††thanks: These authors contributed equally††thanks: These authors contributed
equally

# Pressure-induced creation and annihilation of Weyl points in $T_{d}$\- and
1$T^{\prime\prime}$-Mo0.5W0.5Te2

Bishnu Karki  Central Department of Physics, Tribhuvan University, Kirtipur,
44613, Kathmandu, Nepal  Condensed Matter Physics Research Center, Butwal-11,
Rupandehi, Nepal  Bishnu Prasad Belbase  Central Department of Physics,
Tribhuvan University, Kirtipur, 44613, Kathmandu, Nepal  Condensed Matter
Physics Research Center, Butwal-11, Rupandehi, Nepal  Gang Bahadur Acharya
Central Department of Physics, Tribhuvan University, Kirtipur, 44613,
Kathmandu, Nepal  Institute for Theoretical Solid State Physics, IFW Dresden,
Helmholtzstr. 20, 01069 Dresden, Germany  Sobhit Singh
<EMAIL_ADDRESS>Department of Physics and Astronomy, Rutgers
University, Piscataway, New Jersey 08854, USA  Madhav Prasad Ghimire
<EMAIL_ADDRESS>Central Department of Physics, Tribhuvan
University, Kirtipur, 44613, Kathmandu, Nepal  Condensed Matter Physics
Research Center, Butwal-11, Rupandehi, Nepal  Institute for Theoretical Solid
State Physics, IFW Dresden, Helmholtzstr. 20, 01069 Dresden, Germany

###### Abstract

By means of first-principles density-functional theory calculations, we
investigate the role of hydrostatic pressure on the electronic structure of
$T_{d}$ ($Pmn2_{1}$) and 1$T^{\prime\prime}$ ($Pm$) phases of Weyl semimetal
Mo0.5W0.5Te2, which is a promising material for phase-change memory technology
and superconductivity. We particularly focus on changes occurring in the
distribution of the gapless Weyl points (WPs) within 0 to 45 GPa pressure
range. We further investigate the structural phase transition and lattice
dynamics of the $T_{d}$ and 1$T^{\prime\prime}$ phases within the
aforementioned pressure range. Our calculations suggest that both the $T_{d}$
and 1$T^{\prime\prime}$ phases of Mo0.5W0.5Te2 host four WPs in their full
Brillouin zone at zero pressure. The total number of WPs increases to 44 (36)
with increasing pressure via pair creation up to 20 (15) GPa for the $T_{d}$
(1$T^{\prime\prime}$) phase, and beyond this pressure pair annihilation of WPs
starts occurring leaving only 16 WPs at 45 GPa in both phases. The enthalpy
versus pressure data reveal that the 1$T^{\prime\prime}$ phase is more
favorable below the critical pressure of 7.5 GPa, however, beyond this
critical pressure the $T_{d}$ phase becomes enthalpically favorable. We also
provide the calculated x-ray diffraction spectra along with the calculated
Raman- and infrared-active phonon frequencies to facilitate the experimental
identification of the studied phases.

##  I Introduction

The study of topological materials is of high interest at present due to their
potential applications in the emerging technology Hasan and Kane (2010);
Armitage  _et al._ (2018); Yan and Felser (2017); Manna  _et al._ (2018); Weng
_et al._ (2016); Gao  _et al._ (2019); Burkov (2018); Hasan  _et al._ (2017).
Weyl semimetals (WSMs), one class of topological materials, seem to drag
special attention since their experimental realization in 2015 Lv  _et al._
(2015a, b); Huang  _et al._ (2015); Xu  _et al._ (2015a); Sushkov  _et al._
(2015); Lv  _et al._ (2015c). Breaking of either time-reversal symmetry or
spatial inversion-symmetry, or both, in WSMs results in a particular
electronic band structure possessing crossings of the nondegenerate valance
and conduction bands near the Fermi level forming gapless Weyl points (WPs)
and Weyl cones Murakami (2007). The low-energy electronic excitations near
these WPs behave as the massless Weyl fermions Jia  _et al._ (2016); Weyl
(1929); Wan  _et al._ (2011); Grassano  _et al._ (2018); Ruan  _et al._
(2016); Soluyanov  _et al._ (2015a); Singh  _et al._ (2016, 2018a); Winkler
_et al._ (2019). WSMs are interesting due to their exotic properties such as
WPs acting as the sources and sinks of the Berry curvature in momentum space,
existence of open Fermi-arc states connecting two opposite WPs, extremely
large magnetoresistance Chen  _et al._ (2016); Pei  _et al._ (2017);
Thirupathaiah  _et al._ (2017); Lee  _et al._ (2018), and various quantum Hall
phenomena Xu  _et al._ (2015b, 2016); Belopolski  _et al._ (2016a); Tamai  _et
al._ (2016); Autès  _et al._ (2016a); Zhang  _et al._ (2017); Ghimire  _et
al._ (2019); Singh  _et al._ (2020). Moreover, based on the tilting of the
band crossings near the Fermi level two types of WSMs are reported: (i) type-I
WSMs that preserve the Lorentz invariance and their Fermi surface shrinks to
zero when the Fermi energy (EF) is set at the energy of the WPs, and (ii)
type-II WSMs that violate the Lorentz invariance due to peculiar tilting of
the Weyl cone such that WPs occur at the touching points of an electron and
hole pockets and the Fermi surface never shrinks to absolute zero when EF is
set at the energy of the WPs Yan and Felser (2017); Grassano  _et al._ (2018);
Soluyanov  _et al._ (2015a); Autès  _et al._ (2016b); Chang  _et al._ (2016a);
Koepernik  _et al._ (2016).

WSMs belonging to the transition metal dichalcogenides (TMDs) family are
special, mainly because the s, p, and d orbitals in these systems hybridize to
form bands near the Fermi level, which often yield various fascinating
properties such as distinct quantum phase transitions between different
structures, thermal and optical properties, topological domain walls,
different kinds of Hall effects, and superconductivity Sipos  _et al._ (2008);
Qian  _et al._ (2014); Morris  _et al._ (1972a); Takahashi  _et al._ (2017);
Qi  _et al._ (2016); Chen  _et al._ (2018); Huang  _et al._ (2019). Notably,
dome-shaped superconducting behaviour is observed in MoTe2 with a transition
temperature of 0.01 K that enhances to 8.2 K at 11.7 GPa pressure Qi  _et al._
(2016); Heikes  _et al._ (2018); Morris  _et al._ (1972b); Wei  _et al._
(2014); Hosur  _et al._ (2014). MoTe2 and WTe2 are among the first reported
WSMs with four WPs in their momentum space resulting due to the broken
inversion symmetry Soluyanov  _et al._ (2015b); Li  _et al._ (2018); Bruno
_et al._ (2016); Wu  _et al._ (2016); Li  _et al._ (2017); Lv  _et al._
(2017a); Huang  _et al._ (2016); Deng  _et al._ (2016); Sakano  _et al._
(2017); Sun  _et al._ (2015a); Wang  _et al._ (2016); Jiang  _et al._ (2017);
Liang  _et al._ (2016). The application of external pressure and strain has
been reported to play a vital role in tuning the WSM phase in MoTe2 and WTe2
Sun  _et al._ (2015b); Dissanayake  _et al._ (2019); Aryal and Manousakis
(2019); Xu  _et al._ (2018).

Substitution of W by Mo in WTe2, forming a polymorphic structure of
MoxW1-xTe2, is reported to control the structural phase transition, transport
properties, thermal conductivity, Weyl phase, and superconductivity in this
system Lv  _et al._ (2017b); Yan  _et al._ (2017); Oliver  _et al._ (2017);
Rhodes  _et al._ (2017); Aslan  _et al._ (2018); Schneeloch  _et al._ (2020);
Marchenkov  _et al._ (2019); Tao  _et al._ (2019). Depending on the magnitude
of the applied pressure on MoxW1-xTe2 ($x$ = 0.9, 0.6, and 0.25), the
superconducting transition temperature can be tuned Dahal  _et al._ (2020).
Moreover, tunable WSM phase and Fermi-arc states can be realized in MoxW1-xTe2
($x$ $\leq$ 0.5) by varying $x$ Belopolski  _et al._ (2016b); Chang  _et al._
(2016b); Belopolski  _et al._ (2016c). By means of the temperature-dependent
elastic neutron scattering measurements, Schneeloch et al. Schneeloch  _et
al._ (2020) recently investigated the monoclinic 1$T^{\prime}$ to orthorhombic
$T_{d}$ structural phase transition in Mo1-xWxTe2 as a function of the W
substitution and temperature. They observed that 1$T^{\prime}$ – $T_{d}$ phase
transition is accompanied by an intermediate pseudo-orthorhombic phase
$T_{d}^{*}$, which was first reported by Tao et al. Tao  _et al._ (2019) for
pure MoTe2. Notably, the intermediate $T_{d}^{*}$ phase exists only up to
$x=0.34$ concentration and vanishes for $x>0.34$. Their results further
suggest that at room temperature Mo0.5W0.5Te2 composition favors the $T_{d}$
phase Schneeloch  _et al._ (2020). Marchenkov et al. Marchenkov  _et al._
(2019) experimentally studied the transport and optical properties of a single
crystal Mo0.5W0.5Te2. Their temperature-dependent resistivity data reveal the
metallic nature of the system. Li et al. recently performed dimensionality-
dependent transport measurements on a special Mo0.5W0.5Te2 sample having a
thickness gradient across 2D and 3D regimes and reported strong evidences that
this material is a type-II WSM Li  _et al._ (2021).

The experimental observation of pressure-controlled superconductivity in
Mo0.5W0.5Te2 Dahal  _et al._ (2020) calls for a systematic investigation of
the pressure-induced changes occurring in the crystal structure, electronic
structure, and WSM phase of this system. In this work, we theoretically study
the structural, vibrational, and topological electronic properties of
Mo0.5W0.5Te2 as a function of the hydrostatic pressure within 0 to 45 GPa
pressure range. We explore the aforementioned properties of two candidate
crystal phases of Mo0.5W0.5Te2, $T_{d}$ and 1$T^{\prime\prime}$, as a function
of the applied pressure. Our calculations indicate that both these phases are
dynamically stable within the studied pressure range and both of them host a
number of WPs in their momentum-energy space. Interestingly, WPs in both
phases can be created via pair-creation, destroyed via pair-annihilation, and
shifted in the momentum-energy space by application of an external pressure.
Our work implies that Mo0.5W0.5Te2 inherits a variety of interesting
topological properties at higher pressures and it may provide a novel platform
for realization of Weyltronics. Furthermore, our results suggest an increase
in the superconducting transition temperature at higher pressures, which is in
agreement with recent experimental observations Takahashi  _et al._ (2017);
Dahal  _et al._ (2020).

##  II Computational Details

The density-functional theory (DFT) calculations were performed using the
Projector Augmented Wave (PAW) method as implemented in the Vienna Ab initio
Simulation Package (VASP) Kresse and Furthmüller (1996, 1996); Kresse and
Joubert (1999). Six valence electrons were considered in the PAW
pseudopotentials of Mo, W, and Te. The Perdew-Burke-Ernzerhof generalized-
gradient approximation (GGA-PBE) was used to compute the exchange-correlation
functional Perdew  _et al._ (2008). The GGA-opt86b functional was used for the
van der Waals (vdW) density-functional corrections Klimeš  _et al._ (2011);
Thonhauser  _et al._ (2007); Dion  _et al._ (2004). The reciprocal space was
sampled using a $\Gamma$-centered k-mesh of size 8 $\times$ 12 $\times$ 4
together with a kinetic energy cutoff of 600 eV for plane waves. The energy
convergence and force convergence criteria were set to $10^{-8}$ eV and
$10^{-4}$ eV/Å, respectively. The hydrostatic pressure was applied up to 45
GPa. The crystal structures were fully optimized in inner coordinates as well
in cell parameters for each pressure considering spin-orbit coupling (SOC) and
GGA-opt86b vdW corrections Klimeš  _et al._ (2011); Thonhauser  _et al._
(2007); Dion  _et al._ (2004). All the DFT calculations were carried out in a
twelve-atom unit cell of Mo0.5W0.5Te2. The optimized lattice parameters for
the $T_{d}$ phase at zero pressure are a = 6.301, b = 3.490, and c = 14.076 Å,
and cell angles are $\alpha$ = $\beta$ =$\gamma$ = 90∘, which are in excellent
agreement with the experimental data reported in Ref. Schneeloch  _et al._
(2020).

The phonon calculations were performed using the finite-difference approach
using supercells of size 2$\times$ 3 $\times$ 1\. SOC was considered in all
phonon calculations. The phonopy Togo and Tanaka (2015) packaged was utilized
to plot the phonon dispersions. The MechElastic (Singh  _et al._ , 2021,
2018b) package was used to perform the equation of states analyses using the
enthaply versus pressure data. In order to understand the distribution of Weyl
points in the momentum-energy space, we compute the real-space Wannier
Hamiltonian using the full potential local orbital (FPLO) code Koepernik and
Eschrig (1999). We employed the above-mentioned DFT convergence parameter in
all the FPLO calculations. The Wannier fitting was done using pyfplo Koepernik
and Eschrig (1999) module of the FPLO package considering Mo:4d, 5s, W:5d, 6s,
and Te:5s, 5p as the local orbitals basis. The pyprocar code Herath  _et al._
(2020) was used to investigate the DFT calculated electronic band structures
and VESTA Momma and Izumi (2011) software was used to draw the crystal
structures.

##  III RESULTS AND DISCUSSION

###  I Crystal structures

Figure 1:  (a, b) Crystal structures of the $T_{d}$ and 1$T^{\prime\prime}$
phases of Mo0.5W0.5Te2. Note that 1$T^{\prime\prime}$ has a subtle monoclinic
distortion due to the cell angle $\beta\neq 90^{0}$. (c) Enthalpy versus
pressure phase diagram for $T_{d}$ and 1$T^{\prime\prime}$ phases. The
critical pressure ($P_{C}$) at which structural phase transition occurs is
predicted near 7.5 GPa. Inset: an enlarged view near the crossing point. (d)
The enthalpy difference $\Delta H$ (= $H_{T_{d}}-H_{1T^{\prime\prime}})$ as a
function of pressure. (e, f) Calculated phonon spectra (with-SOC) at 0 and 45
GPa pressure for $T_{d}$ and 1$T^{\prime\prime}$ phases. We would like to note
that we decided to plot the phonon dispersion for the 1$T^{\prime\prime}$
phase along the same high-symmetry k-path as in the $T_{d}$ phase for sake of
better comparison, although 1$T^{\prime\prime}$ phase has a subtle monoclinic
distortion (see SM SM  for details). (g) The calculated x-ray diffraction
(XRD) patterns for the $T_{d}$ and 1$T{{}^{\prime\prime}}$ phases at 0 GPa
pressure. The signature peaks in both phases are marked using magenta color.
The calculated XRD patterns and phonon spectra at higher pressures are
reported in the SM SM .

In order to theoretically investigate the electronic structure of
Mo0.5W0.5Te2, we prepare two candidate crystal structures, as shown in Figs.
1(a,b), by taking the parent $T_{d}$ phase (space group $Pmn2_{1}$) of WTe2
and systematically substituting one W atom by one Mo atom in each vertically-
stacked layer. There are two possible ways to carry out such a substitution.
First, we consider a scenario in which the ordering of Mo and W atoms is
reversed in the adjacent vertically-stacked layers, i.e.,
(Mo-W)$\cdot\cdot\cdot$(W-Mo)$\cdot\cdot\cdot$(Mo-W)$\cdot\cdot\cdot$. This
results in a crystal structure belonging to the space group $Pmn2_{1}$ (no.
31), similar to the parent $T_{d}$ phase, hence we call this structure the
$T_{d}$ phase. Note, this structure retains the vertical glide-mirror
symmetry, as discussed in Ref. Singh  _et al._ (2020). Also, this structure
does not break the orthorhombic symmetry of the parent $T_{d}$ structure after
a free DFT relaxation.

Second, we consider a scenario in which ordering of the Mo and W atoms remains
the same within the adjacent vertically-stacked layers, i.e.,
(Mo-W)$\cdot\cdot\cdot$(Mo-W)$\cdot\cdot\cdot$(Mo-W)$\cdot\cdot\cdot$. Such a
configuration breaks the vertical glide-mirror symmetry Singh  _et al._ (2020)
and results in the space group $Pm$ (no. 6) after a free DFT relaxation of the
unit cell. Note that the optimized structure is slightly distorted from the
parent orthorhombic cell to a distorted monoclinic cell with a monoclinic cell
angle of $\beta\neq 90^{\circ}$, which varies as a function of the applied
hydrostatic pressure, as we discuss below. Although this monoclinic structure
is similar to the 1$T^{\prime}$ phase (space group: $P2_{1}/m$) of TMDs, it is
lower in symmetry due to the broken inversion and vertical glide-mirror
symmetries Singh  _et al._ (2020). Therefore, we decide to call this
monoclinic phase as the 1$T^{\prime\prime}$ phase. We note both the
structures, $T_{d}$ and 1$T^{\prime\prime}$, have broken inversion symmetry,
which is a fundamental requirement for nonmagnetic Weyl semimetals.

Before moving further, let us briefly discuss the possible reason behind the
observed monoclinic distortion in the 1$T^{\prime\prime}$ phase. One could
imagine the presence of an in-plane polarity orientation determined by the
peculiar ordering of Mo and W atoms having different electronegativity within
each vertically-stacked layer, as shown in Figs. 1(a,b). In Fig. 1(a), where
the ordering of Mo-W pairs is reversed as we move in the vertical direction,
the in-plane polarity would reverse its sign in the adjacent vertically-
stacked layers, i.e., $-+-+\cdot\cdot\cdot$ or $+-+-\cdot\cdot\cdot$
(antipolar order). Here, $+/-$ sign denotes the polarity orientation parallel
or antiparallel to the $a$ lattice vector. In such a configuration, a free
relaxation of structure does not require any monoclinic distortion of the
orthorhombic cell due to the perfect cancellation of dipolar-like interactions
along the vertical direction (not strictly speaking, because partial screening
of electric dipoles may occur due to the semimetallic nature of this system).
On the other hand, in Fig. 1(b), where the ordering of Mo-W pairs is the same
in the adjacent vertical layers, the in-plane polarity would be parallel in
the adjacent layers, i.e., $----\cdot\cdot\cdot$ or $++++\cdot\cdot\cdot$,
similar to that of the 1$T^{\prime}$ phase Huang  _et al._ (2019); Singh  _et
al._ (2020). Hence, in order to minimize the total free energy such structures
tend to exhibit a monoclinic distortion due to the sliding of the adjacent
polar layers along the in-plane direction, as discussed in Ref. Singh  _et
al._ (2020).

###  II Pressure-induced effects on the $T_{d}$ and 1$T^{\prime\prime}$
structures

Next, we test the relative stability of the $T_{d}$ and 1$T^{\prime\prime}$
phases as a function of pressure. Our calculations reveal that the enthalpy
difference ($\Delta H$) between these two phases is very small [Figs. 1(c,
d)]. This implies the likelihood of formation of solid solution of the $T_{d}$
and 1$T^{\prime\prime}$ phases at finite temperatures. We find that below the
critical pressure ($P_{C}$ = 7.5 GPa), the 1$T^{\prime\prime}$ phase is more
favorable, whereas above $P_{C}$ the $T_{d}$ phase is preferred. We further
performed the equation of states (EOS) analyses using the Birch, Vinet, and
Birch-Murnaghan models as implemented in the mechelastic package Singh  _et
al._ (2021). The difference between the obtained EOS fitting parameters for
the $T_{d}$ and 1$T^{\prime\prime}$ phases is minimal (see SM SM  for more
details).

The phonon spectra calculated considering the SOC and vdW corrections for both
the $T_{d}$ [Fig. 1(e)] and 1$T^{\prime\prime}$ [Fig. 1(f)] phases demonstrate
the dynamical stability of these phases at zero pressure as well as at higher
pressures (see SM SM ). We notice an increase in the phonon frequencies with
increasing pressure, i.e., phonons harden at higher pressures. No dynamical
instability was observed in both phases within the studied pressure range. A
list of the infrared- and Raman-active phonon frequencies calculated at
different pressures is provided in the SM SM . We hope this could facilitate
the experimental identification of the $T_{d}$ and 1$T^{\prime\prime}$ phases.

Furthermore, we calculate the x-ray diffraction (XRD) patterns of the $T_{d}$
and 1$T^{\prime\prime}$ phases at various pressures using Cu-K$\alpha$ x-ray
of wavelength 1.5406 Å. Fig. 1(g) shows the calculated XRD patterns at zero
pressure (data at higher pressure are provided in SM SM ). Although the
calculated XRD spectrum looks quite similar for both the phases, there are
some signature peaks, marked using magenta stars, which are present in one
phase but absent in another. These peaks can be used to distinguish between
the $T_{d}$ and 1$T^{\prime\prime}$ phases in real crystals.

Figs. 2(a,b) show the pressure dependence of the DFT optimized lattice
parameters for $T_{d}$ and 1$T^{\prime\prime}$ phases. We observe a very
similar trend in the pressure-dependent structural parameters of both phases.
To highlight the observed trend, we plot the normalized the lattice parameters
with respect to the lattice parameters obtained at zero pressure. We observe
maximum change in the $c$ lattice parameter with varying pressure. At 45 GPa,
the $c$ lattice parameter decreases by $\sim$15% for both phases, whereas
relative change in the $a$ and $b$ lattice parameters is less than 10% within
the studied pressure range. Such a behavior is expected owing to the weak vdW
interaction along the out-of-plane $c$ axis. The optimized Mo-W bond length
shows a similar pressure-dependent behavior as the in-plane lattice
parameters. The Mo-W bond length decreases from a value 2.85 Å at zero
pressure to 2.70 Å at 45 GPa. This is almost 5% decreases in the Mo-W bond
length. On the other hand, the maximum compression in the Mo-Te and W-Te bond
lengths is nearly 4% at 45 GPa. Fig. 2(d) shows the variation in the
monoclinic angle $\beta$ as a function of the pressure for the
1$T^{\prime\prime}$ phase. With increasing pressure, the cell angle $\beta$
tends to approach $90^{\circ}$, which implies a decrease in the monoclinic
distortion and preference towards the orthorhombic $T_{d}$ phase at higher
pressures. This is consistent with the data shown in Fig. 1(d).

Since Mo0.5W0.5Te2 is particularly interesting due to its superconducting
properties Dahal  _et al._ (2020), we calculate the density of states at the
Fermi level N(EF) as a function of pressure for both $T_{d}$ and
1$T^{\prime\prime}$ phases, as shown in Fig. 2(c). A systematic increase in
N(EF) with increasing pressure was observed in both phases. We would like to
note that phonons also harden with increasing pressure [see Figs. 1(e,f)].
Thus, an increased N(EF) together with the higher phonon frequencies implies
an enhancement in the effective electron-phonon coupling at higher pressures,
which could substantially increase the superconducting transition temperature
in this system at higher pressures Dynes (1972); Lu  _et al._ (2016); Heikes
_et al._ (2018); Paudyal  _et al._ (2020). This is consistent with recent
experimental observations Dahal  _et al._ (2020); Takahashi  _et al._ (2017).

Figure 2:  (a, b) The optimized lattice parameters normalized to the lattice
parameters obtained at zero pressure ($a_{0}$, $b_{0}$, $c_{0}$), and (c) the
calculated density of states (states/eV) at the Fermi level N(EF) plotted as a
function of pressure for both $T_{d}$ and 1$T^{\prime\prime}$ phases. (d)
Pressure-dependent variation in the monoclinic cell angle ($\beta$) for
1$T^{\prime\prime}$ phase. Note that $\beta=90^{0}$ for the $T_{d}$ phase.

###  III Pressure-tunable Weyl semimetal phase

Figure 3: Band structures of $T_{d}$ phase calculated at different pressure
values. (a) 0 GPa, (b) 5 GPa, (c) 10 GPa and (d) 15 GPa, (e) 25 GPa, and (f)
35 GPa with inclusion of SOC. Red color in the band structure represents the
lowest conduction band and the green color represents the highest valence
band. The dashed horizontal line marks the EF (EF = 0 eV). Figure 4: Band
structures of 1$T^{\prime\prime}$ phase calculated at different pressure
values. (a) 0 GPa, (b) 5 GPa, (c) 10 GPa and (d) 15 GPa, (e) 25 GPa, and (f)
35 GPa with inclusion of SOC. Red color in the band structure represents the
lowest conduction band and the green color represents the highest valence
band. The dashed horizontal line marks the EF. For the better comparison $k$
paths of 1$T^{\prime\prime}$ phase are also chosen same as in the $T_{d}$
phase. Figure 5: (a) Energy states of WPs with respect to EF calculated at
different values of pressure for the $T_{d}$ phase, (b) Stack-bar diagram
showing the creation and annihilation of the WPs from 0 to 45 GPa for the
$T_{d}$ phase. (c) Energy states of WPs corresponding to different pressure
for the 1$T^{\prime\prime}$ phase. (d) Stack-bar diagram showing the creation
and annihilation of the WPs from 0 to 45 GPa for the 1$T^{\prime\prime}$
phase. The total number of WPs at each pressure is written at the top of each
bar.

The band structure calculations of $T_{d}$ and 1$T^{\prime\prime}$ phase for
different pressure is presented in Figs. 3 and 4. The topmost valence band
indicated with green color forms hole pocket and bottom of the conduction band
indicated with red color forms electron pocket nearby the EF along the
$\Gamma$-X, S-$\Gamma$-Y and U-Z-T high-symmetry directions of the Brillouin
zone (BZ). These electron and the hole pockets touch each other at discrete
points in the momentum space forming type-II WPs Soluyanov  _et al._ (2015b).
The change in pressure affects the size of the electron and hole pockets in
the momentum space. With increasing pressure, these pockets come close to each
other producing more number of band crossings, as implied in Figs. 3 and 4. A
similar trend is found for the 1$T^{\prime\prime}$ phase.

TMDs are the first class of materials to host type II WPs. Soluyanov et al.
first reported type II WPs in pure WTe2 located between 0.052 – 0.058 eV above
EF Soluyanov  _et al._ (2015b). 16 WPs are found in the $k$\- space with 8 WPs
at $k_{z}$ = 0 plane and 8 more WPs off the plane ($k_{i}$ $\neq$ 0, $i$ =
$x$, $y$ and $z$). With inclusion of SOC all the WPs at $k_{i}$ $\neq$ 0 are
annihilated leaving only eight of the sixteen WPs. Similarly, Sun et al.,
predicted the type II WSMs in MoTe2 Sun  _et al._ (2015a). Eight WPs are
recorded at two different energies 6 meV and 59 meV above EF.

Here, in our study 50% substitution of Mo on WTe2 without application of
pressure, we report four WPs at an energy state 55 meV above the EF at $k_{z}$
= 0 plane for the $T_{d}$ phase. Contrary to that, a similar number of WPs are
recorded for the 1$T^{\prime\prime}$ phase but at a different energy state of
93 meV below the EF at kz $\neq$ 0 plane. The energy of the WPs obtained for
the $T_{d}$ phase is agreement with the energy of WPs for MoTe2 Sun  _et al._
(2015a) and WTe2 Soluyanov  _et al._ (2015b), whereas the 1$T^{\prime\prime}$
phase hosts WPs below EF. Moreover, the chemical effect is also found to play
a significant role for the change in the total number of the WPs in
Mo0.5W0.5Te2.

Table 1: Location of W1 WPs in the momentum space of the $T_{d}$ and 1$T^{\prime\prime}$ phases at zero pressure WP |  $k_{x}$ $\left(\dfrac{2\pi}{a}\right)$ |  $k_{y}$ $\left(\dfrac{2\pi}{b}\right)$ |  $k_{z}$ $\left(\dfrac{2\pi}{c}\right)$ | Chirality($\chi$)
---|---|---|---|---
W1($T_{d}$) | -0.132 | -0.095 | 0.000 | +1
W1(1$T^{\prime\prime}$) | -0.004 | -0.200 | -0.416 | +1

We obtain a total of four WPs (W1) in the first BZ of the $T_{d}$ and
1$T^{\prime\prime}$ phases of Mo0.5W0.5Te2 at zero pressure. Co-ordinates of
the nonequivalent W1 WPs are given in Table 1; other three WPs are the mirror
reflections of W1 at zero pressure. There is a variation in the total number
and locations of the WPs due to the application of hydrostatic pressure. Below
we discuss the role of the hydrostatic pressure on the Weyl phase of $T_{d}$\-
and $1T^{\prime\prime}$-Mo0.5W0.5Te2.

a) Pressure effects on the $T_{d}$ phase

The variation in the number of the WPs and their energy states for the $T_{d}$
phase is presented through the graphical plots in Figs. 5 (a) and (b). On
increasing the pressure value to 1.5 GPa the new sets of WPs W2 and W3 are
generated along with the initial W1 making the total number of WPs to 20.
Here, we noticed that the pressure shifts the energy state of W1 to EF+75 meV.
The newly created 8 copies of W2 and W3 are found in the energy states of EF+8
meV and EF-7 meV, respectively. No further creation of the WPs upto the
pressure of 8 GPa, simply the position and the energy states of the WPs vary
with the pressure. The energy state of W1 rises to the highest value of EF+139
meV at 8 GPa pressure and decreases on the further increase which reach
nearest to the Fermi level (energy state EF+1 meV) at 35 GPa and shifts below
the Fermi level above that pressure. The pair creation of W4 and W5 each four
in number occurs at 9 GPa which continue to exist up to 10 GPa and get
annihilated above that pressure. The highest number of WPs is observed at 20
GPa and a constant number of 16 WPs is observed after 30 GPa to 45 GPa.

b) Pressure effects on the 1$T^{\prime\prime}$ phase

Pressure also has an impact on the dynamics of WPs in the 1$T^{\prime\prime}$
phase. Pair creation and the annihilation of WPs as the function of pressure
is noticed in the momentum space of the 1$T^{\prime\prime}$ phase. Here, the
W1 WPs continue to exist up to 10 GPa which annihilates on further rise in
pressure. Similarly, the W2 WPs are created at 5 GPa pressure, which exist up
to 40 GPa and then get annihilated. A similar phenomenon of pair creation and
annihilation is noticed for the other energy state of WPs as well, which is
summarized in the Figs. 5 (c) and (d). As compared to the $T_{d}$ phase more
energy states of the WPs are observed in the 1$T^{\prime\prime}$ phase due to
its lower symmetry. In terms of the total number of WPs, the highest number 36
is observed at 15 GPa pressure, whereas a constant number of 16 WPs is noticed
for pressure beyond 40 GPa.

##  IV Summary

In summary, we studied the structural, vibrational, electronic, and
topological Weyl properties of Mo0.5W0.5Te2 by means of first-principles DFT
calculations. We find that there are two possible candidate structures,
$T_{d}$ and 1$T^{\prime\prime}$, for 50:50 Mo:W substitution. We studied the
aforementioned properties of these two phases in 0-45 GPa pressure range. We
find that both these structures are energetically and dynamically stable in
the studied pressure. The calculated x-ray diffraction spectra, and the
infrared- and Raman-active phonon frequencies indicate that these two phases
can be identified in experiments, although they are likely to form solid
solutions due to the subtle difference in their enthalpy at low pressures. The
$T_{d}$ (1$T^{\prime\prime}$) phase is theoretically more favorable at higher
(lower) pressures, the critical pressure being 7.5 GPa. Our calculations
reveal that the density of states and phonon frequencies increase dramatically
with increasing pressure in both phases, which is indicative of larger
electron-phonon coupling at higher pressures and it could substantially
increase the superconducting transition temperature in this system at higher
pressures.

Interestingly, we find that both the $T_{d}$ and 1$T^{\prime\prime}$ phases
host a number of WPs in their momentum-energy space. The total number and
location of WPs can be controlled by varying hydrostatic pressure. Four WPs
are obtained in the $T_{d}$ and 1$T^{\prime\prime}$ phases at zero pressure.
The total number of WPs increases to 44 (36) with increasing pressure, via
pair creation, up to 20 (15) GPa for the $T_{d}$ (1$T^{\prime\prime}$) phase,
and beyond this pressure, pair annihilation of WPs starts occurring leaving
only 16 WPs at 45 GPa in both phases. Therefore, we can conclude that pressure
can tune WPs to the desired location, generate new WPs, and also annihilate
them, thus, providing an ideal platform for realization of Weyltronics in
Mo0.5W0.5Te2.

###### Acknowledgements.

M.P.G. acknowledges the Alexander von Humboldt Foundation, Germany for the
equipment grants and IFW-Dresden for providing the large-scale compute nodes
to Advanced Materials Research Laboratory for scientific computations. S.S.
acknowledges the support from the Office of Naval Research (ONR) grant
N00014-21-1-2107. G.B.A thanks Nepal Academy of Science and Technology for the
PhD fellowship. M.P.G. and G.B.A thanks Manuel Richter for the fruitful
discussion and Ulrike Nitzsche for the technical assistance.

## References

  * Hasan and Kane (2010) M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. 82, 3045 (2010).
  * Armitage  _et al._ (2018) N. P. Armitage, E. J. Mele, and A. Vishwanath, Rev. Mod. Phys. 90, 015001 (2018).
  * Yan and Felser (2017) B. Yan and C. Felser, Annual Review of Condensed Matter Physics 8, 337 (2017).
  * Manna  _et al._ (2018) K. Manna, Y. Sun, L. Muechler, J. Kübler, and C. Felser, Nature Reviews Materials 3, 244 (2018).
  * Weng  _et al._ (2016) H. Weng, X. Dai, and Z. Fang, Journal of Physics: Condensed Matter 28, 303001 (2016).
  * Gao  _et al._ (2019) H. Gao, J. W. Venderbos, Y. Kim, and A. M. Rappe, Annual Review of Materials Research 49, 153 (2019).
  * Burkov (2018) A. Burkov, Annual Review of Condensed Matter Physics 9, 359 (2018).
  * Hasan  _et al._ (2017) M. Z. Hasan, S.-Y. Xu, I. Belopolski, and S.-M. Huang, Annual Review of Condensed Matter Physics 8, 289 (2017).
  * Lv  _et al._ (2015a) B. Lv, N. Xu, H. Weng, J. Ma, P. Richard, X. Huang, L. Zhao, G. Chen, C. Matt, F. Bisti, _et al._ , Nature Physics 11, 724 (2015a).
  * Lv  _et al._ (2015b) B. Lv, H. Weng, B. Fu, X. Wang, H. Miao, J. Ma, P. Richard, X. Huang, L. Zhao, G. Chen, _et al._ , Physical Review X 5, 031013 (2015b).
  * Huang  _et al._ (2015) S.-M. Huang, S.-Y. Xu, I. Belopolski, C.-C. Lee, G. Chang, B. Wang, N. Alidoust, G. Bian, M. Neupane, C. Zhang, _et al._ , Nature communications 6, 1 (2015).
  * Xu  _et al._ (2015a) S.-Y. Xu, I. Belopolski, D. S. Sanchez, C. Zhang, G. Chang, C. Guo, G. Bian, Z. Yuan, H. Lu, T.-R. Chang, _et al._ , Science advances 1, e1501092 (2015a).
  * Sushkov  _et al._ (2015) A. B. Sushkov, J. B. Hofmann, G. S. Jenkins, J. Ishikawa, S. Nakatsuji, S. Das Sarma, and H. D. Drew, Phys. Rev. B 92, 241108 (2015).
  * Lv  _et al._ (2015c) B. Q. Lv, S. Muff, T. Qian, Z. D. Song, S. M. Nie, N. Xu, P. Richard, C. E. Matt, N. C. Plumb, L. X. Zhao, G. F. Chen, Z. Fang, X. Dai, J. H. Dil, J. Mesot, M. Shi, H. M. Weng, and H. Ding, Phys. Rev. Lett. 115, 217601 (2015c).
  * Murakami (2007) S. Murakami, New Journal of Physics 9, 356 (2007).
  * Jia  _et al._ (2016) S. Jia, S.-Y. Xu, and M. Z. Hasan, Nature materials 15, 1140 (2016).
  * Weyl (1929) H. Weyl, journal for physics 56, 330 (1929).
  * Wan  _et al._ (2011) X. Wan, A. M. Turner, A. Vishwanath, and S. Y. Savrasov, Phys. Rev. B 83, 205101 (2011).
  * Grassano  _et al._ (2018) D. Grassano, O. Pulci, A. M. Conte, and F. Bechstedt, Scientific reports 8, 1 (2018).
  * Ruan  _et al._ (2016) J. Ruan, S.-K. Jian, H. Yao, H. Zhang, S.-C. Zhang, and D. Xing, Nature communications 7, 1 (2016).
  * Soluyanov  _et al._ (2015a) A. A. Soluyanov, D. Gresch, Z. Wang, Q. Wu, M. Troyer, X. Dai, and B. A. Bernevig, Nature 527, 495 (2015a).
  * Singh  _et al._ (2016) S. Singh, A. C. Garcia-Castro, I. Valencia-Jaime, F. Muñoz, and A. H. Romero, Phys. Rev. B 94, 161116 (2016).
  * Singh  _et al._ (2018a) S. Singh, Q. Wu, C. Yue, A. H. Romero, and A. A. Soluyanov, Phys. Rev. Materials 2, 114204 (2018a).
  * Winkler  _et al._ (2019) G. W. Winkler, S. Singh, and A. A. Soluyanov, Chinese Physics B 28, 077303 (2019).
  * Chen  _et al._ (2016) F. C. Chen, H. Y. Lv, X. Luo, W. J. Lu, Q. L. Pei, G. T. Lin, Y. Y. Han, X. B. Zhu, W. H. Song, and Y. P. Sun, Phys. Rev. B 94, 235154 (2016).
  * Pei  _et al._ (2017) Q. L. Pei, W. J. Meng, X. Luo, H. Y. Lv, F. C. Chen, W. J. Lu, Y. Y. Han, P. Tong, W. H. Song, Y. B. Hou, Q. Y. Lu, and Y. P. Sun, Phys. Rev. B 96, 075132 (2017).
  * Thirupathaiah  _et al._ (2017) S. Thirupathaiah, R. Jha, B. Pal, J. S. Matias, P. K. Das, P. K. Sivakumar, I. Vobornik, N. C. Plumb, M. Shi, R. A. Ribeiro, and D. D. Sarma, Phys. Rev. B 95, 241105 (2017).
  * Lee  _et al._ (2018) S. Lee, J. Jang, S.-I. Kim, S.-G. Jung, J. Kim, S. Cho, S. W. Kim, J. Y. Rhee, K.-S. Park, and T. Park, Scientific Reports 8, 13937 (2018).
  * Xu  _et al._ (2015b) S.-Y. Xu, I. Belopolski, N. Alidoust, M. Neupane, G. Bian, C. Zhang, R. Sankar, G. Chang, Z. Yuan, C.-C. Lee, _et al._ , Science 349, 613 (2015b).
  * Xu  _et al._ (2016) N. Xu, H. Weng, B. Lv, C. E. Matt, J. Park, F. Bisti, V. N. Strocov, D. Gawryluk, E. Pomjakushina, K. Conder, _et al._ , Nature communications 7, 1 (2016).
  * Belopolski  _et al._ (2016a) I. Belopolski, S.-Y. Xu, D. S. Sanchez, G. Chang, C. Guo, M. Neupane, H. Zheng, C.-C. Lee, S.-M. Huang, G. Bian, N. Alidoust, T.-R. Chang, B. Wang, X. Zhang, A. Bansil, H.-T. Jeng, H. Lin, S. Jia, and M. Z. Hasan, Phys. Rev. Lett. 116, 066802 (2016a).
  * Tamai  _et al._ (2016) A. Tamai, Q. S. Wu, I. Cucchi, F. Y. Bruno, S. Riccò, T. K. Kim, M. Hoesch, C. Barreteau, E. Giannini, C. Besnard, A. A. Soluyanov, and F. Baumberger, Phys. Rev. X 6, 031021 (2016).
  * Autès  _et al._ (2016a) G. Autès, D. Gresch, M. Troyer, A. A. Soluyanov, and O. V. Yazyev, Phys. Rev. Lett. 117, 066402 (2016a).
  * Zhang  _et al._ (2017) W. Zhang, Q. Wu, L. Zhang, S.-W. Cheong, A. A. Soluyanov, and W. Wu, Phys. Rev. B 96, 165125 (2017).
  * Ghimire  _et al._ (2019) M. P. Ghimire, J. I. Facio, J.-S. You, L. Ye, J. G. Checkelsky, S. Fang, E. Kaxiras, M. Richter, and J. van den Brink, Physical Review Research 1, 032044 (2019).
  * Singh  _et al._ (2020) S. Singh, J. Kim, K. M. Rabe, and D. Vanderbilt, Phys. Rev. Lett. 125, 046402 (2020).
  * Autès  _et al._ (2016b) G. Autès, D. Gresch, M. Troyer, A. A. Soluyanov, and O. V. Yazyev, Phys. Rev. Lett. 117, 066402 (2016b).
  * Chang  _et al._ (2016a) G. Chang, S.-Y. Xu, D. S. Sanchez, S.-M. Huang, C.-C. Lee, T.-R. Chang, G. Bian, H. Zheng, I. Belopolski, N. Alidoust, _et al._ , Science Advances 2, e1600295 (2016a).
  * Koepernik  _et al._ (2016) K. Koepernik, D. Kasinathan, D. V. Efremov, S. Khim, S. Borisenko, B. Büchner, and J. van den Brink, Phys. Rev. B 93, 201101 (2016).
  * Sipos  _et al._ (2008) B. Sipos, A. F. Kusmartseva, A. Akrap, H. Berger, L. Forró, and E. Tutiš, Nature materials 7, 960 (2008).
  * Qian  _et al._ (2014) X. Qian, J. Liu, L. Fu, and J. Li, Science 346, 1344 (2014).
  * Morris  _et al._ (1972a) R. Morris, R. Coleman, and R. Bhandari, Physical Review B 5, 895 (1972a).
  * Takahashi  _et al._ (2017) H. Takahashi, T. Akiba, K. Imura, T. Shiino, K. Deguchi, N. K. Sato, H. Sakai, M. S. Bahramy, and S. Ishiwata, Phys. Rev. B 95, 100501 (2017).
  * Qi  _et al._ (2016) Y. Qi, P. G. Naumov, M. N. Ali, C. R. Rajamathi, W. Schnelle, O. Barkalov, M. Hanfland, S.-C. Wu, C. Shekhar, Y. Sun, _et al._ , Nature communications 7, 1 (2016).
  * Chen  _et al._ (2018) F. C. Chen, X. Luo, J. Yan, Y. Sun, H. Y. Lv, W. J. Lu, C. Y. Xi, P. Tong, Z. G. Sheng, X. B. Zhu, W. H. Song, and Y. P. Sun, Phys. Rev. B 98, 041114 (2018).
  * Huang  _et al._ (2019) F.-T. Huang, S. Joon Lim, S. Singh, J. Kim, L. Zhang, J.-W. Kim, M.-W. Chu, K. M. Rabe, D. Vanderbilt, and S.-W. Cheong, Nature Communications 10, 4211 (2019).
  * Heikes  _et al._ (2018) C. Heikes, I.-L. Liu, T. Metz, C. Eckberg, P. Neves, Y. Wu, L. Hung, P. Piccoli, H. Cao, J. Leao, J. Paglione, T. Yildirim, N. P. Butch, and W. Ratcliff, Phys. Rev. Materials 2, 074202 (2018).
  * Morris  _et al._ (1972b) R. C. Morris, R. V. Coleman, and R. Bhandari, Phys. Rev. B 5, 895 (1972b).
  * Wei  _et al._ (2014) H. Wei, S.-P. Chao, and V. Aji, Phys. Rev. B 89, 014506 (2014).
  * Hosur  _et al._ (2014) P. Hosur, X. Dai, Z. Fang, and X.-L. Qi, Phys. Rev. B 90, 045130 (2014).
  * Soluyanov  _et al._ (2015b) A. A. Soluyanov, D. Gresch, Z. Wang, Q. Wu, M. Troyer, X. Dai, and B. A. Bernevig, Nature 527, 495 (2015b).
  * Li  _et al._ (2018) P. Li, W. Wu, Y. Wen, C. Zhang, J. Zhang, S. Zhang, Z. Yu, S. A. Yang, A. Manchon, and X.-x. Zhang, Nature communications 9, 1 (2018).
  * Bruno  _et al._ (2016) F. Y. Bruno, A. Tamai, Q. Wu, I. Cucchi, C. Barreteau, A. De La Torre, S. M. Walker, S. Riccò, Z. Wang, T. Kim, _et al._ , Physical Review B 94, 121112 (2016).
  * Wu  _et al._ (2016) Y. Wu, D. Mou, N. H. Jo, K. Sun, L. Huang, S. L. Bud’ko, P. C. Canfield, and A. Kaminski, Physical Review B 94, 121113 (2016).
  * Li  _et al._ (2017) P. Li, Y. Wen, X. He, Q. Zhang, C. Xia, Z.-M. Yu, S. A. Yang, Z. Zhu, H. N. Alshareef, and X.-X. Zhang, Nature communications 8, 1 (2017).
  * Lv  _et al._ (2017a) Y.-Y. Lv, X. Li, B.-B. Zhang, W. Deng, S.-H. Yao, Y. Chen, J. Zhou, S.-T. Zhang, M.-H. Lu, L. Zhang, _et al._ , Physical review letters 118, 096603 (2017a).
  * Huang  _et al._ (2016) L. Huang, T. M. McCormick, M. Ochi, Z. Zhao, M.-T. Suzuki, R. Arita, Y. Wu, D. Mou, H. Cao, J. Yan, _et al._ , Nature materials 15, 1155 (2016).
  * Deng  _et al._ (2016) K. Deng, G. Wan, P. Deng, K. Zhang, S. Ding, E. Wang, M. Yan, H. Huang, H. Zhang, Z. Xu, _et al._ , Nature Physics 12, 1105 (2016).
  * Sakano  _et al._ (2017) M. Sakano, M. Bahramy, H. Tsuji, I. Araya, K. Ikeura, H. Sakai, S. Ishiwata, K. Yaji, K. Kuroda, A. Harasawa, _et al._ , Physical Review B 95, 121101 (2017).
  * Sun  _et al._ (2015a) Y. Sun, S.-C. Wu, M. N. Ali, C. Felser, and B. Yan, Physical Review B 92, 161107 (2015a).
  * Wang  _et al._ (2016) Z. Wang, D. Gresch, A. A. Soluyanov, W. Xie, S. Kushwaha, X. Dai, M. Troyer, R. J. Cava, and B. A. Bernevig, Physical review letters 117, 056805 (2016).
  * Jiang  _et al._ (2017) J. Jiang, Z. Liu, Y. Sun, H. Yang, C. Rajamathi, Y. Qi, L. Yang, C. Chen, H. Peng, C. Hwang, _et al._ , Nature communications 8, 13973 (2017).
  * Liang  _et al._ (2016) A. Liang, J. Huang, S. Nie, Y. Ding, Q. Gao, C. Hu, S. He, Y. Zhang, C. Wang, B. Shen, _et al._ , arXiv preprint arXiv:1604.01706 (2016).
  * Sun  _et al._ (2015b) Y. Sun, S.-C. Wu, M. N. Ali, C. Felser, and B. Yan, Phys. Rev. B 92, 161107 (2015b).
  * Dissanayake  _et al._ (2019) S. Dissanayake, C. Duan, J. Yang, J. Liu, M. Matsuda, C. Yue, J. A. Schneeloch, J. C. Y. Teo, and D. Louca, npj Quantum Materials 4, 45 (2019).
  * Aryal and Manousakis (2019) N. Aryal and E. Manousakis, Phys. Rev. B 99, 035123 (2019).
  * Xu  _et al._ (2018) N. Xu, Z. W. Wang, A. Magrez, P. Bugnon, H. Berger, C. E. Matt, V. N. Strocov, N. C. Plumb, M. Radovic, E. Pomjakushina, K. Conder, J. H. Dil, J. Mesot, R. Yu, H. Ding, and M. Shi, Phys. Rev. Lett. 121, 136401 (2018).
  * Lv  _et al._ (2017b) Y.-Y. Lv, L. Cao, X. Li, B.-B. Zhang, K. Wang, B. Pang, L. Ma, D. Lin, S.-H. Yao, J. Zhou, Y. B. Chen, S.-T. Dong, W. Liu, M.-H. Lu, Y. Chen, and Y.-F. Chen, Scientific Reports 7, 44587 (2017b).
  * Yan  _et al._ (2017) X.-J. Yan, Y.-Y. Lv, L. Li, X. Li, S.-H. Yao, Y.-B. Chen, X.-P. Liu, H. Lu, M.-H. Lu, and Y.-F. Chen, Applied Physics Letters 110, 211904 (2017).
  * Oliver  _et al._ (2017) S. M. Oliver, R. Beams, S. Krylyuk, I. Kalish, A. K. Singh, A. Bruma, F. Tavazza, J. Joshi, I. R. Stone, S. J. Stranick, _et al._ , 2D Materials 4, 045008 (2017).
  * Rhodes  _et al._ (2017) D. Rhodes, D. A. Chenet, B. E. Janicek, C. Nyby, Y. Lin, W. Jin, D. Edelberg, E. Mannebach, N. Finney, A. Antony, T. Schiros, T. Klarr, A. Mazzoni, M. Chin, Y. c. Chiu, W. Zheng, Q. R. Zhang, F. Ernst, J. I. Dadap, X. Tong, J. Ma, R. Lou, S. Wang, T. Qian, H. Ding, R. M. Osgood, D. W. Paley, A. M. Lindenberg, P. Y. Huang, A. N. Pasupathy, M. Dubey, J. Hone, and L. Balicas,  _Nano Letters_ , Nano Letters 17, 1616 (2017).
  * Aslan  _et al._ (2018) O. B. Aslan, I. M. Datye, M. J. Mleczko, K. Sze Cheung, S. Krylyuk, A. Bruma, I. Kalish, A. V. Davydov, E. Pop, and T. F. Heinz,  _Nano Letters_ , Nano Letters 18, 2485 (2018).
  * Schneeloch  _et al._ (2020) J. A. Schneeloch, Y. Tao, C. Duan, M. Matsuda, A. A. Aczel, J. A. Fernandez-Baca, G. Xu, J. C. Neuefeind, J. Yang, and D. Louca, Phys. Rev. B 102, 054105 (2020).
  * Marchenkov  _et al._ (2019) V. Marchenkov, A. Domozhirova, A. Makhnev, E. Shreder, S. Naumov, V. Chistyakov, J. Huang, and M. Eisterer, Low temperature physics 45, 241 (2019).
  * Tao  _et al._ (2019) Y. Tao, J. A. Schneeloch, C. Duan, M. Matsuda, S. E. Dissanayake, A. A. Aczel, J. A. Fernandez-Baca, F. Ye, and D. Louca, Phys. Rev. B 100, 100101 (2019).
  * Dahal  _et al._ (2020) R. Dahal, L. Z. Deng, N. Poudel, M. Gooch, Z. Wu, H. C. Wu, H. D. Yang, C. K. Chang, and C. W. Chu, Phys. Rev. B 101, 140505 (2020).
  * Belopolski  _et al._ (2016b) I. Belopolski, S.-Y. Xu, Y. Ishida, X. Pan, P. Yu, D. S. Sanchez, H. Zheng, M. Neupane, N. Alidoust, G. Chang, _et al._ , Physical Review B 94, 085127 (2016b).
  * Chang  _et al._ (2016b) T.-R. Chang, S.-Y. Xu, G. Chang, C.-C. Lee, S.-M. Huang, B. Wang, G. Bian, H. Zheng, D. S. Sanchez, I. Belopolski, _et al._ , Nature communications 7, 1 (2016b).
  * Belopolski  _et al._ (2016c) I. Belopolski, D. S. Sanchez, Y. Ishida, X. Pan, P. Yu, S.-Y. Xu, G. Chang, T.-R. Chang, H. Zheng, N. Alidoust, _et al._ , Nature communications 7, 1 (2016c).
  * Li  _et al._ (2021) P. Li, Y. Deng, C.-H. Hsu, C. Zhu, J. Cui, X. Yang, J. Zhou, Y.-C. Hung, J. Fan, Z. Ji, F. Qu, J. Shen, C. Yang, X. Jing, H. Lin, Z. Liu, L. Lu, and G. Liu, Phys. Rev. B 104, 085423 (2021).
  * Kresse and Furthmüller (1996) G. Kresse and J. Furthmüller, Phys. Rev. B 54, 11169 (1996).
  * Kresse and Furthmüller (1996) G. Kresse and J. Furthmüller, Computational Materials Science 6, 15 (1996).
  * Kresse and Joubert (1999) G. Kresse and D. Joubert, Phys. Rev. B 59, 1758 (1999).
  * Perdew  _et al._ (2008) J. P. Perdew, A. Ruzsinszky, G. I. Csonka, O. A. Vydrov, G. E. Scuseria, L. A. Constantin, X. Zhou, and K. Burke, Phys. Rev. Lett. 100, 136406 (2008).
  * Klimeš  _et al._ (2011) J. c. v. Klimeš, D. R. Bowler, and A. Michaelides, Phys. Rev. B 83, 195131 (2011).
  * Thonhauser  _et al._ (2007) T. Thonhauser, V. R. Cooper, S. Li, A. Puzder, P. Hyldgaard, and D. C. Langreth, Phys. Rev. B 76, 125112 (2007).
  * Dion  _et al._ (2004) M. Dion, H. Rydberg, E. Schröder, D. C. Langreth, and B. I. Lundqvist, Phys. Rev. Lett. 92, 246401 (2004).
  * Togo and Tanaka (2015) A. Togo and I. Tanaka, Scripta Materialia 108, 1 (2015).
  * Singh  _et al._ (2021) S. Singh, L. Lang, V. Dovale-Farelo, U. Herath, P. Tavadze, F.-X. Coudert, and A. H. Romero, Computer Physics Communications 267, 108068 (2021).
  * Singh  _et al._ (2018b) S. Singh, I. Valencia-Jaime, O. Pavlic, and A. H. Romero, Phys. Rev. B 97, 054108 (2018b).
  * Koepernik and Eschrig (1999) K. Koepernik and H. Eschrig, Phys. Rev. B 59, 1743 (1999).
  * Herath  _et al._ (2020) U. Herath, P. Tavadze, X. He, E. Bousquet, S. Singh, F. Muñoz, and A. H. Romero, Computer Physics Communications 251, 107080 (2020).
  * Momma and Izumi (2011) K. Momma and F. Izumi, Journal of Applied Crystallography 44, 1272 (2011).
  * (94) See Supporting Information at [href] for a detailed information about the XRD spectra, phonon spectra, Raman- and infrared-active phonon frequencies, DOS, and distribution of WPs calculated at higher pressures.
  * Dynes (1972) R. Dynes, Solid State Communications 10, 615 (1972).
  * Lu  _et al._ (2016) P. Lu, J.-S. Kim, J. Yang, H. Gao, J. Wu, D. Shao, B. Li, D. Zhou, J. Sun, D. Akinwande, D. Xing, and J.-F. Lin, Phys. Rev. B 94, 224512 (2016).
  * Paudyal  _et al._ (2020) H. Paudyal, S. Poncé, F. Giustino, and E. R. Margine, Phys. Rev. B 101, 214515 (2020).


# Planning with Probabilistic Opacity and Transparency: A Computational Model
of Opaque/Transparent Observations

Sumukha Udupa and Jie Fu S. Udupa and J. Fu are with the Dept. of Electrical
and Computer Engineering, University of Florida, Gainesville, Fl 32611 USA.
(e-mail: sudupa<EMAIL_ADDRESS>0000-0002-7462-9843 0000-0002-4470-2827

###### Abstract

Qualitative opacity of a secret is a security property, which means that a
system trajectory satisfying the secret is observation-equivalent to a
trajectory violating the secret. In this paper, we study how to synthesize a
control policy that maximizes the probability of a secret being made opaque
against an eavesdropping attacker/observer, while subject to other task
performance constraints. In contrast to existing belief-based approach for
opacity-enforcement, we develop an approach that uses the observation
function, the secret, and the model of the dynamical systems to construct a
so-called opaque-observations automaton which accepts the exact set of
observations that enforce opacity. Leveraging this opaque-observations
automaton, we can reduce the optimal planning in Markov decision
processes(MDPs) for maximizing probabilistic opacity or its dual notion,
transparency, subject to task constraints into a constrained planning problem
over an augmented-state MDP. Finally, we illustrate the effectiveness of the
developed methods in robot motion planning problems with opacity or
transparency requirements.

##  I Introduction

Opacity is a security and privacy property that evaluates whether an observer
(intruder) can deduce a system’s secret by monitoring its behavior. A system
is opaque if its secret behavior or private information is made uncertain to
an adversarial intruder. Initially introduced for cryptographic protocols by
Mazaré [12], opacity has since then been explored with various notions
depending on the nature of the secret. For instance, state-based opacity
ensures that the observer cannot discern if a secret state has been visited,
language-based opacity enforces that the observer cannot discern if the
trajectory is in a set of secret trajectories[5, 4, 14, 17, 8, 2].

To set the context, consider a scenario involving an autonomous robot (Player
1 or P1) tasked with routine monitoring in a power plant. A sensor network is
deployed for daily activity monitoring but can be vulnerable to eavesdropping
attacks. Should there be an unauthorized Player 2 (P2) observing the sensor
network, P1 must make P2 uncertain if a sequence of waypoints has been visited
to enforce location security. Now, the planning question is how can P1 ensure
maximal opacity to P2, while fulfilling the routine task beyond a specific
threshold?

In this study, we explore language-based opacity and its dual notion,
transparency, in stochastic systems with two players, where one player has
imperfect observations. Quantitative opacity quantifies the security level of
a system [3]. In competitive settings, achieving maximum probabilistic opacity
under task constraints is crucial, while in cooperative scenarios like human-
robot interactions, maximizing probabilistic transparency is essential.

The classical approach for opacity enforcement is to construct a belief-based
planning problem (see Related Work) where P1 tracks P2’s belief state
regarding the current or past state. However, this approach is computational
expensive.

Instead, we present an approach to construct a computational model of opaque
observations, which are observations obtained by state trajectories that
enforce opacity. We use a Finite State Transducer (FST) to encode the
observation function of state trajectories and then the synchronization
product between the FST and secret language, expressed as a Deterministic
Finite Automaton (DFA), to construct another DFA accepting the set of opaque
observations. A product between the Markov decision process (MDP), task DFA,
and the opaque-observations DFA is then computed. We show that the maximally
opaque (maximally transparent) control policy under the task performance
constraints can be solved as a constrained planning problem in this product.

###  I-A Related Work

In the supervisory control of discrete event systems (DES), researchers have
developed opacity-enforcing controllers by modelling the control systems as a
DFA in the presence of a passive observer [13, 14, 24]. Different approaches
have been proposed to either synthesize [6, 20, 21] or verify opacity [9, 15,
19] in deterministic systems. Saboori et al. [16] synthesized a supervisor to
enforce qualitative opacity by restricting system behaviors. Cassez et al. [6]
suggests using a dynamic mask to filter unobservable events, verifying opacity
through solving a 2-player safety game. In [21], Xie et al. proposed a
nondeterministic supervisor to prevent the observer, aware of the supervisor’s
nondeterminism, from determining the secret. Besides enforcing opacity, sensor
attack strategies to compromise opacity has been investigated in [22], where
they proposed an information structure that records state estimates for both
the supervisor and attacker. In game-theoretic approaches to opacity
enforcement, Maubert et al. [11] introduced a game where a player with perfect
observations aims to enforce current state opacity against another player with
imperfect observations. In our previous work [18] we explored qualitative
opacity enforcement in a stochastic environment with both the players having
partial observation. These works primarily address qualitative opacity,
whereas our research examines the probabilistic notions of opacity and
transparency.

In stochastic DESs, several studies delve into quantifying opacity. Saboori et
al. [17] introduced notions of current state opacity for probabilistic finite
automata, along with verification algorithms to verify if the system is
probabilistically opaque for a given threshold. Yin et al. [23] extended this
to the notions of infinite and K-step opacity. Keroglou et al. [8] explored
model-based opacity, where the defender conceals the system model using a
hidden Markov model. Bérard et al. [2] extended language-based opacity to
$\omega$-regular properties on MDP. Bérard et al. define the notions of
symmetrical opacity, and define probabilistic disclosures. They showed that
the quantitative questions relating to opacity become decidable for the class
of $\omega$-regular secrets. Liu et al. [10] studied approximate opacity in
continuous stochastic control systems, developing a verification method for
general continuous-state MDPs with finite abstractions.

Our proposed approach considers probabilistic opacity/transparency-enforcement
in MDPs, distinguishing it from qualitative approaches. In existing stochastic
methods, verification commonly involves tracking current state estimates or
maintaining a belief state for opacity enforcement. In our method, we instead
study an FST-based computational model of opaque observations to directly
encode state trajectories, allowing us to construct another DFA that precisely
accepts the set of opaque observations to enforce opacity/transparency. Our
proposed computational model has the potential for extension to scenarios
involving active attackers, without the need for tracking the belief of state
estimates for the agents (a direction we leave for future work).

#### Organization

In Section II, we introduce preliminaries, and frame the problem. Section III
presents our computational model. In Section IV, we use the computational
model to solve an optimal planning problem. Section V illustrates the planning
problem’s application in a gridworld scenario. Finally, we conclude in Section
VI.

##  II Preliminaries and Problem Formulation

#### Notations

Let $\Sigma$ be a finite set of symbols, called the alphabet. We denote a set
of all $\omega$-regular words as $\Sigma^{\omega}$ obtained by concatenating
the elements in $\Sigma$ infinitely many times. A sequence of symbols
$w=\sigma_{0}\sigma_{1}\cdots\sigma_{n}$ with $\sigma_{i}\in\Sigma$ for any
$0\leq i\leq n$, is a finite word. The set $\Sigma^{*}$ is the set of all
finite words that can be generated with $\Sigma$. The length of a word is
denoted by $|w|$. For a word $w=uv$, $u$ is a prefix of $v$ for
$u\in\Sigma^{\ast}$ and $v\in\Sigma^{\omega}$. Let $\mathbf{R}$ denote the set
of real numbers. Given a finite set $Z$, the set of probability distributions
over $Z$ is represented as $\mathcal{D}(Z)$. Given $d\in\mathcal{D}(Z)$, the
support of $d$ is $\mathsf{Supp}(d)=\\{z\in Z\mid d(z)>0\\}$.

###  II-A Markov decision process

We consider the interaction of P1 with a stochastic environment, monitored by
P2 with partial observations, modeled as a terminating MDP (without the reward
function).

######  Definition 1.

A probabilistic transition system (i.e. , an MDP without the reward function)
is a tuple

$M=(S,A,P,\mu_{0},s_{\top},s_{\bot},\mathcal{AP},O,L)$

where $S$ is a finite set of states, $A$ is a finite set of actions.
$s_{\top}\in S$ is the _initiating state_ , which is a unique start state.
$s_{\bot}\in S$ is the _terminating state_ , which is a unique sink state. The
set $A$ is the action set, which includes a special _initiating action_ and a
_terminating action_ $a_{\top}$ and $a_{\bot}$ respectively. $a_{\top}$ is
only enabled in $s_{\top}$, while $a_{\bot}$ is enabled for any $s\in
S\setminus\\{s_{\top},s_{\bot}\\}$. $P:S\times A\rightarrow\mathcal{D}(S)$ is
a probabilistic transition function, with $P(s^{\prime}|s,a)$ representing the
probability of reaching state $s^{\prime}$ given action $a$ at state $s$.
$\mu_{0}$ is the initial distribution on states $S\setminus\\{s_{\top}\\}$
such that $\mu_{0}(s)$ represents the probability of $s$ being the initial
state. For $s_{\top}\in S$, $P(s_{\top},a_{\top},s)=\mu_{0}(s)$. That is, if
P1 selects the initiating action $a_{\top}$, then an initial state is reached
surely. For any $s\in S\setminus\\{s_{\top},s_{\bot}\\}$,
$P(s,a_{\bot},s_{\bot})=1$. That is, if P1 selects the terminating action
$a_{\bot}$, then a terminating state $s_{\bot}$ is reached surely. Finally,
for any $a\in A\setminus\\{a_{\top},a_{\bot}\\}$, $P(s_{\bot},a,s_{\bot})=1$,
i.e. , all allowed actions form a self-loop. $\mathcal{AP}$ is a set of atomic
propositions, $L:S\rightarrow 2^{\mathcal{AP}}$ is the labeling function that
maps a state to a set of atomic propositions that evaluate true at that state.
The initiating state is labeled with a “starting” symbol, $\rtimes$, i.e. ,
$L(s_{\top})=\rtimes$. The terminating state is labeled the “ending” symbol
$\ltimes$, i.e. , $L(s_{\bot})=\ltimes$. $O\subseteq 2^{S}$ is the set of all
finite observables.

A finite play $\rho=s_{\top}a_{\top}s_{0}a_{0}s_{1}\ldots
s_{n}a_{\bot}s_{\bot}$ is a sequence of interleaving states and player’s
actions, where for all $i\geq 0$, $P(s_{i+1}\mid s_{i},a_{i})>0$. Let
$\mathsf{Plays}(M)\subseteq(S\times A)^{\ast}S$ be the set of finite plays
that can be generated from the game $M$. The labeling of a finite play,
denoted by $L(\rho)$, is defined as $L(\rho)=\rtimes L(s_{0})L(s_{1})\ldots
L(s_{n})\ltimes$ i.e. , the labeling function omits the actions from the play
and applies to states only.

A randomized, finite-memory strategy is a function
$\pi:\mathsf{Plays}(M)\rightarrow\mathcal{D}(A)$ mapping a play to a
distribution over actions. A randomized, Markov strategy is a function
$\pi:S\rightarrow\mathcal{D}(A)$ mapping the current state to a distribution
over actions. The stochastic process generated by applying a policy $\pi$ to
the MDP $M$ is denoted as $M^{\pi}$, with transition dynamics $P^{\pi}(s\mid
s_{\top})=\mu_{0}(s)$ for the initiating state and $P^{\pi}(s^{\prime}\mid
s)=\sum_{a\in A}P(s,a,s^{\prime})\cdot\pi(a\mid s)$ for all $s\in
S\setminus\\{s_{\top}\\}$.

#### Information structure

We assume that  1. P1 has perfect observation of states and actions.  2. P2
has partial observation of states and no observation of P1’s actions.

######  Definition 2 (Transition-Observation function).

The transition-observation function of P2 is $\mathsf{Obs}:S\times A\times
S\rightarrow O$ mapping a transition $(s,a,s^{\prime})$ to an observation
$o\in O$. The observation function of P2 is extended to plays in $M$ as
$\mathsf{Obs}:\mathsf{Plays}(M)\rightarrow O^{\ast}$ such that for any play
$\rho=\rho_{1}\cdot s_{n}a_{n}s_{n+1}$,
$\mathsf{Obs}(\rho)=\mathsf{Obs}(\rho_{1})\cdot\mathsf{Obs}(s_{n}a_{n}s_{n+1})$
which is the observation of the prefix $\rho_{1}$ concatenated with the
observation of the last transition. Two plays $\rho_{1},\rho_{2}$ are
observation-equivalent iff $\mathsf{Obs}(\rho_{1})=\mathsf{Obs}(\rho_{2})$.

The inverse observation function of P2
$\mathsf{Obs}^{-1}:\mathsf{Obs}(M)\rightarrow 2^{\mathsf{Plays}(M)}$ maps each
$\eta\in\mathsf{Obs}(M)$ to the set
$\mathsf{Obs}^{-1}(\eta)=\\{\rho\in\mathsf{Plays}(M)\mid\mathsf{Obs}(\rho)=\eta\\}$.
Given a play $\rho\in\mathsf{Plays}(M)$, we denote the set of plays that are
observation-equivalent to $\rho$ in P2’s perspective by $[\rho]_{2}$.

Objective and secret in temporal logic In the game arena, P1 aims to achieve a
temporal objective $\psi$ specified by a Linear Temporal Logic over Finite
Traces (LTLf) formula, while P2 observes P1. P1 also holds a secret LTLf
formula $\varphi$ with the goal of concealing whether $\varphi$ is satisfied
or not from P2. The two formulas $\psi$ and $\varphi$ may be different.

The syntax of the LTLf formula is given as follows.

######  Definition 3 (LTLf [7]).

An (LTLf) formula over $\mathcal{AP}$ is defined inductively as follows:

$\varphi:=p\mid\neg\varphi\mid\varphi_{1}\land\varphi_{2}\mid\varphi_{1}\lor\varphi_{2}\mid\bigcirc\,\varphi\mid\varphi_{1}\mbox{$\,{\sf
U}\,$}\varphi_{2}\mid\Diamond\,\varphi\mid\Box\,\varphi,$

where $p\in\mathcal{AP}$; $\neg$, $\land$ and $\lor$ are the Boolean operators
negation, conjunction and disjunction, respectively; and $\bigcirc\,$, $\,{\sf
U}\,$, $\Diamond\,$ and $\Box\,$ denote the temporal modal operators for next,
until, eventually and always, respectively.

The operator $\bigcirc\,\varphi$ specifies that formula $\varphi$ holds at the
next time instant. $\varphi_{1}\mbox{$\,{\sf U}\,$}\varphi_{2}$ denotes that
there exists a future time instant at which $\varphi_{2}$ holds, and that
$\varphi_{1}$ holds at all time instants up to and including that future
instant. The formula $\Diamond\,\varphi$ specifies that $\varphi$ holds at
some future instant, and $\Box\,\varphi$ specifies that $\varphi$ holds at all
current and future time instants. See [7] for detailed semantics of LTLf.

For any LTLf formula $\varphi$ over $AP$, a set of words
$\mathsf{Words}(\varphi)=\\{w\subseteq(2^{AP})^{*}\mid w\models\varphi\\}$
that satisfy the formula is associated.

######  Definition 4 (DFA).

A DFA is a tuple $\mathcal{A}=(Q,\Sigma,\delta,{\iota},F)$ with a finite set
of states $Q$, a finite alphabet $\Sigma$, a deterministic transition function
$\delta:Q\times\Sigma\rightarrow Q$, extended recursively as $\delta(q,\sigma
u)=\delta(\delta(q,\sigma),u)$ for $\sigma\in\Sigma,u\in\Sigma^{\ast}$.
${\iota}$ is the initial state, and $F\subseteq Q$ is a set of accepting
states.

A Nondeterministic Finite Automaton (NFA) is defined similarly to a DFA, with
a nondeterministic transition function $\delta:Q\times\Sigma\rightarrow
2^{Q}$, where for every $q\in Q$ and $\sigma\in\Sigma$, transition occurs to a
set of states $X\in 2^{Q}$.

Using the method of De Giacomo and Vardi [7], we convert the LTLf formulae
into DFA that accepts $\mathsf{Words}(\varphi)$ with
$\Sigma=2^{\mathcal{AP}}$. We have the DFA accepting all the words satisfying
the secret $\varphi$ as
$\mathcal{A}_{s}=(Q_{s},2^{\mathcal{AP}},\delta_{s},{\iota}_{s},F_{s})$, and
the DFA accepting all words satisfying the task specification $\psi$ as
$\mathcal{A}$. Both DFAs are considered to be complete 111An incomplete DFA
can be completed by adding a sink state and redirecting all undefined
transitions to that sink state..

###  II-B Probabilistic opacity and transparency

We introduce the definition of symmetric opacity as in [3].

######  Definition 5 (Symmetrical Opacity).

An LTLf formula $\varphi$ is opaque to P2 with respect to a play
$\rho\in\mathsf{Plays}(M)$ iff either of the following cases hold:  1.
$L(\rho)\models\varphi$; and there exists at least one observation-equivalent
play $\rho^{\prime}\in[\rho]_{2}$ with $L(\rho^{\prime})\not\models\varphi$;
or  2. $L(\rho)\not\models\varphi$; and there exists at least one observation-
equivalent play $\rho^{\prime}\in[\rho]_{2}$ with $L(\rho)\models\varphi$.

We now define transparency as a dual property of opacity.

######  Definition 6 (Symmetrical Transparency).

An LTLf formula $\varphi$ is transparent to P2 with respect to a play
$\rho\in\mathsf{Plays}(M)$ iff either of the following cases hold:  1.
$L(\rho)\models\varphi$; and for any observation-equivalent play
$\rho^{\prime}\in[\rho]_{2}$, $L(\rho^{\prime})\models\varphi$, or  2.
$L(\rho)\not\models\varphi$; and for any observation-equivalent play
$\rho^{\prime}\in[\rho]_{2}$ with $L(\rho)\not\models\varphi$.

We define the set of opaque and non-opaque runs for the secret $\varphi$ and
observation function $\mathsf{Obs}$ as follows.

  * •

The set of opaque runs is as given below.

$\mathcal{S}_{H}=\\{\rho\in\mathsf{Plays}(M)\mid\exists\rho^{\prime}\in[\rho]_{2},(L(\rho)\not\models\varphi\land\\\
L(\rho^{\prime})\models\varphi)\lor(L(\rho)\models\varphi\land
L(\rho^{\prime})\not\models\varphi)\\}$ (1)

  * •

The set of transparent runs is

$\mathcal{S}_{T}=\\{\rho\in\mathsf{Plays}(M)\mid\forall\rho^{\prime}\in[\rho]_{2},\\\
(L(\rho)\models\varphi\implies L(\rho^{\prime})\models\varphi)\\\
\lor(L(\rho)\not\models\varphi\implies L(\rho^{\prime})\not\models\varphi)\\}$
(2)

It is easy to see $\mathcal{S}_{T}\cap\mathcal{S}_{H}=\emptyset$.

######  Lemma 1.

$\mathcal{S}_{H}\cup\mathcal{S}_{T}=\mathsf{Plays}(M)$.

The proof follows from the definition and thus is omitted.

The following definition of quantitative opacity has been introduced in [2]
for probabilistic systems.

######  Definition 7 (Probabilistic opacity and transparency).

Given the MDP $M$, observation function $\mathsf{Obs}$, the secret $\varphi$
and policy $\pi$, the probabilistic opacity of $\varphi$ in the stochastic
process $M^{\pi}$ is

$PH(\varphi;M^{\pi},\mathsf{Obs})=\Pr(\mathcal{S}_{H};M^{\pi})$

where $\Pr(\mathcal{S}_{H};M^{\pi})$ denotes the probability of an opaque run
given $M^{\pi}$. The probabilistic transparency of $\varphi$ in $M^{\pi}$ is

$PT(\varphi;M^{\pi},\mathsf{Obs})=\Pr(\mathcal{S}_{T};M^{\pi})$

We also define the set of runs that satisfy the task specification as given
below.

$\mathcal{S}_{\psi}=\\{\rho\in\mathsf{Plays}(M)\mid L(\rho)\models\psi\\}$ (3)

######  Problem 1.

Given the MDP $M$, secret $\varphi$, task $\psi$ and the observation function
$\mathsf{Obs}$, compute a strategy $\pi^{\ast}$ for P1 that maximizes the
probabilistic opacity of $\varphi$ while ensuring that the probability of
satisfying the task $\psi$ is greater than $\epsilon$. Formally,

$\displaystyle\max PH(\varphi;M^{\pi},\mathsf{Obs})$ (4)
$\displaystyle\textrm{s.t.}\Pr(\mathcal{S}_{\psi};M^{\pi})\geq\epsilon.$

By replacing the maximization with minimization, and Lemma 1, the solution of
the optimization problem is a policy that maximizes the probabilistic
transparency of $\varphi$ under a task performance constraint.

To illustrate our definitions, we introduce a running example.

$s_{1}$start$s_{2}$$s_{3}$$s_{4}$$s_{5}$$s_{6}$$s_{7}$$a,b(0.5)$$a,b(0.5)$$a(0.2)$$a(0.8)$$b(0.4)$$b(0.6)$$a(0.2)$$b(0.3)$$a(0.4)$,
$b(0.2)$$b(0.5)$$a(0.4)$$a,b$$a(0.3)$, $b(1)$$a(0.2)$$a(0.5)$$a(0.5)$,
$b(1)$$a(0.5)$$a,b(0.5)$$a,b(0.5)$ Figure 1: MDP for the illustrative example.
The colored boxes represent the observation partition of the state space.

######  Example 1 (Part I).

Consider the MDP in Fig.1 with $7$ states, $s_{1}$ through $s_{7}$ ($s_{1}$ as
the initial state i.e. , $\mu_{0}(s_{1})=1$). Here, $s_{4}$ is a sink, and
edges are labeled with actions and transition probabilities. The transition-
observation of a transition to P2, $\mathsf{Obs}(s,a,s^{\prime})$, is a set of
states that are observation equivalent to $s^{\prime}$ (actions are not
observable). The MDP has observation-equivalent states sets $\\{s_{1}\\}$,
$\\{s_{2},s_{3}\\}$, $\\{s_{4}\\}$, $\\{s_{5},s_{6}\\}$ and $\\{s_{7}\\}$.
P1’s task is to eventually reach $s_{4}$ ($\psi=\Diamond\,s_{4}$), and secret
formula is satisfied if P1 reaches the state $s_{6}$,
($\varphi=\Diamond\,s_{6}$). P1 must compute a strategy for maximal
probabilistic opacity of $\varphi$ while ensuring to satisfy $\psi$ with a
probability $\geq\epsilon$. We can also consider the planning problem where P1
is to maximize the probabilistic transparency with respect to $\varphi$ under
the task performance constraint.

##  III Main results

In this section, we present a computational model to obtain the opaque
observations of P2. Using this computational model, we formulate constrained
MDPs to solve Problem 1.

###  III-A A computational model for opaque observations

To solve Problem 1, we need to construct the set $\mathcal{S}_{H}$ of opaque
runs, which is measurable. However, the set of possible runs in $M$ can be
large or infinite, such an enumeration procedure cannot terminate and
therefore we cannot construct $\mathcal{S}_{H}$ for planning purpose. We
present a method to construct a DFA that accepts the set of opaque
observations.

######  Definition 8 (Opaque observations).

Given the secret $\varphi$, an observation $\rho_{o}\in O^{\ast}$ is opaque
iff there is an opaque run $\rho\in\mathcal{S}_{H}$ and
$\rho_{o}=\mathsf{Obs}(\rho)$. The set of _opaque observations_ is
$\mathsf{OpqObs}$.

First, it is seen that the observation function
$\mathsf{Obs}:\mathsf{Plays}(M)\rightarrow O^{\ast}$ can be expressed as an
FST whose input strings are $\mathsf{Plays}(M)$ and output strings are the
observations of $\mathsf{Plays}(M)$.

######  Definition 9 (Finite-state transducer representation for the
observation function).

The FST encoding of the observation function
$\mathsf{Obs}:\mathsf{Plays}(M)\rightarrow O^{\ast}$ is a tuple

$\mathcal{A}_{o}=\langle S,\Sigma_{I},\Sigma_{O},T_{o}\rangle$

where

  * •

$S$ is the finite set of states of the MDP.

  * •

$\Sigma_{I}=S\times A\times S$ is the set of input symbols, which are the
transitions in the MDP.

  * •

$\Sigma_{O}=O\cup\\{\rtimes,\ltimes\\}$ is the set of output symbols, which
are the observations in the MDP and two symbols $\rtimes$ and $\ltimes$
marking the beginning and ending of a word.

  * •

$T_{o}:S\times\Sigma_{I}\rightarrow S\times\Sigma_{O}$ is a deterministic
transition function that maps a given pair of state and input to a next state
and an output. The transition function is constructed as follows:

    * –

if $s_{\top}\in S$ is the initiating state, the input $(s_{\top},a_{\top},s)$
is enabled if $\mu_{0}(s)>0$ in the MDP $M$, then

$T_{o}(s_{\top},(s_{\top},a_{\top},s))=(s,\rtimes),$

    * –

if $s\in S$, then the input $(s,a,s^{\prime})$ is enabled if
$P(s,a,s^{\prime})>0$ in the MDP $M$,

$T_{o}(s,(s,a,s^{\prime}))=(s^{\prime},o),$

where $o=\mathsf{Obs}(s,a,s^{\prime})$.

    * –

for any state $s\in S$, if $a_{\bot}$ action is enabled, then

$T_{o}(s,(s,a_{\bot},s_{\bot}))=(s_{\bot},\ltimes).$

A transition from state $t$ to $t^{\prime}$ given input $\sigma\in\Sigma_{I}$
and output $o\in\Sigma_{o}$ is also denoted
$t\xrightarrow[o]{\sigma}t^{\prime}$. Given a sequence of inputs, also called
input word $w=\sigma_{0}\sigma_{1}\ldots\sigma_{n}$, the transducer generates
a sequence of transitions, or a run
$\rho=t_{0}\xrightarrow[o_{0}]{\sigma_{0}}t_{1}\xrightarrow[o_{1}]{\sigma_{1}}t_{2}\ldots\xrightarrow[o_{n-1}]{\sigma_{n-1}}t_{n}$
with the output uniquely determined as $w_{o}=o_{0}o_{1}\ldots o_{n}$.

By construction, it is noted that for any play
$\rho=s_{\top}a_{\top}s_{0}\ldots s_{n}a_{\bot}s_{\bot}$, the corresponding
input to the FST is

$\mathsf{In}(\rho)=(s_{\top},a_{\top},s_{0})(s_{0},a_{0},s_{1})(s_{1},a_{1},s_{2})\ldots\\\
(s_{n-1},a_{n-1},s_{n})(s_{n},a_{\bot},s_{\bot}).$

The output is

$\mathsf{Out}(\rho)=\rtimes\mathsf{Obs}(s_{0},a_{0},s_{1})\mathsf{Obs}(s_{1},a_{1},s_{2})\ldots\\\
\mathsf{Obs}(s_{n-1},a_{n-1},s_{n})\ltimes.$

The corresponding run is $\mathsf{Run}(\rho)=s_{\top}s_{0}s_{1}s_{2}\ldots
s_{n}s_{\bot}.$

######  Example 2 (Part II).

Continuing the Example 1, a fraction of FST encoding of the observation
function is shown in Fig. 2. Each of the states in the transducer represent
the states in the MDP shown in Fig. 1 along with initiating ($s_{\top}$) and
terminating ($s_{\bot}$) states. Transitions are labeled with input and output
symbols. For instance, the transition from $s_{1}$ to $s_{2}$ is labeled
$(s_{1},a,s_{2})$ as input and $[s_{2},s_{3}]$ as output symbols. Each state
also has a transition to the terminating state $s_{\bot}$.

$s_{\top}$start$s_{1}$$s_{2}$$s_{3}$$s_{4}$$s_{5}$$s_{\bot}$$(s_{\top},a_{\top},s_{1})\backslash[\rtimes]$$(s_{1},a,s_{2})\backslash[s_{2},s_{3}]$$(s_{1},b,s_{2})\backslash[s_{2},s_{3}]$$(s_{1},b,s_{3})\backslash[s_{2},s_{3}]$$(s_{1},a,s_{3})\backslash[s_{2},s_{3}]$$(s_{2},a,s_{2})\backslash[s_{2},s_{3}]$$(s_{2},a,s_{3})\backslash[s_{2},s_{3}]$$(s_{2},b,s_{5})\backslash[s_{5},s_{6}]$$(s_{2},b,s_{4})\backslash[s_{4}]$$(s_{3},a,s_{3})\backslash[s_{2},s_{3}]$$(s_{3},b,s_{5})\backslash[s_{5},s_{6}]$$(s_{3},a,s_{5})\backslash[s_{5},s_{6}]$$(s_{5},a,s_{4})\backslash[s_{4}]$$(s_{4},a,s_{4})\backslash[s_{4}]$$(s_{4},a_{\bot},s_{\bot})\backslash[\ltimes]$$(s_{3},a_{\bot},s_{\bot})\backslash[\ltimes]$
Figure 2: A fragment of FST encoding the observation function.

The next step is to compute a product between the FST of the observation
function and the DFA accepting the secret.

######  Definition 10 (Product FST).

Given the FST representation of the observation function $\mathcal{A}_{o}$,
and the secret DFA $\mathcal{A}_{s}$, the product of the FST and DFA is a
tuple,

$\mathcal{A}_{H}=\langle
Q_{h},\Sigma_{I},\Sigma_{O},T_{h},{\iota}_{h},F_{h}\rangle$

  * •

$Q_{h}=S\times Q_{s}$ is the set of states.

  * •

${\iota}_{h}=(s_{\top},{\iota}_{s})$ is the initial state.

  * •

$T_{h}:Q_{h}\times\Sigma_{I}\rightarrow Q_{h}\times\Sigma_{O}$ is the
transition function and is defined as follows. We consider the following
cases: For $\sigma\in S\times A\times S$, if
$s\xrightarrow[o]{\sigma}s^{\prime}$ in $\mathcal{A}_{o}$ and
$q\xrightarrow[]{a}q^{\prime}$ in $\mathcal{A}_{s}$, then,

    1. 1.

For $\sigma=(s_{\top},a_{\top},s)$, and
$s_{\top}\xrightarrow[\rtimes]{\sigma}s$, let

$(s_{\top},{\iota}_{s})\xrightarrow[\rtimes]{\sigma}(s,q)$

where $q=\delta({\iota}_{s},L(s))$.

    2. 2.

For $\sigma=(s,a_{\bot},s_{\bot})$, and
$s\xrightarrow[\ltimes]{\sigma}s_{\bot}$, let

$(s,q)\xrightarrow[\ltimes]{\sigma}(s_{\bot},q).$

    3. 3.

For any other $\sigma$, let

$(s,q)\xrightarrow[o]{\sigma}(s^{\prime},q^{\prime})\text{ if and only if
}\delta_{s}(q,L(s^{\prime}))=q^{\prime}.$

    4. 4.

Any state $(s,q)$ where $s=s_{\bot}$ is a sink state.

  * •

$F_{h}=\\{(s_{\bot},q)\mid q\in F_{s}\\}$ is a set of accepting states.

From the above construction, we establish a mapping
$\mathfrak{R}:\mathsf{Plays}(M)\rightarrow\mathsf{Run}(\mathcal{A}_{H})$
between plays in MDP $M$ and runs in the product of the FST and DFA
$\mathcal{A}_{H}$ as follows: For a play $\rho=s_{\top}a_{\top}s_{0}\ldots
s_{n}a_{\bot}s_{\bot}\in\mathsf{Plays}(M)$, there is a run (i.e. , a state
sequence augmented with automata states) $\hat{\rho}$ such that
$\hat{\rho}=(s_{\top},{\iota}_{s})(s_{0},q_{0})\ldots(s_{n},q_{n})(s_{\bot},q_{n})$
where $q_{i}=\delta(q_{i-1},L(s_{i}))$ for $i\geq 1$ and
$q_{0}=\delta({\iota}_{s},L(s_{0}))$.

######  Lemma 2.

A run $\rho\in\mathsf{Plays}(M)$ is such that $L(\rho)\models\varphi$ iff
$\mathfrak{R}(\rho)$ ends in $(s_{\bot},q)$ where $q\in F_{s}$.

###### Proof.

By construction, consider a run $\rho=s_{\top}a_{\top}\ldots
a_{\bot}s_{\bot}\\\ \in\mathsf{Plays}(M)$, if $\mathfrak{R}(\rho)$ ends in
$(s_{\bot},q)$ with $q\in F_{s}$, then $L(s_{\top}s_{0}\ldots s_{n}s_{\bot})$
is accepted by $\mathcal{A}_{s}$. Thus, $L(\rho)\models\varphi$. ∎

We denote the last/final state of $\mathfrak{R}(\rho)$ as
$\mathsf{Last}(\rho)$. We then compute two NFAs from the product transducer.

######  Definition 11.

Given the product FST $\mathcal{A}_{H}=\langle
Q_{h},\Sigma_{I},\Sigma_{O},T_{h},{\iota}_{h},F_{h}\rangle$, the _output
language_ accepted by $\mathcal{A}_{H}$ is obtained as the set
$L_{O}(\mathcal{A}_{H})=\\{\mathsf{Out}(\rho)\mid\rho\in\Sigma_{I}^{\ast},\mathsf{Last}(\rho)\in\\{s_{\bot}\\}\times
F_{s}\\}$.

The NFA accepting $L_{O}(\mathcal{A}_{H})$ is obtained from $\mathcal{A}_{H}$
by removing the input symbols and use the output symbols on each transition as
the input symbol.

######  Lemma 3.

For any $\rho\in\mathsf{Plays}(M)$, if $\rho\models\varphi$,
$\mathsf{Out}(\rho)\in L_{O}(\mathcal{A}_{H})$ and if
$\rho\not\models\varphi$, $\mathsf{Out}(\rho)\notin L_{O}(\mathcal{A}_{H})$.

###### Proof.

By construction, $\mathsf{Last}(\rho)=(s_{\bot},q)$ where $q\in F_{s}$. From
Lemma 2, we have that $L(\rho)\models\varphi$. ∎

Also, let $\mathcal{A}_{H}^{\dagger}=\langle
Q_{h},\Sigma_{I},\Sigma_{O},T_{h},{\iota}_{h},F_{h}^{\dagger}\rangle$ where
$F_{h}^{\dagger}=\\{(s_{\bot},q)\mid q\in Q_{s}\setminus F_{s}\\}$. The output
language $L_{O}(\mathcal{A}_{H}^{\dagger})$ and its NFA is obtained in a
similar way.

######  Theorem 1.

The set of _opaque runs_ is $Z=\mathsf{Obs}^{-1}(L_{O}(\mathcal{A}_{H})\cap
L_{O}(\mathcal{A}_{H}^{\dagger}))$.

###### Proof.

From Def. 11, for any $u\in L_{O}(\mathcal{A}_{H})$, there exists an input
string $\rho^{\prime}$ such that $L(\rho^{\prime})\models\varphi$ and
$\mathsf{Out}(\rho^{\prime})=u$. For any $u\in
L_{O}(\mathcal{A}^{\dagger}_{H})$, there exists an input string
$\rho^{\prime\prime}$ such that $L(\rho^{\prime\prime})\models\neg\varphi$ and
$\mathsf{Out}(\rho^{\prime})=u$. Thus, if $\mathsf{Obs}(\rho)=u\in
L_{O}(\mathcal{A}_{H})\cap L_{O}(\mathcal{A}^{\dagger}_{H})$, then
$\rho,\rho^{\prime},\rho^{\prime\prime}$ are observation-equivalent. On
observing $\mathsf{Obs}(\rho)$
($=\mathsf{Obs}(\rho^{\prime})=\mathsf{Obs}(\rho^{\prime\prime})$), P2 cannot
know if $L(\rho)\models\varphi$ or not.

Next, we aim to show that any run not in $Z$ must be transparent. For any
$\rho\notin Z$, $\mathsf{Obs}(\rho)\notin L_{O}(\mathcal{A}_{H})\cap
L_{O}(\mathcal{A}_{H}^{\dagger})$. Since $L_{O}(\mathcal{A}_{H})\cup
L_{O}(\mathcal{A}_{H}^{\dagger})=\mathsf{Obs}(\mathsf{Plays}(M))$ includes all
possible observations, it is either (I)
$\mathsf{Obs}(\rho)\in(L_{O}(\mathcal{A}_{H})\setminus
L_{O}(\mathcal{A}_{H}^{\dagger}))$ or (II)
$\mathsf{Obs}(\rho)\in(L_{O}(\mathcal{A}^{\dagger}_{H})\setminus
L_{O}(\mathcal{A}_{H}))$. In Case (I), the observation informs that $\varphi$
is satisfied by $\rho$ because any run violating $\varphi$ is not observation-
equivalent to $\rho$. In case (II), the observation informs P2 that $\varphi$
is violated because any run satisfying $\varphi$ is not observation-equivalent
to $\rho$.

∎

$p_{1}$start$p_{2}$$\neg s_{4}$$s_{4}$$True$

$q_{1}$start$q_{2}$$\neg s_{6}$$s_{6}$$True$

Figure 3:  (a) DFA for the task specification $\psi=\Diamond\,s_{4}$. (b) DFA
for the secret specification $\varphi=\Diamond\,s_{6}$.
$(s_{\top},q_{1})$start$(s_{1},q_{1})$$(s_{2},q_{1})$$(s_{3},q_{1})$$(s_{6},q_{2})$$(s_{\bot},q_{1})$$(s_{\bot},q_{2})$$(s_{\top},a_{\top},s_{1})\backslash[\rtimes]$$(s_{1},a,s_{2})\backslash[s_{2},s_{3}]$$(s_{1},b,s_{2})\backslash[s_{2},s_{3}]$$(s_{1},a_{\bot},s_{\bot})\backslash[\ltimes]$$(s_{1},a,s_{3})\backslash[s_{2},s_{3}]$$(s_{1},b,s_{3})\backslash[s_{2},s_{3}]$$(s_{2},a_{\bot},s_{\bot})\backslash[\ltimes]$$(s_{3},a_{\bot},s_{\bot})\backslash[\ltimes]$$(s_{3},b,s_{6})\backslash[s_{5},s_{6}]$$(s_{6},a_{\bot},s_{\bot})\backslash[\ltimes]$$1$$1$
Figure 4: A fragment of product FST.

Corollary 1, as it follows from Theorem 1, facilitates the computation of the
set of opaque observations.

######  Corollary 1.

The set of opaque observations is $\mathsf{OpqObs}=L_{O}(\mathcal{A}_{H})\cap
L_{O}(\mathcal{A}_{H}^{\dagger}).$

We compute the intersection product of NFAs accepting the languages
$L_{O}(\mathcal{A}_{H})$ and $L_{O}(\mathcal{A}^{\dagger}_{H})$ in the usual
manner as in [1]. This results in another NFA, which can be determinized into
a DFA as in [1]. We call this DFA
$\mathcal{A}^{opaque}=(\hat{Q},\Sigma_{O},\hat{\delta},\hat{\iota},\hat{F})$
as the _opaque-observation DFA_ as it accepts precisely the set of
observations that are opaque to P2 (follows from Corollary 1).

######  Example 3 (Part III).

Continuing with Example 1, the task specification for P1 is
$\psi=\Diamond\,s_{4}$ and secret is $\varphi=\Diamond\,s_{6}$, represented by
the DFAs shown in the Fig. 3(a) and Fig. 3(b).

The product FST is constructed with the secret DFA and the FST representation
of the observation function. A fragment of the product FST is shown in Fig. 4.
Nodes represent the transducer state augmented with the DFA state (e.g.,
$(s_{1},q_{1})$ where $s_{1}$ is transducer and $q_{1}$ is the DFA states).
The transitions include input and output symbols, like the transducer
representation of the observation function.

With the product FST, we construct an NFA accepting $L_{O}(\mathcal{A}_{H})$
as shown in Fig. 5, where the state $(s_{\bot},q_{2})$ is the final accepting
state. Likewise, NFA for accepting $L_{O}(\mathcal{A}_{H}^{\dagger})$, with
$(s_{\bot},q_{1})$ as the final accepting state. The edges in the NFA are
labeled with the observations received by P2.

$(s_{\top},q_{1})$start$(s_{1},q_{1})$$(s_{2},q_{1})$$(s_{3},q_{1})$$(s_{5},q_{1})$$(s_{6},q_{2})$$(s_{\bot},q_{1})$$(s_{\bot},q_{2})$$\rtimes$$[s_{2},s_{3}]$$[s_{2},s_{3}]$$\ltimes$$[s_{2},s_{3}]$$[s_{2},s_{3}]$$\ltimes$$[s_{5},s_{6}]$$[s_{5},s_{6}]$$\ltimes$$[s_{2},s_{3}]$$[s_{5},s_{6}]$$\ltimes$$[s_{5},s_{6}]$$\ltimes$$1$$1$
Figure 5: A fragment of NFA for $L_{O}(\mathcal{A}_{H})$.

Next, we make use of the above computed opaque-observation DFA in our planning
algorithm.

##  IV Planning Algorithm

Finally, we can compute a product MDP for solving the optimal
opacity/transparency planning problem:

######  Definition 12 (Product MDP).

Given the MDP $M=(S,A,P,\mu_{0},s_{\top},s_{\bot},\mathcal{AP},L)$, the task
DFA $\mathcal{A}=(Q,\Sigma,\delta,{\iota},F)$, and the opaque-observation DFA
$\mathcal{A}^{opaque}=(\hat{Q},\Sigma_{O},\hat{\delta},\hat{\iota},\hat{F}$)
with reward, the product MDP ($M\times\mathcal{A}\times\mathcal{A}^{opaque}$)
is a tuple,

$\mathcal{M}=(V,A,\mathcal{P},v_{0},R_{1},R_{2})$

where

  * •

$V=\\{(s,q,\hat{q})\mid s\in S,q\in Q,\hat{q}\in\hat{Q}\\}$ is the set of
states where each state includes a state from the MDP $M$, a state from task
DFA $\mathcal{A}$ and a state from the opaque-observations DFA
$\mathcal{A}^{opaque}$ respectively.

  * •

$A$ is the set of actions.

  * •

$\mathcal{P}:V\times A\rightarrow\mathcal{D}(V)$ is the probabilistic
transition function. For each state $v=(s,q,\hat{q})\in V$, an action $a\in A$
enabled from $s$ and state
$v^{\prime}=(s^{\prime},q^{\prime},\hat{q}^{\prime})\in V$,
$\mathcal{P}(v,a,v^{\prime})=P(s,a,s^{\prime})$ if
$q^{\prime}=\delta(q,L(s^{\prime}))$ and
$\hat{q}^{\prime}=\hat{\delta}(\hat{q},\mathsf{Obs}(s,a,s^{\prime}))$.

  * •

$v_{0}=(s_{\top},q_{0},\hat{q}_{0})\in V$ is the initial state, where,
$q_{0}={\iota}$ and $\hat{q}_{0}=\hat{\iota}$.

  * •

$R_{1}:V\times A\times V\rightarrow\mathbf{R}$ is the reward function for the
task specification. For a state $(s,q,\hat{q})\in V$, action $a\in A$ and
$(s^{\prime},q^{\prime},\hat{q}^{\prime})\in V$, the reward is given by

$R_{1}((s,q,\hat{q}),a,(s^{\prime},q^{\prime},\hat{q}^{\prime}))=\begin{cases}1,&\text{if}\
q\notin F,q^{\prime}\in F\\\ 0,&\text{otherwise}.\end{cases}$

  * •

$R_{2}:V\times A\rightarrow\mathbf{R}$ is the reward function for enforcing
opacity. For a state $(s,q,\hat{q})\in V$, action $a\in A$, the reward is
given by

$R_{2}((s,q,\hat{q}),a)=\begin{cases}1,&\text{if}\
\hat{q}\in\hat{F},a=a_{\bot}\\\ 0,&\text{otherwise}.\end{cases}$

By construction, for any policy, the accumulated rewards $R_{1}$ relates to
the probability of satisfying the task specification
($\Pr(\mathcal{S}_{\psi};M^{\pi})$) and the accumulated rewards $R_{2}$
relates to the probability of enforcing opacity
($PH(\varphi;M^{\pi},\mathsf{Obs})$). Since $M$ specifies that $a_{\bot}$ is
unavailable in $s_{\bot}$, the agent can only receive the reward once.

We now show that the above product MDP can be formulated as a constrained
linear program. We formulate the LP problem using the occupancy measures. The
optimal opacity enforcement problem (Problem 1) can be solved using the
following LP:

$\displaystyle\max_{m}\sum_{v\in V,a\in A}R_{2}(v,a)m(v,a).$ (5) s.t.
$\displaystyle\sum_{a\in A}m(v,a)=\sum_{v^{\prime}\in V,a^{\prime}\in
A}m(v^{\prime},a^{\prime})\mathcal{P}(v^{\prime},a^{\prime},v)+\nu(v)$
$\displaystyle\forall v\in V.$ $\displaystyle m(v,a)\geq 0.$
$\displaystyle\sum_{v\in V,a\in A,v^{\prime}\in
V}\mathcal{P}(v,a,v^{\prime})R_{1}(v,a,v^{\prime})m(v,a)\geq\epsilon.$

where $m(v,a)$ is the occupancy measure that represents the probability of
taking an action $a$ from $v$, $\nu$ is the initial distribution of
$\mathcal{M}$ such that $\nu(v_{0})=1$. Since the agent can receive the reward
for both $R_{1},R_{2}$ functions only once, for any policy, the total reward
(without discounting) is bounded. Using the solution of LP, the optimal policy
for P1 $\pi^{\ast}$ is obtained as
$\pi^{\ast}(v,a)=\dfrac{m(v,a)}{\sum_{a^{\prime}\in A}m(v,a^{\prime})}$.

##  V Case Study

In this section, we showcase our optimal planning algorithm on Example 1 and
then in a power plant monitoring scenario. The LPs are solved using the Gurobi
solver on an Intel Core i7 CPU @ 3.2 GHz with 32 GB RAM.

######  Example 4 (Part IV).

We now setup the LP formulation for Example 1. With the opaque-observations
DFA $\mathcal{A}^{opaque}$, MDP $M$ and task DFA $\mathcal{A}$, we construct
the product MDP $\mathcal{M}$. With $\mathcal{M}$, we set up the LP
formulations to maximize probabilistic opacity, and maximize probabilistic
transparency.

#### Discussion

Table I presents results for varying thresholds to maximize opacity, and Table
II shows the results for varying thresholds to maximize transparency.
Experimental validations of $PH(\varphi)$, $PT(\psi)$, and the probability of
task specification are tabulated for P1’s policy, obtained by running it for
$5000$ instances.

For comparison, a policy that uniformly chooses an action at each state for P1
was applied for $5000$ runs yielded a task satisfaction probability of
$0.3402$ and $PH(\varphi)$ of $0.275$.

Observations from the LP solutions revealed interesting scenarios. In $s_{5}$,
opacity enforcement can be achieved by choosing the terminating action when P2
observes $[s_{5},s_{6}]$. This policy is generated when the threshold is
$0.4$. Increasing the threshold prioritizes task satisfaction, leading to a
policy requiring P1 to take the actions $b$ and then $a$ to prevent P2 from
observing $[s_{4}]$. Likewise, in $s_{7}$ with a threshold $0.8$, the policy
recommends playing action $a$ with probability $0.786$ and terminate with a
probability $0.214$. However, this could result in P2 observing $[s_{6}]$,
thus not enforcing opacity.

Threshold ($\epsilon$) | Max. $PH(\varphi)$ | Exp. $PH(\varphi)$ | Exp. Task
---|---|---|---
0.4 | 0.7 | 0.6966 | 0.3974
0.6 | 0.6 | 0.6036 | 0.5936
0.8 | 0.4 | 0.4032 | 0.7924
TABLE I: Results of LP for the running example enforcing opacity. Threshold ($\epsilon$) | Max. $PT(\varphi)$ | Exp. $PT(\varphi)$ | Exp. Task
---|---|---|---
0.4 | 0.9828 | 0.9833 | 0.3980
0.6 | 0.9742 | 0.9751 | 0.5966
0.8 | 0.9658 | 0.9667 | 0.7951
TABLE II: Results for the running example enforcing transparency.

###  V-A Opacity enforcement

Consider a power plant as represented in the $6\times 6$ gridworld shown in
Fig.6. The plant ($\mathbf{C}$) is in cell $8$, a control center
($\mathbf{A}$) in cell $34$ and two data centers ($\mathbf{B}$) in cells $16$
and $25$. Security alarms are placed in the cells $1,11,13,15,27$ and $35$. A
robot is performing a routine maintenance task on the plant, specified by the
LTLf formula $\psi=\Diamond\,\mathbf{C}$.

The robot moves in four compass directions. It enters the intended cell with a
probability $p=0.6$, and the neighboring cells with probability $(1-p)/2$.
Cells $17$ and $23$ have bouncing walls. If the robot hits into the boundary
or bouncing walls, it stays in its previous cell. Entering cells with security
alarms triggers them and disables the robot.

P2 observes the robot through a set of binary range sensors (1,2,3,4, yellow,
green, indigo and purple resply.) and a precision sensor (5, blue). The binary
sensors return a value $\mathsf{True}$ when the robot is in the range and
$\mathsf{False}$ otherwise, while the precision sensors return the exact
position of the robot when the robot is in range. The coverage of each of the
sensor is shown in Fig. 6. P2 also deploys as a dynamic sensor flying between
cells $34$ and $16$, with a $0.65$ probability to move to the next cell and
$0.35$ to stay. The drone’s camera provides precision sensing for the drone’s
cell and the north cell (e.g., at $34$, it precisely monitors $34$ and $28$).

We consider the scenario where the robot, controlled by P1, aims to enforce
opacity under task constraints. P1 has the task $\psi=\Diamond\,\mathbf{C}$.
Additionally, P1 must perform a secret task specified by the LTLf formula
$\varphi=\Diamond\,\mathbf{B}\land\Diamond\,\mathbf{A}$. The opaque-
observations DFA, with $585$ states is constructed in an average time of
$90.1576s$ for the considered configuration. Subsequently, the product MDP,
with $7596$ states is constructed in $2.9703s$. The LP for optimal policy is
solved in an average time of $3099.1386s$.

Table IV summarizes opacity enforcement probabilities at various task
satisfaction thresholds. As the task specification threshold increases, the
probability of enforcing opacity decreases under the solved policy. For lower
thresholds, the policy suggests terminating the game after satisfying the
secret. While with a higher threshold, the robot is compelled to take the
specific path through cells $16$, $10$, and $9$ or $4$ to reach the target in
cell $8$. Notably, taking the path to cell $16$ from $22$ reveals to P2 that
cell $16$ has been reached, compromising opacity. Thus, opacity is only
enforced when reaching cell $34$ (or $\mathbf{A}$) is concealed.

To assess the impact of sensors on probabilistic opacity, we conduct
experiments under the following conditions:  1. No drone in environment,  2.
The drone flies between the cells $34$, $22$ and $25$.  3. No sensor
$\mathbf{3}$ in the environment.  The results are tabulated in Table III,
specifically for a threshold of $0.4$ (due to page limitations). From Table
III, it is evident that in the absence drone or sensor $\mathbf{3}$, P1
achieves better opacity enforcement. As P2’s sensing capabilities decrease,
the probability of opacity enforcement increases as expected.

Experiment | Prob. Opacity | Exp. Opacity
---|---|---
No drone | $0.7058$ | $0.7102$
Drone covers larger area | $0.6154$ | $0.6179$
No Sensor $\mathbf{3}$ | $0.7950$ | $0.7894$
TABLE III: Comparison of opacity enforcement (with threshold $0.4$). Figure 6: Gridworld depiction of power plant (sensors numbered in red). Threshold ($\epsilon$) | Prob. Opacity | Exp. Opacity | Exp. Task
---|---|---|---
0.4 | 0.6766 | 0.6678 | 0.4014
0.6 | 0.5333 | 0.5401 | 0.6021
0.7 | 0.4611 | 0.4625 | 0.7056
0.8 | 0.3448 | 0.3359 | 0.8046
0.95 | 0.1079 | 0.1085 | 0.9504
TABLE IV: Enforced probabilistic opacity and experimental validation.

###  V-B Transparency enforcement

Consider a case where P1 enforces transparency, in a setup similar to the
opacity enforcement case, shown in the Fig. 7. The environment includes one
data center ($\mathbf{B}$) in the cell $25$ and a control center
($\mathbf{A}$) in the cell $34$. Three sensors are present: sensor
$\mathbf{1}$ a static precision sensor that monitors the cells $30$ and $31$,
sensor $\mathbf{2}$ a static binary sensor that monitors the cells $2,8,9,14$,
and sensor $\mathbf{3}$ a static binary sensor covering the cells $10$ and
$16$. Additionally, the drone, operates between the cells $34$ and $22$ with
probabilities of flying to next cell with probability $0.65$ and stay with
$0.35$.

Figure 7: Gridworld depiction for enforcing transparency.

The robot maintains the previous task specification
$\psi=\Diamond\,\mathbf{C}$, and P1’s specification for transparency on P2 is
$\varphi=\Diamond\,\mathbf{B}\land\Diamond\,\mathbf{A}$.

Table V summarizes transparency enforcement probabilities at various task
specification threshold. It also presents the results for P1 following the
optimal policy.

Threshold ($\epsilon$) | Prob. Transp. | Exp. Transp. | Exp. Task
---|---|---|---
0.4 | 0.8795 | 0.8784 | 0.4022
0.6 | 0.6802 | 0.6814 | 0.5999
0.7 | 0.5581 | 0.5623 | 0.7012
0.8 | 0.4100 | 0.4105 | 0.8064
0.95 | 0.1204 | 0.1198 | 0.9503
TABLE V: Enforced probabilistic transparency and validation.

##  VI Conclusion

We introduced constrained probability planning methods for MDPs to optimize
opacity/transparency under task constraints. Our approach starts with building
computation models based on FST to capture the observation function of the
observer that maps the input languages — state trajectories into the output
languages — observed trajectories. Then, by performing the product operations
between the FST and the DFA accepting the secret, we can derive another DFA
that accepts all possible observations that enforce the opacity of the secret
to the observer. Thus, we can formulate the probabilistic planning with
opacity as a constrained MDP, augmented with the task state and the state in
the DFA of the opaque observations. The dual problem of transparency, can be
solved by replacing the maximization with a minimization in the objective
function for the constrained MDP. Through experimental analysis, we
investigated the impact of sensors and sensor configurations on opacity and
transparency enforcement. The construction of opaque/transparent observations
can be extended for optimizing opacity in partially observable systems or
games with partial observations, involving both collaborative and competitive
interactions.

## References

  * [1] C. Baier and J.-P. Katoen.  Principles of model checking.  MIT press, 2008.
  * [2] B. Bérard, K. Chatterjee, and N. Sznajder.  Probabilistic opacity for markov decision processes.  Information Processing Letters, 115(1):52–59, 2015.
  * [3] B. Bérard, J. Mullins, and M. Sassolas.  Quantifying opacity.  Mathematical Structures in Computer Science, 25(2):361–403, 2015.
  * [4] J. W. Bryans, M. Koutny, L. Mazaré, and P. Y. Ryan.  Opacity generalised to transition systems.  International Journal of Information Security, 7:421–435, 2008.
  * [5] J. W. Bryans, M. Koutny, and P. Y. Ryan.  Modelling opacity using petri nets.  Electronic Notes in Theoretical Computer Science, 121:101–115, 2005.
  * [6] F. Cassez, J. Dubreil, and H. Marchand.  Synthesis of opaque systems with static and dynamic masks.  Formal Methods in System Design, 40:88–115, 2012.
  * [7] G. De Giacomo and M. Y. Vardi.  Linear temporal logic and linear dynamic logic on finite traces.  In Proceedings of the Twenty-Third international joint conference on Artificial Intelligence, pages 854–860. ACM, 2013.
  * [8] C. Keroglou and C. N. Hadjicostis.  Probabilistic system opacity in discrete event systems.  Discrete Event Dynamic Systems, 28:289–314, 2018.
  * [9] F. Lin.  Opacity of discrete event systems and its applications.  Automatica, 47(3):496–503, 2011.
  * [10] S. Liu, X. Yin, D. V. Dimarogonas, and M. Zamani.  On approximate opacity of stochastic control systems.  arXiv preprint arXiv:2401.01972, 2024.
  * [11] B. Maubert, S. Pinchinat, and L. Bozzelli.  Opacity issues in games with imperfect information.  arXiv preprint arXiv:1106.1233, 2011.
  * [12] L. Mazaré.  Using unification for opacity properties.  Proceedings of the 4th IFIP WG1, 7:165–176, 2004.
  * [13] A. Saboori.  Verification and enforcement of state-based notions of opacity in discrete event systems.  University of Illinois at Urbana-Champaign, 2010.
  * [14] A. Saboori and C. N. Hadjicostis.  Notions of security and opacity in discrete event systems.  In IEEE Conference on Decision and Control, pages 5056–5061, 2007.
  * [15] A. Saboori and C. N. Hadjicostis.  Verification of initial-state opacity in security applications of des.  In 9th International Workshop on Discrete Event Systems, pages 328–333. IEEE, 2008.
  * [16] A. Saboori and C. N. Hadjicostis.  Opacity-enforcing supervisory strategies via state estimator constructions.  IEEE Transactions on Automatic Control, 57(5):1155–1165, 2012.
  * [17] A. Saboori and C. N. Hadjicostis.  Current-state opacity formulations in probabilistic finite automata.  IEEE Transactions on Automatic Control, 59(1):120–133, 2014.
  * [18] S. Udupa, H. Rahmani, and J. Fu.  Opacity-enforcing active perception and control against eavesdropping attacks.  In International Conference on Decision and Game Theory for Security, pages 329–348. Springer, 2023.
  * [19] Y.-C. Wu and S. Lafortune.  Comparative analysis of related notions of opacity in centralized and coordinated architectures.  Discrete Event Dynamic Systems, 23(3):307–339, 2013.
  * [20] Y.-C. Wu and S. Lafortune.  Synthesis of insertion functions for enforcement of opacity security properties.  Automatica, 50(5):1336–1348, 2014.
  * [21] Y. Xie, X. Yin, and S. Li.  Opacity enforcing supervisory control using nondeterministic supervisors.  IEEE Transactions on Automatic Control, 67(12):6567–6582, 2021.
  * [22] J. Yao, X. Yin, and S. Li.  Sensor deception attacks against initial-state privacy in supervisory control systems.  In 2022 IEEE 61st Conference on Decision and Control (CDC), pages 4839–4845. IEEE, 2022.
  * [23] X. Yin, Z. Li, W. Wang, and S. Li.  Infinite-step opacity and k-step opacity of stochastic discrete-event systems.  Automatica, 99:266–274, 2019.
  * [24] Y. Zhou, Z. Chen, and Z. Liu.  Verification and enforcement of current-state opacity based on a state space approach.  European Journal of Control, 71:100795, 2023.

  *[FST]: Finite State Transducer
  *[DFA]: Deterministic Finite Automaton
  *[MDP]: Markov decision process
  *[LTLf]: Linear Temporal Logic over Finite Traces
  *[NFA]: Nondeterministic Finite Automaton
  *[ DFA]: Deterministic Finite Automaton