Variable substitution

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Variable substitution

The preceding sections have demonstrated the use of AFL in writing audio rendering rules for conveying complex structures. However, oral communication takes more time than written communication, and the listener has to retain a lot more information than a person perusing printed text. This disadvantage is most felt when listening to complex mathematics. It takes time to speak complicated expressions and sometimes the listener has forgotten the beginning of an expression by the time it has been fully rendered.

[conjecture1589]

Thus, given Faa De Bruno's formula:

[equation1591]

We see it as an equation with a derivative on the LHS and a double summation on the RHS. We then see that the inner summation has a complicated constraint and that the summand is a fraction. Finally, we read the entire expression.

The steps enumerated above are carried out implicitly by the eye, making it difficult to identify the atomic actions involved. Yet, it is clear that we rely on this type of breaking up or ``chunking'' of complex expressions when understanding them. In fact, most of visual mathematical notation is an attempt at aiding this process of grouping subexpressions together in a meaningful manner -even in the visual setting, writing out Faa De Bruno's formula in a fully linearized manner (e.g., the TeX encoding) makes comprehension difficult, if not impossible.

In the audio setting, the listener does not have the luxury of being able to view both the top level structure as well as the leaves of the formula when listening to a straight rendering of the expression. This means that AsTeR needs to take over part of the work that was described as being implicit in the visual context -AsTeR needs to recognize and convey the same kind of grouping that the experienced reader perceives in the visual notation.

We call this process rendering with variable substitution. Thus, given a complex expression of the form [tex2html_wrap5782], where the [tex2html_wrap5784] are complex math expressions, AsTeR recognizes this top-level structure and produces the rendering, ``Fraction x over y, where x is [tex2html_wrap5786] and y is [tex2html_wrap5788]''. In the following subsections, we enumerate the conditions under which such variable substitution is performed. Based on these, we have implemented a variable substitution rendering style. The listener can activate this style and have AsTeR perform variable substitution where appropriate.

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TV Raman
Thu Mar 9 20:10:41 EST 1995