Algorithm

In mathematics, computer science, and related subjects, an 'algorithm' is an effective method for solving a problem expressed as a finite sequence of instructions. Algorithms are used for calculation, data processing, and many other fields. (In more advanced or abstract settings, the instructions do not necessarily constitute a finite sequence, and even not necessarily a sequence; see, e.g., "nondeterministic algorithm".)

Each algorithm is a list of well-defined instructions for completing a task. Starting from an initial state, the instructions describe a computation that proceeds through a well-defined series of successive states, eventually terminating in a final ending state. The transition from one state to the next is not necessarily deterministic; some algorithms, known as randomized algorithms, incorporate randomness.

A partial formalization of the concept began with attempts to solve the Entscheidungsproblem (the "decision problem") posed by David Hilbert in 1928. Subsequent formalizations were framed as attempts to define "effective calculability"^[1] or "effective method";^[2] those formalizations included the Gödel–Herbrand–Kleene recursive functions of 1930, 1934 and 1935, Alonzo Church's lambda calculus of 1936, Emil Post's "Formulation 1" of 1936, and Alan Turing's Turing machines of 1936–7 and 1939.

The adjective "continuous" when applied to the word "algorithm" can mean: 1) 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 2) An algorithm in the form of a differential equation that operates continuously on the data, running on an analog computer.^[3]

 Etymology

Al-Khwārizmī (in Arabic : خوارزمی), Muslim astronomer and mathematician, wrote a treatise in the Arabic language in 825 AD, On Calculation with Hindu–Arabic numeral system. (See algorism). It was translated from Arabic into Latin in the 12th century as Algoritmi de numero Indorum (al-Daffa 1977), whose title is supposedly likely intended to mean "Algoritmi on the numbers of the Indians", where "Algoritmi" was the translator's rendition of the author's name; but people misunderstanding the title treated Algoritmi as a Latin plural and this led to the word "algorithm" (Latin algorismus) coming to mean "calculation method". The intrusive "th" is most likely due to a false cognate with the Greek ἀριθμός (arithmos) meaning "numbers".

 Why algorithms are necessary: an informal definition

For a detailed presentation of the various points of view around the definition of "algorithm" see Algorithm characterizations. For examples of simple addition algorithms specified in the detailed manner described in Algorithm characterizations, see Algorithm examples.

While there is no generally accepted formal definition of "algorithm," an informal definition could be "a process that performs some sequence of operations." For some people, a program is only an algorithm if it stops eventually. For others, a program is only an algorithm if it stops before a given number of calculation steps.

A prototypical example of an algorithm is Euclid's algorithm to determine the maximum common divisor of two integers.

We can derive clues to the issues involved and an informal meaning of the word from the following quotation from Boolos & Jeffrey (1974, 1999) (boldface added):

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^[4]

The term "enumerably infinite" means "countable using integers perhaps extending to infinity." 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 chosen from 0 to infinity. Thus we might expect an algorithm to be an algebraic equation such as y = m + n — two arbitrary "input variables" m and n that produce an output y. As we see in Algorithm characterizations — the word algorithm implies much more than this, something on the order of (for our addition example):

Precise instructions (in 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 munch arbitrary input integers/symbols m and n, symbols + and = ... and (reliably, correctly, "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. That notion 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 with our 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.

 Formalization

Algorithms are essential to the way computers process information. Many computer programs contain algorithms that specify 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".^[5]

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".^[6]

Typically, when an algorithm is associated with processing information, data is read from an input source, written to an output device, and/or stored for further processing. Stored data is 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 any such computational process, the algorithm must be rigorously defined: specified in the way it applies in all possible circumstances that could arise. That is, 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 will always be critical 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, this discussion of the formalization of an algorithm has assumed the premises of imperative programming. This is the most common conception, and it attempts to describe a task in discrete, "mechanical" means. Unique to this conception of formalized algorithms is the assignment operation, setting the value of a variable. It derives from the intuition of "memory" as a scratchpad. There is an example below of such an assignment.

For some alternate conceptions of what constitutes an algorithm see functional programming and logic programming .

 Termination

Some writers restrict the definition of algorithm to procedures that eventually finish. In such a category Kleene places the "decision procedure or decision method or algorithm for the question".^[7] Others, including Kleene, include procedures that could run forever without stopping; such a procedure has been called a "computational method"^[8] or "calculation procedure or algorithm (and hence a calculation problem) in relation to a general question which requires for an answer, not yes or no, but the exhibiting of some object".^[9]

Minsky makes the pertinent observation, in regards to determining whether an algorithm will eventually terminate (from a particular starting state):

But if the length of the process isn't known in advance, then "trying" it may not be decisive, because if the process does go on forever—then at no time will we ever be sure of the answer.^[5]

As it happens, no other method can do any better, as was shown by Alan Turing with his celebrated result on the undecidability of the so-called halting problem. There is no algorithmic procedure for determining of arbitrary algorithms whether or not they terminate from given starting states. The analysis of algorithms for their likelihood of termination is called termination analysis.

See the examples of (im-)"proper" subtraction at partial function for more about what can happen when an algorithm fails for certain of its input numbers—e.g., (i) non-termination, (ii) production of "junk" (output in the wrong format to be considered a number) or no number(s) at all (halt ends the computation with no output), (iii) wrong number(s), or (iv) a combination of these. Kleene proposed that the production of "junk" or failure to produce a number is solved by having the algorithm detect these instances and produce e.g., an error message (he suggested "0"), or preferably, force the algorithm into an endless loop.^[10] Davis (1958) does this to his subtraction algorithm—he fixes his algorithm in a second example so that it is proper subtraction and it terminates.^[11] Along with the logical outcomes "true" and "false" Kleene (1952) also proposes the use of a third logical symbol "u" — undecided^[12] — thus an algorithm will always produce something when confronted with a "proposition". The problem of wrong answers must be solved with an independent "proof" of the algorithm e.g., using induction:

We normally require auxiliary evidence for this [that the algorithm correctly defines a mu recursive function], e.g., in the form of an inductive proof that, for each argument value, the computation terminates with a unique value.^[13]

 Expressing algorithms

Algorithms can be expressed in many kinds of notation, including natural languages, pseudocode, flowcharts, 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 and control tables are structured ways to express algorithms that avoid many of the ambiguities common in natural language statements, while remaining independent of a particular implementation language. Programming languages are primarily intended for expressing algorithms in a form that can be executed by a computer, but are 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 more at finite state machine and state transition table), as flowcharts (see more at state diagram), or as a form of rudimentary machine code or assembly code called "sets of quadruples" (see more at Turing machine).

Sometimes it is helpful in the description of an algorithm to supplement small "flow charts" (state diagrams) with natural-language and/or arithmetic expressions written inside "block diagrams" to summarize what the "flow charts" are accomplishing.

Representations of algorithms are generally classed into three accepted levels of Turing machine description:^[14]

• 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."

• 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 Algorithm examples.

 Computer algorithms

In computer systems, an algorithm is basically an instance of logic written in software by software developers to be effective for the intended "target" computer(s), in order for the software on the target machines to do something. For instance, if a person is writing software that is supposed to print out a PDF document located at the operating system folder "/My Documents" at computer drive "D:" every Friday at 10 pm, they will write an algorithm that specifies the following actions: "If today's date (computer time) is 'Friday,' open the document at 'D:/My Documents' and call the 'print' function". While this simple algorithm does not look into whether the printer has enough paper or whether the document has been moved into a different location, one can make this algorithm more robust and anticipate these problems by rewriting it as a formal CASE statement^[15] or as a (carefully crafted) sequence of IF-THEN-ELSE statements.^[16] For example the CASE statement might appear as follows (there are other possibilities):

CASE 1: IF today's date is NOT Friday THEN exit this CASE instruction ELSE

CASE 2: IF today's date is Friday AND the document is located at 'D:/My Documents' AND there is paper in the printer THEN print the document (and exit this CASE instruction) ELSE

CASE 3: IF today's date is Friday AND the document is NOT located at 'D:/My Documents' THEN display 'document not found' error message (and exit this CASE instruction) ELSE

CASE 4: IF today's date is Friday AND the document is located at 'D:/My Documents' AND there is NO paper in the printer THEN (i) display 'out of paper' error message and (ii) exit.

Note that CASE 3 includes two possibilities: (i) the document is NOT located at 'D:/My Documents' AND there's paper in the printer OR (ii) the document is NOT located at 'D:/My Documents' AND there's NO paper in the printer.

The sequence of IF-THEN-ELSE tests might look like this:

TEST 1: IF today's date is NOT Friday THEN done ELSE TEST 2:

TEST 2: IF the document is NOT located at 'D:/My Documents' THEN display 'document not found' error message ELSE TEST 3:

TEST 3: IF there is NO paper in the printer THEN display 'out of paper' error message ELSE print the document.

These examples' logic grants precedence to the instance of "NO document at 'D:/My Documents' ". Also observe that in a well-crafted CASE statement or sequence of IF-THEN-ELSE statements the number of distinct actions—4 in these examples: do nothing, print the document, display 'document not found', display 'out of paper' – equals the number of cases.

Given unlimited memory, a computational machine with the ability to execute either a set of CASE statements or a sequence of IF-THEN-ELSE statements is Turing complete. Therefore, anything that is computable can be computed by this machine. This form of algorithm is fundamental to computer programming in all its forms (see more at McCarthy formalism).

 Implementation

Most algorithms are intended to be implemented as computer programs. 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.

 An animation of the quicksort algorithm sorting an array of randomized values. The red bars mark the pivot element; at the start of the animation, the element farthest to the right hand side is chosen as the pivot.

One of the simplest algorithms is to find the largest number in an (unsorted) list of numbers. The solution necessarily requires looking at every number in the list, but only once at each. From this follows a simple algorithm, which can be stated in a high-level description English prose, as:

High-level description:

1. Assume the first item is largest.
2. Look at each of the remaining items in the list and if it is larger than the largest item so far, make a note of it.
3. The last noted item is the largest in the list when the process is complete.

(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:

Algorithm LargestNumber

  Input: A non-empty list of numbers L.

  Output: The largest number in the list L.

  largest ← L₀

  for each item in the list (Length(L)≥1), do

    if the item > largest, then

      largest ← the item

  return largest

• "←" is a loose shorthand for "changes to". For instance, "largest ← item" means that the value of largest changes to the value of item.
• "return" terminates the algorithm and outputs the value that follows.

For a more complex example of an algorithm, see Euclid's algorithm for the greatest common divisor, one of the earliest algorithms known.

 Algorithmic analysis

Main article: Analysis of algorithms

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, the algorithm above has a time requirement of O(n), using the big O notation with n as the length of the list. At all times the algorithm only needs to remember two values: the largest number found 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 will usually outperform a brute force sequential search when used for table lookups on sorted lists.

 Formal versus empirical

Main articles: Empirical algorithmics, Profiling (computer programming), and Program optimization

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.

Empirical testing is useful because it may uncover unexpected interactions that affect performance. For instance an algorithm that has no locality of reference may have much poorer performance than predicted because it 'thrashes the cache'. Benchmarks may be used to compare before/after potential improvements to an algorithm after program optimization.

 Classification

There are various ways to classify algorithms, each with its own merits.

 By implementation

One way to classify algorithms is by implementation means.^{[original research?]}

• Recursion or iteration: A recursive algorithm is one that invokes (makes reference to) itself repeatedly until a certain 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 in 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.^[17] 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 has a well defined change in the algorithm.
• Serial or 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. 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 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. 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 close to the true solution. Approximation may use either a deterministic or a random strategy. Such algorithms have practical value for many hard problems.

 By design paradigm

Another^{[original research?]} 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 will include many different types of algorithms. Some commonly found paradigms include:

• Brute-force or exhaustive search. This is the naïve method of trying every possible solution to see which is best.^[18]
• 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, that 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 will be more complex than decrease and conquer algorithms. An example of decrease and conquer algorithm is the binary search algorithm.
• Dynamic programming. When a problem shows optimal substructure, 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, 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. 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, but the difference is that solutions to the subproblems do not have to be known at each stage; instead a "greedy" choice can be made of what looks best for the moment. The greedy method extends the solution with the best possible decision (not all feasible decisions) at an algorithmic stage based on the current local optimum and the best decision (not all possible decisions) made in a previous stage. It is not exhaustive, and does not give accurate answer to many problems. But when it works, it will be the fastest method. The most popular greedy algorithm is finding the minimal spanning tree as given by Kruskal.
• Linear programming. When solving a problem using linear programming, specific inequalities involving the inputs are found and then an attempt is made to maximize (or minimize) some linear function of the inputs. Many problems (such as the maximum flow for directed graphs) can be stated in a linear programming way, and then be solved by a 'generic' algorithm such as the simplex algorithm. A more complex variant of linear programming is called integer programming, where the solution space is restricted to the integers.
• Reduction. 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.
• 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. This category also includes search algorithms, branch and bound enumeration and backtracking.

1. Randomized algorithms are those that make some choices randomly (or pseudo-randomly); for some problems, it can in fact be proven that the fastest solutions must involve some randomness. There are two large classes of such algorithms:
1. Monte Carlo algorithms return a correct answer with high-probability. E.g. RP is the subclass of these that run in polynomial time)
2. Las Vegas algorithms always return the correct answer, but their running time is only probabilistically bound, e.g. ZPP.
2. In optimization problems, heuristic algorithms do not try to find an optimal solution, but an approximate solution where the time or resources are limited. They are not practical to find perfect solutions. An example of this would be local search, tabu search, or simulated annealing algorithms, a class of heuristic probabilistic algorithms that vary the solution of a problem by a random amount. The name "simulated annealing" alludes to the metallurgic term meaning the heating and cooling of metal to achieve freedom from defects. The purpose of the random variance is to find close to globally optimal solutions rather than simply locally optimal ones, the idea being that the random element will be decreased as the algorithm settles down to a solution. Approximation algorithms are those heuristic algorithms that additionally provide some bounds on the error. Genetic algorithms attempt to find solutions to problems by mimicking biological evolutionary processes, with a cycle of random mutations yielding successive generations of "solutions". Thus, they emulate reproduction and "survival of the fittest". In genetic programming, this approach is extended to algorithms, by regarding the algorithm itself as a "solution" to a problem.

 By field of study

See also: List of algorithms

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, 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

See also: Complexity class and Parameterized complexity

Algorithms can be classified by the amount of time they need to complete compared to their input size. There is a wide variety: some algorithms complete in linear time relative to input size, some do so in an exponential amount of time or even worse, and some never halt. Additionally, 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.

 By computing power

Another^{[dubious – discuss]} way to classify algorithms is by computing power. This is typically done by considering some collection (class) of algorithms. A recursive class of algorithms is one that includes algorithms for all Turing computable functions. Looking at classes of algorithms allows for the possibility of restricting the available computational resources (time and memory) used in a computation. A subrecursive class of algorithms is one in which not all Turing computable functions can be obtained. For example, the algorithms that run in polynomial time suffice for many important types of computation but do not exhaust all Turing computable functions. The class of algorithms implemented by primitive recursive functions is another subrecursive class.

Burgin (2005, p. 24) uses a generalized definition of algorithms that relaxes the common requirement that the output of the algorithm that computes a function must be determined after a finite number of steps. He defines a super-recursive class of algorithms as "a class of algorithms in which it is possible to compute functions not computable by any Turing machine" (Burgin 2005, p. 107). This is closely related to the study of methods of hypercomputation.

Notes

1. Kleene 1943 in Davis 1965:274
2. Rosser 1939 in Davis 1965:225
3. Adaptation and learning in automatic systems, page 54, Ya. Z. Tsypkin, Z. J. Nikolic, Academic Press, 1971, ISBN 978-0-12-702050-1
4. Boolos and Jeffrey 1974,1999:19
5. Minsky 1967:105
6. Gurevich 2000:1, 3
7. Kleene 1952:136
8. Knuth 1997:5
9. Boldface added, Kleene 1952:137
10. Kleene 1952:325
11. Davis 1958:12–15
12. Kleene 1952:332
13. Minsky 1967:186
14. Sipser 2006:157
15. Kleene 1952:229 shows that "Definition by cases" is primitive recursive. CASES requires that the list of testable instances within the CASE definition to be (i) mutually exclusive and (ii) collectively exhaustive i.e. it must include or "cover" all possibility. The CASE statement proceeds in numerical order and exits at the first successful test; see more at Boolos–Burgess–Jeffrey Fourth edition 2002:74
16. An IF-THEN-ELSE or "logical test with branching" is just a CASE instruction reduced to two outcomes: (i) test is successful, (ii) test is unsuccessful. The IF-THEN-ELSE is closely related to the AND-OR-INVERT logic function from which all 16 logical "operators" of one or two variables can be derived; see more at Propositional formula. Like definition by cases, a sequence of IF-THEN-ELSE logical tests must be mutually exclusive and collectively exhaustive over the variables tested.
17. Kowalski 1979
18. Sue Carroll, Taz Daughtrey. Fundamental Concepts for the Software Quality Engineer. pp. 282 et seq.. http://books.google.com/books?id=bz_cl3B05IcC&pg=PA282. 
19. Davis 2000:18
20. Bolter 1984:24
21. Bolter 1984:26
22. Bolter 1984:33–34, 204–206)
23. All quotes from W. Stanley Jevons 1880 Elementary Lessons in Logic: Deductive and Inductive, Macmillan and Co., London and New York. Republished as a googlebook; cf Jevons 1880:199–201. Louis Couturat 1914 the Algebra of Logic, The Open Court Publishing Company, Chicago and London. Republished as a googlebook; cf Couturat 1914:75–76 gives a few more details; interestingly he compares this to a typewriter as well as a piano. Jevons states that the account is to be found at Jan . 20, 1870 The Proceedings of the Royal Society.
24. Jevons 1880:199–200
25. All quotes from John Venn 1881 Symbolic Logic, Macmillan and Co., London. Republished as a googlebook. cf Venn 1881:120–125. The interested reader can find a deeper explanation in those pages.
26. Bell and Newell diagram 1971:39, cf. Davis 2000
27. * Melina Hill, Valley News Correspondent, A Tinkerer Gets a Place in History, Valley News West Lebanon NH, Thursday March 31, 1983, page 13.
28. Davis 2000:14
29. van Heijenoort 1967:81ff
30. van Heijenoort's commentary on Frege's Begriffsschrift, a formula language, modeled upon that of arithmetic, for pure thought in van Heijenoort 1967:1
31. Dixon 1906, cf. Kleene 1952:36–40
32. cf. footnote in Alonzo Church 1936a in Davis 1965:90 and 1936b in Davis 1965:110
33. Kleene 1935–6 in Davis 1965:237ff, Kleene 1943 in Davis 1965:255ff
34. Church 1936 in Davis 1965:88ff
35. cf. "Formulation I", Post 1936 in Davis 1965:289–290
36. Turing 1936–7 in Davis 1965:116ff
37. Rosser 1939 in Davis 1965:226
38. Kleene 1943 in Davis 1965:273–274
39. Kleene 1952:300, 317
40. Kleene 1952:376
41. Turing 1936–7 in Davis 1965:289–290
42. Turing 1936 in Davis 1965, Turing 1939 in Davis 1965:160
43. Hodges, p. 96
44. Turing 1936–7:116)
45. Turing 1936–7 in Davis 1965:136
46. Turing 1939 in Davis 1965:160

 References

• Axt, P. (1959) On a Subrecursive Hierarchy and Primitive Recursive Degrees, Transactions of the American Mathematical Society 92, pp. 85–105
• Bell, C. Gordon and Newell, Allen (1971), Computer Structures: Readings and Examples, McGraw-Hill Book Company, New York. ISBN 0-07-004357-4}.
• Blass, Andreas; Gurevich, Yuri (2003). "Algorithms: A Quest for Absolute Definitions". Bulletin of European Association for Theoretical Computer Science 81. http://research.microsoft.com/~gurevich/Opera/164.pdf.  Includes an excellent bibliography of 56 references.
• Boolos, George; Jeffrey, Richard (1974, 1980, 1989, 1999). Computability and Logic (4th ed.). Cambridge University Press, London. ISBN 0-521-20402-X. : cf. Chapter 3 Turing machines where they discuss "certain enumerable sets not effectively (mechanically) enumerable".
• Burgin, M. Super-recursive algorithms, Monographs in computer science, Springer, 2005. ISBN 0-387-95569-0
• 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. 91–109
• Church, Alonzo (1936a). "An Unsolvable Problem of Elementary Number Theory". The American Journal of Mathematics 58 (2): 345–363. doi:10.2307/2371045.  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.
• Church, Alonzo (1936b). "A Note on the Entscheidungsproblem". The Journal of Symbolic Logic 1 (1): 40–41. doi:10.2307/2269326.  Church, Alonzo (1936). "Correction to a Note on the Entscheidungsproblem". The Journal of Symbolic Logic 1 (3): 101–102. doi:10.2307/2269030.  Reprinted in The Undecidable, p. 110ff. Church shows that the Entscheidungsproblem is unsolvable in about 3 pages of text and 3 pages of footnotes.
• Daffa', Ali Abdullah al- (1977). The Muslim contribution to mathematics. London: Croom Helm. ISBN 0-85664-464-1. 
• Davis, Martin (1965). The Undecidable: Basic Papers On Undecidable Propositions, Unsolvable Problems and Computable Functions. New York: Raven Press. ISBN 0486432289.  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, Martin (2000). Engines of Logic: Mathematicians and the Origin of the Computer. New York: W. W. Nortion. ISBN 0393322297.  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.
•  This article incorporates public domain material from the NIST document "algorithm" by Paul E. Black (Dictionary of Algorithms and Data Structures).
• Dennett, Daniel (1995). Darwin's Dangerous Idea. New York: Touchstone/Simon & Schuster. ISBN 0684802902. 
• Yuri Gurevich, Sequential Abstract State Machines Capture Sequential Algorithms, ACM Transactions on Computational Logic, Vol 1, no 1 (July 2000), pages 77–111. Includes bibliography of 33 sources.
• Kleene C., Stephen (1936). "General Recursive Functions of Natural Numbers". Mathematische Annalen 112 (5): 727–742. doi:10.1007/BF01565439.  Presented to the American Mathematical Society, September 1935. Reprinted in The Undecidable, p. 237ff. 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).
• Kleene C., Stephen (1943). "Recursive Predicates and Quantifiers". American Mathematical Society Transactions 54 (1): 41–73. doi:10.2307/1990131.  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).
• Kleene, Stephen C. (First Edition 1952). Introduction to Metamathematics (Tenth Edition 1991 ed.). North-Holland Publishing Company. ISBN 0720421039.  Excellent—accessible, readable—reference source for mathematical "foundations".
• Knuth, Donald (1997). Fundamental Algorithms, Third Edition. Reading, Massachusetts: Addison–Wesley. ISBN 0201896834. 
• Kosovsky, N. K. Elements of Mathematical Logic and its Application to the theory of Subrecursive Algorithms, LSU Publ., Leningrad, 1981
• Kowalski, Robert (1979). "Algorithm=Logic+Control". Communications of the ACM 22 (7): 424–436. doi:10.1145/359131.359136. ISSN 0001-0782. 
• 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 60-51085.]
• Minsky, Marvin (1967). Computation: Finite and Infinite Machines (First ed.). Prentice-Hall, Englewood Cliffs, NJ. ISBN 0131654497.  Minsky expands his "...idea of an algorithm—an effective procedure..." in chapter 5.1 Computability, Effective Procedures and Algorithms. Infinite machines."
• Post, Emil (1936). "Finite Combinatory Processes, Formulation I". The Journal of Symbolic Logic 1 (3): 103–105. doi:10.2307/2269031.  Reprinted in The Undecidable, p. 289ff. 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.
• Rosser, J.B. (1939). "An Informal Exposition of Proofs of Godel's Theorem and Church's Theorem". Journal of Symbolic Logic 4.  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)
• Sipser, Michael (2006). Introduction to the Theory of Computation. PWS Publishing Company. ISBN 053494728X. 
• Stone, Harold S. (1972). Introduction to Computer Organization and Data Structures (1972 ed.). McGraw-Hill, New York. ISBN 0070617260.  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).
• Turing, Alan M. (1936–7). "On Computable Numbers, With An Application to the Entscheidungsproblem". Proceedings of the London Mathematical Society, Series 2 42: 230–265. doi:10.1112/plms/s2-42.1.230. . 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.
• Turing, Alan M. (1939). "Systems of Logic Based on Ordinals". Proceedings of the London Mathematical Society, Series 2 45: 161–228. doi:10.1112/plms/s2-45.1.161.  Reprinted in The Undecidable, p. 155ff. Turing's paper that defined "the oracle" was his PhD thesis while at Princeton USA.
• United States Patent and Trademark Office (2006), 2106.02 **>Mathematical Algorithms< - 2100 Patentability, Manual of Patent Examining Procedure (MPEP). Latest revision August 2006

 Secondary references

• Bolter, David J. (1984). Turing's Man: Western Culture in the Computer Age (1984 ed.). The University of North Carolina Press, Chapel Hill NC. ISBN 0807815640. , ISBN 0-8078-4108-0 pbk.
• Dilson, Jesse (2007). The Abacus ((1968,1994) ed.). St. Martin's Press, NY. ISBN 031210409X. , ISBN 0-312-10409-X (pbk.)
• van Heijenoort, Jean (2001). From Frege to Gödel, A Source Book in Mathematical Logic, 1879–1931 ((1967) ed.). Harvard University Press, Cambridge, MA. ISBN 0674324498. , 3rd edition 1976[?], ISBN 0-674-32449-8 (pbk.)
• Hodges, Andrew (1983). Alan Turing: The Enigma ((1983) ed.). Simon and Schuster, New York. ISBN 0671492071. , ISBN 0-671-49207-1. Cf. Chapter "The Spirit of Truth" for a history leading to, and a discussion of, his proof.

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