Semantic Algebras

The semantic algebras used are mostly the same as for the traditional denotational semantics of imperative languages.

• Nat as usual
• StorableValue = Nat+ErrValue where ErrValue= Unit. For simplicity we are assuming that all the variables are integer
• we need to provide a representation for objects; these are easily represented as list of name-value pairs, where the name is the attribute name and the value is the corresponding value. A more effective way is to actually represent the object as a function which maps identifiers (i.e., attributes) to their values. A special value, UndefAttr can be used to identify attributes that do not belong to the object. Thus:
  
  Object = Id $\mapsto$ AttrValue
  where
  
  AttrValue = StorableValue + UndefAttr
• the Store will give a meaning to each identifier. Each identifier can represent either a normal variable or an object: Store = Nat + Object + ErrValue

  Furthermore, in order to build the objects when they are created we need to maintain a description of the class hierarchy. Since the hierarchy is in this case largely statical (i.e., class structure is not modified after the creation of the class), we can use the following approach:
  

  Hierarchy = Id $\mapsto$ (Class + ErrValue)
  where the algebra Class is defined as
  
  Class = Id $\mapsto$ (ErrValue+UndefAttr)
  The intuition is that each class is described by a function which maps the attributes of the class to ErrValue and the identifiers that do not correspond to any attribute of the class to UndefAttr.