Slide 6.1: Denotational semantics
Slide 6.3: Syntactic and semantic domains
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Denotational Semantics (Cont.)

Another Example
In many programming languages, a program can be viewed as something that receives input and produces output. Thus the semantics of a program can be represented by a function from input to output, and a semantic function for programs would look like this 
     P: Program → ( Input → Output )
The semantic domain to which P maps programs is a set of functions, namely, the functions from Input to Output, which we represent by Input → Output, and the semantic value of a program is a function. For example, if p represents the C program: 
     main( ) {
       int  x;
       scanf ( "%d", &x );
       printf( "%d\n", x );
       return( 0 );
     }
that inputs an integer and outputs the same integer, then p denotes the identity function f from integers to integers: P(p) = f, where f: Integer → Integer is given by f(x) = x. Very often semantic domains in denotational descriptions will be function domains, and values of semantic functions will be functions themselves. To simplify the notation of these domains, we will often assume that the function symbol ‘→’ is right associative and leave off the parentheses from domain descriptions. Thus, becomes