2.9.1 Unification

The symbol is a transformational operator which transforms number systems into interval number systems. Floating point interval arithmetic, , can be rewritten as ; while real interval arithmetic, , can be rewritten as . As denotes a number system,   denotes an interval number system.

Interval arithmetic has been generalized through this simplification. Consider the number system which denotes an interval system where the endpoints are extended integers:

Infinities are useful in the underlying number system since intervals may need to describe arbitrarily distant numbers. Without them, some interval operators are forced to be only partially defined.

Consider the interval number system ; the previous example had . The interval inclusion property for n-ary function g is clearly stated as:

The argument is considered to vary over the domain of g. This property is equivalent to the inclusion property for both real and floating point intervals.

The interval extension of an n-ary function g is defined as:

The demotions and are used since the derived interval endpoints will need to be ``rounded out'' to ensure the endpoints are valid and of the correct type. The argument is considered to vary over the domain of g. Demotions are not needed if the underlying number system is no poorer than the number system which the result of g belongs to, as was seen when g was a real valued function and the interval system was .

Jeff Tupper

March 1996