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2.9.1 Unification

The symbol tex2html_wrap_inline32671 is a transformational operator which transforms number systems into interval number systems. Floating point interval arithmetic, tex2html_wrap_inline32517 , can be rewritten as tex2html_wrap_inline32675 ; while real interval arithmetic, tex2html_wrap_inline32653 , can be rewritten as tex2html_wrap_inline32679 . As tex2html_wrap_inline32169 denotes a number system,   tex2html_wrap_inline31473 denotes an interval number system.

Interval arithmetic has been generalized through this simplification. Consider the number system tex2html_wrap_inline32685 which denotes an interval system where the endpoints are extended integers:

math6799

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 tex2html_wrap_inline32687 ; the previous example had tex2html_wrap_inline32689 . The interval inclusion property for n-ary function g is clearly stated as:

math6803

The argument tex2html_wrap_inline32213 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:

math6808

The demotions tex2html_wrap_inline32705 and tex2html_wrap_inline32707 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 tex2html_wrap_inline32213 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 tex2html_wrap_inline32653 .


next up previous notation contents
Next: 2.9.2 Three Valued Logic Up: 2.9 Generalized Interval Arithmetic Previous: 2.9 Generalized Interval Arithmetic
Jeff TupperMarch 1996