The I - math is a variation of the -analytics. The particulars of the I - math, known as I - terms, are developed recursively from a given set V of factors; the I - terms take one of the accompanying structures: x variable x.M reflection, where M is a I - term and x FV(M) MN application, where M and N are I - terms. The arrangement of free factors FV(M), -correspondence and -decrease are characterized along these lines to the relating -math definitions. (I, B, C and S. Demonstrate why these combinators are not to the point of communicating the typical -math. [6 marks] (b) Demonstrate the interpretation utilizing the I - term x.y.(xMMy), where M is the character work z.z. [3 marks] (c) Define the Church numerals for the standard -analytics, and recognize those numerals which are not I - terms. [3 marks] (d) By adjusting the meaning of the Church numerals, characterize I - terms n for every n > 0 to such an extent that n MN = Mn (N) for n > 1 and inconsistent I - terms M, N 0 m n = n for erratic m, n > 0. Show that these uniformities are without a doubt fulfilled. [Hint: the I - term displayed in (b) may assist with settling a specific case.] [8 marks]

The twofold trees, indicated by B, whose branch hubs contain normal numbers, are created by the sentence structure B ::= Leaf | Br (n, B, B) where n ranges over regular numbers. Albeit this question concerns the encoding of double trees as -terms, you might utilize the encodings of other notable information structures, for example, booleans and matches, gave you express the properties accepted. Give an encoding of parallel trees as -terms by characterizing as -terms (a) Leaf and Br, used to build the -terms relating to paired trees; (b) isLeaf, which tests whether a -term compares to a leaf or a branch hub; (c) esteem, fstsubtree and sndsubtree, used to distinguish separately the regular number and the two subtrees at a branch hub. Legitimize your response by portraying the way of behaving of isLeaf, esteem, fstsubtree and sndsubtree: for instance, the decrease isLeaf(Leaf) genuine depicts part of the way of behaving of isLeaf. [8 marks] Consider the capacity treeadd characterized inductively on the design of parallel trees by treeadd (m, Leaf) = Leaf treeadd (m, Br (n, B1, B2)) = Br (m + n,treeadd (m, B1),treeadd (m, B2)) Give and legitimize a -term which encodes treeadd, utilizing the -term Y f.(x.f(xx))(x.f(xx)). [6 marks] Give the -term for the endless twofold tree whose branch hubs comprise of zeros at even profundities and ones at odd profundities, as presented beneath: