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A Tutorial Introduction to the Lambda Calculus

A Tutorial Introduction to the Lambda CalculusRa ul Rojas FU Berlin, WS-97/98 AbstractThis paper is a short and painless Introduction to the Calculus . Originallydeveloped in order to study some mathematical properties of effectively com-putable functions, this formalism has provided a strong theoretical foundationfor the family of functional programming languages. We show how to performsome arithmetical computations using the Calculus and how to define recur-sive functions, even though functions in Calculus are not given names and thuscannot refer explicitly to DefinitionThe Calculus can be called thesmallest universal programming language of theworld. The Calculus consists of a single transformation rule (variable substitution)and a single function definition scheme.

It is understood here that sis the rst argument to be substituted during the evalu-ation and zthe second. Using this notation, the rst natural numbers can be de ned as 1 sz:s(z) 2 sz:s(s(z)) 3 sz:s(s(s(z))) and so on. 4. Our rst interesting function is the successor function. This can be …

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