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Ta b l e 7 . 1 0 Formula for solving the second degree equation ax 2
+ bx + c =
0
= b ± b 2
4 ac
x
2 a
In mathematical expressions, two, or three, or sometime more types of parentheses
are used. Let us consider the formula of Table 7.10 giving the solutions of a second
degree equation.
The same formula can be represented by the Latex text [176], reported in Table
7.11. This language developed by Leslie Lamport, is based on the language TEX,
which was introduced by Donald Knuth in 1978. It was essentially a formalism pro-
viding abstract formulae expressing structures of scientific texts and of scientific
formulae. In this context, the attribute “abstract” means that such formulae are built
by textual characters (including some special characters), but they intend to repre-
sent the logical structure of documents and of mathematical formulae, and leave to
different levels the specific stylistic choices of a concrete syntactical realization (see
[42, 41] for abstract syntax representations of linguistic texts).
Ta b l e 7 . 1 1 Latex text which produces the formula of Table 7.10
b 2
$$ x
= \
frac
{−
b
\
pm
\
sqrt
{
4 ac
}}{
2 a
}
$$
One of the basic ideas of TEX is the use of a textual formalism for expressing
parentheses and operation symbols. This idea, developed by many people, in dif-
ferent contexts of programming language formats, led to the definition of Mark-up
languages , based on the general notion of tag . Tags are a generalization of paren-
thesized expressions. An open marker is a symbol put between two angles, and it
uniquely corresponds to its closing marker, according the following structure:
<
marker
>
content
</
marker
>.
Mark-up languages are the basis of data transfer protocols (for example, HTML,
XML), where information at different levels is enclosed in different types of paren-
thesized expression (XML stands for “eXtensible Markup Language”). When a no-
tion of link is added to tags, by means of keys , which univocally identify tag occur-
rences, then any graph can be represented by a suitable tag expression. In fact, let us,
for example, insert inside the left marker an additional information, after the special
symbol #, which denotes a key of the tag. In this manner, we can easily express the
pointer structure of Fig. 7.13 in the XML style of Table 7.13.
Trees are special graphs, sequences are special trees (where any node different
from a leaf has only one son), and a multiset is a special sequence (the sequence of
multiplicities, with respect to an ordering of the elements). Moreover, membranes
which realize hypermultisets, that is, iterated multiset aggregations, are naturally
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