Computer Programming & Formal Systems by P. Braffort, D. Hirschberg

By P. Braffort, D. Hirschberg

A made of seminars held within the IBM international exchange eu schooling middle of Blaricum (Holland) in 1961, the 1st of which used to be devoted to a normal survey of non-numerical functions of pcs while the second one was once extra particularly enthusiastic about a few elements of the idea of formal platforms.

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Fuctoriul(n-1))) cannot serve as a name for this function because it is not clear that the occurrence of “factorial” in the expression refers to the function defined by the expression as a whole. Therefore, for recursive functions we adopt an additional convention. ,x n ) =e where any occurrences of the function letter f within e stand for the function being defined. The letter f is a dummy variable. The factorial function then has the name - lubeZ(fuctoriul,A( (n),(n = 0 + 1,T + n fuctoriuZ(n- l ) ) ) ) , and since factorial and n are dummy variables the expression lubel(g,A((r),(r= 0 -+ 1,T + r - g ( r - 1 ) ) ) ) represents the same function.

We shall call this method of proving two functions equivalent by the name of recursion induction. We shall develop some of the properties of the elementary functions of integers in order to illustrate proof by recursion induction. We recall the definitions m + n = (n = 0 + m,T + m‘+n-) mn = (n = 0 + O,T + m+mn-) Th. 1. m + O Proof m 0 + =m = (0 = 0 - m. + m,T + m’ + 0-) Only the definition of addition and the properties of conditional expressions were used in this proof. + + Th. 2. (m n)’ = m‘ n Proof Definef(m,n) = (n = 0 + m‘,T + f(m,n-)).

U ~ : ( A O B ) O C +- A O ( B Q C ) 1. &:AC x BC + ( A X B)c 5. d3:ABX Ac +- AB@C 6. s~:(AB)C+- ABxC We shall denote the null set (containing no elements) by 0 and the set A BASIS FOR A MATHEMATICAL THEORY OF COMPUTATION 51 consisting of the integers from 1 to n by n. x A (n terms, associate to left by convention) .. Suppose we write the recursive equation S={A} e A x S . We can interpret this as defining the set of sequences of elements of A as follows : 1. Interpret A as denoting the null sequence.

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