algorithm a clerical or effective procedure that can be applied to any of a class of certain symbolic inputs and that will in a finite time and number of steps eventuate in a result in a corresponding symbolic output. A function for which an algorithm (sometimes more than one) can be given is an algorithmic function. The following are common examples: (a) given n, finding the nth prime number; (b) differentiating a polynomial; (c) finding the greatest common divisor of x and y (the Euclidean algorithm); and (d) given two numbers x, y, deciding whether x is a multiple of y. When an algorithm is used to calculate values of a numerical function, as in (a), (b), and (c), the function can also be described as algorithmically computable, effectively computable, or just computable. Algorithms are generally agreed to have the following properties – which made them essential to the theory of computation and the development of the Church-Turing thesis – (i) an algorithm can be given by a finite string of instructions, (ii) a computation device (or agent) can carry out or compute in accordance with the instructions, (iii) there will be provisions for computing, storing, and recalling steps in a computation, (iv) computations can be carried out in a discrete and stepwise fashion (in, say, a digital computer), and (v) computations can be carried out in a deterministic fashion (in, say, a deterministic version of a Turing machine). See also CHURCH’s THESIS, COMPUTABILITY , COMPUTER THEOR. F.A.