II. Notes on Data Structuring * - Cornell University

NOTES ON DATA STRUCTURING 143
to the same data type as itself; in the same way that a recursive procedure
can call itself recursively from more than one place in its own body. As in
the case of recursive procedures such a structure can conveniently be defined
by writing the name of the type being defined actually inside its own definition;
or in the case of mutually recursive definition, in the definition of some
preceding type.
The most obvious examples of recursive data structures are to be found
in the description of arithmetic or logical expressions, programming lan-
guages, where the recursion reflects the possibility of nesting one expression
inside another. For example, an arithmetic expression might be defined as
follows:
"An expression is a series of terms, each of which consists of a sign
(+ or-) followed by a sequence of factors. Each factor except the first
consists of a sign (x or/) followed by a primary. A primary is either a
constant, a variable, or an expression surrounded by brackets. An initial
plus sign in an expression may be omitted."
A structured data type whose values comprise such expressions may be
defined using only techniques already familiar, plus recursion:
type expression = sequence term;
type term = (addop: operator; f: sequence factor);
type factor = (mulop: operator; p: primary);
type primary = (const:(val:real),
var: (id: identifier),
bracketed: (e: expression));
type operator = (plus, minus, times, div);
This definition expresses the abstract structure of an arithmetic expression,
but not the details of its concrete representation as a string of characters.
For example, it does not specify the symbols used for brackets or operators,
nor does it state whether an infix, prefix or postfix notation is used for them.
It does not state how the three kinds of primary are to be distinguished.
It does not even represent the optional omission of plus on the first term of
an expression, and the necessary omission of x on the first factor of a term.
Apart from this degree of abstraction and representation-independence, this
type definition would correspond to a set of BNF syntax equations'
(expression) :: = (term) [ (addop) (term) ]
(term):: = (primary) ] (term)(mulop)(primary)
(expression) ( addop ) ( term )
(primary) :: = (unsigned real number) I (variable) I
((expression))