COP 5021 Lecture -*- Outline -*-

* Denotational Semantics (2.2)

*** denotational example
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   DENOTATIONAL SEMANTICS OF EXPRESSIONS
           FOR WHILE LANGUAGE

Syntax (1.2):

  x,y \in Var
  n \in NumericLiteral
  a \in AExp
  b \in BExp
  op_a \in Op_a
  op_b \in Op_b
  op_r \in Op_r

  a ::= x | n | a1 op_a a2
  b ::= true | false | not b | b1 opb b2
        | a1 op_r a2

Domains:
        Z = { ..., -1, 0, 1, ... }
        T = { true, false }
  s \in State = Var -> Z

Meaning Functions:

  A : Aexp -> State -> Z

  A[[x]]s = s(x)
  A[[n]]s = N(n)
  A[[a1 op_a a2]]s
     = (O_a[[op_a]]) (A[[a1]]s) (A[[a2]]s)

where

  N : NumericLiteral -> Z
  N[[n]] = the value of literal n

  O_a : Op_a -> (Z x Z -> Z)

  O_a[[+]] = \n1 n2 . n1 + n2
  ...

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        E.g., in C or C++, N[[015]] = 13, since 015 is in octal.

     Q:  Why is the state a parameter to the meaning of arithmetic
         expressions?
         it's needed to evaluate variable references

     Q: How would you define B : Bexp -> (State -> T)?

     Q: How would you define a tagged union type with operations left,
     right, and case?