The abstract functions of section~#secsimilarities#27610> violate Scheme's
basic grammar in two ways. First, the names of functions and primitive
operations are used as arguments in applications. An argument, though, is
an expression, and the class of expressions does not contain primitive
operations and function names. It does contain variables, but we agreed that
they are only those variables mentioned in variable definitions and as
function parameters. Second, parameters are used as if they were functions,
that is, the first position of applications. But the grammar of
section~#secsynsem#27611> allows only the names of functions and primitive
operations in this place.
#tabular27613#
<#27686#>Figure: Scheme with functions as values<#27686#>
Spelling out the problem suggests the necessary changes. First, we should
include the names of functions and primitive operations in the definition of
<#71299#><#65208#>;SPMlt;<#27688#>exp<#27688#>;SPMgt;<#65208#><#71299#>. Second, the first position in an application should allow other
things than function names and primitive operations; at a minimum, it must
allow variables that play the role of function parameters. In anticipation
of other uses of functions, we agree on allowing expressions in that
position.
Here is a summary of the three changes:
#tabular27690#
Figure~#figgrammarprog#27701> displays the entire Scheme grammar, with all
the extensions we have encountered so far. It shows that the accommodation of
abstract functions does not lengthen the grammar, but makes it
simpler.
The same is true of the evaluation rules. Indeed, they don't change at
all. What changes is the set of values. To accommodate functions as
arguments of functions, the simplest change is to say that the set of values
includes the names of functions and primitive operations:
#tabular27704#
Put differently, if we now wish to decide whether we can apply the
substitution rule for functions, we must still ensure that all arguments are
values, but we must recognize that function names and primitive operations
count as values, too.
<#27725#>Exercise 20.1.1<#27725#>
Assume the <#27727#>Definitions<#27727#> window in DrScheme contains <#65226#><#27728#>(define<#27728#><#27729#> <#27729#><#27730#>(f<#27730#>\ <#27731#>x)<#27731#>\ <#27732#>x)<#27732#><#65226#>. Identify the values among the following expressions:
- <#65227#><#27734#>(cons<#27734#>\ <#27735#>f<#27735#>\ <#27736#>empty)<#27736#><#65227#>
- <#65228#><#27737#>(f<#27737#>\ <#27738#>f)<#27738#><#65228#>
- <#65229#><#27739#>(cons<#27739#>\ <#27740#>f<#27740#>\ <#27741#>(cons<#27741#>\ <#27742#>10<#27742#>\ <#27743#>(cons<#27743#>\ <#27744#>(f<#27744#>\ <#27745#>10)<#27745#>\ <#27746#>empty)))<#27746#><#65229#>
Explain why they are values and why the remaining expressions are
not values.~ 
Solution<#65230#><#65230#>
<#27753#>Exercise 20.1.2<#27753#>
Argue why the following sentences are legal definitions:
- <#65231#><#27756#>(define<#27756#>\ <#27757#>(f<#27757#>\ <#27758#>x)<#27758#>\ <#27759#>(x<#27759#>\ <#27760#>10))<#27760#><#65231#>
- <#65232#><#27761#>(define<#27761#>\ <#27762#>(f<#27762#>\ <#27763#>x)<#27763#>\ <#27764#>f)<#27764#><#65232#>
- <#65233#><#27765#>(define<#27765#>\ <#27766#>(f<#27766#>\ <#27767#>x<#27767#>\ <#27768#>y)<#27768#>\ <#27769#>(x<#27769#>\ <#27770#>'<#27770#><#27771#>a<#27771#>\ <#27772#>y<#27772#>\ <#27773#>'<#27773#><#27774#>b))<#27774#><#65233#>
Solution<#65234#><#65234#>
<#27781#>Exercise 20.1.3<#27781#>
Develop <#65235#><#27783#>a-function=?<#27783#><#65235#>. The function determines whether two functions
from numbers to numbers produce the same results for <#65236#><#27784#>1.2<#27784#><#65236#>,
<#65237#><#27785#>3<#27785#><#65237#>, and <#65238#><#27786#>-5.7<#27786#><#65238#>.
Can we hope to produce <#65239#><#27787#>function=?<#27787#><#65239#>, which determines whether two
functions from numbers are equal?~ Solution<#65240#><#65240#>