Changes

{| align="center" cellpadding="8" width="90%"

{| align="center" cellpadding="8" width="90%"

| <math>x = x \operatorname{I}</math>

| <math>x = x\operatorname{I}</math>

|}

|}

One is asked to find a ''pure interpretant'' for <math>\operatorname{I},</math> that is, an equivalent term in <math>\langle \operatorname{K}, \operatorname{S} \rangle,</math> the ''combinatory algebra'' generated by <math>\operatorname{K}</math> and <math>\operatorname{S},</math> that does as <math>\operatorname{I}</math> does.

One is asked to find a ''pure interpretant'' for <math>\operatorname{I},</math> that is, an equivalent term in <math>\langle \operatorname{K}, \operatorname{S} \rangle,</math> the ''combinatory algebra'' generated by <math>\operatorname{K}</math> and <math>\operatorname{S},</math> that does as <math>\operatorname{I}</math> does.

A handle on the problem can be gotten by observing the following relationships:

Observe:

  x = (xK)(xK) = x(K(KS))

 

 

x = (x\operatorname{K})(x\operatorname{K}) = x(\operatorname{K}(\operatorname{K}\operatorname{S}))

  I = K(KS)

\operatorname{I} = \operatorname{K}(\operatorname{K}\operatorname{S})

and so K(KS) constitutes a syntactic algorithm for I.

Thus the sequence of operations indicated by <math>\operatorname{K}(\operatorname{K}\operatorname{S})</math> is a <math>\langle \operatorname{K}, \operatorname{S} \rangle</math> proxy for <math>\operatorname{I}.</math>

</pre>

===Step 2===

===Step 2===