=Paper=
{{Paper
|id=Vol-2344/paper8
|storemode=property
|title=Logic Graphs: A complete visualization method for logical languages based on Ch. S. Peirce’s existential graphs
|pdfUrl=https://ceur-ws.org/Vol-2344/paper8.pdf
|volume=Vol-2344
|authors=Dmitry Mouromtsev,Ildar Baimuratov
|dblpUrl=https://dblp.org/rec/conf/micsecs/MouromtsevB18
}}
==Logic Graphs: A complete visualization method for logical languages based on Ch. S. Peirce’s existential graphs==
https://ceur-ws.org/Vol-2344/paper8.pdf
Logic Graphs: A complete visualization method
for logical languages based on Ch. S. Peirce’s
existential graphs
Dmitry Mouromtsev and Ildar Baimuratov
ITMO University, 49 Kronverksky Pr., 197101 St. Petersburg, Russia
d.muromtsev@gmail.com, baimuratov.i@gmail.com
Abstract. Contemporary tools of graph data visualization do not allow
to work with logic expressions completely. The purpose of our research is
to develop a complete and convenient method for its visualization. The
developed prototype is intended for realization in the Ontodia library.
We analyze the tools of Ontodia with respect to description logic syntax
and adopt required tools from Ch. S. Pierce’s existential graphs system.
In addition, we propose to use schemes of expression for prompting to
a user, optimizing data structure and explicating an ontology structure.
We illustrate the developed method, named logic graphs, on a simple
demo-ontology.
Keywords: Visualization · Description Logic · Existential graphs.
1 Introduction
The purpose of our research is to develop a complete and convenient method
for logic expressions visualization. Contemporary tools of graph data visualiza-
tion do not allow to work with logic expressions completely, while necessity to
work with them in visual format is obvious. Providing to a user an opportunity
to observe ontologies constructed by axioms would be extremely useful. The
developed prototype should be suitable for realization in the Ontodia library [1].
2 Tools overview
We present an overview of the existing ontology visualization tools.
VOWL. In the current realization of VOWL [2] it is impossible to represent a
composition of two and more logic operators. For example, consider an axiom
W oman ≡ P erson u ¬M an (1)
which means “A woman is a person, who is not a man”. Its visualization is given
at the Fig. 1. In this axiom the negation of the concept Man is also a part of
conjunction, while at the visualization only the negation is represented. Besides,
VOWL language is not satisfactory, as logic relations are not visualized, but only
symbolically depicted, this in substance does not differ from a formula.
2 D.I. Mouromtsev, I.R. Baimuratov
Fig. 1. “A woman is a person, who is not a man” in VOWL
Venn diagrams. First, Venn diagrams [3] do not allow to represent logic rela-
tions unambiguously. For example, a conjunction c1 u c2 can be represented by
three different diagrams, that are given at the Fir. 2. Second, it is possible to
Fig. 2. c1 and c2 in Venn diagrams
visualize only concepts, but not roles with Venn diagrams.
Logic Graphs 3
Graphol. Graphol language [4] is not satisfactory as well, because, as in VOWL,
logic relations are not interpreted. Besides, Graphol language is overcomplicated
and hard to perceive. As an example, visualization of an axiom
CheezeyP izza ≡ P izza u ∃hasT opping.CheezeyT opping (2)
which means “CheezeyPizza is a pizza, which has cheezy topping”, from the
Pizza ontology [5] is given at the Fig 3.
Fig. 3. “CheezeyPizza is a pizza, which has cheezy topping” in Graphol
Therefore, it is necessary to develop a complete, explicit, but simple visual-
ization method for OWL language and description logic.
3 Logic Graphs
We consider visualization method completeness with respect to syntax of de-
scription logic, which is used to formulate ontology axioms. Basic description
logic syntax [6] consists of notations given at the Table 1.
Ontodia already has visualization tools for concepts, roles, inclusion, equivalence
and assertion. Developing our method, we intend to save all features presented in
Ontodia and, therefore, we use all existing visualization tools without changes. In
result, it is sufficient to find a way to visualize conjunction, disjunction, negation
and universal and existential restrictions.
The proposed method is based on the existential graphs, suggested by Ch. S.
Pierce [7]. In the existential graphs system sheets of assertion are considered. A
sheet of assertion denotes a space of true expressions, therefore, expressions sit-
uated on the sheet are conjuncted. Besides, there are cuts, that means inversion
4 D.I. Mouromtsev, I.R. Baimuratov
Table 1. Basic description logic syntax
Symbol Description Example Reading
C Concept c1 Concept c1
R Role op1 Role op1
u Conjunction c1 u c2 c1 and c2
t Disjunction c1 t c2 c1 or c2
¬ Negation ¬c1 Not c1
∀ Universal restriction ∀op1.c1 All op1-successors are in c1
∃ Existential restriction ∃op1.c1 An op1-successors exists in c1
v Inclusion c1 v c2 All c1 are c2
≡ Equivalence c1 ≡ c2 c1 is equivalent to c2
: Assertion a : c1, (a, b) : op1 a is a c1, a is op1-related to b
of the expression space truth-value, therefore, if an expression is situated in a
cut, it is negated. Finally, there are curves to denote relations. Visualization of
description logic expressions with existential graphs is illustrated in the Fig. 4.
Though existential graphs are sufficient for visualizing description logic, prac-
tically it is not convenient as existential graphs for complex expression are over-
complicated and hardly interpretable. It can be seen on the graph 6 already.
Therefore, we propose to develop a new system, which inherit current visualiza-
tion tools of Ontodia and extends it with required elements of the existential
graphs. The resulting system, named logic graphs, adjusted to Ontodia design,
is given at the Table 5.
Summing up, the proposal extends visualization tools of Ontodia with only
two elements: filling for negation and embedded concepts for conjunction. All
roles are considered as existentially restricted by default. Other required opera-
tions are expressed by means of these two.
Logic graphs are applicable for visualizing other logical systems. The original
existential graphs system, described in the Fig. 4, consists of Alpha and Beta
subsystems. The existential graphs 1-6 relate to Alpha subsystem, which is iso-
morphic to propositional logic. As logic graphs system inherit existential graphs
1-6 of Alpha as graphs 1-3 and its derivatives, logic graphs are applicable for vi-
sualizing propositional logic as well. However, they are more convenient as logic
graphs inherit simpler inclusion and equivalence representation from Ontodia.
Schemes. For interactive constructing of ontologies from axioms we propose to
use axiom schemes – logic structure of a formula regardless of situated concepts.
Axiom schemes serve to several purposes:
– Prompting to user, in which node of an ontology he should place a selected
concept. Elements of a schema are not clickable, after constructing is finished,
all additional elements are hidden.
– Decreasing the amount of data displayed at the data panel. One schema could
be associated with different axioms. In this case it is sufficient to display at
Logic Graphs 5
1. c1 2. ¬c1
3. c1 u c2 4. c1 t c2
5. c1 v c2 6. c1 ≡ c2
7. ∃op1.c1 8. ∀op1.c1
9. a : c1 10. (a, b) : op1
Fig. 4. Existential graphs
1 c1 2 ¬c1
3 c1 u c2 4 c1 t c2
5 c1 v c2 6 c1 ≡ c2
7 ∃op1.c1 8 ∀op1.c1
9 a : c1 10 (a, b) : op1
Fig. 5. Logic graphs
6 D.I. Mouromtsev, I.R. Baimuratov
the data panel only scheme and different axioms would be formed in result
of filling it with concepts.
– Explication of ontology structure and the process of its construction. As user
would have to construct an ontology by filling schemes, during this process
he would get an understanding of what is structural elements of an ontology
and how they are connected.
Concepts on the canvas has two states: “out of context” and “in context”. A
concept is out of context, if it is out of a scheme, and it is “in context” otherwise.
Concepts, which are out of context, can have only one instance at the canvas,
and multiple instances otherwise.
In addition, we propose two interactivity modes, that differ in organization
of the data panel. In the Mode 1 concepts from expressions that have identical
structure are grouped by its schemes. This approach allows to compress data at
the data panel, but still allows duplication. In the Mode 2 all concepts are listed
in the separate list. In this case concept duplication is excluded.
4 Examples
To illustrate the method we made a list of axioms:
1. SP bW oman ≡ P erson u ¬M an u ∃LiveIn.SP b (“Saint-Petersburg woman
is a person, who is not a man and lives in Saint-Petersburg”);
2. M skW oman ≡ P erson u ¬M an u ∃LivsIn.M sk (“Moscow woman is a
person, who is not a man and lives in Moscow”).
3. P erson ≡ M an t W oman (“Person is a man or a woman”).
4. W oman v ¬M an (“All women are not men”).
5. N ativeSP bW oman2ndGen v ∀HasP arent.BornInSP b (“All native Saint-
Petersburg women in second generation has only those parents, who were
born in Saint-Petersburg”).
6. P arentOf SP bP erson ≡ ∃HasChild.∃LiveIn.SP b (“Parent of an Saint-
Petersburg person is a person, who has a child, who lives in Saint-Petersburg”).
Their visualization is schematically represented below.
Modes. First, consider difference between the Modes 1 and 2. Expressions about
SPbWoman and M skW oman have identical structure, therefore, in the Mode 1
concepts from these expressions are grouped into one schema. Other expressions
have different schemes and there is a concept group for each axiom in the class
tree. This approach allows to reduce concept duplication in the tree. For example,
concepts M an and P erson from the two axioms with identical structure (about
SP bW oman and M skW oman) are not duplicated. It is notable, that it is one
and the same concept M an in two different groups and it has one and the same
IRI.
In the Mode 2 the user has two lists in the concept tree: a list of unique
concepts and a list of schemes. The difference is illustrated at the Fig. 6. Further
only Mode 1 is considered.
Logic Graphs 7
Mode 1 Mode 2
Fig. 6. Modes
Step 1. After data initilized, a user has a list of accessible schemes in the
class tree. The concepts Domain 1 and 2 are auxiliary and appears after axioms
being parsed. They are necessary as domains of the corresponding roles are not
specified in the formula. At this step all elements are clickable, it is possible to
drag them on the canvas. See Fig. 7.
Step 2. The user can drag all concepts from the tree. At this step all of them are
out of context. After the user dragged Schema 1 on the canvas, the structure of
the schema appears. Elements of a schema are not clickable, they prompt to the
user, which concepts it is able to drag in the context. As the schema embraces
Axiom 1 and 2, all their concepts are specified. It is notable that though the
concept M an is negated in the Schema 1, it is not filled being out of context,
but it is into the schema, where the concept is negated. See Fig. 8.
Step 3. The user fills the Schema 1 with accessible concepts in arbitrary order.
As soon as required domain and range for the role LiveIn are added, the con-
nection automatically appears. After the user dragged all required concepts into
the schema, it transforms into a particular axiom, depending on the concepts
situated, schema borders disappear. In this case, the Schema 1 resulted in the
Axiom 2. After that, the user can drag the concept M an on the canvas out of
context from another schema. See Fig. 9.
Step 4. All schemes are illustrated. See Fig. 10.
8 D.I. Mouromtsev, I.R. Baimuratov
Fig. 7. Step 1
Fig. 8. Step 2
Logic Graphs 9
Fig. 9. Step 3
Fig. 10. Step 4
10 D.I. Mouromtsev, I.R. Baimuratov
5 Conclusion
Summing up, the developed method allows to visualize semantics of logical ex-
pressions completely, unambiguously and in a simple way. It was illustratd on
a demo ontology. The future work will consider the possibility of visualizing
expressions of different logic languages.
References
1. Ontodia Homepage, http://ontodia.org/. Last accessed 29 Jan 2019
2. VOWL Homepage, http://vowl.visualdataweb.org/. Last accessed 29 Jan 2019
3. Venn J.: I. On the diagrammatic and mechanical representation of propositions and
reasonings. The London, Edinburgh, and Dublin Philosophical Magazine and Journal
of Science 10(59), 1–18 (1880)
4. Graphol Homepage, http://www.obdasystems.com/graphol. Last accessed 13 Mar
2019
5. Pizza Ontology, https://protege.stanford.edu/ontologies/pizza/pizza.owl. Last ac-
cessed 29 Jan 2019
6. Baader F., Calvanese D., McGuinness D., Nardi D., Patel-Schneider P. (eds.): The
Description Logic Handbook: Theory, Implementation, Applications. Cambridge Uni-
versity Press, Cambridge, UK (2003)
7. Peirce Ch.: Manuscript 514. 1909. Transcribed by Michel Balat with commentary
by J. F. Sowa. http://www.jfsowa.com/peirce/ms514.htm. Last accessed 29 Jan 2019