which is deleted in many contexts. (The labialization plus is sometimes interpreted as an underlying phoneme .) This essentially causes the wholesale loss of the short-long vowel distinction, with the original long vowels remaining as half-long , phonemically , which are used to represent both short and long vowels in borrowings from Literary Arabic. Most spoken dialects have monophthongized original to in most circumstances, including adjacent to emphatic consonants, while keeping them as the original diphthongs in others e.g. . In most of the Moroccan, Algerian and Tunisian (except Sahel and Southeastern) Arabic dialects, they have subsequently merged into original . Consonants
In most dialects, there may be more or fewer phonemes than those listed in the chart above. For example, is considered a native phoneme in most Arabic dialects except in Levantine dialects like Syrian or Lebanese where is pronounced and is pronounced . or () is considered a native phoneme in most dialects except in Egyptian and a number of Yemeni and Omani dialects where is pronounced . | [
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or and are distinguished in the dialects of Egypt, Sudan, the Levant and the Hejaz, but they have merged as in most dialects of the Arabian Peninsula, Iraq and Tunisia and have merged as in Morocco and Algeria. The usage of non-native and depends on the usage of each speaker but they might be more prevalent in some dialects than others. The Iraqi and Gulf Arabic also has the sound and writes it and with the Persian letters and , as in "plum"; "truffle". Early in the expansion of Arabic, the separate emphatic phonemes and coalesced into a single phoneme . Many dialects (such as Egyptian, Levantine, and much of the Maghreb) subsequently lost fricatives, converting into . Most dialects borrow "learned" words from the Standard language using the same pronunciation as for inherited words, but some dialects without interdental fricatives (particularly in Egypt and the Levant) render original in borrowed words as . Another key distinguishing mark of Arabic dialects is how they render the original velar and uvular plosives , (Proto-Semitic ), and :
retains its original pronunciation in widely scattered regions such as Yemen, Morocco, and urban areas of the Maghreb. It is pronounced as a glottal stop in several prestige dialects, such as those spoken in Cairo, Beirut and Damascus. But it is rendered as a voiced velar plosive in Persian Gulf, Upper Egypt, parts of the Maghreb, and less urban parts of the Levant (e.g. Jordan). | [
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In Iraqi Arabic it sometimes retains its original pronunciation and is sometimes rendered as a voiced velar plosive, depending on the word. Some traditionally Christian villages in rural areas of the Levant render the sound as , as do Shiʻi Bahrainis. In some Gulf dialects, it is palatalized to or . It is pronounced as a voiced uvular constrictive in Sudanese Arabic. Many dialects with a modified pronunciation for maintain the pronunciation in certain words (often with religious or educational overtones) borrowed from the Classical language. is pronounced as an affricate in Iraq and much of the Arabian Peninsula but is pronounced in most of North Egypt and parts of Yemen and Oman, in Morocco, Tunisia, and the Levant, and , in most words in much of the Persian Gulf. usually retains its original pronunciation but is palatalized to in many words in Israel and the Palestinian Territories, Iraq, and countries in the eastern part of the Arabian Peninsula. Often a distinction is made between the suffixes ('you', masc.) and ('you', fem. ), which become and , respectively. In Sana'a, Omani, and Bahrani is pronounced . Pharyngealization of the emphatic consonants tends to weaken in many of the spoken varieties, and to spread from emphatic consonants to nearby sounds. In addition, the "emphatic" allophone automatically triggers pharyngealization of adjacent sounds in many dialects. As a result, it may difficult or impossible to determine whether a given coronal consonant is phonemically emphatic or not, especially in dialects with long-distance emphasis spreading. | [
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(A notable exception is the sounds vs. in Moroccan Arabic, because the former is pronounced as an affricate but the latter is not.) Grammar
Literary Arabic
As in other Semitic languages, Arabic has a complex and unusual morphology (i.e. method of constructing words from a basic root). Arabic has a nonconcatenative "root-and-pattern" morphology: A root consists of a set of bare consonants (usually three), which are fitted into a discontinuous pattern to form words. For example, the word for 'I wrote' is constructed by combining the root 'write' with the pattern 'I Xed' to form 'I wrote'. Other verbs meaning 'I Xed' will typically have the same pattern but with different consonants, e.g. 'I read', 'I ate', 'I went', although other patterns are possible (e.g. 'I drank', 'I said', 'I spoke', where the subpattern used to signal the past tense may change but the suffix is always used). | [
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From a single root , numerous words can be formed by applying different patterns:
كَتَبْتُ 'I wrote'
كَتَّبْتُ 'I had (something) written'
كَاتَبْتُ 'I corresponded (with someone)'
أَكْتَبْتُ 'I dictated'
اِكْتَتَبْتُ 'I subscribed'
تَكَاتَبْنَا 'we corresponded with each other'
أَكْتُبُ 'I write'
أُكَتِّبُ 'I have (something) written'
أُكَاتِبُ 'I correspond (with someone)'
أُكْتِبُ 'I dictate'
أَكْتَتِبُ 'I subscribe'
نَتَكَتِبُ 'we correspond each other'
كُتِبَ 'it was written'
أُكْتِبَ 'it was dictated'
مَكْتُوبٌ 'written'
مُكْتَبٌ 'dictated'
كِتَابٌ 'book'
كُتُبٌ 'books'
كَاتِبٌ 'writer'
كُتَّابٌ 'writers'
مَكْتَبٌ 'desk, office'
مَكْتَبَةٌ 'library, bookshop'
etc. Nouns and adjectives
Nouns in Literary Arabic have three grammatical cases (nominative, accusative, and genitive [also used when the noun is governed by a preposition]); three numbers (singular, dual and plural); two genders (masculine and feminine); and three "states" (indefinite, definite, and construct). The cases of singular nouns (other than those that end in long ā) are indicated by suffixed short vowels (/-u/ for nominative, /-a/ for accusative, /-i/ for genitive). The feminine singular is often marked by ـَة /-at/, which is pronounced as /-ah/ before a pause. Plural is indicated either through endings (the sound plural) or internal modification (the broken plural). Definite nouns include all proper nouns, all nouns in "construct state" and all nouns which are prefixed by the definite article اَلْـ /al-/. | [
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Indefinite singular nouns (other than those that end in long ā) add a final /-n/ to the case-marking vowels, giving /-un/, /-an/ or /-in/ (which is also referred to as nunation or tanwīn). Adjectives in Literary Arabic are marked for case, number, gender and state, as for nouns. However, the plural of all non-human nouns is always combined with a singular feminine adjective, which takes the ـَة /-at/ suffix. Pronouns in Literary Arabic are marked for person, number and gender. There are two varieties, independent pronouns and enclitics. Enclitic pronouns are attached to the end of a verb, noun or preposition and indicate verbal and prepositional objects or possession of nouns. The first-person singular pronoun has a different enclitic form used for verbs (ـنِي /-nī/) and for nouns or prepositions (ـِي /-ī/ after consonants, ـيَ /-ya/ after vowels). Nouns, verbs, pronouns and adjectives agree with each other in all respects. However, non-human plural nouns are grammatically considered to be feminine singular. Furthermore, a verb in a verb-initial sentence is marked as singular regardless of its semantic number when the subject of the verb is explicitly mentioned as a noun. Numerals between three and ten show "chiasmic" agreement, in that grammatically masculine numerals have feminine marking and vice versa. Verbs
Verbs in Literary Arabic are marked for person (first, second, or third), gender, and number. | [
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They are conjugated in two major paradigms (past and non-past); two voices (active and passive); and six moods (indicative, imperative, subjunctive, jussive, shorter energetic and longer energetic), the fifth and sixth moods, the energetics, exist only in Classical Arabic but not in MSA. There are also two participles (active and passive) and a verbal noun, but no infinitive. The past and non-past paradigms are sometimes also termed perfective and imperfective, indicating the fact that they actually represent a combination of tense and aspect. The moods other than the indicative occur only in the non-past, and the future tense is signaled by prefixing سَـ or سَوْفَ onto the non-past. The past and non-past differ in the form of the stem (e.g., past كَتَبـ vs. non-past ـكْتُبـ ), and also use completely different sets of affixes for indicating person, number and gender: In the past, the person, number and gender are fused into a single suffixal morpheme, while in the non-past, a combination of prefixes (primarily encoding person) and suffixes (primarily encoding gender and number) are used. The passive voice uses the same person/number/gender affixes but changes the vowels of the stem. The following shows a paradigm of a regular Arabic verb, كَتَبَ 'to write'. In Modern Standard, the energetic mood (in either long or short form, which have the same meaning) is almost never used. | [
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Derivation
Like other Semitic languages, and unlike most other languages, Arabic makes much more use of nonconcatenative morphology (applying many templates applied roots) to derive words than adding prefixes or suffixes to words. For verbs, a given root can occur in many different derived verb stems (of which there are about fifteen), each with one or more characteristic meanings and each with its own templates for the past and non-past stems, active and passive participles, and verbal noun. These are referred to by Western scholars as "Form I", "Form II", and so on through "Form XV" (although Forms XI to XV are rare). These stems encode grammatical functions such as the causative, intensive and reflexive. Stems sharing the same root consonants represent separate verbs, albeit often semantically related, and each is the basis for its own conjugational paradigm. As a result, these derived stems are part of the system of derivational morphology, not part of the inflectional system. Examples of the different verbs formed from the root كتب 'write' (using حمر 'red' for Form IX, which is limited to colors and physical defects):
Form II is sometimes used to create transitive denominative verbs (verbs built from nouns); Form V is the equivalent used for intransitive denominatives. The associated participles and verbal nouns of a verb are the primary means of forming new lexical nouns in Arabic. | [
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This is similar to the process by which, for example, the English gerund "meeting" (similar to a verbal noun) has turned into a noun referring to a particular type of social, often work-related event where people gather together to have a "discussion" (another lexicalized verbal noun). Another fairly common means of forming nouns is through one of a limited number of patterns that can be applied directly to roots, such as the "nouns of location" in ma- (e.g. 'desk, office' < 'write', 'kitchen' < 'cook'). The only three genuine suffixes are as follows:
The feminine suffix -ah; variously derives terms for women from related terms for men, or more generally terms along the same lines as the corresponding masculine, e.g. 'library' (also a writing-related place, but different from , as above). The nisbah suffix -iyy-. This suffix is extremely productive, and forms adjectives meaning "related to X". It corresponds to English adjectives in -ic, -al, -an, -y, -ist, etc. The feminine nisbah suffix -iyyah. This is formed by adding the feminine suffix -ah onto nisba adjectives to form abstract nouns. For example, from the basic root 'share' can be derived the Form VIII verb 'to cooperate, participate', and in turn its verbal noun 'cooperation, participation' can be formed. This in turn can be made into a nisbah adjective 'socialist', from which an abstract noun 'socialism' can be derived. Other recent formations are 'republic' (lit. | [
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"public-ness", < 'multitude, general public'), and the Gaddafi-specific variation 'people's republic' (lit. "masses-ness", < 'the masses', pl. of , as above). Colloquial varieties
The spoken dialects have lost the case distinctions and make only limited use of the dual (it occurs only on nouns and its use is no longer required in all circumstances). They have lost the mood distinctions other than imperative, but many have since gained new moods through the use of prefixes (most often /bi-/ for indicative vs. unmarked subjunctive). They have also mostly lost the indefinite "nunation" and the internal passive. The following is an example of a regular verb paradigm in Egyptian Arabic. Writing system
The Arabic alphabet derives from the Aramaic through Nabatean, to which it bears a loose resemblance like that of Coptic or Cyrillic scripts to Greek script. Traditionally, there were several differences between the Western (North African) and Middle Eastern versions of the alphabet—in particular, the faʼ had a dot underneath and qaf a single dot above in the Maghreb, and the order of the letters was slightly different (at least when they were used as numerals). However, the old Maghrebi variant has been abandoned except for calligraphic purposes in the Maghreb itself, and remains in use mainly in the Quranic schools (zaouias) of West Africa. Arabic, like all other Semitic languages (except for the Latin-written Maltese, and the languages with the Ge'ez script), is written from right to left. | [
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