# Graphol: A Graphical Language for Ontology Modeling Equivalent to OWL 2

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## Abstract

**:**

## 1. Introduction

#### 1.1. Introducing the Graphol Language

#### 1.2. Paper Organization and Contributions

## 2. Related Work

## 3. Preliminaries

- -
- A nonempty interpretation domain ${\mathsf{\Delta}}^{I}={\mathsf{\Delta}}_{O}^{\phantom{\rule{3.33333pt}{0ex}}I}\cup {\mathsf{\Gamma}}_{V}$, where ${\mathsf{\Delta}}_{O}^{\phantom{\rule{3.33333pt}{0ex}}I}$ is the domain of objects, and ${\mathsf{\Gamma}}_{V}$ is the set of values previously introduced (indeed, in every interpretation each value is intepreted by itself);
- -
- An interpretation function ${\xb7}^{I}$ that assigns an element of ${\mathsf{\Delta}}_{O}^{\phantom{\rule{3.33333pt}{0ex}}I}$ to each constant in ${\mathsf{\Gamma}}_{O}$, and interprets each DL expression as shown in Table 1.

## 4. The Graphol Language

#### 4.1. Graphol Syntax

**Definition**

**1.**

- 1.
- A concept expression can be:
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- A concept node (in this case the sink is the node itself);
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- A domain or range restriction node, with label “exists”, “forall”, “$(x,-)$”, or “$(-,y)$”, taking as input a role expression and a concept expression (in this case the sink is the domain or range restriction node);
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- A domain or range restriction node, with label “self”, taking as input a role expression (in this case, the sink is the domain or range restriction node);
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- A domain restriction node, with label “exists”, “forall”, “$(x,-)$”, or “$(-,y)$”, taking as input an attribute expression and a value-domain expression (in this case, the sink is the domain restriction node);
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- A union or intersection node taking as input at least two concept expressions (in this case, the sink is the union or intersection node);
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- A complement node taking as input a concept expression (in this case, the sink is the complement node);
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- A one-of node taking as input at least an individual node (in this case, the sink is the one-of node).

- 2.
- A role expression can be:
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- A role node (in this case, the sink is the node itself);
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- An inverse node taking as input a role expression (in this case, the sink is the inverse node);
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- A complement node taking as input a role expression (in this case, the sink is the complement node);
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- A chain node taking as input n role expressions, with $n\le 2$, each associated to a label $1\le i\le n$ and such that there are no two input edges with the same label (in this case, the sink is the chain node).

- 3.
- An attribute expression can be:
- -
- An attribute node (in this case, the sink is the node itself);
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- A complement node taking as input an attribute expression (in this case, the sink is the complement node).

- 4.
- A value-domain expression can be:
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- A value-domain node (in this case, the sink is the node itself);
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- A range restriction node with label “exists” taking as input an attribute expression (in this case, the sink is the range restriction node);
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- A union or intersection node taking as input at least two value-domain expressions (in this case, the sink is the union or intersection node);
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- A complement node taking as input a value-domain expression (in this case, the sink is the complement node);
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- A one-off node taking as input at least a value node (in this case, the sink is the one-off node).

#### 4.2. Graphol Semantics

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- If $\mathsf{sk}\left({E}_{G}\right)$ is a concept, role, attribute, value-domain, or individual/value node with label S, then $\mathsf{\Lambda}\left({E}_{G}\right)=S$ (we are considering concept, role, attribute, value-domain, individual, and value alphabets as pairwise disjoint. Moreover, if $S=$“Top”, we assume that $\mathsf{\Lambda}$ returns the corresponding DL universal predicate, which depends on the form of $\mathsf{sk}\left({E}_{G}\right)$. Analogously if $S=$“Bottom”);
- -
- If $\mathsf{sk}\left({E}_{G}\right)$ is a domain restriction node with label “exists” (resp., “forall”, “$(x,-)$”, “$(-,y)$”), and $\mathsf{ar}\left({E}_{G}\right)=\{{\u03f5}_{RA},{\u03f5}_{CV}\}$, where either ${\u03f5}_{RA}$ is a Graphol role expression and ${\u03f5}_{CV}$ is a Graphol concept expression or ${\u03f5}_{RA}$ is a Graphol attribute expression and ${\u03f5}_{CV}$ is a Graphol value-domain expression, then $\mathsf{\Lambda}\left({E}_{G}\right)=\exists \mathsf{\Lambda}\left({\u03f5}_{RA}\right).\mathsf{\Lambda}\left({\u03f5}_{CV}\right)$ (resp., $\mathsf{\Lambda}\left({E}_{G}\right)=\forall \mathsf{\Lambda}\left({\u03f5}_{RA}\right).\mathsf{\Lambda}\left({\u03f5}_{CV}\right)$, $\mathsf{\Lambda}\left({E}_{G}\right)=\ge x\phantom{\rule{3.33333pt}{0ex}}\mathsf{\Lambda}\left({\u03f5}_{RA}\right).\mathsf{\Lambda}\left({\u03f5}_{CV}\right)$, $\mathsf{\Lambda}\left({E}_{G}\right)=\le y\phantom{\rule{3.33333pt}{0ex}}\mathsf{\Lambda}\left({\u03f5}_{RA}\right).\mathsf{\Lambda}\left({\u03f5}_{CV}\right)$);
- -
- if $\mathsf{sk}\left({E}_{G}\right)$ is a range restriction node with label “exists” (resp. “forall”, “$(x,-)$”, “$(-,y)$”), and $\mathsf{ar}\left({E}_{G}\right)=\{{\u03f5}_{R},{\u03f5}_{C}\}$, where ${\u03f5}_{R}$ is a Graphol role expression and ${\u03f5}_{C}$ is a Graphol concept expression, then $\mathsf{\Lambda}\left({E}_{G}\right)=\exists {\left(\mathsf{\Lambda}\left({\u03f5}_{R}\right)\right)}^{-}.\mathsf{\Lambda}\left({\u03f5}_{C}\right)$ (resp. $\mathsf{\Lambda}\left({E}_{G}\right)=\forall {\left(\mathsf{\Lambda}\left({\u03f5}_{R}\right)\right)}^{-}.\mathsf{\Lambda}\left({\u03f5}_{C}\right)$, $\mathsf{\Lambda}\left({E}_{G}\right)=\ge x\phantom{\rule{3.33333pt}{0ex}}{\left(\mathsf{\Lambda}\left({\u03f5}_{R}\right)\right)}^{-}.\mathsf{\Lambda}\left({\u03f5}_{C}\right)$, $\mathsf{\Lambda}\left({E}_{G}\right)=\le y\phantom{\rule{3.33333pt}{0ex}}{\left(\mathsf{\Lambda}\left({\u03f5}_{R}\right)\right)}^{-}.\mathsf{\Lambda}\left({\u03f5}_{C}\right)$);
- -
- If $\mathsf{sk}\left({E}_{G}\right)$ is a domain (resp., range) restriction node with label “self”, and $\mathsf{ar}\left({E}_{G}\right)=\left\{{\u03f5}_{RA}\right\}$, where ${\u03f5}_{RA}$ is a Graphol role expression, then $\mathsf{\Lambda}\left({E}_{G}\right)=\exists \mathsf{\Lambda}\left({\u03f5}_{RA}\right).Self$ (resp., $\mathsf{\Lambda}\left({E}_{G}\right)=\exists {\left(\mathsf{\Lambda}\left({\u03f5}_{RA}\right)\right)}^{-}.Self$);
- -
- If $\mathsf{sk}\left({E}_{G}\right)$ is a range restriction node with label “exists” and $\mathsf{ar}\left({E}_{G}\right)=\left\{{\u03f5}_{A}\right\}$, where ${\u03f5}_{A}$ is a Graphol attribute expression, then $\mathsf{\Lambda}\left({E}_{G}\right)=\exists {\left(\mathsf{\Lambda}\left({\u03f5}_{A}\right)\right)}^{-}$;
- -
- If $\mathsf{sk}\left({E}_{G}\right)$ is a union (resp. intersection or a one-of ) node and $\mathsf{ar}\left({E}_{G}\right)=\{{\u03f5}^{1},\dots ,{\u03f5}^{n}\}$, then $\mathsf{\Lambda}\left({E}_{G}\right)={\u2a06}_{i=1}^{n}\mathsf{\Lambda}\left({\u03f5}^{i}\right)$ (resp. $\mathsf{\Lambda}\left({E}_{G}\right)={\sqcap}_{i=1}^{n}\mathsf{\Lambda}\left({\u03f5}^{i}\right)$, $\mathsf{\Lambda}\left({E}_{G}\right)=\{\mathsf{\Lambda}\left({\u03f5}^{1}\right),\dots ,\mathsf{\Lambda}\left({\u03f5}^{n}\right)\}$);
- -
- if $\mathsf{sk}\left({E}_{G}\right)$ is a complement node and $\mathsf{ar}\left({E}_{G}\right)=\left\{\u03f5\right\}$, then $\mathsf{\Lambda}\left({E}_{G}\right)=\neg \mathsf{\Lambda}\left(\u03f5\right)$;
- -
- If $\mathsf{sk}\left({E}_{G}\right)$ is an inverse node and $\mathsf{ar}\left({E}_{G}\right)=\left\{{\u03f5}_{R}\right\}$, then $\mathsf{\Lambda}\left({E}_{G}\right)={\left(\mathsf{\Lambda}\left({\u03f5}_{R}\right)\right)}^{-}$;
- -
- If $\mathsf{sk}\left({E}_{G}\right)$ is a chain node and $\mathsf{ar}\left({E}_{G}\right)=\{{\u03f5}_{R}^{1},\cdots {\u03f5}_{R}^{n}\}$, where every ${\u03f5}_{R}^{i}$, where $1\le i\le n$, is a Graphol role expression that is connected to $\mathsf{sk}\left({E}_{G}\right)$ through an input edge with label i, then $\mathsf{\Lambda}\left({E}_{G}\right)=\mathsf{\Lambda}\left({\u03f5}_{R}^{1}\right)\circ \mathsf{\Lambda}\left({\u03f5}_{R}^{2}\right)\circ \cdots \circ \mathsf{\Lambda}\left({\u03f5}_{R}^{n}\right)$.

#### 4.3. Shortcuts

#### 4.4. Graphol and OWL 2

**Proviso.**Role expressions given as input to domain or range restriction nodes, to self nodes, inverse nodes, complement nodes, or chain nodes can be only basic role expressions. Attribute expressions given as input to domain restriction nodes can be only attribute nodes. Role and attribute expressions having the complement node as sink cannot be the source of any inclusion edge. Role expressions having the chain node as sink cannot be the target of any inclusion edge and can be included only in basic role expressions. Attribute expressions in input to range restriction nodes can be only attribute nodes. Graphol inclusion assertions between value-domain expressions must involve at least a value-domain node (i.e., the source or the target of the assertion must be an atomic data type). Finally, value-domains and values are as in OWL 2, and the ontology must obey the same global restrictions imposed on OWL 2 (e.g., only regular role hierarchies are allowed and only simple roles can occur in cardinality restrictions) [66,73].

**Theorem**

**1.**

**Proof.**

**Theorem**

**2.**

**Proof.**

#### 4.5. Example

## 5. Comparison with UML Class Diagrams

## 6. User Evaluation Study

#### 6.1. Setup of the Study

#### 6.2. Objectives of the Study

- Evaluate the difficulty of using the Graphol language for ontology comprehension and editing by users very experienced in conceptual modeling and (in some cases) with basic skills in logics and ontologies.
- Evaluate the difficulty of approaching and learning the Graphol language for users with only basic knowledge of conceptual modeling and little or no experience with ontologies, both in isolation, and in comparison with another graphical ontology language that is heavily based on a formalism that is already familiar to them.

#### 6.3. Participants

#### 6.4. Ontology Models

#### 6.5. Language for Comparison

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- OWLGrEd concepts are represented as UML classes, without operations;
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- OWLGrEd attributes are represented as UML class attributes, but with different default cardinalities;
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- OWLGrEd roles are represented as UML binary associations with the arrow indicating the direction, from the domain to the range. OWLGrEd roles are thus typed in both the domain and the range;
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- OWLGrEd cardinalities on roles are represented as UML cardinalities on roles, with the possibility of further refining the cardinality;
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- OWLGrEd cardinality restrictions on attributes are represented as UML cardinalities on attributes;
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- OWLGrEd inclusions between concepts are represented as UML ISAs between classes;
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- OWLGrEd generalizations are represented as UML generalizations, using a special graphical symbol;
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- OWLGrEd uses the OWL 2 Manchester syntax to specify expressions which denote complex concepts;
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- OWLGrEd role restrictions are represented as red arrows from the concept which is included in the restriction to the concept that qualifies the restriction, labeled with the name of the restricted role.

#### 6.6. Tasks

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- Was the question clear?Not clear at all $\hspace{1em}\square -\hspace{1em}\square -\hspace{1em}\square -\hspace{1em}\square -\hspace{1em}\square $ Very clear
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- Were you able to easily answer this question?Not at all $\hspace{1em}\square -\hspace{1em}\square -\hspace{1em}\square -\hspace{1em}\square -\hspace{1em}\square $ Absolutely

#### 6.7. Structure of the Study

**Beginners:**

- Introduction and briefGrapholtutorial (15 min): a general introduction to the purpose of the experiment, and a brief tutorial on ontologies and on the Graphol language.
- Brief OWLGrEd tutorial (15 min): a brief tutorial on the OWLGrEd language.
- Brief user background questionnaire (5 min): participants had to answer a brief questionnaire in which they were asked to provide some personal background information, as well as to rate their knowledge of conceptual modeling and ontologies on a scale from 1 to 5 (with 1 indicating extremely low and 5 extremely high expertise), to indicate how many years they had of experience with ontologies (if any), whether they were familiar with some of the more popular ontology editors and knowledge representation and conceptual modeling formalisms, and whether they had any experience with ontologies in real-life scenarios or in manually creating or editing ontologies.
- LUBM comprehension tasks (40 min): each user was asked to answer ten questions on the LUBM model they were provided. Half of the users were provided a Graphol version of the LUBM model, and half an OWLGrEd version.
- Pizza comprehension tasks (40 min): each user was asked to answer ten questions on the Pizza model they were provided. Half of the users (those which were provided the OWLGrEd version of the LUBM model) were provided a Graphol version of the Pizza model, and half an OWLGrEd version.
- Ex-post survey (10 min): after all the comprehension tasks were completed, we asked the participants to compile a short survey regarding their experience. The survey required the users to rate, on a scale from 0 to 4, the general difficulty of the comprehension tasks, the difficulty of learning the Graphol and OWLGrEd symbols, the difficulty of using Graphol and OWLGrEd to read ontologies, and, optionally, to indicate aspects of Graphol and OWLGrEd which they particularly liked, or that they would like to see improved.

**Experts:**

- Introduction and briefGrapholtutorial (30 min): participants were given the same introductory tutorial on ontologies and on the Graphol language as the beginners, with the addition of some more complex features on the Graphol language which were featured in the expert questionnaire but not the beginner questionnaire.
- Brief user background questionnaire (5 min): we asked the participants to fill out the same background questionnaire given to the beginners.
- Grapholcomprehension tasks (35 min): after completing the introductory part on Graphol, each user was asked to answer ten questions on the Graphol model of the Pizza ontology they were provided.
- Grapholediting tasks (35 min): we asked each participant to perform ten editing tasks on the Graphol model of the Family ontology they were provided.
- Ex-post survey (5 min): after carrying out both the comprehension and editing tasks, the users were asked to fill out a brief survey, analogous to the one given to the beginners.

- -
- All participants, in support of their tasks, were provided with documentation regarding the languages in play for that specific task. Specifically, the questionnaire included some cheat sheets which recapped the symbols of the Graphol and the OWLGrEd language (the latter only for tests carried out by beginners) and their meaning, along with some examples of the representation of some of the most common ontology expressions and assertions in the two formalisms. Additionally, users were provided with a printout of the slides of the introductory tutorials.
- -
- The order in which the tasks were presented in the questionnaires was intentionally random, i.e., not linked to the expected difficulty of each task. This choice was made in order to compensate for a potential bias given by the learning curve of familiarizing with Graphol or OWLGrEd during the course of the tasks. In other words, we wanted to avoid facilitating the participants by allowing them to face easier questions at the beginning of each task, and more difficult ones at the end, when they would probably have gained familiarity with the language.
- -
- The experimental design method we chose for the comparative study between Graphol and OWLGrEd is the within-subjects method, common in HCI [75]. This choice, as opposed to the between-subjects technique, was made mainly due to the limited number of participants to the experiment. Therefore, each user was asked to complete the comprehension tasks both for the Graphol language and for the OWLGrEd language. In order to avoid the transfer of learning effects between tasks, we split the ten users into two groups of five, and asked the first group to first carry out the comprehension tasks on the Graphol version of the LUBM model, and then the comprehension tasks on the OWLGrEd version of the Pizza model, and the second group to do the opposite.

#### 6.8. Study Results

#### 6.9. Post-Questionnaire Analysis

## 7. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 3.**Example of a disjoint concept hierarchy represented in Graphol with (

**left-hand side figure**) and without (

**right-hand side figure**) the disjoint union node.

**Figure 4.**Example of an existential restriction in Graphol with (

**left-hand side figure**) and without (

**right-hand side figure**) the compact notation.

**Figure 5.**Example of globally functional role represented without the compact notation (

**left-hand side**), and examples (

**right-hand side**) of globally functional role, inverse functional role, functional and inverse functional role, and functional attribute (resp., top left, top right, bottom left, and bottom right) represented with the compact notation.

**Figure 10.**Correctness results for user tests. In boxplots (

**a**,

**b**) the scale is 0–20 (sum of correctness scores for five students on questions graded from 0 to 4); in boxplot (

**c**) the scale is 0–32 (sum of correctness scores for eight students on questions graded from 0 to 4).

Construct | Syntax | Semantics |
---|---|---|

Atomic concept | A | ${A}^{I}\subseteq {\mathsf{\Delta}}_{O}^{\phantom{\rule{3.33333pt}{0ex}}I}$ |

Atomic role | P | ${P}^{I}\subseteq {\mathsf{\Delta}}_{O}^{\phantom{\rule{3.33333pt}{0ex}}I}\times {\mathsf{\Delta}}_{O}^{\phantom{\rule{3.33333pt}{0ex}}I}$ |

Atomic attribute | U | ${U}^{I}\subseteq {\mathsf{\Delta}}_{O}^{\phantom{\rule{3.33333pt}{0ex}}I}\times {\mathsf{\Gamma}}_{V}$ |

Atomic value-domain | T | ${T}^{I}\subseteq {\mathsf{\Gamma}}_{V}$ |

Universal concept | ${\top}_{C}$ | ${\mathsf{\Delta}}_{O}^{\phantom{\rule{3.33333pt}{0ex}}I}$ |

Universal role | ${\top}_{R}$ | ${\mathsf{\Delta}}_{O}^{\phantom{\rule{3.33333pt}{0ex}}I}\times {\mathsf{\Delta}}_{O}^{\phantom{\rule{3.33333pt}{0ex}}I}$ |

Universal attribute | ${\top}_{A}$ | ${\mathsf{\Delta}}_{O}^{\phantom{\rule{3.33333pt}{0ex}}I}\times {\mathsf{\Gamma}}_{V}$ |

Universal value-domain | ${\top}_{D}$ | ${\mathsf{\Gamma}}_{V}$ |

Empty concept, role attribute, value-domain | ${\perp}_{C},{\perp}_{R},{\perp}_{A},{\perp}_{D}$ | ∅ |

Unqualified role existential restriction | $\exists R$ | $\{\phantom{\rule{3.33333pt}{0ex}}o\mid \exists {o}^{\prime}.\phantom{\rule{0.166667em}{0ex}}(o,{o}^{\prime})\in {R}^{I}\phantom{\rule{3.33333pt}{0ex}}\}$ |

Qualified role existential restriction | $\exists R.C$ | $\{\phantom{\rule{3.33333pt}{0ex}}o\mid \exists {o}^{\prime}.\phantom{\rule{0.166667em}{0ex}}(o,{o}^{\prime})\in {R}^{I}\wedge {o}^{\prime}\in {C}^{I}\phantom{\rule{3.33333pt}{0ex}}\}$ |

Qualified role universal restriction | $\forall R.C$ | $\{\phantom{\rule{3.33333pt}{0ex}}o\mid \forall {o}^{\prime}.\phantom{\rule{0.166667em}{0ex}}(o,{o}^{\prime})\in {R}^{I}\to {o}^{\prime}\in {C}^{I}\}$ |

Qualified maximum cardinality role restriction | $\le n\phantom{\rule{0.166667em}{0ex}}R.C$ | $\{\phantom{\rule{3.33333pt}{0ex}}o\mid \u266f\{{o}^{\prime}\mid (o,{o}^{\prime})\in {R}^{I}\wedge {o}^{\prime}\in {C}^{I}\phantom{\rule{3.33333pt}{0ex}}\}\le n\}$ |

Qualified minimum cardinality role restriction | $\ge n\phantom{\rule{0.166667em}{0ex}}R.C$ | $\{\phantom{\rule{3.33333pt}{0ex}}o\mid \u266f\{{o}^{\prime}\mid (o,{o}^{\prime})\in {R}^{I}\wedge {o}^{\prime}\in {C}^{I}\phantom{\rule{3.33333pt}{0ex}}\}\ge n\}$ |

Self restriction | $\exists R.Self$ | $\{\phantom{\rule{3.33333pt}{0ex}}o\mid (o,o)\in {R}^{I}\}$ |

Unqualified attribute existential restriction | $\exists V$ | $\{\phantom{\rule{3.33333pt}{0ex}}o\mid \exists v.\phantom{\rule{0.166667em}{0ex}}(o,v)\in {V}^{I}\phantom{\rule{3.33333pt}{0ex}}\}$ |

Qualified attribute existential restriction | $\exists V.F$ | $\{\phantom{\rule{3.33333pt}{0ex}}o\mid \exists v.\phantom{\rule{0.166667em}{0ex}}(o,v)\in {V}^{I}\wedge v\in {F}^{I}\phantom{\rule{3.33333pt}{0ex}}\}$ |

Qualified attribute universal restriction | $\forall V.F$ | $\{\phantom{\rule{3.33333pt}{0ex}}o\mid \forall v.\phantom{\rule{0.166667em}{0ex}}(o,v)\in {V}^{I}\to v\in {F}^{I}\}$ |

Qualified maximum cardinality attribute restriction | $\le n\phantom{\rule{0.166667em}{0ex}}V.F$ | $\{\phantom{\rule{3.33333pt}{0ex}}o\mid \u266f\{v\mid (o,v)\in {V}^{I}\wedge v\in {F}^{I}\phantom{\rule{3.33333pt}{0ex}}\}\le n\}$ |

Qualified minimum cardinality attribute restriction | $\ge n\phantom{\rule{0.166667em}{0ex}}V.F$ | $\{\phantom{\rule{3.33333pt}{0ex}}o\mid \u266f\{v\mid (o,v)\in {V}^{I}\wedge v\in {F}^{I}\phantom{\rule{3.33333pt}{0ex}}\}\ge n\}$ |

One-of (concept) | $\{{c}_{1},\cdots ,{c}_{n}\}$ | $\{{c}_{1}^{I},\cdots ,{c}_{n}^{I}\}$ |

One-of (value-domain) | $\{{w}_{1},\cdots ,{w}_{n}\}$ | $\{{w}_{1},\cdots ,{w}_{n}\}$ |

Attribute range | $\exists {V}^{-}$ | $\{\phantom{\rule{3.33333pt}{0ex}}v\mid \exists o.\phantom{\rule{0.166667em}{0ex}}(o,v)\in {V}^{I}\phantom{\rule{3.33333pt}{0ex}}\}$ |

Inverse role | ${P}^{-}$ | $\{\phantom{\rule{3.33333pt}{0ex}}(o,{o}^{\prime})\mid ({o}^{\prime},o)\in {P}^{I}\phantom{\rule{3.33333pt}{0ex}}\}$ |

Role chain | $R\circ R$ | $\{\phantom{\rule{3.33333pt}{0ex}}(o,{o}^{\prime})\mid \exists {o}^{\u2033}.\phantom{\rule{0.166667em}{0ex}}(o,{o}^{\u2033})\in {R}_{1}^{I}\wedge ({o}^{\u2033},{o}^{\prime})\in {R}_{2}^{I}\}$ |

Concept negation | $\neg C$ | ${\mathsf{\Delta}}_{O}^{\phantom{\rule{3.33333pt}{0ex}}I}\setminus {C}^{I}$ |

Role negation | $\neg R$ | $({\mathsf{\Delta}}_{O}^{\phantom{\rule{3.33333pt}{0ex}}I}\times {\mathsf{\Delta}}_{O}^{\phantom{\rule{3.33333pt}{0ex}}I})\setminus {R}^{I}$ |

Attribute negation | $\neg V$ | $({\mathsf{\Delta}}_{O}^{\phantom{\rule{3.33333pt}{0ex}}I}\times {\mathsf{\Gamma}}_{V})\setminus {V}^{I}$ |

Value-domain negation | $\neg F$ | $\left({\mathsf{\Gamma}}_{V}\right)\setminus {F}^{I}$ |

Concept conjunction | ${C}_{1}\sqcap \cdots \sqcap {C}_{n}$ | ${C}_{1}^{I}\cap \cdots \cap {C}_{n}^{I}$ |

Concept disjunction | ${C}_{1}\bigsqcup \cdots \bigsqcup {C}_{n}$ | ${C}_{1}^{I}\cup \cdots \cap {C}_{n}^{I}$ |

**Table 2.**Correspondence between Graphol and DL for concept and role expressions of depth 0 or 1. C, ${C}_{1}$, and ${C}_{2}$ denote atomic concepts, and R, ${R}_{1}$, and ${R}_{2}$ denote atomic roles.

Graphol | DL | |
---|---|---|

Atomic concept | C | |

Role domain restriction | $\exists R.C\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\forall R.C$ $\ge xR.C\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\le yR.C$ | |

Role range restriction | $\exists {R}^{-}.C\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\forall {R}^{-}.C$ $\ge x{R}^{-}.C\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\le y{R}^{-}.C$ | |

Attribute domain restriction | $\exists V.F\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\forall V.F$ $\ge xV.F\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\le yV.F$ | |

Concept intersection | ${C}_{1}\sqcap {C}_{2}$ | |

Concept union | ${C}_{1}\bigsqcup {C}_{2}$ | |

Concept complement | $\neg C$ | |

One-of (concept) | $\{a,b,c\}$ | |

Self restriction | $\exists R.Self$ | |

Atomic role | P | |

Role inverse | ${R}^{-}$ | |

Role complement | $\neg R$ | |

Chain | ${R}_{1}\circ {R}_{2}$ |

**Table 3.**Correspondence between Graphol and DL for attribute and value-domain expressions of depth 0 or 1. V denotes an atomic attribute, and F, ${F}_{1}$, and ${F}_{2}$ denote an atomic value-domain.

Graphol | DL | |
---|---|---|

Atomic attribute | U | |

Attribute complement | $\neg U$ | |

Atomic value-domain | T | |

Attribute range existential restriction | $\exists {V}^{-}$ | |

Value-domain intersection | ${F}_{1}\sqcap {F}_{2}$ | |

Value-domain union | ${F}_{1}\bigsqcup {F}_{2}$ | |

Value-domain complement | $\neg F$ | |

One-of (value-domain) | {“1”, “2”, “3”} |

**Table 4.**Statistics of the participants (“Beg.” indicates statistics for beginners, “Exp.” indicates statistics for experts): for Education, 1 = Bachelor’s degree, 2 = Master’s degree, 3 = Ph.D; conceptual modeling and ontology knowledge are on a scale from 1 to 5, with 1 indicating no knowledge.

Age | Education | Conceptual Modeling | Ontology Knowledge | |||||
---|---|---|---|---|---|---|---|---|

Knowledge | ||||||||

Beg. | Exp. | Beg. | Exp. | Beg. | Exp. | Beg. | Exp. | |

Min | 22 | 27 | 1 | 2 | 2 | 3 | 1 | 1 |

Max | 28 | 47 | 2 | 3 | 4 | 5 | 2 | 3 |

Median | 24.5 | 31 | 1 | 2 | 3 | 4 | 1 | 2.5 |

Mean | 24.7 | 34.2 | 1.1 | 2.4 | 3.3 | 4.2 | 1.4 | 2.4 |

St.dev. | 2.3 | 6.7 | 0.3 | 0.5 | 0.7 | 0.7 | 0.5 | 0.7 |

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**MDPI and ACS Style**

Lembo, D.; Santarelli, V.; Savo, D.F.; De Giacomo, G.
Graphol: A Graphical Language for Ontology Modeling Equivalent to OWL 2. *Future Internet* **2022**, *14*, 78.
https://doi.org/10.3390/fi14030078

**AMA Style**

Lembo D, Santarelli V, Savo DF, De Giacomo G.
Graphol: A Graphical Language for Ontology Modeling Equivalent to OWL 2. *Future Internet*. 2022; 14(3):78.
https://doi.org/10.3390/fi14030078

**Chicago/Turabian Style**

Lembo, Domenico, Valerio Santarelli, Domenico Fabio Savo, and Giuseppe De Giacomo.
2022. "Graphol: A Graphical Language for Ontology Modeling Equivalent to OWL 2" *Future Internet* 14, no. 3: 78.
https://doi.org/10.3390/fi14030078