In mathematics and computer science, an algorithm () is a sequence of mathematically rigorous instructions, typically used to solve a class of specific problems or to perform a computation.[Blair, Ann, Duguid, Paul, Goeing, Anja-Silvia and Grafton, Anthony. Information: A Historical Companion, Princeton: Princeton University Press, 2021. p. 247]
In contrast, a heuristic is an approach to problem-solving that may not be fully specified or may not guarantee correct or optimal results, especially in problem domains where there is no well-defined correct or optimal result.[David A. Grossman, Ophir Frieder, Information Retrieval: Algorithms and Heuristics, 2nd edition, 2004, ] For example, social media recommender systems rely on heuristics in such a way that, although widely characterized as "algorithms" in 21st-century popular media, cannot deliver correct results due to the nature of the problem.
As an effective method, an algorithm can be expressed within a finite amount of space and time["Any classical mathematical algorithm, for example, can be described in a finite number of English words" (Rogers 1987:2).] and in a well-defined formal language[Well defined with respect to the agent that executes the algorithm: "There is a computing agent, usually human, which can react to the instructions and carry out the computations" (Rogers 1987:2).] for calculating a function.["an algorithm is a procedure for computing a function (with respect to some chosen notation for integers) ... this limitation (to numerical functions) results in no loss of generality", (Rogers 1987:1).] Starting from an initial state and initial input (perhaps Empty string),["An algorithm has zero or more inputs, i.e., Quantity which are given to it initially before the algorithm begins" (Knuth 1973:5).] the instructions describe a computation that, when executed, proceeds through a finite["A procedure which has all the characteristics of an algorithm except that it possibly lacks finiteness may be called a 'computational method (Knuth 1973:5).] number of well-defined successive states, eventually producing "output"["An algorithm has one or more outputs, i.e. quantities which have a specified relation to the inputs" (Knuth 1973:5).] and terminating at a final ending state. The transition from one state to the next is not necessarily deterministic; some algorithms, known as randomized algorithms, incorporate random input.[Whether or not a process with random interior processes (not including the input) is an algorithm is debatable. Rogers opines that: "a computation is carried out in a discrete stepwise fashion, without the use of continuous methods or analogue devices ... carried forward deterministically, without resort to random methods or devices, e.g., dice" (Rogers 1987:2).]
Etymology
Around 825 AD, Persian scientist and polymath
Al-Khwarizmi wrote
kitāb al-ḥisāb al-hindī ("Book of Indian computation") and
kitab al-jam' wa'l-tafriq al-ḥisāb al-hindī ("Addition and subtraction in Indian arithmetic").
In the early 12th century, Latin translations of said al-Khwarizmi texts involving the Hindu–Arabic numeral system and
arithmetic appeared, for example
Liber Alghoarismi de practica arismetrice, attributed to John of Seville, and
Liber Algorismi de numero Indorum, attributed to Adelard of Bath.
Hereby,
alghoarismi or
algorismi is the Latinization of Al-Khwarizmi's name; the text starts with the phrase
Dixit Algorismi, or "Thus spoke Al-Khwarizmi".
Around 1230, the English word
algorism is attested and then by
Geoffrey Chaucer in 1391, English adopted the French term.
In the 15th century, under the influence of the Greek word ἀριθμός (
arithmos, "number";
cf. "arithmetic"), the Latin word was altered to
algorithmus.
Definition
One informal definition is "a set of rules that precisely defines a sequence of operations",
[Stone 1973:4] which would include all
(including programs that do not perform numeric calculations), and (for example) any prescribed
bureaucratic procedure
[
]
or
Cookbook recipe.
[
] In general, a program is an algorithm only if it stops eventually
[Stone requires that "it must terminate in a finite number of steps" (Stone 1973:7–8).]—even though infinite loops may sometimes prove desirable. define an algorithm to be a set of instructions for determining an output, given explicitly, in a form that can be followed by either a computing machine or a human who could only carry out specific elementary operations on symbols
.[Boolos and Jeffrey 1974,1999:19]
The concept of algorithm is also used to define the notion of decidability—a notion that is central for explaining how come into being starting from a small set of 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 Implementation as . 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.
History
Ancient algorithms
Since antiquity, step-by-step procedures for solving mathematical problems have been attested. This includes in Babylonian mathematics (around 2500 BC),
Egyptian mathematics (around 1550 BC),
Indian mathematics (around 800 BC and later),
[Hayashi, T. (2023, January 1). Brahmagupta. Encyclopedia Britannica.] The Ifa Oracle (around 500 BC), Greek mathematics (around 240 BC),
and Arabic mathematics (around 800 AD).
The earliest evidence of algorithms is found in the Babylonian mathematics of ancient Mesopotamia (modern Iraq). A clay tablet found in Shuruppak near Baghdad and dated to described the earliest division algorithm.
Algorithms for arithmetic are also found in ancient Egyptian mathematics, dating back to the Rhind Mathematical Papyrus .
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.
Computers
Weight-driven clocks
Bolter credits the invention of the weight-driven clock as "The key invention of". In particular, he credits the
verge escapement mechanism
[Bolter 1984:24] that provides us with the tick and tock of a mechanical clock. "The accurate automatic machine"
[Bolter 1984:26] led immediately to "mechanical
automata theory" 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.
[Bolter 1984:33–34, 204–206.] 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.
Electromechanical relay
Bell and Newell (1971) indicate that the
Jacquard loom (1801), a precursor to
(punch cards, 1887), and "telephone switching technologies" were the roots of a tree leading to the development of the first computers.
[Bell and Newell diagram 1971:39, cf. Davis 2000] 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 () was in use, as was the use of Hollerith cards in the 1890 U.S. census. Then came the
teleprinter () with its punched-paper use of
Baudot code on tape.
Telephone-switching networks of 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".[Melina Hill, Valley News Correspondent, A Tinkerer Gets a Place in History, Valley News West Lebanon NH, Thursday, March 31, 1983, p. 13.] The mathematician Martin Davis supported the particular importance of the electromechanical relay.[Davis 2000:14]
Formalization
In 1928, a partial formalization of the modern concept of algorithms began with attempts to solve the
Entscheidungsproblem (decision problem) posed by
David Hilbert. Later formalizations were framed as attempts to define "effective calculability"
[Kleene 1943 in Davis 1965:274] or "effective method".
[Rosser 1939 in Davis 1965:225] Those formalizations included the Gödel–
Jacques 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 1936–37 and 1939.
Representations
Algorithms can be expressed in many kinds of notation, including natural languages,
pseudocode,
,
DRAKON, programming languages or
(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 statements based on natural language. Programming languages are primarily intended for expressing algorithms in a form that can be executed by a computer, but they are also often used as a way to define or document algorithms.
Turing machines
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 also be classified into three accepted levels of Turing machine description: high-level description, implementation description, and formal description.
[Sipser 2006:157] A high-level description describes qualities of the algorithm itself, ignoring how it is implemented on the Turing machine.
An implementation description describes the general manner in which the machine moves its head and stores data in order to carry out the algorithm, but does not give exact states.
In the most detail, a formal description gives the exact state table and list of transitions of the Turing machine.
Flowchart representation
The graphical aid called a
flowchart offers a way to describe and document an algorithm (and a computer program corresponding to it). 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.
[cf Tausworthe 1977]
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 that adds up the elements of a list of
n numbers would have a time requirement of , 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 , if the space required to store the input numbers is not counted, or 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 ) outperforms a sequential search (cost ) when used for lookup table on sorted lists or arrays.
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.
[Haitham Hassanieh, Piotr Indyk, Dina Katabi, and Eric Price, " ACM-SIAM Symposium On Discrete Algorithms (SODA) , Kyoto, January 2012. See also the sFFT Web Page .] Speedups of this magnitude enable computing devices that make extensive use of image processing (like digital cameras and medical equipment) to consume less power.
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, such as divide-and-conquer or dynamic programming within operation research. 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.
Structured programming
Per the Church–Turing thesis, any algorithm can be computed by a model known to be
Turing complete. In fact, it has been demonstrated that Turing completeness requires only four instruction types—conditional GOTO, unconditional GOTO, assignment, HALT. However, 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".
[John G. Kemeny and Thomas E. Kurtz 1985 Back to Basic: The History, Corruption, and Future of the Language, Addison-Wesley Publishing Company, Inc. Reading, MA, .] Tausworthe augments the three Böhm-Jacopini canonical structures:
[Tausworthe 1977:101] SEQUENCE, IF-THEN-ELSE, and WHILE-DO, with two more: DO-WHILE and CASE.
[Tausworthe 1977:142] An additional benefit of a structured program is that it lends itself to proofs of correctness using mathematical induction.
[Knuth 1973 section 1.2.1, expanded by Tausworthe 1977 at pages 100ff and Chapter 9.1]
Legal status
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), so 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. The patenting of software is controversial,
and there are criticized patents involving algorithms, especially
data compression algorithms, such as
Unisys's LZW patent. Additionally, some cryptographic algorithms have export restrictions (see export of cryptography).
Classification
There are various ways to classify algorithms, each with its own merits.
By implementation
One way to classify algorithms is by implementation means.
|
int gcd(int A, int B) {
if (B == 0)
return A;
else if (A > B)
return gcd(A-B,B);
else
return gcd(A,B-A);
}
|
| Recursive C implementation of Euclid's algorithm from the above flowchart |
- 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. Iteration algorithms use repetitive constructs like Program 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.
- 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. An for such an environment is called a serial algorithm, as opposed to parallel algorithms or distributed algorithms. Parallel algorithms are algorithms that take advantage of computer architectures where multiple processors can work on a problem at the same time. Distributed algorithms are algorithms that use multiple machines connected to a computer network. Parallel and distributed algorithms divide the problem into more symmetrical or asymmetrical subproblems and collect the results back together. For example, a CPU would be an example of a parallel algorithm. 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, but 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. The 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.
- Quantum algorithm
- Quantum algorithms 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
- Brute force is a method of problem-solving that involves systematically trying every possible option until the optimal solution is found. This approach can be very time-consuming, as it requires going through every possible combination of variables. However, it is often used when other methods are not available or too complex. Brute force can be used to solve a variety of problems, including finding the shortest path between two points and cracking passwords.
- 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 recursion) until the instances are small enough to solve easily. One such example of divide and conquer is mergesort. 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 graph theory. A graph exploration algorithm specifies rules for moving around a graph and is useful for such problems. This category also includes , 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.
[For instance, the volume of a convex polytope (described using a membership oracle) can be approximated to high accuracy by a randomized polynomial time algorithm, but not by a deterministic one: see ] Whether randomized algorithms with polynomial time complexity can be the fastest algorithm 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 algorithms. 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.
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.
George B. Dantzig and Mukund N. Thapa. 2003.
Linear Programming 2: Theory and Extensions. Springer-Verlag. 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. 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 optimum, 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 coding, 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. 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.
Examples
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.
-
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''
See also
-
Abstract machine
-
ALGOL
-
Algorithm aversion
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Algorithm engineering
-
Algorithm characterizations
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Algorithmic bias
-
Algorithmic composition
-
Algorithmic entities
-
Algorithmic synthesis
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Algorithmic technique
-
Algorithmic topology
-
Garbage in, garbage out
-
Introduction to Algorithms (textbook)
-
Government by algorithm
-
List of algorithms
-
List of algorithm general topics
-
Regulation of algorithms
-
Theory of computation
-
Computability theory
-
Computational complexity theory
-
Computational mathematics
Notes
Bibliography
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Bell, C. Gordon and Newell, Allen (1971), Computer Structures: Readings and Examples, McGraw–Hill Book Company, New York. .
-
Includes a bibliography of 56 references.
-
,
-
: cf. Chapter 3 Turing machines where they discuss "certain enumerable sets not effectively (mechanically) enumerable".
-
-
Campagnolo, M.L., Cris Moore, 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
-
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, Alan 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, George Boole, Gottlob Frege, Georg Cantor, David 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.
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, 3rd edition 1976?, (pbk.)
-
, . Cf. Chapter "The Spirit of Truth" for a history leading to, and a discussion of, his proof.
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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).
-
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).
-
-
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Kosovsky, N.K. Elements of Mathematical Logic and its Application to the theory of Subrecursive Algorithms, LSU Publ., Leningrad, 1981
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A.A. Markov (1954) Theory of algorithms. Translated Imprint Moscow, Academy of Sciences of the USSR, 1954 i.e., 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
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Minsky expands his "...idea of an algorithm – an effective procedure..." in chapter 5.1 Computability, Effective Procedures and Algorithms. Infinite machines.
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Reprinted in The Undecidable, pp. 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.
-
-
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)
-
-
-
-
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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.
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United States Patent and Trademark Office (2006), 2106.02 **>Mathematical Algorithms: 2100 Patentability, Manual of Patent Examining Procedure (MPEP). Latest revision August 2006
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Further reading
External links
- Algorithm repositories
