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Sometimes an algorithm cannot exist for solving an infinite class of problems, particularly when some further restriction is made upon the accepted method. For instance, two problems from Euclid’s time requiring the use of only a compass and a straightedge (unmarked ruler)—trisecting an angle and constructing a square with an area equal to a given circle—were pursued for centuries before they were shown to be impossible. At the turn of the 20th century, the influential German mathematician Navy Paris trainers hot sale cheap online buy cheap the cheapest SryaZOp4eD
proposed 23 problems for mathematicians to solve in the coming century. The second problem on his list asked for an investigation of the consistency of the axioms of arithmetic. Most mathematicians had little doubt of the eventual attainment of this goal until 1931, when the Austrian-born logician Kurt Gödel demonstrated the surprising result that there must exist arithmetic propositions (or questions) that cannot be proved or disproved. Essentially, any such proposition leads to a determination procedure that never ends (a condition known as the halting problem). In an unsuccessful effort to ascertain at least which propositions are unsolvable, the English mathematician and logician Alan Turing rigorously defined the loosely understood concept of an algorithm. Although Turing ended up proving that there must exist undecidable propositions, his description of the essential features of any general-purpose algorithm machine, or Turing machine , became the foundation of computer science . Today the issues of decidability and computability are central to the design of a computer program—a special type of algorithm.

Learn More in these related Britannica articles:

computer science: Data structures and algorithms
A major area of study in computer science has been the storage of data for efficient search and retrieval. The main memory of a computer is linear, consisting of a sequence of memory cells that are numbered 0, 1, 2,… in order. Similarly, the…
mathematics: Mathematics in the 9th century
…name, , yielded the word .…
operations research: Deriving solutions from models
…the solution is called an algorithm.…
automata theory: Probabilistic questions
…identify an automaton with an algorithm, or rule of computation, in which the output of the automaton was a logically determined function of the explicitly expressed input. From the time of the invention of the all-mechanical escapement clock in Europe toward the end of the 13th century, through the mechanistic…
diagnosis: Formulating a diagnosis
An algorithm is a sequence of alternate steps that can be taken to solve problems—that is, a decision tree. Starting with a chief complaint or key clue, the physician moves along this decision tree, directed one of two ways by each new piece of information, and…
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