II. Notes on Data Structuring * - Cornell University

144 C. A. R. HOARE
Note how we have used sequences to replace the recursion wherever
possible. In fact this can be done whenever a type name occurs recursively
only once at the beginning or at the end of its definition. For example"
type expression = sequence term;
might have been formulated recursively:
type expression =
(empty: (), non-empty: (first: term; final: expression)).
A similar alternative formulation permits while loops to be expressed as
recursive procedures.
The construction of values of a recursively defined type requires no new
operators or transfer functions; all that is needed is recursive use of the
methods defined for the other relevant structuring methods. For example,
the expression
3/(b - 2)
could be specified by the cumbersome construction"
[term (plus, [factor (times, primary (const (3))),
])
factor (div, primary (bracketed (
[term (plus, [factor (times, primary (var ("b")))]),
term (minus, [factor (times, primary (const (2)))])])))
1.
An effective method of getting the computer itself to translate expressions
into abstract structures will be given as an example in (9.2).
Another familiar example of recursively defined data is the family tree.
A family tree (excluding information about marriage) can be defined by
associating with each person the family trees of all his/her offspring. We
assume that certain additional personal details are required to be held"
type family = (head:person; offspring :sequence family);
A person with no children is an ultimate component of the family tree,
and may be represented:
family (Tom, [])
A family with three children may be represented:
family (Jill, [family (Tom, []),
family (Joanna, []),
family (Matthew, [])]).