↑ “Definition of ALGORITHM”. Merriam-Webster Online Dictionary (bằng tiếng Anh). Bản gốc lưu trữ ngày 14 tháng 2 năm 2020. Truy cập ngày 14 tháng 11 năm 2019.
↑ "Any classical mathematical algorithm, for example, can be described in a finite number of English words" (Rogers 1987:2).
↑ 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).
↑ "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).
↑ "An algorithm has zero or more inputs, i.e., quantities which are given to it initially before the algorithm begins" (Knuth 1973:5).
↑ "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).
↑ "An algorithm has one or more outputs, i.e. quantities which have a specified relation to the inputs" (Knuth 1973:5).
↑ 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).
↑ Chabert, Jean-Luc (2012). A History of Algorithms: From the Pebble to the Microchip. Springer Science & Business Media. tr. 7–8. ISBN9783642181924.
↑ Simanowski, Roberto (2018). The Death Algorithm and Other Digital Dilemmas. Untimely Meditations 14. Cambridge, Massachusetts: MIT Press. tr. 147. ISBN9780262536370. Bản gốc lưu trữ ngày 22 tháng 12 năm 2019. Truy cập ngày 27 tháng 5 năm 2019. [...] the next level of abstraction of central bureaucracy: globally operating algorithms. Đã bỏ qua tham số không rõ |translator-last= (trợ giúp); Đã bỏ qua tham số không rõ |translator-first= (trợ giúp)
↑ Dietrich, Eric (2001) [1999]. “Algorithm”. Trong Wilson, Robert Andrew; Keil, Frank C. The MIT Encyclopedia of the Cognitive Sciences. MIT Cognet library. Cambridge, Massachusetts: MIT Press. tr. 11. ISBN9780262731447. Truy cập ngày 22 tháng 7 năm 2020. An algorithm is a recipe, method, or technique for doing something.
↑ Stone simply requires that "it must terminate in a finite number of steps" (Stone 1973:7–8).
↑ Knuth 1973:7 states: "In practice we not only want algorithms, we want good algorithms... one criterion of goodness is the length of time taken to perform the algorithm... other criteria are the adaptability of the algorithm to computers, its simplicity, and elegance, etc."
↑ Stone 1973:7–8 states that there must be, "...a procedure that a robot [i.e., computer] can follow in order to determine precisely how to obey the instruction". Stone adds finiteness of the process, and definiteness (having no ambiguity in the instructions) to this definition.
↑ In his essay "Calculations by Man and Machine: Conceptual Analysis" Seig 2002:390 credits this distinction to Robin Gandy, cf Wilfred Seig, et al., 2002 Reflections on the foundations of mathematics: Essays in honor of Solomon Feferman, Association for Symbolic Logic, A.K. Peters Ltd, Natick, MA.
↑ cf Gandy 1980:126, Robin Gandy Church's Thesis and Principles for Mechanisms appearing on pp. 123–148 in J. Barwise et al. 1980 The Kleene Symposium, North-Holland Publishing Company.
↑ A "robot": "A computer is a robot that performs any task that can be described as a sequence of instructions." cf Stone 1972:3
↑ Lambek's "abacus" is a "countably infinite number of locations (holes, wires etc.) together with an unlimited supply of counters (pebbles, beads, etc). The locations are distinguishable, the counters are not". The holes have unlimited capacity, and standing by is an agent who understands and is able to carry out the list of instructions" (Lambek 1961:295). Lambek references Melzak who defines his Q-machine as "an indefinitely large number of locations... an indefinitely large supply of counters distributed among these locations, a program, and an operator whose sole purpose is to carry out the program" (Melzak 1961:283). B-B-J (loc. cit.) add the stipulation that the holes are "capable of holding any number of stones" (p. 46). Both Melzak and Lambek appear in The Canadian Mathematical Bulletin, vol. 4, no. 3, September 1961.
↑ If no confusion results, the word "counters" can be dropped, and a location can be said to contain a single "number".
↑ "We say that an instruction is effective if there is a procedure that the robot can follow in order to determine precisely how to obey the instruction." (Stone 1972:6)
↑ cf Minsky 1967: Chapter 11 "Computer models" and Chapter 14 "Very Simple Bases for Computability" pp. 255–281 in particular
↑ Knuth 1973 section 1.2.1, expanded by Tausworthe 1977 at pages 100ff and Chapter 9.1
↑ Heath 1908:300; Hawking's Dover 2005 edition derives from Heath.
↑ " 'Let CD, measuring BF, leave FA less than itself.' This is a neat abbreviation for saying, measure along BA successive lengths equal to CD until a point F is reached such that the length FA remaining is less than CD; in other words, let BF be the largest exact multiple of CD contained in BA" (Heath 1908:297)
↑ For modern treatments using division in the algorithm, see Hardy and Wright 1979:180, Knuth 1973:2 (Volume 1), plus more discussion of Euclid's algorithm in Knuth 1969:293–297 (Volume 2).
↑ Euclid covers this question in his Proposition 1.
↑ While this notion is in widespread use, it cannot be defined precisely.
↑ Knuth 1973:13–18. He credits "the formulation of algorithm-proving in terms of assertions and induction" to R W. Floyd, Peter Naur, C.A.R. Hoare, H.H. Goldstine and J. von Neumann. Tausworth 1977 borrows Knuth's Euclid example and extends Knuth's method in section 9.1 Formal Proofs (pp. 288–298).
