Thinking with Diagrams '97 Position Statement
[This is a slightly modified and updated version
of my Position Statement for the
Thinking with Diagrams '97 workshop]
My interests in diagrammatic representation and reasoning have their origin
in the idea of computer-aided visual communication I had called in early
eighties by the name of "iconics" .
After moving with my work into other (though related) areas, I have returned
to the just emerging field of diagrammatic reasoning in early nineties.
To put my understanding of the subject in order, first I wrote
the survey paper . Subsequently, I am working,
till now, mostly on applications of diagrams in Qualitative Physics
, proposing recently a diagrammatic representation
of interval space and interval relations [4, 5].
My understanding of main issues of this field
can be summarized as follows:
Diagrams are a kind of analogical (or direct) knowledge representation
mechanism that is characterized by a parallel (though not necessarily
isomorphic) correspondence between the structure of the representation
and the structure of the represented .
E.g., relative positions and distances of certain marks on a map
are in direct correspondence to relative positions and distances
of the cities they represent, whereas in a propositional representation,
its parts or relationships between them need not correspond explicitly
to any parts and relations within the thing denoted.
The analogical representation can be said to model or depict
the thing represented, whereas the propositional representation
rather describes it. A similar distinction can be made regarding
the method of retrieving information from the representation.
The needed information can usually be simply observed (or measured)
in the diagram, whereas it must be inferred from the descriptions
of the facts and axioms comprising the propositional representation.
Knowledge representation and reasoning schemes originating in logic
(specifically, predicate calculus), called propositional, were for long
predominant in AI, as opposed to analogical schemes, including
diagrammatic representations. This was only partially due to limitations
of early computers and software in handling visual (or graphical) data.
Possibly even more important cause was that most of the researchers
in computer science and AI were (and still are) of mathematical background.
And mathematics has been generally ruled by an implicit dogma stating
that propositional reasoning using logic is the ultimate tool of precise
and formal thinking. Many mathematicians, especially logicians
(but even researchers in geometry!), tend to use diagrams rarely,
sometimes as heuristics to prompt certain trains of inference,
but mostly only as informal aids to understanding for uninitiated.
On occasion, some of them explicitly stated that the diagram has
no proper place in the proof as such. This state of affairs seems
to change now - as recent research has shown, it is quite possible
to formalize diagrammatic reasoning so as to make it no less precise
and formal than logical reasoning [7, 8].
The main advantages of diagrammatic representations
I would like to summarize as follows [2, 9]:
Diagrams are (at least) two-dimensional:
Richer possibilities of specifying and structuring in two dimensions
(topological, metric, morphological: by connectivity, proximity, shape,
etc.), compared to those available in a one-dimensional (linear) strings
of symbols in propositional formulations, lead to the reduction of:
the size of problem search space, i.e. the amount of data to be considered at the given inference step, and
the search costs, due to direct access to related elements eliminating the need of search for, say, matching symbolic labels.
Diagrams represent analogically:
They permit explicit representation and direct retrieval of information
(especially structural and spatial relations) that can be represented
only implicitly in other types of representations and then has to be
computed (or inferred), sometimes at great cost, to make it explicit for use.
Also, onstruction of an analogical representation (e.g. a diagram)
for the given set of facts usually causes the emergence of certain
new entities, properties of the problem elements, and relations between them
that follow from the given facts. These so-called "implicit facts"
or "emergent properties" are a kind of ready-made inferences that
can be directly read from the diagram at little or no cost.
Visual processing is easy:
Humans possess a well developed apparatus for making easy
perceptual inferences on a diagram. However, this does not yet apply
in full to computers which are still somewhat better at brute-force
number crunching, rather than at visual data analysis.
The main criteria characterizing usefullness of a diagrammatic language
can be summarized as follows [2, 10,
The visual language must be able to express all the facts
and only the facts we want to visualize. The first requirement is obvious;
the second takes into account the "implicit facts" feature -
the visual representation should not, as a by-product of representing
the wanted facts, represent implicitly unwanted (false, incorrect) facts.
It determines how easy it is to state the facts in the language
and how easy it is to perceive the facts from the representation.
Effectiveness is relative with respect to the user of the representation -
other rules apply when it is to be used by humans, other when by computers.
Human visual perception criteria are quite well known and codified
in extensive human factors and graphic design literature.
E.g., it is well known that color and shading are far less effective
in representing accurately numerical quantities than position on a scale
or length of a line. Much less can be said about the preferences
of computers, although some results, both negative and positive,
It takes into account what kind of operations the user wants to perform
with the representation, or what kind of tasks the representation
will be used to aid. Examples of different goals might be,
e.g., accurate value lookup (then representing the values as marks
on a numerical scale is advisable) versus general magnitude comparison
(then representing the values by size of graphical entities might be better).
- Z. Kulpa. Iconics: computer-aided visual communication.
- In: Levialdi S. (Ed.): Digital Image Analysis. (Proc. 2nd Conference
on Image Analysis and Processing. Fasano, Italy, 1982).
Pitman, London (1983), 280-282.
- Z. Kulpa. Diagrammatic representation and reasoning.
- Machine GRAPHICS & VISION 3(1/2),
[PostScript file available (380KB, gzipped)]
- M. Kleiber, Z. Kulpa. Computer-assisted hybrid reasoning
in simulation and analysis of physical systems.
- CAMES, 2(3),
[PostScript file available
- Z. Kulpa. Diagrammatic representation of interval space in proving
theorems about interval relations.
Reliable Computing, 3(3): 209-217, 1997.
Also presented at INTERVAL'96 Conference (Würzburg, Germany, September 30 - October 2, 1996).
[PostScript file of the Extended Abstract
(2 pp., 2 figs.) available (170KB, gzipped)]
- Z. Kulpa. Diagrammatic representation for a space of intervals.
- Machine GRAPHICS & VISION 6(1):
(Special Issue on Diagrammatic Representation and Reasoning)
[PostScript file available (620KB, gzipped)]
- A. Sloman. Afterthoughts on analogical representations.
- Proc. 1st Workshop on Theoretical Issues in Natural Language Processing
(TINLAP-1), Cambridge, MA (1975), 164-171. (Reprinted in: Brachman R.J.,
Levesque H.J. (Eds.): Readings in Knowledge Representation.
Morgan Kaufmann, San Mateo, CA (1985), 432-439).
- D. Wang, J.R. Lee. Visual reasoning: its formal semantics and applications.
- J. Visual Languages and Computing, 4(4),
- G. Allwein, J. Barwise, Eds. Logical Reasoning with Diagrams.
- Oxford Univ. Press, Oxford (1996).
- J.H. Larkin, H.A. Simon. Why a diagram is (sometimes) worth
ten thousand words.
- Cognitive Science, 11, (1987), 65-99.
- J. Mackinlay, M.R. Genesereth. Expressiveness and language choice.
- Data & Knowledge Engineering, 1, (1985), 17-29.
- S.F. Roth, J. Mattis. Data characterization for intelligent
- In: Human Factors in Computing Systems - VII
(Proc. of the Conf. on Computer-Human Interaction (CHI'90)).
ACM Press, (1990), 193-200.
- J. Bertin. Semiology of Graphics: Diagrams, Networks, Maps.
- University of Wisconsin Press (1983).
Back to my diagrammatics page
Original version: Jan 17, 1997.
Revised and updated: Jun 8, 1999.