# Low-temperature thermal expansion of the topological material candidates
$\beta$-PtBi2 and $\beta$-Bi2Pd

V. F. Correa<EMAIL_ADDRESS>P. Pedrazzini  D. G. Franco  A. J.
Rosa  B. Rubrecht  N. Haberkorn  Centro Atómico Bariloche and Instituto
Balseiro, CNEA, CONICET and U. N. de Cuyo, 8400 San Carlos de Bariloche,
Argentina  Leibniz IFW Dresden, Helmholtzstraße 20, Dresden, Germany
Institute of Advanced Materials (INAM), Universitat Jaume I, 12006 Castelló,
Spain  Institute for Solid State and Material Physics, TU Dresden, 01062
Dresden, Germany

###### Abstract

We report on the low-temperature ($T<$ 120 K) thermal expansion of the
bismuth-based topological semimetal $\beta$-PtBi2 and topological
superconductor $\beta$-Bi2Pd candidates. The linear thermal-expansion
coefficient of tetragonal $\beta$-Bi2Pd shows a pronounced anisotropy between
the $a$\- and $c$-axis while the volume thermal-expansion coefficient
$\alpha_{V}$($\beta$-Bi2Pd) is considerable larger than
$\alpha_{V}$($\beta$-PtBi2). The coefficient $\alpha_{V}$($\beta$-PtBi2)
nearly matches the experimental specific heat, from which a Debye temperature
$\theta_{D}=$ 199 K is obtained. On the other hand,
$\alpha_{V}$($\beta$-Bi2Pd) reasonably fits the Debye model with $\theta_{D}=$
138 K, extracted from the low-temperature specific heat. An almost constant
Grüneisen parameter $\Gamma\approx$ 2 is obtained for both compounds. No
magnetostriction is observed in any of both compounds up to $\mu_{0}H=$ 16 T.
We compare our results with other Bi-based topological materials.

###### keywords:

crystal growth , thermal expansion , topological materials

##  1 Introduction

Topological matter (TM) has received a considerable interest in recent years
[1, 2, 3, 4, 5]. Unlike traditional states of matter, the character of
topological phases are not determined by any physicochemical property of the
underlying material but, ultimately, by the presence or lack of some internal
symmetry. The very peculiarity of the electronic-topological materials is that
their bulk properties differ from those at the surface [1].

In topological insulators (TI), a crossing between the valence and conduction
bands occurs. The residual spin-orbit interaction may open a considerable full
gap at the Fermi level, ultimately giving rise to metallic surface states
while the bulk material remains insulating [1]. The metallic character of the
surface cannot be destroyed by passivation or partial doping. It depends only
on certain symmetries, particularly the time-inversion symmetry [1]: they are
said to be symmetry-protected surface states. On the other hand, in
topological semimetals (TSM) the gap at the Fermi level is not complete. At
high-symmetry locations of the reciprocal space (Dirac points) gapless
touching bands can result in linearly dispersive energies, as it occurs in
graphene. Unlike graphene, however, TSM are three-dimensional systems. Again,
distinctive metallic topology-protected surface states appear [4].

Other gapped systems can also show robust metallic surface states. This is the
case of topological superconductors (TSC), where the bulk of the system
remains gapped and superconductor while the surface shows symmetry-protected
metallic states filled up with very particular quasi-particles known as
Majorana fermions [2, 3, 6].

The list of proposed topological compounds is quite extensive. The first
experimental realization of a TI was in HgTe/CdTe quantum wells [7, 8] while
Bi1-xSbx [9], Bi2Se3 [10, 11], Bi2Te3 [10], Sb2Te3 [10], and the rare earth
Heusler compounds LnPtBi (Ln=Y, La, and Lu) [12, 13, 14] can be listed as
intrinsic TI. On the other hand, some TSM candidates are WTe2 [15], MoTe2
[16], PtBi2 [17], TaAs [18], TaP [19], NbAs [20], NbP [21], and Cd3As2 [22]
among many others. Although far from conclusive, the first proposed TSC was
Sr2RuO4 [23]. Some other compounds with experimental results supporting
topological superconductivity are CuxBi2Se3 [24], Sn1-xInxTe [25], BiPd [26],
and TaC [27]. An almost universal feature of topological compounds is the
presence of a metalloid, a chalcogen or close element (P, Sn, Bi) in their
unit formula.

Besides the natural interest in TM from the purely theoretical and fundamental
side, some topological systems are also promising candidates for applications
in spintronics and high-speed electronics [28]. Most experimental and
theoretical studies have been directed to the low-temperature electronic
properties. On the other hand, lattice studies have received poor or null
attention. A very small electron-lattice coupling explains this fact.
Nonetheless, and mostly with prospective applications in mind, it is important
to know the lattice behavior as the temperature changes. Large differential
thermal expansions should always be avoided in devices.

In this work we report on the low-temperature ($T<$ 120 K) linear and volume
thermal expansion measurements of the bismuth-based binary compounds
$\beta$-PtBi2 and $\beta$-Bi2Pd. The former is a TSM candidate that shows one
of the largest non-saturating transverse magnetoresistances [17]. The latter
is a superconductor with one of the highest critical temperatures ($T_{c}$ =
5.4 K) among the TSC candidates [29]. We show that although both systems have
comparable unit-cell volumes, their volume thermal-expansion coefficients
differ substantially but basically track the specific heat. The linear
thermal-expansion coefficient of tetragonal $\beta$-Bi2 is markedly
anisotropic.

The rest of this manuscript is organized as follows. Section 2 is devoted to
the crystal synthesis and characterization of our samples. Section 3 presents
the thermal-expansion results and a comparison with other bismuth-based
systems in the literature. Finally, in section 4, we present the conclusions.

##  2 Crystal synthesis and characterization

Figure 1: (a) X-ray diffraction scan of a $\beta$-PtBi2 crystal corresponding
to the (hhh) family planes. Inset: crystal image with facets labeled. (b)
Temperature dependence of the magnetic susceptibility under an applied field
$H$ = 5000 Oe. (c) Temperature dependence of the electrical resistivity.
Inset: magnetoresistance $\left(R(H)-R(0)\right)/R(0)$ as a function of the
applied magnetic field $H$. (d) Specific heat divided by temperature versus
square temperature along with a polynomial fit (see text for details).

Single crystals of $\beta$-PtBi2 and $\beta$-Bi2Pd were grown by the self-flux
technique. Initial mixtures of pure elements (Bi: $>$ 99.999$\%$; Pt: $>$
99.98$\%$; Pd: $>$ 99.9$\%$) are placed in previously evacuated quartz tubes
that are then sealed after incorporating a small amount of Ar. Mixtures are
homogeneized at high temperature and then cooled down following specific
ramps. Both $\beta$ phases are metastable at room temperature so the initial
mixture must be quenched to obtain good crystalline samples. Crystals
decompose when temperature is raised. In particular, $\beta$-Bi2Pd starts to
transform into the stable $\alpha$-Bi2Pd phase at temperatures as low as 60
∘C, which critically affects the sample quality [30]. Energy-dispersive x-ray
spectroscopy (EDS) and Cu $K_{\alpha}$ x-ray diffraction (XRD) scans confirm
the correct stoichiometry and crystal structure of the samples. $\beta$-PtBi2
crystallizes in the cubic $Pa\bar{3}$ structure, while $\beta$-Bi2Pd does it
in the tetragonal $I4/mmm$ structure.

Magnetization (M) experiments were performed in a Quantum Design MPMS
magnetometer while a standard four-probe setup was used in the electrical
transport measurements. Specific heat was measured in a PPMS using a standard
relaxation technique.

Figure 2: (a) X-ray diffraction scan of a $\beta$-Bi2Pd crystal corresponding
to the (00l) family planes. Inset: crystal image. (b) Temperature dependence
of the magnetic susceptibility under an applied field $H$ = 10000 Oe. (c)
Temperature dependence of the electrical resistivity. Lower inset:
superconducting transition from resistivity at $T_{c}=$ 5.1 K. Upper inset:
Magnetic moment as a function of temperature showing a Meissner state below
$T_{c}=$ 5.1 K. (d) Specific heat divided by temperature versus temperature at
zero field and $H=$ 6 kOe along with a polynomial fit (see text for details).
Inset: Electronic contribution to the specific heat $c_{p}^{el}$.

###  2.1 $\beta$-PtBi2

The initial mixture (Pt:Bi = 3-4:97-96) is homogeneized at 500 ∘C for 4-10
hours, then slowly cooled down to 300 ∘C in seven days. At this final
temperature, the grown crystals are separated from the residual Bi-rich flux
using a centrifugue (alumina wool is used as a filter) in a very rapid
process, which also acts as a quench [31]. Well faceted
($\left\\{100\right\\}$ and $\left\\{111\right\\}$) mm-sized crystals of up to
1 gram are obtained, as seen in Fig. 1(a). If a slower cooling ramp is used
(several weeks), smaller crystals are obtained.

A typical XRD scan corresponding to the (hhh) planes is shown in Fig. 1(a).
Additional peaks coming from Cu $K_{\beta}$ and W $L_{\alpha 1}$ and
$L_{\alpha 2}$ (typical for copper tubes) are also observed for the very
intense (222) and (333) reflections. Low intensity wide peaks around 40∘ and
60∘ come from the grease used to attach the crystal to the sample holder. An
average (from several batches) lattice parameter $a=$ 6.696(5) Å is
calculated, which agrees with the reported value [17].

Fig. 1(b) displays the magnetic susceptibility $\chi=M/H$ under an applied
field $H$ = 5000 Oe. An almost temperature-independent diamagnetic response is
observed, which could arise not only from the ion cores, but also from the
conduction electrons (Landau diamagnetism) as in doped semiconductors, given
the low electron effective masses reported in this compound. Comparable $\chi$
values are measured in the trigonal $\gamma$-PtBi2 polymorph [32].

The temperature dependence of the electrical resistivity is depicted in Fig.
1(c). A metallic behavior is observed down to 2 K. The residual resistivity
$\rho\left(0\right)\approx$ 0.065 $\mu\Omega\cdot$cm and the residual
resistivity ratio $RRR=\rho(300K)/\rho(0)\approx$ 515 are comparable to those
reported in the literature [17, 31]. Our samples quality is further confirmed
by the magnetoresistance $MR=\left(R(H)-R(0)\right)/R(0)$ shown in the inset
of Fig. 1(c), where at $T=$ 2 K, clear Shubnikov de Haas oscillations are
observed at $\mu_{0}H>$ 10 T. On the other hand, the $MR$ has a parabolic
field dependence in the whole field range ($\mu_{0}H=$ 16 T) and reaches a
magnitude of $MR$ (9T) = 7700, about 46 % of the highest reported value [17].

Specific heat $c_{p}$ divided by temperature as a function of square
temperature is shown in Fig. 1(d). The experimental data is fitted with the
polynomial expression $c_{p}/T=\gamma+\beta T^{2}+\epsilon T^{4}$ to give a
Sommerfeld coefficient $\gamma$ = 2 mJ mol-1 K-2 and $\beta=$ 7.4 $\cdot$ 10-4
J mol-1 K-4. According to the Debye model $\beta$ = 234
3$N_{a}\,k_{B}\,\theta_{D}^{-3}$ ($N_{a}$ and $k_{B}$ are the Avogadro number
and the Boltzmann constant, respectively), from which a Debye temperature
$\theta_{D}=$ 199(1) K is obtained. This value is considerable higher than in
$\gamma$-PtBi2 ($\theta_{D}=$ 145 K) [33]. The layered structure of this
trigonal structure, as compared with the highly compact cubic structure of
$\beta$-PtBi2 may explain this difference.

###  2.2 $\beta$-Bi2Pd

The initial mixture (Bi:Pd = 2:1) is homogeneized at 900 ∘C for 4 hours, then
cooled down to 490 ∘C in four days. From this temperature the still liquid
mixture is further cooled down to 395 ∘C in eight days. At this temperature
the quartz capsule is quenched in cold water [34]. Large plateled-like
crystals of several mm are obtained as seen in Fig. 2(a). The large face is
parallel to (00l). A typical XRD scan corresponding to this family of planes
is shown in Fig. 2(a). No spurious peaks other than Cu $K_{\beta}$ reflections
are observed. Lattice parameters $c=$ 12.98(2) Å and $a=$ 3.369(6) Å are
calculated from (00l) and (h0h) scans, in good agreement with previously
reported values [29, 34].

Fig. 2(b) displays the magnetic susceptibility $\chi$ under an applied field
$H$ = 10000 Oe along the $a$\- and $c$-axis. An almost temperature independent
diamagnetic response is observed at all temperatures, in agreement with
previous measurements [35]. An important magnetic anisotropy can be seen as
well. The temperature dependence of the electrical resistivity is depicted in
the main panel of Fig. 2(c). A metallic behavior is observed. The residual
resistivity $\rho\left(0\right)\approx$ 23 $\mu\Omega\cdot$cm and $RRR\approx$
3 are average among the reported values [36, 37]. At $T_{c}=$ 5.1 K, the
system becomes superconducting (see lower inset of Fig. 2(c)). This is
confirmed by zero-field-cooling magnetic moment measurements ($H=$ 3 Oe) that
show a clear Meissner state below $T_{c}$ as shown in the upper inset of Fig.
2(c). Again, this critical temperature is average but lower than the higher
reported value (= 5.4 K) [29].

Bulk superconductivity is supported by the specific heat as seen in Fig. 2(d).
Zero-field $c_{p}$ shows a transition around 5 K that is suppressed by a
moderate magnetic field ($H=$ 6 kOe). $c_{p}$(6 kOe) is fitted with the
polynomial expression $c_{p}/T=\gamma+\beta T^{2}+\epsilon T^{4}+\delta T^{6}$
to give a Sommerfeld coefficient $\gamma$ = 7.37 mJ mol-1 K-2 and $\beta=$
22.2 $\cdot$ 10-4 J mol-1 K-4, from which a $\theta_{D}=$ 138(1) K is
calculated. Both $\gamma$ and $\theta_{D}$ agree with reported values [29, 37,
38]. The electronic contribution to the specific heat, $c_{p}^{el}$, is
obtained after subtracting the non-linear terms of the polynomial fit.
$c_{p}^{el}$ is displayed in the inset of Fig. 2(d) showing a nearly
compensated entropy at $T_{c}$.

##  3 Thermal expansion

Thermal expansion was measured with a high-resolution ($\Delta L\leq$ 1 Å)
capacitive dilatometer [39]. Several mm-sized samples were cut and polished
along high-symmetry crystallographic directions. Fig. 3(a) shows the
temperature dependence of the linear-thermal-expansion coefficient
$\alpha_{L}=\frac{1}{L}\left(\frac{\partial L}{\partial T}\right)_{T}$ for
both $\beta$-PtBi2 (measured along the [111] direction) and $\beta$-Bi2Pd
(measured along the two tetragonal axis, [100] and [001]). The dip around 50 K
in $c$-axis $\alpha_{L}$($\beta$-Bi2Pd) is associated with an experimental
noisy condition rather than an intrinsic feature.

The first thing to remark is that $\alpha_{L}$ is larger in $\beta$-Bi2Pd
along any direction than in $\beta$-PtBi2. This gives rise to a sizeable
difference in the volume thermal-expansion coefficient
$\alpha_{V}=\frac{1}{V}\left(\frac{\partial V}{\partial T}\right)_{T}$ and the
concomitant volume change of both compounds, as seen in Fig. 3(b) and its
inset. Here, $\alpha_{V}$($\beta$-PtBi2) = 3$\cdot\alpha_{111}$ and
$\alpha_{V}$($\beta$-Bi2Pd) = $\alpha_{001}$ \+ 2$\cdot\alpha_{100}$, given
the cubic and tetragonal crystal symmetry, respectively. The difference is
interesting since both compounds have similar unit-cell volumes,
$V_{u}$($\beta$-PtBi2) = 75.06 Å3 and $V_{u}$($\beta$-Bi2Pd) = 73.66 Å3.

Figure 3: (a) Temperature dependence of the linear thermal-expansion
coefficients $\alpha_{L}$. (b) Temperature dependence of the volume thermal-
expansion coefficients $\alpha_{V}$ (symbols). The solid curve corresponds to
the experimental $c_{p}$ ($\beta$-PtBi2) while the dash curve is a fit to
$\alpha_{V}$($\beta$-Bi2Pd) with the Debye model using a Debye temperature
$\theta_{D}=$ 138 K . Inset: volume expansivity $\Delta V(T)/V$.

$c_{p}$($\beta$-PtBi2) is also shown in Fig. 3(b). The match with $\alpha_{V}$
is excellent in the whole temperature range indicating a nearly constant
Grüneisen parameter $\Gamma=\frac{N_{a}V_{u}B_{s}\alpha_{V}}{c_{p}}$, provided
the isentropic bulk modulus $B_{s}$ has a negligible temperature dependence,
as it is usual. Taking $B_{s}=$ 107-137.9 GPa from the literature [40, 41],
$\Gamma$(100 K) = 2.19-2.83 is obtained in very close agreement with the
expectation for lattice vibrations ($\approx$ 2) [42]. We have not measured
$c_{p}$($\beta$-Bi2Pd) at high temperature. Instead,
$\alpha_{V}$($\beta$-Bi2Pd) is fitted with the expression for the specific
heat given by the Debye model using $\theta_{D}=$ 138 K, obtained from the
low-T $c_{p}$. The fit is good below 30 K and it overestimates the
experimental $\alpha_{V}$ in the intermediate range (30 K $<T<$ 100 K). Taking
$c_{p}$ (100 K) = 68 J mol-1 K-1 and $B_{s}=$ 63.8 GPa from the literature
[38, 43], $\Gamma$(100 K) = 1.75 is calculated. The validity of the Debye fit
is further confirmed by the fact that the ratio $\alpha_{V}/T$ has a parabolic
temperature dependence below 6 K in both compounds (not shown here).

No sign of the superconducting transition is observed in
$\alpha_{L}$($\beta$-Bi2Pd), even though the magnetization of the same sample
shows a clear Meissner effect. From the Ehrenfest equation for second-order
phase transitions, a jump in the thermal-expansion coefficient
$\Delta\alpha_{V}=\frac{dT}{dp}\frac{\Delta c_{p}}{N_{a}V_{u}T_{c}}$ is
expected. Here, $\frac{dT}{dp}$ = 2.5 $\cdot$ 10-10 K Pa-1 [38, 44] and
$\Delta c_{p}$ = 5.7 $\cdot$ 10-2 J mol-1 K-1 (from Fig. 2(d)) are the slope
of the transition curve in the temperature-pressure ($T$ \- $p$) phase diagram
and the jump in the specific heat $c_{p}$ at the transition, respectively.
From these experimental values a jump $\Delta\alpha_{V}=$ 6.3 $\cdot$ 10-8 K-1
is expected, which is comparable to the experimental resolution of our
dilation experiments explaining why the transition is undetected.

No magnetic field dependence of the lattice lengths was observed in any of
both compounds up to $\mu_{0}H=$ 16 T, which imposes an upper limit for the
linear magnetostriction, $\left|\Delta L/L\right|$ (16 T) $<$ 4 $\cdot$ 10-7.
This fact confirms the insignificant electron-lattice coupling common to the
non-magnetic TM.

As mentioned above and due to this small electron-lattice coupling,
expansivity data for TM candidates are scarce. In Table 1 we compare our
results with those from other Bi-based TI found in the literature.
$\alpha_{L}$ ($\beta$-Bi2Pd) shows a significant anisotropy between
$\alpha_{a}$ and $\alpha_{c}$ (see Fig. 3(a)) as Bi2Te3 does, while
$\alpha_{V}$($\beta$-Bi2Pd) is larger than what is found in any other
compound. On the other hand, $\alpha_{V}$($\beta$-PtBi2) is considerable
smaller and comparable to what is observed in Bi2Se3.

| Structure | Type |  $\theta_{D}$ (K) |  $\alpha$$\left[\right.$100 K$\left.\right]$ (10-5 K-1)
---|---|---|---|---
$\beta$-PtBi2 | cubic | TSM | 199a |  $\alpha_{a}$ = 0.93a
$\alpha_{V}$ = 2.8a
$\beta$-Bi2Pd | tetragonal | TSC | 138a |  $\alpha_{a}$ = 1.22a
$\alpha_{c}$ = 1.78a
$\alpha_{V}$ = 4.2a
Bi1-xSbx | rhombohedral | TI |  |  $\alpha_{a}\lesssim$ 1.2b
|  $\alpha_{c}\gtrsim$ 1.2b
|  $\alpha_{V}\sim$ 3.5b
Bi2Se3 | rhombohedral | TI | 182.3c |  $\alpha_{a}$ = 0.7d
$\alpha_{c}$ = 1.2d
$\alpha_{V}$ = 2.6d
Bi2Te3 | rhombohedral | TI | 164.9c |  $\alpha_{a}$ = 0.9e
$\alpha_{c}$ = 1.9e
$\alpha_{V}$ = 3.7e
Table 1: Debye temperature $\theta_{D}$ and thermal-expansion coefficients
$\alpha$ for Bi-based topological material candidates. Superscript letters
correspond to References: a = this work, b = [45], c = [46], d = [47], e =
[48]

##  4 Conclusions

Low-temperature thermal-expansion ($T<$ 120 K) has been measured in single
crystals of cubic $\beta$-PtBi2 and tetragonal $\beta$-Bi2Pd. Both systems
have received much attention in recent years due to the potential topological
nature of their electronic properties.

The linear thermal-expansion coefficient of superconducting $\beta$-Bi2Pd
shows a pronounced anisotropy between the $a$\- and $c$-axis. The volume
thermal-expansion coefficient $\alpha_{V}$ of $\beta$-Bi2Pd is considerable
larger than that of the semimetal $\beta$-PtBi2. The coefficient
$\alpha_{V}$($\beta$-PtBi2) nearly matches the experimental specific heat,
from which a Debye temperature $\theta_{D}=$ 199 K is obtained. On the other
hand, $\alpha_{V}$($\beta$-Bi2Pd) reasonably fits the Debye model with
$\theta_{D}=$ 138 K, extracted from the low-temperature specific heat.

An almost temperature independent Grüneisen parameter $\Gamma\approx$ 2 is
obtained for both compounds in good agreement with the expectation for lattice
vibrations. No magnetostriction is observed in any of both compounds up to
$\mu_{0}H=$ 16 T.

##  5 Acknowledgments

This work was supported by CONICET grant number PIP2021-11220200101796CO,
ANPCyT grant number PICT2019-02396, Universidad Nacional de Cuyo (SIIP) grant
number 06/C559. P.P. acknowledges support from Deutsche Forschungsgemeinschaft
(DFG) through SFB 1143 (project id 247310070). N.H. is member of the INN,
CNEA-CONICET, Argentina.

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7th Workshop on Foundational Ontology (FOUST) co-located with FOIS 2023, 19-20
July, 2023, Sherbrooke, Québec, Canada. Corresponding author: D. F.
Santamaria.

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# A behaviouristic approach to representing processes and procedures in the
OASIS 2 ontology

Giampaolo Bella  Department of Mathematics and Computer Science, University of
Catania, Viale Andrea Doria 6 - 95125 - Catania, Italy  Gianpietro Castiglione
Daniele Francesco Santamaria

(2023)

###### Abstract

Foundational ontologies devoted to the effective representation of processes
and procedures are not widely investigated at present, thereby limiting the
practical adoption of semantic approaches in real scenarios where the precise
instructions to follow must be considered. Also, the representation ought to
include how agents should carry out the actions associated with the process,
whether or not agents are able to perform those actions, the possible roles
played as well as the related events.

The OASIS 2 ontology [1, 2] provides an established model to capture agents
and their interactions but lacks means for representing processes and
procedures carried out by agents. This motivates the research presented in
this article, which delivers an extension of the OASIS 2 ontology to combine
the capabilities for representing agents and their behaviours with the full
conceptualization of processes and procedures. The overarching goal is to
deliver a foundational OWL ontology that deals with agent planning, reaching a
balance between generality and applicability, which is known to be an open
challenge.

###### keywords:

Semantic Web Ontology OWL Agent Process Procedure Event Process Procedure

##  1 Introduction

The notions of process and procedure are broadly known in the literature, even
outside the computer science field, although they bear subtle differences.
According to ISO 9001-2015 definitions, a process is _a set of correlated or
interactive activities that transform inputs into outputs_ describing _the
specified way of carrying out an activity_ , whereas a procedure describes
_how to carry out all or part of a process_. Hence, we can argue that a
process may be composed of many procedures.

Representing processes and procedures through foundational _Web Ontology
Language_ (OWL) ontologies is necessary both for easing the exchange of
process descriptions among systems and to provide a machine-understandable
interpretation of the related activities. One of the main benefits of using
Semantic Web technologies in such a context is the reasoning capability that
permits the inference of new facts from existing data and the verification of
the knowledge base. For example, in the context of operating systems, _race
conditions_ [3] could lead to problems related to the lock of resources by
concurrent processes. A formalization of the processes, procedures and agents
together with semantic reasoning could reduce the ambiguity that may occur
when agents attempt to perform more operations at the same time.

Defining an ontological architecture for processes and procedures aims at
improving so-called “automated planning and acting” [4]. One of the most
remarkable attempts to standardize the planning problem is the PDDL (Planning
Domain Definition Language) [5], which separates the planning problem into a)
domain description and b) problem description. For instance, the PDDL in its
current version introduces _derived predicates_ for dependency modelling of
properties between objects. Therefore, processes and procedures benefit from
the adoption of ontological representations because ontologies can fully
address both planning sub-problems.

Although many ontological approaches are available in the literature, they
suffer from the lack of a complete and general approach to effectively
represent processes and procedures, especially one that a) combines the
representation of processes and procedures with agents and their commitments,
b) models the events generated during the executions of processes and c)
accounts for the roles that are played by the committer agents.

The current paper is motivated by the delivery of such a model. We start by
considering the ontological foundations for the domain of multi-agent systems.
Specifically, we take into account OASIS 2 [1, 2], a foundational OWL 2
ontology that leverages the behaviouristic approach derived from the _Theory
of Agents_ and the related mentalistic notions. The behaviouristic approach is
an effective way of semantically describing agents by characterizing their
capabilities. Agents are enabled to report the set of activities that they can
perform, the types of data required to execute those activities as well as the
expected outputs, through the description of their behaviours. Agents’
implementation details are abstracted away to make the discovery of agents
transparent, automatic, and independent of the underlying technical
infrastructure. In consequence, agent commitments are clearly described, and
the entire evolution of its environment is unambiguously represented,
searchable, and accessible: agents may join a collaborative environment in a
plug-and-play fashion, as there is no more need for third-party interventions.

Our work rests on the observation that OASIS 2 has lacked a specific
characterization of processes and procedures so far, even though it models so-
called _plans_ , which are ways of depicting a sequence of (planned) actions
to be tackled once. Therefore, plans cannot be applied again once they are
performed by agents. In this paper, we extend OASIS 2 to deal with general
specifications of processes and procedures that can be consumed by agents,
including events and the played role, that can be practically leveraged in
real scenarios such as the one concerning race conditions.

The paper is organized as follows. Section 2 presents the related work through
a comparison with the OASIS 2 approach; Section 3 introduces the model of
OASIS 2 devoted to the representation of agents and their behaviours. Section
4 presents the novel extension of OASIS 2 that deals with the notions of
processes and procedures. Section 5 closes the paper with some hints for
future outcomes.

##  2 Related Work

In DOLCE [6], the concepts of processes and events are presented as special
types of perdurants, whereas functions and roles are formalized in some
extensions of the ontology [7, 8]. Aware of the general definitions provided
by DOLCE, OASIS 2 focuses on the behaviouristic approach which is leveraged to
provide agents with novel means for representing complex planning and related
actions, dealing also with events. Specifically, roles are conceived by OASIS
2 as ways of enabling agents with additional behavioural capabilities, whereas
events are represented to be aligned with the definition of agent behaviours.
From this point of view, OASIS 2 introduces a different conceiving of events
and roles since it describes agents in terms of their capabilities. A full
mapping of OASIS 2 in DOLCE is feasible and one of our planned future works.

Wang et al. introduce a model for processes related to water quality
monitoring [9]. The model is strictly focused on the description of
observational processes concerning water pollution monitoring, thus being
limited to general applicability outside of the domain.

Within CIDOC-CRM [10], there is a work-in-progress extension called CRMact
[11] that defines the classes and properties for planning future activities
and events. However, CRMact is mainly focused on cultural heritage and
documentation records and hence is not generally applicable.

Concerning business processes, it is worth mentioning several works. Thomas et
al. propose an ontological approach for representing business processes
together with a system architecture prototype, exploiting the proposed model
[12]. Greco et al. use ontologies as facilities within a framework for
assisting designers in the realization and analysis of complex processes [13].
Corea et al. present an approach to verify whether a business process is
compliant with given business rules combining logic programming and ontologies
[14], while Calvanese et al. propose an approach for using Ontology-Based Data
Access (OBDA) to govern data-aware processes, and in particular, those
executed over a relational database that issue calls to external services to
acquire new information and update data [15]. Finally, a comparison among
ontologies related to business processes for task monitoring, measurements and
evaluation strategies, and for modelling process information was built [16],
[17], [18].

However, the downside of limiting ontological models to business processes
lies in the absence of relationships between agents and their commitments that
instead represent the core of agent-oriented representation. That implies the
inability of finding agents/services with specific capabilities, invoking
them, and enabling their interoperability. Notably, the benefits of process-
oriented representations, such as the facilitation mechanisms for the search
and selection of process models, are also offered by the agent-oriented ones
as long as they are sufficiently general although flexible.

##  3 Preliminaries on OASIS 2

The first version of OASIS [19] is a foundational ontology that leverages the
behaviouristic approach to characterize agents in terms of the actions they
are able to perform, including purposes, goals, responsibilities, information
about the world they observe and maintain, and their internal and external
interactions. It models the executions and assignments of tasks, restrictions
and constraints used to establish agent responsibilities and authorizations.

In the recent past, OASIS has been extended and applied to deal with so-called
_Ontological Smart Contracts_ [20] and with the ontological models for smart
contracts on the blockchain [21]. OASIS is also part of the POC4COMMERCE
project [22], funded by the NGI-ONTOCHAIN consortium [23].

The last version of OASIS is OASIS 2 [24, 1]111The OASIS 2 ontology can be
reached at [2], which extends OASIS with some new features, such as the
entrustment of agents, and reshapes the model adopted for the representation
of agents and their commitments.

Inspired by the _Tropos_ methodology [25] which devises from Agent Oriented
Programming (AOP), OASIS 2 represents agents through three essential and
publicly shared mental states, namely (expected) _behaviours_ , _goals_ and
_tasks_. Behaviours represent the mental state of the agent associated with
its ability to modify its environment or, in general, act or do something.
Goals describe mental attitudes representing preferred progressions of a
particular system that the agent has chosen to put effort into bringing about
[26]. Tasks depict how to carry on such progressions and describe atomic
operations that agents perform.

Agents and their interactions are represented by carrying out three main
steps, namely: a) an optional step that consists of modelling descriptions of
general abstract behaviours, called _templates_ , conceptual characterization
of behaviours from which concrete agent behaviours are drawn; b) modelling
concrete agent behaviours, possibly, drawn by agent templates; c) modelling
actions and associating them with the corresponding behaviours. The first
step, not mandatory, consists in defining the agent’s behaviour template,
namely a higher-level description of the behaviour of abstract agents that can
be leveraged to define concrete behaviours of real agents; for example, a
template is designed to describe the abstract behaviour consisting in
obtaining and releasing locks on resources. Additionally, templates are useful
to guide developers in the definition of the behaviours of their specific
agents. To describe abstract agent’s capabilities to perform actions, an agent
template comprises three main elements, namely behaviour, goal and task. The
latter constitutes the most simple (atomic) operations that agents are able to
actually perform including, possibly, input and output parameters required to
accomplish them. The second step consists of representing concrete agent
behaviours either by relying on a template or by defining it from scratch. In
both cases, concrete behaviours are modelled analogously to those of
templates, where the models of outstanding features are replaced with actual
characteristics. Behaviours drawn by shared templates are associated with them
in order to depict the behaviour inheritance relationship. In the last step,
actions performed by agents are described as direct consequences of some
behaviours and are associated with the behaviours of the agent that performed
them. To describe such an association, OASIS 2 introduces _plan executions_.
Plan executions describe the actions performed by an agent, associating them
with one of its behaviours. Associations are carried out by connecting the
description of the performed action to the behaviour from which the action has
been drawn: actions are hence described by suitable graphs that retrace the
model of the agent’s behaviour.

Plans can be additionally either submitted to agents as requests for
performing some actions or they can be assigned by specific agents called
_entruster agents_.

In OASIS 2, agent templates are defined according to the UML class diagram in
Fig. 1.

Figure 1: Diagram of agent templates in OASIS 2

To consider how both abstract and concrete agents perform actions, the
description of agents comprises three main elements, namely _behaviour_ ,
_goal_ , and _task_. Agent tasks, in their turn, describe atomic operations
that agents perform, including possibly input and output parameters required
to accomplish them. Those elements in OASIS 2 are introduced by way of the
following OWL classes:

  * •

Agent: This class comprises all the individuals representing agents. Instances
of such a class are connected with one or more instances of the class
Behaviour using the OWL object-property hasBehaviour.

  * •

Behaviour: Behaviours can be seen as collectors comprising all the goals that
an agent may achieve. Instances of Behaviour are connected with one or more
instances of the class GoalDescription by means of the object-property
consistsOfGoalDescription.

  * •

GoalDescription: Goals represent containers embedding all the tasks that the
agent can achieve. Instances of GoalDescription comprised by a behaviour may
also satisfy dependency relationships introduced by the object-property
dependsOn. Goals are connected with the tasks that form them and are
represented by instances of the class TaskDescription through the object-
property consistsOfTaskDescription.

  * •

TaskDescription: This class describes atomic operations that agents perform.
Atomic operations are the most simple actions that agents are able to execute
and, hence, they represent what agents can do within their environment. Atomic
operations may depend on other atomic operations when the object-property
dependsOn is specified. Atomic operations whose dependencies are not
explicitly expressed are intended to be performed in any order.

The core of agent behaviour revolves around the description of atomic
operations represented by instances of the class TaskDescription that
characterizes the mental state corresponding to commitments. In their turn,
instances of the class TaskDescription are related to the following five
elements that identify the operation:

  * •

An instance of the class TaskOperator, characterizing the mental state
corresponding to the action to be performed. Instances of TaskOperator are
connected either by means of the object-property refersExactlyTo or
refersAsNewTo to instances of the class Action. The latter class describes
physical actions represented by means of entity names in the form of infinite
verbs (e.g., produce, sell). Specifically, the object-property refersExactlyTo
is used to connect the task operator with a precise action having a specific
IRI, whereas refersAsNewTo is used to connect a task operator with an entity
representing an action of which only a general abstract description is given
(for example, an action for which only the type is known).

In the latter case, the entity representing the action is also defined as an
instance of the TemplateThing: such instances are used to define entities that
represent templates for the referred element and that describe the
characteristics that such element should satisfy. TemplateThing is the class
used to characterize all the individuals involved in the definition of
behaviour templates and to distinguish them from the entities representing
concrete behaviours, plans or actions, thus eliminating the need of having
separated models for those aspects.

In order to specify the classes of which the entity must be an instance, it is
eventually possible to connect such entity by means of the object-property
refersAsInstanceOf with the individual instances of the desired classes.

  * •

Possibly, an instance of the class TaskOperatorArgument, connected using the
object-property hasTaskOperatorArgument and representing additional
specifications or subordinate characteristics of task operators (e.g., on,
off, left, right). Instances of TaskOperatorArgument are referred to the
operator argument by using either the object-property refersAsNewTo or
refersExactlyTo.

  * •

An instance of the class TaskObject, connected by means of the object-property
hasTaskObject and representing the template of the object recipient of the
action performed by the agent (e.g., price). Instances of TaskObject are
referred to the action recipient by specifying either the object-property
refersAsNewTo or refersExactlyTo.

  * •

Input parameters and output parameters are introduced by instances of the
classes TaskInputParameter and TaskOutputParameter, respectively. Instances of
TaskDescription are related to instances of the classes TaskInputParameter and
TaskOutputParameter by means of the object-properties hasTaskInputParameter
and hasTaskOutputParameter, respectively. Instances of TaskInputParameter and
of TaskOutputParameter are referred to the parameter by specifying either the
object-property refersAsNewTo or refersExactlyTo. Moreover, the classes
TaskInputParameter and TaskOutputParameter are also subclasses of
TaskParameter.

Finally, in the case of agent behaviour templates, instances of Agent,
Behaviour, GoalDescription, TaskDescription, TaskOperator,
TaskOperatorArgument, TaskObject TaskInputParameter, and TaskOutputParameter
are also instances of TemplateThing.

Figure 2: Example of an OASIS 2 agent template

Fig. 2 illustrates the case study mentioned above where a lock on a resource
can be obtained and released by agents. Specifically, Fig. 2 presents a
template describing an abstract agent that is able to request a lock on a
resource. The agent template comprises a single behaviour, constituted by a
single goal that in its turn comprises a single task. The task, which
represents the ability to request a lock, provides four elements:

  * •

request_lock_task_operator, representing the mental state associated with the
behaviour’s action (the task operator), which in its turn is associated with
the individual request, the latter describing the capability of requesting
something.

  * •

request_lock_task_operator_argument, introducing an additional feature (the
operator argument) associated with the action and represented by the
individual lock. The argument describes the fact that the request action is
referred to a lock. Task operator and its argument describe together the
capability of requesting a lock;

  * •

request_lock_task_object_template, representing the recipient of the
operation, which is related to an instance of the class Lock by means of the
object-property refersAsNewTo. Such an instance comprises all the features
that the recipient of the adopting action should own: the concrete actions
implementing the behaviour template for requesting a lock are supposed to
effectively request a lock with the desired features;

  * •

_request_lock_task_output_template_ , representing the input of the operation,
namely a resource on which the lock is requested.

In the second step, concrete agent behaviours are defined either by
instantiating one or more templates or from scratch. In OASIS 2, the modelling
pattern of concrete behaviours has a structure analogous to one of the
behaviour templates, illustrated above, with the difference that individuals
used to define a concrete behaviour, instead of being instances of the class
TemplateThing, are instances of the class BehaviourThing. The latter class is
devoted to describe all the mental states associated with concrete behaviours
of real agents that induce actions.

Concrete behaviours may be connected with the template they are drawn from. In
order to describe the fact that concrete agents inherit their behaviours from
a commonly shared template, the instances related to the concrete behaviours
are connected with the instances of the template through the sub-properties of
the object-property overloads as follows. The association is carried out by
connecting the instances of the classes:

  * •

Behaviour, by means of overloadsBehaviour;

  * •

GoalDescription, by means of overloadsGoalDescription;

  * •

TaskDescription, by means of overloadsTaskDescription;

  * •

TaskObject, by means of overloadsTaskObject;

  * •

TaskOperator, by means of overloadsTaskOperator;

  * •

TaskInputParameter, by means of overloadsTaskInputParameter;

  * •

TaskOutputParameter, by means of the object-property
overloadsTaskOutputParameter.

As the last step, agent commitments devised from behaviours are introduced to
describe agent actions. In OASIS 2, commitments are represented by adopting
the same pattern presented for abstract behaviours with the difference that
instances of the class TemplateThing are instead modelled as instances of the
class ExecutionThing and the agent responsible for the execution of the action
is related with the plan representing the commitment by means of the object-
property performsPlanExecution, subproperty of performs. The class
ExecutionThing is introduced to characterize all the entities involved in the
definition of concrete and already performed actions and to distinguish them
from the ones introduced for templates, behaviours and plans.

In order to relate agent commitments with the behaviour from which they are
drawn, subproperties of the object-property drawnBy are introduced.
Specifically, planExecutionDrawnBy connects the instance of GoalDescription of
the agent action to its analogue of agent behaviour; much in the same way,
goalExecutionDrawnBy connects the instance of the class GoalDescription of the
commitment with its analogue, while taskExecutionDrawnBy, taskObjectDrawnBy,
taskOperatorDrawnBy, taskInputParameterDrawnBy, and taskOutputParameterDrawnBy
are introduced for TaskDescription, TaskObject, TaskOperator,
TaskInputParameter, and TaskOutputParameter, respectively.

Usually, agents proposing plans identify the behaviours responsible for their
realization beforehand, in such a way as to completely describe and trace how
agent intentions are realized. In this case, the entities representing the
submitted plan are related to the entities describing the responsible
behaviour by means of suitable subproperties of the object-property
submittedTo, relating instances of PlanningThing with instances of
BehaviourThing as follows: a) planDescriptionSubmittedTo, for instances of
Behaviour; b) goalDescriptionSubmittedTo, for instances of GoalDescription; c)
taskDescriptionSubmittedTo, for instances of TaskDescription; d)
taskObjectSubmittedTo, for instances of TaskObject; e)
taskOperatorSubmittedTo, for instances of TaskOperator; f)
taskInputParameterSubmittedTo, for instances of TaskInputParameter; g)
taskOutputParameterSubmittedTo, for instances of TaskOutputParameter.

In a similar way, plans are also related to the agent’s actions realizing
them. For this purpose, the subproperties of the object-property hasExecution
are introduced, namely hasPlanExecution, hasGoalExecution, hasTaskExecution,
hasTaskObjectExecution, hasTaskOperatorExecution,
hasTaskInputParameterExecution, and hasTaskOutputParameterExecution.
Analogously, actions are connected with the behaviour responsible for their
execution by means of suitable subproperties of the object-property
executionDrawnBy.

##  4 Processes and procedures in OASIS 2

Since the first version, OASIS was already capable of describing how agent
activities are carried out by suitably combining behaviours, plans and
actions. Plans, in particular, can be used to describe in detail how inputs
are processed to be turned into outputs, but they are not sufficient to
describe general complex processes. Plans require the presence of a committer
agent beforehand and can be applied only once. This is because the mental
state related to the desire or wish to perform actions does not abstract from
the committer agent. Plans can be leveraged to represent single applications
of procedures but they should be combined properly. Then, plans can be
associated with the behaviours responsible for their executions thus carrying
out the actions required to accomplish the procedures. The approach is
illustrated in Fig. 3. Leveraging the ISO definition, processes defined by
agents are constituted by procedures. In the behaviouristic vision, the
procedures states are associated with plans that are executed thus leading to
actions. Actions derive from agent behaviours that in their turn can rely on
specific templates. Agents performing actions are provided with behaviours
either intrinsically or through a played role. Finally, actions could lead to
events either incidentally or as foreseen by procedures. In the first case,
events are conceived as noteworthy and extraordinary happenings that have not
been previously planned, in the second case, as recurrent situations.

Figure 3: Process and procedure in OASIS 2

For example, agents may play the role of resource manager, hence they are able
to request and release locks on resources according to such role. A process
describing the steps required to request and release locks is formalized, so
that they can tackle a plan for each step. Plans are executed so that the
resource is released and modified. Actions are performed thanks to agent
behaviours provided by the resource manager’s role. During the execution of
one of those actions, the system exceptionally sends a message to the
committer agent. We will partially see this scenario together with the
presentation of the model.

In light of the above considerations, OASIS 2 introduces the following novel
OWL classes:

  * •

Process, which encompasses the procedures describing how activities are
carried out and;

  * •

Procedure, the subclass of the class Activity, which introduces the set of
plans required to accomplish the procedure itself and to be realized through
actions. In OASIS 2 procedures are constituted by one or more ProcedureState,
each one connected to a specific plan describing how the activity is carried
out. Specifically, we identify two types of ProcedureState: a)
TerminatingProcedureState, which includes InitialProcedureState and
FinalProcedureState, and b) NonTerminatingProcedureState.

Procedures are constituted by _procedure states_ that describe single steps of
the procedure. _Initial procedure states_ describe the beginning of the
procedure, while _final procedure states_ its termination. Moreover, the
initial procedure state coincides with the final procedure state in the case
of a single-step procedure. Finally, _non-terminating procedure states_
describe the intermediate steps to be performed, including all the steps
between the initial state and the final state. The schema for processes and
procedures is illustrated in Fig. 4. The subproperties of the object-property
procedureConsistsOfProcedureState are used to suitably connect a procedure
with its procedure states. Specifically, the object-properties
procedureConsistsOfInitialProcedureState and
procedureConsistsOfFinalProcedureState (both subproperty of
procedureConsistsOfTerminatingProcedureState) connect the procedure with its
initial and final procedure state, respectively, while the object-property
procedureConsistsOfNonTerminatingProcedureState connects the procedure with
its non-terminating states. The initial procedure state is connected with the
subsequent non-terminating procedure states by means of the object-property
hasNextNonTerminatingProcedureState. The latter property is also used to
connect non-terminating procedure states with their subsequent non-terminating
procedure states. Whenever the next procedure state is the final state, the
object-property hasFinalProcedureState is adopted in its place. Both
hasNextNonTerminatingProcedureState and hasFinalProcedureState are defined as
subproperty of the object-property hasNext.

Finally, since the intended meaning of processes and related procedures is to
give a description of how activities should be carried out, the instances
introduced so far are also instances of the class PlanningThing. To complete
the representation of processes, it is now sufficient to connect an instance
of the class Process with the ones of the class Procedure that model the
process activities. In case a process is constituted by more than one
procedure, the latter can be sorted by connecting them through the object-
property hasNextProcedure.

Figure 4: The OASIS 2 model for processes and procedures

To describe how process activities should be carried out, we introduce a
behaviour as in Fig. 1, where the instances of TemplateThing are instead
defined as instances of PlanningThing. The choice is motivated by the fact
that templates describe abstract behaviours that are instantiated to introduce
concrete behaviours, whereas plans describe actions that agents wish to
perform or see accomplished: this is exactly the case of process, where a set
of actions must be tackled in order to achieve the desired end. Additionally,
plans allow one to describe in detail what actions are required to accomplish
the activity associated with the procedure state, the input parameters and the
expected output, without the need of selecting an agent responsible for those
actions beforehand. To associate a procedure state with the planned behaviour,
OASIS 2 introduces the object-property isDescribedBy.

The process concerning the case study considered at the beginning of the
section is depicted in Fig. 5. The process describes the modification of a
resource which requires the acquisition and release of an exclusive lock
before and after the modification, respectively.

Figure 5: Example of process in OASIS 2

The process consists of a single procedure that, in its turn, consists of
three distinct states, an initial state, a non-terminating state, and a final
state. The initial state is associated with a behaviour describing how to
request a lock on a resource. The behaviour follows the template in Fig. 2,
where instances of the class TemplateThing are replaced with instances of the
class PlanningThing as stated in Section 3. Analogously, the non-terminating
state is associated with a behaviour describing how to modify the resource,
while the final state is associated with the behaviour describing how to
release the lock.

Once the process is fully described, it is possible to model its application.
To do so, a model retracing the structure of a process (see Fig. 4) is
introduced, where instances of the class PlanningThing are instead defined as
instances of the class ExecutionThing. In a similar way to TemplateThing and
PlanningThing that are introduced to represent descriptions of abstract
behaviours and planned actions, respectively, ExecutionThing is conceived for
those mental states that represent actions already committed. In the case of
processes, the class provides a means for characterizing the realization of
the process, the achievement of the related activities, and the commitment of
the planned actions. Moreover, to uniquely relate the elements of the process
with the ones of its realization, two subproperties of the object-property
drawnBy are introduced. Specifically, processDrawnBy and procedureDrawnBy are
introduced to connect the instances of Process and Procedure, respectively.
Hence, it is clear how to trace back to the agent’s behaviour responsible for
the execution of a process, thus unambiguously identifying actors and actions
of arbitrarily complex environments.

For instance, the process of the example in Fig. 5 can be realized as
partially depicted in Fig. 6 that illustrates the modification process on a
resource. The schema retraces the one introduced for the process, where
instances of the class PlanningThing are suitably replaced by instances of the
class ExecutionThing. Specifically, the process realization introduces three
behaviour executions, one for each plan of the process, namely a)
requestLockBehaviourExec, representing the execution of the requestLockPlan
plan; b) modifyResourceBehaviourExec, representing the execution of the
modifyResourcePlan plan; c) releaseLockBehaviourExec, representing the
execution of the releaseLockPlan plan.

Moreover, each behaviour of the process is connected with its analogous of the
process realization by means of the object-property hasPlanExecution. Finally,
each plan execution can be connected with the responsible behaviour by means
of the object-property planExecutionDrawnBy (subproperty of executionDrawnBy,
see Section 3).

Figure 6: Example of process realization in OASIS 2

As stated above, in OASIS 2 specific events can be associated with procedure
states. In OASIS 2 events are semantically represented by instances of the
class Event, while the procedure state is related to the triggered event by
means of the object-property triggersEvent. The phenomenon associated with the
event is introduced by means of OASIS 2 actions. These are in fact
sufficiently general and powerful to potentially cover up any type of
phenomenon springing from an event that is fully within the vision of a
behaviouristic approach.

In the case of the case study, when the system is triggered to send a message
to selected users when a resource is modified, for example, because an error
occurred, an action such as the one in Fig. 7 is introduced. Other information
such as the type of event, its duration, or the time window in which it
happens, can be additionally specified.

Figure 7: Example of event in OASIS 2

Finally, it is admissible that agents perform procedures exclusively according
to the specific role that they play. Roles permit the separation of the
concerns of behaviours intrinsically owned by agents from the ones temporarily
at their disposal to execute a process. This implies that the agent is able to
perform an action only when such a role is played. Hence, the behaviour
responsible for performing those actions is not strictly associated with the
agent but with the role. To represent the scenario described above and to
model the playing of roles, OASIS 2 introduces roles as depicted in Fig. 8.

Figure 8: Roles in OASIS 2

In OASIS 2 roles, introduced as instances of the class Role, provide agents
with behaviours by means of the object-property providesBehaviour. In its
turn, a role is associated with the agent playing it by means of the object-
property playRole. Whenever agents cease to play a role, the associated
behaviours are no longer available. The end of a role is specified by making
the instance of Role deprecated, hence the instance of the class
DeprecatedThing. This implies that actions are associated with role-oriented
behaviours until the corresponding role is not deprecated. Concerning the case
study, an agent playing the role consisting in requesting a lock on a resource
is depicted in Fig. 9. The role is introduced as a fresh instance of the class
Role, specifically provided for the agent’s representational needs. The role
types, introduced as an instance of the class RoleType, are defined
accordingly to the domain to be described, hence they are out of the OASIS 2
scopes: in the case of the case study, a specific role type, called
resource_consumer_role, is introduced to depict the roles dedicated for
consuming resources.

Figure 9: Example of playing roles in OASIS 2

##  5 Conclusions

This paper introduced an extension of the OASIS 2 ontology and, therefore,
inherits the behaviouristic approach to semantically represent agents and
their commitments through the formalization of their mental states, namely,
_behaviour_ , _goal_ and _task_. In particular, the proposed extension of
OASIS 2 deals with the modelling of processes and procedures, thereby
providing a general although practical way of representing processes and
procedures to be tackled by agents through their behaviours. Now that such a
foundational ontology is available, our semantic representation capabilities
are substantially enhanced. In consequence, all application scenarios where
specific instructions must be followed have fallen into scope.

For example, how agents reach a consensus, together with the modelling of
their behaviours, is one of the future challenges. An application of OASIS 2
is foreseen for the characterisation of security directives, aiming at a
structural solution for translating security documents to a mathematically-
driven world. The approach targets the NIS 2 directive but other similar
directives can be addressed. In addition, we intend to apply OASIS 2 to
represent security constraints for cybersecurity threat contexts, in
particular for the purpose of semantically representing authentication and
confidentiality properties for agents.

Also, we shall consider how to integrate OASIS 2 with the PDDL and with the
main frameworks such as JADE [27], all with the aim of automatically
generating agents and artefacts. Similarly, an integration with _CArtAgO_
[28], a framework for building shared computational worlds, appears to be
valuable. The horizons of the semantic representation of agents have
substantially expanded but retain the vast potential for yet more expansion in
the near future.

## Acknowledgments

Gianpietro Castiglione acknowledges a studentship by Intrapresa S.r.l. and
Italian “Ministero dell’Università e della Ricerca” (D.M. n. 352/2022).

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