output data from given input data. Computability theory, which studies computability of functions from inputs to outputs, and for which Turing machines were invented, reflects this practice. Since the 1970s, interactive use of computers became much more common. In principle, it is possible to model this by having an external agent read from the tape and write to it at the same time as a Turing machine, but this rarely matches how interaction actually happens; therefore, when describing interactivity, alternatives such as I/O automata are usually preferred. History Historical background: computational machinery Robin Gandy (1919–1995)—a student of Alan Turing (1912–1954), and his lifelong friend—traces the lineage of the notion of "calculating machine" back to Charles Babbage (circa 1834) and actually proposes "Babbage's Thesis": Gandy's analysis of Babbage's analytical engine describes the following five operations (cf. p. 52–53): The arithmetic functions +, −, ×, where − indicates "proper" subtraction if . Any sequence of operations is an operation. Iteration of an operation (repeating n times an operation P). Conditional iteration (repeating n times an operation P conditional on the "success" of test T). Conditional transfer (i.e., conditional "goto"). Gandy states that "the functions which can be calculated by (1), (2), and (4) are precisely those which are Turing computable." (p. 53). He cites other proposals for "universal calculating machines" including those of Percy Ludgate (1909), Leonardo Torres y Quevedo (1914), Maurice d'Ocagne (1922), Louis Couffignal (1933), Vannevar Bush (1936), Howard Aiken (1937). However: The Entscheidungsproblem (the "decision problem"): Hilbert's tenth question of 1900 With regard to Hilbert's problems posed by the famous mathematician David Hilbert in 1900, an aspect of problem #10 had been floating about for almost 30 years before it was framed precisely. Hilbert's original expression for No. 10 is as follows: By 1922, this notion of "Entscheidungsproblem" had developed a bit, and H. Behmann stated that By the 1928 international congress of mathematicians, Hilbert "made his questions quite precise. First, was mathematics complete ... Second, was mathematics consistent ... And thirdly, was mathematics decidable?" (Hodges p. 91, Hawking p. 1121). The first two questions were answered in 1930 by Kurt Gödel at the very same meeting where Hilbert delivered his retirement speech (much to the chagrin of Hilbert); the third—the Entscheidungsproblem—had to wait until the mid-1930s. The problem was that an answer first required a precise definition of "definite general applicable prescription", which Princeton professor Alonzo Church would