Defining function for arithmetic expressions

Question

I've got these arithmetic expressions

a  ::=  n  |  x  |  a1 + a2  |  a1 ? a2  |  a1 − a2  

which are a part of language.

An undefined element is allowed ⊥ which extends the set of intergers to Z∪{⊥}.

Problem:

I need to define the function

A: Aexp → (State → Z∪{⊥})

by defining equations for each form of the arithmetic expression a.

Now I've tried to come up with an solution but I am pretty sure I am wrong and I gotta admit I am pretty stuck.

Some help understanding this problem would be great.

Answer 1

Hint: Here is one example

$A($ a1 - a2$, s) = minus(A($ a1$,s),A($ a2$,s))$

or alternatively

$A($ a1 - a2$) = \lambda s . minus(A($ a1$,s),A($ a2$,s))$

where $minus$ is the substraction in $\mathbb Z\cup\{\bot\}$