binary method

The method for computing a sum x+y depends on the types of x and y. For example, the method for adding an integer x and a polynomial y differs from the method for adding two integers modulo 111. Because both the type of x and the type of y must enter into the selection of the method, we refer to these methods as binary methods. Each binary method is a function of two variables, and is stored either in the class of x or in the class of y. See also lookup.

Let's assume that X is the class (or type) of x, and that Y is the class of y. The way to install a method for the addition of an instance x of class X to an instance y of class Y is with a statement of the form

               X + Y := (x,y) -> ...

where ... represents the body of the function, consisting of suitable code for the operation at hand.

The method installed by the code above is automatically inherited by subclasses of X and Y. Here is a brief description of the way this works. Suppose X is the parent of P and Y is the parent of X. When a sum p+q is evaluated where the class of p is P and the class of q is Q, then the binary method for P+Q is applied, unless there isn't one, in which case the binary method for P+Y is applied, unless there isn't one, in which case the binary method for X+Q is applied, unless there isn't one, in which case the binary method for P+Q is applied. In general this search for a binary method continues all the way up the chain of parents to the topmost ancestor of everything, which is called Thing.

As an extreme example of inheritance, the code

               Thing + Thing := (x,y) -> ...

will install a binary method for adding any two things, which will take effect as a last resort whenever more a specifically defined method isn't found.

The new function also uses a ternary lookup table to find the initialization function for the new thing, and should be thought of as a ternary operator. The initialization function for a new expression created by

              new Z of x from y

              lookup(NewMethod,Z,X,Y)

and can be installed with

              new Z of X from Y := (z,y) -> ...

where z denotes the new hash table of class Z and parent x provided to the routine by the system.

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