4.1.4 Syntax

A generous syntax is used to descibe relations to be graphed. This generosity does not burden graphing, as the generous syntax may be built up using another, more basic syntax. Here, we will show how some more luxurious elements may be built from a basic syntax.

General comparisons, namely , may be emulated by exploiting the following identities:

Both g and h are arbitrary functions of x and y.

Conjunctions and disjunctions may be emulated by exploiting the following identities:

Equations which contain partial functions may be modified so that the equation itself is total. The following identities may be used for and :

similar identities may be used for other partial functions.

Logical negation may now be emulated by exploiting the following identity:

where g is a total function.

Some of the more exotic forms of syntax may be emulated by exploiting the following identities:

Note that different emulation strategies may produce differing bounds when evaluated with an interval arithmetic. For example, exploiting the identity

rather than the identity given earlier, allows one to produce formulas which result in sharper bounds when evaluated using an interval arithmetic.

Although is used above, it may be emulated by exploiting the following identity:

Both and may be emulated as well. At some point, a set of basic operators must be defined. This emulation scheme may be foiled by providing a restrictive set of basic operators. Since it is not difficult to implement the more luxurious operators, we choose to provide them directly.

Jeff Tupper

March 1996