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λ-Calculus: The Other Turing Machine

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journal contribution
posted on 01.01.2004 by Guy E. Blelloch, Robert Harper

The early 1930s were bad years for the worldwide economy, but great years for what would eventually be called Computer Science. In 1932, Alonzo Church at Princeton described his λ-calculus as a formal system for mathematical logic,and in 1935 argued that any function on the natural numbers that can be effectively computed, can be computed with his calculus [4]. Independently in 1935, as a master’s student at Cambridge, Alan Turing was developing his machine model of computation. In 1936 he too argued that his model could compute all computable functions on the natural numbers, and showed that his machine and the λ-calculus are equivalent [6]. The fact that two such different models of computation calculate the same functions was solid evidence that they both represented an inherent class of computable functions. From this arose the so-called Church-Turing thesis, which states, roughly, that any function on the natural numbers can be effectively computed if and only if it can be computed with the λ-calculus, or equivalently, the Turing machine. Although the Church-Turing thesis by itself is one of the most important ideas in Computer Science, the influence of Church and Turing’s models go far beyond the thesis itself.