Chapter 17
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CALCULATION OF ICONS and attempts to predict the future
Visualizations - the flesh and blood of thinking.
Rudolf Arnheim
VISUAL logical calculus
Include another "side of the spotlight" and try to look at the
visual syntax of a dragon with the positions of mathematical
logic. Our eyes will open an unusual picture. It turns out that
any abstract dragon scheme is a theorem that is strictly deduced
(proved) of the two axioms, which are the primitive and the
blank-blank-silhouette.
- What is this axiom? - Has the right to surprise the reader. -
It's just a picture-mole rats! A dragon-circuit does not look
like a theorem! Who and what they have to prove? Perhaps it was
a joke or a metaphor.
- Not at all, not a metaphor. It will be shown that the visual
syntax DRAGON built as a logical calculus (called the "calculus
of icons"). This calculus can be seen as part of visual
mathematical logic. The latter concept is not traditional.
Mathematical logic and its basic concepts (calculus, a logical
conclusion, and so on. D.) Formed within the text paradigm. In
this chapter, apparently for the first time entered the visual
analogues of these concepts and their basis of calculation of
construction icons.
KNOWN ABOUT mathematical logic
The principal achievement of mathematical logic is the
development of modern axiomatic method, which is characterized
by three features:
* explicit formulation of assumptions (axioms) to develop the
theory of (formal system);
* explicit wording of the rules of inference by which derived
from axioms theorem theory;
* the use of formal languages ??for presentation of the theory
of the theory.
The main object of study in mathematical logic are logical
calculus. The concept includes the calculation of basic
components, such as:
1. formal language, which is given by the alphabet and syntax,
2. axioms,
3. rules of inference.
Thus, the calculation allows knowing the axioms and rules of
inference, we obtain (t. E. Deduce prove) all theorems of the
theory, the theorem as an axiom, recorded only in a formal
language.
Recall that in the framework of mathematical logic three terms
logical calculus, a formal system and the theory can be regarded
as synonyms. Consequently, Theorem calculus theorems of the
formal system and theorems of the theory - it is one and the
same.
ON A common misconception
There are two approaches to the formalization of human
knowledge: visual (graphic, pictorial), and text. This problem
is related to a curious contradiction. On the one hand, the
advantage of the graphics to the human-readable text is
generally accepted as the human brain it is mainly focused on
visual perception, and people get information when reviewing
graphic images faster than reading text. I. Velbits cue rightly
points out: "The text - the most common and least informative in
the sense of clarity and speed perception of the presentation of
information", and drawing - "the most advanced integrated form
of knowledge representation."
On the other hand, the theoretical development of the principles
of visual formalization of knowledge is still not adequately
deployed. The reason for the backlog to be found in the history
of science, in particular the peculiarities of the development
of mathematics and logic.
In these disciplines for a long time (sometimes explicitly,
often implicitly) assumed that the results of mathematical and
logical formalization of knowledge in most cases should be in
the form of text (but not images). For example, Stephen Kleene
wrote: "As a formal theory of the structure is no longer a
system of meaningful sentences and phrases, the system
considered as a sequence of words, which, in turn, are sequences
of letters ... The symbolic language characters will usually
correspond to a whole words, not letters, but a sequence of
characters corresponding to the phrases, will be called
"formula" ... theory of evidence ... suggests ... the
construction of arbitrarily long sequences of characters. "
From these considerations it is evident that the wedge (as well
as many other authors) puts in the Research Center of
textformalization and completely loses sight of the totality of
the problems associated with visual formalization.
Analysis of the literature on this topic shows that the majority
of scientists based on the implicit assumption that scientific
knowledge-ing - is primarily a "text" knowledge that the most
adequate (or even the only possible) form for the presentation
of the results of scientific research is consistency and
formalized formalized phrases, t. e. the text (rather than
visual images). The basis for this assumption is erroneous view,
which can be characterized as "absolute principle of the text."
The principle of absolute text
The essence of it can be expressed, for example, in the form of
the following considerations.
+--------------------------------------------------------------+
| The progress of science ensures the success of |
| logical-mathematical formalization and development of new |
| scientific concepts and principles, rather than improving |
| the picture. Formulas and words express the essence of |
| scientific thought patterns - it's just to illustrate the |
| scientific text, they facilitate the understanding of the |
| already finished formed scientific thought, but are not |
| involved in its formation. In short, the language of science |
| - a formula and proposals, but not images. The science is |
| the essence, the core, on which depends the success of the |
| scientific creativity and obtaining new scientific results |
| (it is expressed in logical-mathematical formalism, |
| scientific concepts and opinions expressed in words).And |
| there are auxiliary tasks (training beginners, the exchange |
| of information between scientists) - here's the pictures and |
| help facilitating mutual understanding. In addition, the |
| drawings are optional, free and non-strict form, it is |
| impossible to formalize. Therefore, the formalization of |
| scientific knowledge is incompatible with the use of |
| drawings.Thus, the drawings have something external to |
| science. Improving the language of drawings and scientific |
| progress - two different things, they are not related. |
+--------------------------------------------------------------+
There are a number of works, indirectly prove that the principle
of absolute text is wrong and harmful. Today more and more
scientists have come to the conclusion that visual formalization
of knowledge can not be regarded as something secondary for
scientific knowledge, because it is part of the very fabric of
the mental process of learning and "may mediate the deepest and
creative steps of scientific knowledge."
However, in mathematical logic visual methods, to our knowledge,
has not yet been widely used. In other words, mathematical logic
to this day remains a stronghold of the text of thinking and
methods of text formalization of knowledge. This fact plays a
negative role, interfering to put the last point in the
dock-gations fallacy of "absolute principle of the text."
Next we will try a particular example to demonstrate the
principle of calculation icons to visualize at least some
sections, or, more accurately say, questions of mathematical
logic.
IMAGING CONCEPTS mathematical logic
We need a definition of two concepts: visual logic output (video
output) and visual logical calculus (videoischislenie). To
facilitate the study of the material already familiar to the
reader to use the method of confrontation, by placing in the
left column of Table. 6 well-known "text" concept, and the right
- a new definition of "visual" concept.
+--------------------------------------------------------------+
| ]The definition of | ]The definition of "video |
| "inference" | output" (visual inference) |
|-----------------------------+--------------------------------|
| |Video output in videoischislenii|
|Deduction in V is a sequence|V is a sequence of C [1,] ...|
|of C [1,] ..., C [n] formulas|C[n] videoformul, such that for|
|such that for any [i] the|any i videoformula C [1] is|
|formula C [i] is either an|either videoaksioma|
|axiom of the calculus of V,|videoischisleniya of V, or a|
|or a direct consequence of|direct consequence of the|
|the above formulas by one of|previous videoformul one of the|
|the rules of inference.|rules of the video output.|
|Formula C [n] is called a|Videoformula C [n] is called|
|theorem of V, if there is a|videoteoremoy videoischisleniya|
|derivation in V, which is the|V, if there is a video output in|
|latest formula C [n |V, which is the last|
| |videoformuloy C [n |
+--------------------------------------------------------------+
]It is easy to observe that the new definition (right) almost
exactly coincides with the classical (left); The only difference
is the addition of the prefix "video."
+--------------------------------------------------------------+
| The definition of "logical | The definition of |
| calculus" | "videoischislenie" (visual |
| | logical calculus) |
|-----------------------------+--------------------------------|
| |Videoischislenie can be |
|The logical calculus can be |presented as a formal system in |
|represented as a formal |the form of four |
|system of four | |
| | V = <u, S [0,] A, F> |
| V = <u, S [0,] A, F> | |
| |And where - many icons (visual |
|And where - set of basic |alphabet letters); S [0] - a set|
|elements (letters of the |of rules of visual syntax, based|
|alphabet); [0] S - set of |on which of the icons are built |
|syntax rules, on which the |properly constructed |
|letters are constructed |videoformuly; A - the set of |
|well-formed formulas; A - the|well-formed videoformul, whose |
|set of well-formed formulas, |elements are called |
|elements of which are called |videoaksiomami; F - the rules of|
|axioms; F - inference rules |the video output that the set A |
|that allow the set A to |let you receive the new |
|receive new well-formed |well-formed videoformuly - |
|formulas - Theorem |videoteoremy.(Many theories |
| |denote T.) |
+--------------------------------------------------------------+
Building on this approach, and relying on the "text"
determination log-cal calculation, by analogy to introduce the
concept of "videoischislenie" (tab. 7).
CALCULATION OF ICONS
So, we have identified the desired visual concepts of
mathematical logic. With their help, we can construct a calculus
icons.
* ? Many icons and (visual alphabet letters) given thesis 1
(see. Ch. 15) and shown in Fig. 1.
* Many S [o] visual syntax rules described in Sec. 15 theses
2-37.
* A set of visual axioms includes only two elements: the
primitive and the blank-blank-silhouette (Fig. 115).
Further, we call them primitive and axiom-the axiom
silhouette.
* The set T covering all videoteoremy calculus V, is nothing
more than a set of abstract dragon schemes. Note that the
set T does not include the axiom, since the latter contain
blank critical points and, therefore, equivalent to let the
operator. The set T is divided into two parts: a set of
primitives of T [1] and T [2] set of silhouettes.
* Many of the rules F the video output is also divided into
two parts F [1] and F [2.] Lots F [1] allows you to display
all the theorems primitives belonging to the set T [1] of a
single axiom-primi*tiva. It contains five rules of
inference: Enter atom, adding options, transplant vines,
lateral connection, removing kon*tsa primitive. These rules
are described in the listing 10, 21, 28, 30, 31, 34, Ch. 15.
* Many F [2] gives the possibility to display all the theorems
of [T-2] silhouettes of a single axiom-silhouette. It
contains eight inference rules: entering the atom, adding
options, adding branches, grafting vines, creepers ground,
lateral connection, removing the last branch, an auxiliary
input. Inference rules for the silhouette described in
theses 10, 21, 28-33, 35 Ch. 15.
This construction calculation finishes icons.
It is known that the study of calculus syntax of the
mathematical logic. In addition, the latter is engaged in
semantic study of formal languages, the basic concept of
semantics is the concept of truth.
In calculating icons trivial semantics. Various videoformuly
(block diagram) can be true or false. Videoformula called true
if it - or axiom, or deduced from the axioms using the rules of
inference (ie. E. Is a theorem), and false otherwise. Thus, all
well-formed abstract dragon design (theorems) are true.
Conversely, properly constructed schemes that do not meet the
rules of visual language DRAGON, are false. Examples of false
schemes are shown in Fig. 131 and 132 in the left column.
ONCE AGAIN ABOUT skewer method
Earlier we defined a skewer method as a theory of visual
structured programming. In this chapter, an opportunity to
greatly enrich this concept and consider it as a calculus of
icons, including the interpretation of the latter.
To emphasize the theoretical nature of the skewer method, it is
advisable to slightly change the terminology. In particular, the
use of the name DRAGON related to the practical development of
specific programming languages, for theoretical purposes would
be inappropriate. Therefore, we make the change of terms.
Skewer diagram - abstract dragon diagram. We emphasize that the
skewer diagram is by definition abstract, t. E. Completely
devoid of text.
Skewer-language - the language of the skewer-circuits. To
skewer-language set only visual syntax, word syntax is not
defined.
CHART skewers as an abstract model of the program
It has been said that videoprogrammirovaniya characterized by
"splitting syntax." Syntax S splits into a visual syntax S
[0,]which determines the rules for constructing a skewer
diagrams and text syntax S [1,] which defines the rules of the
alphabet tekstoelementov and recording statements within the
text icons. From this, it can be said that the skewer The
programconsists of two parts: B [0] and B [1,] where B [0] -
skewer diagram syntax S [0;] IN [1] - text portion of the
program, ie. E. The total text kept all the icons of the
program, defines the syntax S [1.
]One is struck by the similarity between the undoubted skewer
charts and diagrams programs. Noticing this analogy and
repeating - with some almost obvious changes - the general
outline of reasoning adopted in schematology, we can conclude
that the skewer in the diagram [0] describes is not a single
program, but a whole class of programs, ie. E. A poliprogrammoy
and skewer-language serves multiyazykom - poliprogrammirovaniya
language.
Class diagrams skewer is a subclass of large-block schemes, the
degree of abstraction which occupies an intermediate position
somewhere between Martyniuk schemes and standard circuits.
Relationship between skewer charts and diagrams of programs is
fundamental, and raises a number of interesting problems, in
particular, to the fact that "the task effectivization broadcast
programs grows into the problem of automation of designing
quality programs."
From the point of view of the theory videoprogrammirovaniya,
graph-scheme used in the (text) theoretical programming, have
the disadvantage - as conventional flowchart application
programming, they are informal. Although the works of Ershov
made a step toward formalizing the graph-schemes, but his
solution is not satisfactory, because the syntax used Ershov
visual graph-scheme does not yield an unambiguous strictly
deterministic visual configuration (topology) graph-schemes and,
consequently, It does not provide a unique solution of visual
problems.
However, Yershov and did not set out such tasks. However, for
our purposes is strictly formalizing syntax visual flowcharts
(including flow-chart) plays a fundamental role.
CONVERSION skewer skewers schema in program
We emphasize once again that we have constructed language
(skewer-language) - this is not a programming language and the
language programs of large-block schemes, ie. E.
Poliprogrammirovaniya language. However, it can be easily
converted into a programming language, which can be done in many
ways. To do this, you must also set a text syntax and semantics
of S [1] Q [1] text operators placed in the icons skewer charts.
For example, if you take the text syntax and semantics of the
corresponding Pascal obtain visual programming language that can
be called "spit-Pascal." Similarly, you can build a
language-BASIC skewer, skewer-si and so on. D.
Using the terminology schematology, we can say that the program
has a skewer skewer interpreted diagram, however, the concept of
interpretation in this case is markedly different from the
classic. Detailed consideration is beyond the scope of this
book, we restrict ourselves to a brief remark. To specify the
interpretation of the skewer-circuit and turn it into a skewer
program, it is necessary, first, to extend the definition
skewer-language and turn it into a programming language that
describes the syntax and semantics of S [1] Q [1] text
operators. Secondly, the text should include specific statements
recorded in accordance with the syntax of S [1] and placed in
the icons skewer-circuit to [0.] This will set the text of the
skewerin [one] program in. Thus, the interpretation of the
skewer-circuit is defined as a triple <S [1,] Q [1,] B [1>.
]This implies the following obvious remark.As a skewer-language
is an abstract model of any imperative programming language
(Empire language) insofar Empire language is interpreted by a
skewer-language. This interpretation of the skewer-language,
transforming it into a specific language Empire, defined as a
pair < S [1] , Q [1] >.
Skewers METHOD AND EVIDENCE right program
According to R. Anderson, "the goal of many studies in the field
of proof-of-program ... is the mechanization of such evidence."
D. Grice points out that "the evidence must stay ahead of the
construction program" [2]11 . By combining both requirements, we
find that the automatic proof-of-building program should be
ahead. It is easy to verify that the skewer method provides
partial fulfillment of this requirement. In fact, in the
beginning of the chapter, it was shown that any well-formed
skewer chart is strictly proven theorem. The algorithms DRAGON
editor coded numbered icons, so any skewer diagram built with
his help, the true, that is. E. A properly constructed. This
means that the dragon-editor provides a 100% automatic
proof-of-skewer schemes guaranteeing fundamental impossibility
of visual syntax errors. As a skewer, skewer scheme is
part-program, said equivalent proof of partial correctness
skewer program.
At the beginning of the chapter, we asked a funny question: if
the Dragon scheme - a theorem, who are supposed to prove? The
answer is simple. They had not been required to prove, as they
proved once and for all because the work Dragon editor is built
as a realization of calculation icons.
Now, add a spoonful of tar in a barrel of honey. Unfortunately,
this method enables us to prove the correctness of the
skewer-circuit only. It is only a small part of the total work
to be performed to prove the correctness of the program is 100%.
However, there is a small consolation: a partial proof of the
correctness of the program with the help of the Dragon Editor
done without any human intervention, and achieved for free, as
additional labor costs, time and resources are required. A gift
horse in the teeth not look.
POSSIBLE THEORY visual programming?
Although videoprogrammirovanie - a relatively new trend in this
field for a considerable number of interesting applications
development. However, the theoretical visual programming is in
its infancy. In the available literature, the author was able to
find only a few lines, which can to some extent be interpreted
as a program for future research in the field of theory
videoprogrammirovaniya: "For visual programming is necessary to
conduct rigorous scientific studies, mathematical definitions
and models - the majority of developments in this area is yet
empirical character. Perspective can be use in the graphical
user interface technology of artificial intelligence, which is
commonly used to describe the application area. The system of
knowledge representation may include a set of visual primitives,
their symbolic description and rules of inference conclusions. "
As probably the reader noticed, in this paper, addressing a
similar problem (the problem of the withdrawal of formal
conclusions by performing operations on visual features, which
are mainly used icons skewer charts), we went to a somewhat
different way. The difference is as follows. The authors cited
the work speak of "symbolic descriptions" visual primitives,
meaning text rules of inference conclusions adopted in the
traditional text of mathematical logic. However, even in the
construction of A. Ershov calculating equivalent transformation
schemes Yanov first attempt to move away from "pure text"
mathematical logic, using the formulas of inference rules is not
only symbolic descriptions, and graphics.However, the method
Ershov due to defects in the visual syntax not entirely formal.
In this book, the development of ideas Ershov went in two
directions. Firstly, mentioned defects are eliminated, which
made the formalization of an abstract syntax flowcharts
comprehensive and rigorous. Second, it was launched and enforced
the idea of ??complete abandonment of the symbolic description
of visual features (intraengine binary representation does not
count).
We can assume that the above principles of visualization of
mathematical logic, implemented using the concepts of visual
calculation and visual inference can be useful for a more
complete and rigorous formalization of not only the language of
abstract block diagrams (skewer-language), but other visual
languages knowledge representation and visual programming.
Assumptions about the future imperative programming languages
Summarizing the material, the author could not resist the
temptation to look to the future and in order to express their
preliminary discussion possibly erroneous assumptions about the
development imperative languages, which are presented below in
the form of eight theses.
* Despite sharp criticism from John Backus and other
researchers von Neumann (mandatory) language still are
widely used and continue to hold strong, and in some areas -
the dominant position. It is logical to assume that this or
about this situation will continue in the future. A similar
position is taken by other authors, according to which the
mandatory languages ??"in the foreseeable future will remain
a dominant position in practical programming."
* In the coming century due to the further reduction of the
unit cost of equipment, many personal computer screen,
apparently to increase the size of the desk, which will
facilitate the visualization of programming by allowing
direct work with the drawings A1 or A0 PC screen on the
principle of WYSIWYG - What You See Is What You Get (What
you see is what you have). According to the hypothesis
developed, this will allow better use of the solid angle and
structure of the human field of vision, finally put an end
to the systematic underutilization of the rich possibilities
of the human eye, use the powerful reserves of simultaneous
perception and thus significantly increase the speed and
efficiency of the brain programmers and users. Given these
considerations and the seriousness of the problem of
productivity in programming, we assume that the expected
increase in the size of screens will provide a powerful
incentive for the large-scale replacement of the text in the
visual imperative languages.
* If we assume that the rendering imperative languages ??is
inevitable, it is advisable to carry out its not
spontaneously, but according to a pre-planned and
coordinated plan, one of the goals which should be
considered as a partial unification of languages.
* In this regard, the question arises: is it possible to un
References
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