or algorism, n. 1. a step-by-step procedure by which an operation can be carried out without any exercise of intelligence, and so, for example, by a machine; formally, a recursive specification of a procedure by which a given type of problem can be solved in a finite number of mechanical steps. Simple algorithms familiar in elementary arithmetic are those used in the extraction of square roots and in long division. The problem of how much of mathematics can be described in such terms is the subject of computability theory, and Hilbert's program was in essence an attempt to demonstrate that all of mathematics was recoverable from algorithms that operate upon strings of mathematical symbols. See automata theory. See also Gödel's theorem, Turing machine. 2. a recursive definition enabling any member of an infinite sequence of terms to be generated by its repeated application.