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Tue Apr 12 08:13:54 PDT 2005

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    Object-oriented vs algebraic documentation

      There are two ways to define a set of objects by setting up rules relating the objects, or by specifying the 'contents' of the objects themselves.

    1. Example::=following
      Net
      1. Two ways of defining the complex numbers: c1 and c2.

      2. C1::=
        Net{
          rp,ip::Real.
        1. arg::angle= /tan(ip/rp).
        2. abs::Real~Negative= /(_^2)(rp^2+ip^2).
          }.
      3. c1::=$ C1.

      4. C2::=
        Net{
        1. c2::Sets. rp,ip::c2->Real
        2. arg::c2->angle= /tan((ip)/(rp)).
        3. abs::c2->Real~Negative= /(_^2)((rp)^2+(ip)^2).
          }.

        First, note that c2 satisfies C:

      5. |-c2 in $ C.

        Consider the category of classes that satisfy C It might be thought that c1---c2. However, by definition of "$", c1 is the initial class satisfying C. Class c2 can be any class that satisfies C. Because c1 is initial, c1->c2.

        Consider

      6. C3::=C2 and Net{for z:c2, z.abs=1}.

        Clearly if c3=$ C3 then c1->c2<==c3<-<c1.

        The C1 kind of description is "object-oriented" and the C2 form "algebraic".


      (End of Net)

      In any case many would argue that one a class of objects in not really a class unless there is a set of operations defined that operate on the objects: object = structure+constraint+operations.

      Notes on MATHS Notation

      Special characters are defined in [ intro_characters.html ] that also outlines the syntax of expressions and a document.

      Proofs follow a natural deduction style that start with assumptions ("Let") and continue to a consequence ("Close Let") and then discard the assumptions and deduce a conclusion. Look here [ Block%20Structure in logic_2_Proofs ] for more on the structure and rules.

      The notation also allows you to create a new network of variables and constraints, and give them a name. The schema, formal system, or an elementary piece of documentation starts with "Net" and finishes "End of Net". For more, see [ notn_13_Docn_Syntax.html ] for these ways of defining and reusing pieces of logic and algebra in your documents.

    End