Denotational Semantics

Overview and example
Language of Propositional Logic
Language of Simple Expressions
Language Main Program and Error Values
Language with State (Imperative)
Language with Combined Expressions and Commands
Language with Function Calls

Example: language of simple expressions

Syntactic Categories

  I in Ident  (identifiers)
  E in Exp    (expressions)

   E  ::=  0 | 1 | I | (E1 + E2) | (E1 - E2) 
        |  let I = E1 in E2 end

          Int = { ..., -2, -1, 0, 1, 2, ... }   (integers)
   rho in Env = Ident --> Int                   (environment, values for variables)

   EE : Exp --> Env --> Int

   EE [0] rho  =  0
   EE [1] rho  =  1
   EE [I] rho  =  rho(I)
   EE [E1 + E2] rho  =  plus ( EE [E1] rho, EE [E2] rho )
   EE [E1 - E2] rho  =  minus ( EE [E1] rho, EE [E2] rho )

     note here the "+" and "-" on the left is SYNTAX. 
     The "plus" and "minus" on the right are operations defined on 
     corresponding value domains (here, Int, since EE generates Int)
     For this reason, you can think of the Value Domains as algebras...
     i.e., sets of values and operations on them, i.e., ADTs

   EE [[ let I = E1 in E2 end ]] rho  =  EE [[ E2 ]] (rho[I |-> EE [[E1]] rho ])