Chapter 9
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Visualization of logical formulas
There is a form of visual presentation, catchy, understandable
to all from childhood. This form is the schedule.
Valery Venda
FUNCTIONS AND IMAGING
- Where can I buy a puppy?
- In our town, they are sold on the market, but today the market
is closed. In addition, puppies are sold every day. Puppies are
quite expensive and some plain - do not know if they like you.
From overheard conversation clear that the purchase can be made
to a puppy and only when four conditions are satisfied (Fig.
55):
* the market is open (this condition is denoted by P);
* the buyer has the money (Q);
* Puppies are on sale (R);
* puppy like (S).
As a result, we obtain the logical function
X = P and Q and R and S
wherein X is "You can buy a puppy?" (Fig. 55).
In traditional programming languages, values ??of logical
variables are considered a pair (TRUE, FALSE) or (1, 0). From an
ergonomic point of view, such an approach can not be considered
successful. In fact, the use of "helluva lot of erudite" words
TRUE and FALSE or mysterious numbers 1 and 0 in the example of
the puppies (and any other specific example) is frivolous,
misleading and does not promote an understanding of the issue.
To improve matters, as the values ??of the logical variables and
logical functions much better to choose a simple and clear words
"yes" and "no", the semantics of which is self-explanatory and
understandable even to a child. On this basis, in the language
DRAGON logic functions, variables and expressions are regarded
as so-ations questions. Logic function and is defined as a
function that takes the value of "yes" if all of the logical
variables are set to "yes." In other cases, the function gets
the value of "no" (Fig. 55) ^1.
There are two ways the image features and languages ??DRAGON:
text and visual. In the first case, an icon of a "question"
within which they write a logical expression consisting of
logical variables connected by logical AND operation marks (Fig.
56 left). In another case, a vertical paint N Icons "issue",
where N - number of logical variables, with each icon recorded a
boolean variable (Fig. 56 right).
The visual formula in Fig. 56 shows that both methods are
equivalent. Examples Fig. 57 confirm this. Recommended best
practice is a visual way, as it more evident and makes it easier
to find errors in complex algorithms. It should be stressed that
the text method is not prohibited, but use it with caution and
only in those cases when the user is convinced of its ability to
guarantee the absence of errors. Experience shows that most
people choose a more visual way easy. However, trained
professionals who are familiar with the basics of mathematical
logic, sometimes prefer a text method. Such people can be
advised to master both.
Visualization function or
Logical OR function is set to "yes" if at least one logical
variable is "yes." The function is set to "no" if all of the
logical variables have a value of "no" (Fig. 58).
In the language of DRAGON OR function can be written in two
ways. If you select text mode, the icon draws a "question" that
contains a logical expression (Fig. 59 left). When the visual
method uses several icons "issue," which combined the lower
output; each icon writing a boolean variable (Fig. 59 right).
From Fig. 59 and 60 it is evident that both methods are
equivalent.
Visualization function is not
The function W = Z function is not called if the logical
variables Z and W are inverted values, t. E. Satisfy the
conditions:
if Z = yes, W = no;
if Z = no, W = yes.
The visual formula in Fig. 61 show that the sign of logical
negation can be excluded from the dragon-circuit if interchanged
the words "yes" and "no" on the outputs of the icon "question"
(the icon located in the fork of the shoulders, should be left
in place).
Exercises in Fig. 62-66 will help the reader to consolidate the
material.
Visualization of complex logic functions
Consider the function
X = (A & B and C) or (D and E andF) (1)
Fig. 67 shows a visual way to write this function. The figure
shows that the formula (1) is divided into three parts:
1) A and B and C; 2) D and E and F; 3) the "OR" operation.
Function A and C and is represented by three icons A, B, C
located on the same vertical. Similarly, paint the D and E and
B.Bunch "or" being implemented by means of lines that combine
lower yields icons C and F at the point K (Fig. 67).
In formula (1), some members recorded without logical negation
(A, B, D, E), others - negated (B, F). Members without negation
transformed into icons of A, B, D, E, whose lower outlet is
marked with the word "yes." Members of the denial of the icons
in the match, and F, where the lower output is marked with the
word "no" (Fig. 67). Other examples of algorithms that compute
complex logic functions are shown in Fig. 68-74.
These considerations allow us to formulate two theorems.
Theorem 1. Dragon diagram containing logical connectives AND,
OR, NOT inside icons "issue" can always be converted into an
equivalent diagram of a dragon that does not contain these
bundles.
Theorem 2. If some fragment of a dragon-circuit has one input,
two outputs, and contains only the icons of "question", the
first output function calculates the X, the second output
computes the logical negation of X (Fig. 67-73).
The proof of the theorems to the reader.
CONCLUSIONS
1. In complex logic algorithms are often used conditional
statements with logical expressions. Experience shows that
such operators are often difficult to understand, which
often leads to mistakes.
2. The language used DRAGON visual logic expressions, allowing,
if desired, to completely eliminate the logical connectives
AND, OR, NOT of conditional statements.
3. Visualization of logical formulas in many practically
important cases, significantly facilitates the understanding
and reduces the likelihood of errors.
References
Visible links
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