# Holographic Thermodynamics of an Enhanced Charged AdS Black Hole in String
Theory’s Playground

Behnam Pourhassan1,2 Sareh Eslamzadeh1 İzzet Sakallı3 and Sudhaker Upadhyay4,5

###### Abstract

In this paper, we consider an $\alpha^{\prime}$ corrected Reissner-Nordström
AdS black hole to study thermodynamics. We study the $P-V$ criticality and
thermodynamical stability of the black hole. We obtained a first-order phase
transition, which may be interpreted as the large/small black hole phase
transition. Therefore, we obtained a van der Waals behaviour and obtained
critical points. Finally, we calculate quantum work used to resolve the
information loss paradox.

##  1 Introduction

Strominger and Vafa observed that the microscopic entropy of 5-dimensional
extremal black holes in string theory matches with the macroscopic entropy at
the first order [1]. Beyond the first order, the macroscopic entropy requires
corrections to the superstring theory effective action, and corresponding
black hole solutions are valid to the level of approximation needed. Another
way to phrase this is that while most black-hole solutions in string theory
adhere only to the effective field theories at the zeroth order, the precision
of stringy black hole solutions can be improved by incorporating higher-order
terms in $\alpha^{\prime}$ corrections. A modified version of black holes,
accounting for corrections associated with string theory, where
$\alpha^{\prime}$ represents the string length scale, is an intriguing topic
in theoretical physics. The inclusion of higher-order terms in
$\alpha^{\prime}$ is crucial as they signify genuine deviations from matter
coupled Einstein gravity due to the non-zero string length. The validity of
microscopic entropy can only be considered circumstantial without explicit
knowledge of complete $\alpha^{\prime}$ corrected black hole solutions and
subsequent entropy computation using the action.

The first-order $\alpha^{\prime}$ correction to a family of solutions of the
heterotic superstring effective action, which are parallel to solitonic
5-branes with Kaluza-Klein monopoles, is studied [2]. It is demonstrated that
this family of $\alpha^{\prime}$ corrected solutions remains invariant under
$\alpha^{\prime}$ corrected $T$-duality transformations. Additionally, the
first-order $\alpha^{\prime}$ correction to the 4-charge Reissner-Nordström
black hole beyond the near-horizon limit in the heterotic superstring
effective action is examined [3]. In this context, the Wald entropy is found
to be in agreement with the entropy obtained through microstate counting.
Furthermore, the first-order $\alpha^{\prime}$ correction to non-extremal
4-dimensional dyonic Reissner-Nordström black holes with equal electric and
magnetic charges is also discussed in the context of heterotic superstring
effective field theory [4].

In this work, we consider a 4-dimensional $\alpha^{\prime}$ corrected
Reissner-Nordström AdS black hole. We first study the effect of
$\alpha^{\prime}$ correction, charge parameter and cosmological constant on
the metric function. To be more precise, the decrease in charge parameter
creates one coincidence horizon and two separated horizons. As the charge of
the black hole decreases, the outer horizons of the Reissner-Nordström AdS
black hole with/without $\alpha^{\prime}$ correction have more overlap. With
the decrease in cosmological constant, the outer horizons disappear but the
inner horizon, which is constructed due to the existence of $\alpha^{\prime}$
correction, exists for any cosmological constant. The increasing
$\alpha^{\prime}$ widens the separation of two outer horizons and the inner
horizon radius bigger. We also study the thermodynamics of the black hole.
Incidentally, the existence of $\alpha^{\prime}$ prevents the outer horizon’s
final temperature from zero, but it doesn’t affect the critical mass of the
black hole. With increasing $\alpha^{\prime}$ correction, the unstable phase
is gradually corrected, and the heat capacity becomes continuously positive.
However, $\alpha^{\prime}$ correction causes instability in the final
evaporation stage when the black hole mass approaches the critical mass.
Finally, we compute quantum work for this black hole. The quantum work is
represented by a unitary information preserving process, which is the dual
picture of an $\alpha^{\prime}$ corrected Reissner-Nordström AdS black hole.
We find that an $\alpha^{\prime}$ corrected Reissner-Nordström AdS black hole
is unstable through a first-order phase transition and will evaporate at the
final state.

The paper is organized in the following manner. In section 2, we focus on
$\alpha^{\prime}$ corrected 4- dimensional Reissner-Nordström AdS black hole
solution and discuss the horizon structure. In section 3, we study the
thermodynamics of black holes by calculating Hawking temperature and entropy.
Section 4 is devoted to calculating quantum work for the $\alpha^{\prime}$
corrected Reissner-Nordström AdS black hole. Finally, we conclude the results
in the last section.

##  2 Black hole solution

A 4-dimensional Reissner-Nordström AdS black hole of mass $M$ with the
$\alpha^{\prime}$ correction given by the following metric [5],

$ds^{2}=N^{2}f(r)dt^{2}-\frac{dr^{2}}{f(r)}-r^{2}(d\theta^{2}+\sin^{2}\theta
d\varphi^{2}),$ (2.1)

with

$N^{2}=1+\alpha^{\prime}\frac{p^{2}}{8r^{4}},$ (2.2)
$f(r)=1-\frac{2M}{r}+\frac{r^{2}}{l^{2}}+\frac{p^{2}}{2r^{2}}-\alpha^{\prime}\frac{p^{2}}{4r^{4}}\left[1-\frac{3M}{2r}+\frac{11p^{2}}{40r^{2}}\right],$
(2.3)

where $p$ is a physical constant related to the black hole charges, and $l$,
at $4D$ space-time, is the AdS radius, which is related to the cosmological
constant via

$\Lambda=-\frac{3}{l^{2}}.$ (2.4)

We look into the structure of the $\alpha^{\prime}$ corrected Reissner-
Nordström AdS black hole, and we study the effect of this correction on the
metric. According to equation (2.3), we check the metric dependency on $p$,
$\alpha^{\prime}$ and, $\Lambda$. In Figure 1, we plot $f(r)$ versus radius
with different $p$. Here, we find that the decrease in $p$ creates one
coincidence horizon and two separated horizons. Hence, comparing the
$\alpha^{\prime}$ corrected Reissner-Nordström AdS black hole with Reissner-
Nordström AdS black hole shows the existence of $\alpha^{\prime}$ correction
causes for any content of the charge, ever less than $p_{cri}=\sqrt{2}M$,
there will be at least one horizon. Moreover, as the charge of the black hole
decreases, the outer horizons of the Reissner-Nordström AdS black hole
with/without $\alpha^{\prime}$ correction have more overlap. In Figure 2, we
fix the black hole’s charge to the critical charge. As $\Lambda$ decreases,
outer horizons disappear, but the inner horizon, which is constructed due to
the existence of $\alpha^{\prime}$ correction, exists for any cosmological
constant. In the following, we fix the mass and charge of the black hole, and
we probe the effect of the $\alpha^{\prime}$ parameter. As shown in Figure 3,
increasing $\alpha$ (increasing $a$) makes two outer horizons more separated
and the inner horizon’s radius bigger.

Figure 1:  $f(r)$ versus of $r$ with different $p$ for the $\alpha^{\prime}$
corrected Reissner-Nordström AdS black hole. Plots have been depicted with
$p=1.5,1.4,1.3,1.2$ from left to right. Dashed line shows the Reissner-
Nordström black hole without correction ($\alpha^{\prime}=0$). As $p$
decreases, two separated horizons are created. There exists critical $p$ for
which we have one coincidence horizon. Plots have been depicted with
$M=1,\alpha^{\prime}=M^{2}(a=1),\Lambda=-0.1$.

Figure 2:  $f(r)$ versus of $r$ with different $\Lambda$ for the
$\alpha^{\prime}$ corrected Reissner-Nordström AdS black hole. Plots have been
depicted with $\Lambda=-0.1,-0.3,-0.5,-0.7$ from left to right. Dashed line
shows the Reissner-Nordström black hole without correction
($\alpha^{\prime}=0$). As $\Lambda$ decreases, outer horizons disappear. Plots
have been depicted with $M=1,\alpha^{\prime}=M^{2}(a=1),p\simeq 1.4$. Figure
3: $f(r)$ versus of $r$ with different $\alpha^{\prime}=aM^{2}$ for the
$\alpha^{\prime}$ corrected Reissner-Nordström AdS black hole.

The area of the event horizon for the black hole in terms of horizon radius
$r_{h}$ is given by

$A_{h}=4\pi r_{h}^{2},$ (2.5)

and the black hole volume is

$V=\frac{4}{3}\pi r_{h}^{3}.$ (2.6)

##  3 Thermodynamics

We calculate the Hawking temperature of this kind of black hole based on the
famous formula between the surface gravity of the event horizon and the
temperature that has been given by [6]

$T=\frac{1}{4\pi}\Big{[}\frac{1}{\sqrt{-g_{tt}g_{rr}}}\frac{dg_{tt}}{dr}\Big{]}_{r_{h}}.$
(3.1)

Considering the metric 2.3, we find the horizons of the $\alpha^{\prime}$
corrected Reissner-Nordström AdS black hole, and we calculate the temperature
of the outer horizon, $r_{h}$, as follows,

$T=\frac{l^{2}\left(-10p^{2}r_{h}\left(15\alpha^{\prime}M+8r_{h}^{3}-8\alpha^{\prime}r_{h}\right)+160Mr_{h}^{5}+33\alpha^{\prime}p^{4}\right)+160r_{h}^{8}}{80\sqrt{2}\pi
l^{2}r_{h}^{7}\sqrt{\frac{\alpha^{\prime}p^{2}}{r_{h}^{4}}+8}},$ (3.2)

where $r_{h}$ is the largest root of $f(r)=0$ using the equation (2.3). Also,
we illustrate it in Figures 4 and 5. One can see in Figure 4 that decreasing
the mass and charge of the black hole, the temperature of the
$\alpha^{\prime}$ corrected Reissner-Nordström AdS black hole increases until
it reaches a maximum and after that decreases to a finite temperature with
finite mass and charge. To research about the effect of $\alpha^{\prime}$, we
plot Figure 5 with different $a$. The most important result is that the
existence of $\alpha^{\prime}$ prevents the outer horizon’s final temperature
from zero, but it doesn’t affect the critical mass of the black hole.

Figure 4: The behaviour of temperature versus the mass and the charge of the
$\alpha^{\prime}$ corrected Reissner-Nordström AdS black hole. Left plot has
been depicted with $\alpha^{\prime}=2M^{2}$ and right one with
$\alpha^{\prime}=5M^{2}$ while $\Lambda=-0.1$. Increasing $\alpha^{\prime}$
predicts less temperature, also, less mass and charge for the remnant of the
black hole.

Figure 5: The behavior of temperature versus the mass for the
$\alpha^{\prime}$ corrected Reissner-Nordström AdS black hole. Plots have been
depicted with $a=0,1,2,3,4$ ($\alpha^{\prime}=aM^{2}$) and $\Lambda=-0.1$
while $p<p_{cri}$.

We consider the entropy of the $\alpha^{\prime}$ corrected Reissner-Nordström
AdS black hole as a

$S=\pi r_{h}^{2},$ (3.3)

which $r_{h}$ is the largest root of the equation (2.3) as a outer horizon. To
investigate the heat capacity,

$C=T\frac{dS}{dT},$ (3.4)

we may approximate in two regions as,

$\displaystyle
C=\begin{array}[]{cc}T\frac{dS}{dM}\left(\frac{dT}{dM}\right)^{-1},&\left(\frac{dT}{dM}\gg\frac{dT}{dq}\right)\\\
T\frac{dS}{dq}\left(\frac{dT}{dq}\right)^{-1},&\left(\frac{dT}{dM}\ll\frac{dT}{dq}\right).\end{array}$
(3.7)

So, we employ the first condition of equation (3.7) and present the results of
our calculations in Figure 6. As expected for the Reissner-Nordstr”om AdS
black hole, there exists an unstable thermodynamic phase ($C<0$) when $r_{h}$
falls within a specific range. The black hole is in a stable thermodynamic
phase ($C>0$) at the final stage of evaporation [7] (See dashed blue curve in
Figure 6). A notable outcome is that when we introduce $\alpha^{\prime}$
corrections and increase their values, the thermodynamically unstable phase
gradually corrects itself, and the heat capacity becomes continuously
positive. Conversely, as depicted in the right panel of Figure 6, this type of
correction induces instability in the final evaporation stage as the black
hole mass approaches the critical mass. Additionally, the larger the value of
$\alpha^{\prime}$, the more phase transitions are predicted.

Figure 6: The behavior of temperature versus the mass for the
$\alpha^{\prime}$ corrected Reissner-Nordström AdS black hole. Plots have been
depicted with $a=0,1,2,3,4$ ($\alpha^{\prime}=aM^{2}$) and $\Lambda=-0.1$
while $p<p_{cri}$.

Eventually, we will investigate the effect of the $\alpha^{\prime}$ correction
on the Van der Waals equation. According to the outer horizon of the metric
(2.3), we derive the mass of the black hole as follows

$M=\frac{l^{2}\left(-11\alpha^{\prime}p^{4}+40p^{2}\left(2r_{h}^{4}-\alpha^{\prime}r_{h}^{2}\right)+160r_{h}^{6}\right)+160r_{h}^{8}}{20l^{2}\left(16r_{h}^{5}-3\alpha^{\prime}p^{2}r_{h}\right)}.$
(3.8)

Hence, the black hole pressure and its conjugate variable, the black hole
volume, are given by

$P=\frac{3}{8\pi l^{2}},\hskip 28.45274ptV=\frac{4}{3}\pi r_{h}^{3}.$ (3.9)

In this case, the first law of thermodynamics is modified as,

$dM=TdS+\Phi dq+VdP+Ad\alpha^{\prime},$ (3.10)

where $\Phi$ is the corrected electrostatic potential conjugate to the black
hole charge, while $A$ is a thermodynamics variable conjugate to the
correction parameter. Therefore, the thermodynamics relations read as,

$\displaystyle\left(\frac{\partial S}{\partial
M}\right)_{q,\alpha^{\prime}}=\frac{1}{T},$ $\displaystyle\left(\frac{\partial
S}{\partial q}\right)_{M,\alpha^{\prime}}=-\frac{\Phi}{T},$
$\displaystyle\left(\frac{\partial
S}{\partial\alpha^{\prime}}\right)_{M,q}=-\frac{A}{T}.$ (3.11)

Based on equation (3.9), if we suppose the cosmological constant has not been
allowed to vary, $P$ is treated as a constant. Moreover, for a fixed
$\alpha^{\prime}$ and a fixed charge, we derive the equation of state for the
$\alpha^{\prime}$ corrected Reissner-Nordström AdS black hole concerning
equations (3.2), (3.8), and (3.9) as follows

$P=\frac{\frac{10\pi
T}{v}\sqrt{1+\frac{2p^{2}\alpha^{\prime}}{v^{4}}}\Big{(}v^{4}-3p^{2}\alpha^{\prime}\Big{)}-5v^{2}+10p^{2}+\frac{66p^{2}\alpha^{\prime
2}}{v^{8}}-\frac{60p^{4}\alpha^{\prime
2}}{v^{6}}-\frac{20p^{4}\alpha^{\prime}}{v^{4}}}{10\pi(v^{4}-7p^{2}\alpha^{\prime})},$
(3.12)

where, $v$, is the specific volume with $v=2l_{Pl}^{2}r_{h}$. Besides, one can
find the critical point in the condition

$\frac{\partial{P}}{\partial{v}}=0,\hskip
28.45274pt\frac{\partial^{2}{P}}{\partial{v^{2}}}=0.$ (3.13)

We present the $P-V$ plot in Figure 7 for the $\alpha^{\prime}$-corrected
Reissner-Nordstr”om AdS black hole per critical temperature. As demonstrated,
increasing $\alpha^{\prime}$ reduces the oscillatory component of the curves.
In other words, increasing $\alpha^{\prime}$ leads to phase transitions
occurring over smaller changes in the volume of the black hole. Additionally,
we maintain $\alpha^{\prime}$ constant and illustrate decreasing temperature
from top to bottom in Figure 8. If the temperature exceeds the critical
temperature $T_{c}$, no phase transition occurs. When the temperature falls
between $T_{c}$ and $T_{0}$, the behavior of the black hole resembles that of
a Van der Waals gas, exhibiting an inflection point. Finally, if the
temperature is below $T_{0}$, the pressure becomes negative for certain values
of $r_{+}$. As expected, in the absence of $\alpha^{\prime}$ correction, we
recover the equation of state for a charged AdS black hole, as can be found in
various references, such as Ref. [7, 8],

$P=\frac{T}{v}-\frac{1}{2\pi v^{2}}+\frac{p^{2}}{\pi v^{4}}.$ (3.14)

Figure 7: $P-V$ diagram of the $\alpha^{\prime}$ corrected Reissner-Nordström
AdS black hole for various $\alpha^{\prime}$ ($\alpha^{\prime}=aM^{2}$) per
critical temperature.  Figure 8: $P-V$ diagram of the $\alpha^{\prime}$
corrected Reissner-Nordström AdS black hole per various temperature. Plots
have been depicted with $\alpha^{\prime}=1.5M^{2}$.

##  4 Quantum work

Black hole entropy, at quantum scales, is related to the microstates. Hence, a
change in the black hole microstates yields a change in the entropy. So, one
can express the change in the quantum corrected entropy of the Reissner-
Nordström AdS black hole as

$\Delta S=S(r_{h2})-S(r_{h1}).$ (4.1)

This change in the entropy and the first law of thermodynamics [9] can be used
to obtain a change in the internal energy of the Reissner-Nordström AdS black
hole.
The internal energy of $\alpha^{\prime}$ corrected Reissner-Nordström AdS
black hole is calculated as following,

$E=\frac{X_{E}}{120l^{2}r_{h}^{5}\left(r_{h}^{4}-112\alpha^{\prime}p^{2}\right)\left(16r_{h}^{4}-3\alpha^{\prime}p^{2}\right)\sqrt{\frac{\alpha^{\prime}p^{2}}{r_{h}^{4}}+8}},$
(4.2)

where

$\displaystyle X_{E}$ $\displaystyle=$
$\displaystyle-480\alpha^{\prime}p^{2}r_{h}^{12}\left(224\sqrt{\frac{\alpha^{\prime}p^{2}}{r_{h}^{4}}+8}-775\sqrt{\frac{64\alpha^{\prime}p^{2}}{r_{h}^{4}}+2}\right)$
(4.3) $\displaystyle-$ $\displaystyle
480r_{h}^{16}\left(16\sqrt{\frac{64\alpha^{\prime}p^{2}}{r_{h}^{4}}+2}-2\sqrt{\frac{\alpha^{\prime}p^{2}}{r_{h}^{4}}+8}\right)$
$\displaystyle-$ $\displaystyle 161280\alpha^{\prime
2}r_{h}^{8}p^{4}\sqrt{\frac{64\alpha^{\prime}p^{2}}{r_{h}^{4}}+2}$
$\displaystyle+$ $\displaystyle
320r_{h}^{14}l^{2}\left(19\sqrt{\frac{\alpha^{\prime}p^{2}}{r_{h}^{4}}+8}-8\sqrt{\frac{64\alpha^{\prime}p^{2}}{r_{h}^{4}}+2}\right)$
$\displaystyle+$ $\displaystyle 1584p^{8}l^{2}\alpha^{\prime
3}\left(8192\sqrt{\frac{\alpha^{\prime}p^{2}}{r_{h}^{4}}+8}-\sqrt{\frac{64\alpha^{\prime}p^{2}}{r_{h}^{4}}+2}\right)$
$\displaystyle+$ $\displaystyle 20\alpha^{\prime
2}p^{4}r_{h}^{6}\left(790080\sqrt{\frac{\alpha^{\prime}p^{2}}{r_{h}^{4}}+8}-1158\sqrt{\frac{64\alpha^{\prime}p^{2}}{r_{h}^{4}}+2}\right)$
$\displaystyle+$ $\displaystyle
2\alpha^{\prime}p^{4}r_{h}^{8}\left(632287\sqrt{\frac{\alpha^{\prime}p^{2}}{r_{h}^{4}}+8}-30800\sqrt{\frac{64\alpha^{\prime}p^{2}}{r_{h}^{4}}+2}\right)$
$\displaystyle+$ $\displaystyle
80p^{2}\alpha^{\prime}r_{h}^{10}\left(-4431\sqrt{\frac{\alpha^{\prime}p^{2}}{r_{h}^{4}}+8}+1542\sqrt{\frac{64\alpha^{\prime}p^{2}}{r_{h}^{4}}+2}\right)$
$\displaystyle+$ $\displaystyle
80p^{2}r_{h}^{12}\left(16\sqrt{\frac{64\alpha^{\prime}p^{2}}{r_{h}^{4}}+2}-506\sqrt{\frac{\alpha^{\prime}p^{2}}{r_{h}^{4}}+8}\right)$
$\displaystyle+$ $\displaystyle\alpha^{\prime
3}p^{6}r_{h}^{2}\left(-983040\sqrt{\frac{\alpha^{\prime}p^{2}}{r_{h}^{4}}+8}+1920\sqrt{\frac{64\alpha^{\prime}p^{2}}{r_{h}^{4}}+2}\right)$
$\displaystyle+$ $\displaystyle\alpha^{\prime
2}p^{6}r_{h}^{4}\left(2571\sqrt{\frac{64\alpha^{\prime}p^{2}}{r_{h}^{4}}+2}-23148128\sqrt{\frac{\alpha^{\prime}p^{2}}{r_{h}^{4}}+8}\right).$

Hence, we can obtain the change in internal energy as

$\Delta E=E(r_{h2})-E(r_{h1}).$ (4.4)

In the final stage, to study the amount of quantum work, we need to derive
Helmholtz free energy of the black hole, we have to consider the modified
Smarr-Gibbs-Duhem formula for a $\alpha^{\prime}$ corrected Reissner-Nordström
AdS black hole as follows

$M=2TS-2PV+\Phi Q+A\alpha^{\prime}.$ (4.5)

Comparing the first law of thermodynamics, equation (3.10), with Smarr
formula, equation (4.5), we identify the Helmholtz free energy of the black
hole as mentioned in Ref. [10] as a

$F=E-TS.$ (4.6)

As a result, we find Helmholtz free energy of the $\alpha^{\prime}$ corrected
Reissner-Nordström AdS black hole as follows

$F=\frac{X_{F}}{960l^{2}r_{h}^{5}\left(r_{h}^{4}-112\alpha^{\prime}p^{2}\right)\left(16r_{h}^{4}-3\alpha^{\prime}p^{2}\right)X_{1}},$
(4.7)

where

$\displaystyle X_{F}=$
$\displaystyle-480\Bigl{(}r_{h}^{16}\Bigl{(}-16X_{1}+128X_{2}+48\sqrt{2}\Bigr{)}+\alpha^{\prime}p^{2}r_{h}^{12}\Bigl{(}1792X_{1}-6200X_{2}-5397\sqrt{2}\Bigr{)}$
$\displaystyle+336\alpha^{\prime
2}p^{4}r_{h}^{8}\Bigl{(}8X_{2}+7\sqrt{2}\Bigr{)}+l^{2}\Bigl{(}-2560r_{h}^{14}\Bigl{(}-19X_{1}+8X_{2}+3\sqrt{2}\Bigr{)}$
$\displaystyle-1584\alpha^{\prime
3}p^{8}\Bigl{(}-65536X_{1}+8X_{2}+7\sqrt{2}\Bigr{)}\Bigr{)}$
$\displaystyle+160p^{2}l^{2}r_{h}^{10}\Bigl{(}5385\sqrt{2}\alpha^{\prime}-17724\alpha^{\prime}X_{1}+6168\alpha^{\prime}X_{2}+r_{h}^{2}\Bigl{(}-2024X_{1}+64X_{2}+24\sqrt{2}\Bigr{)}\Bigr{)}$
$\displaystyle-8l^{2}\alpha^{\prime}p^{4}r_{h}^{6}\Bigl{(}20205\sqrt{2}\alpha^{\prime}-15801600\alpha^{\prime}X_{1}+23160\alpha^{\prime}X_{2}$
$\displaystyle+r_{h}^{2}\Bigl{(}-1264574X_{1}+61600X_{2}+53820\sqrt{2}\Bigr{)}\Bigr{)}$
$\displaystyle+3l^{2}\alpha^{\prime
2}p^{6}r_{h}^{2}\Bigl{(}1920\Bigl{(}7\sqrt{2}\alpha^{\prime}-4096\alpha^{\prime}X_{1}+8\alpha^{\prime}X_{2}\Bigr{)}$
$\displaystyle+r_{h}^{2}\Bigl{(}-185185024X_{1}+20568X_{2}+17953\sqrt{2}\Bigr{)}\Bigr{)}\Bigr{)},$
(4.8)

with

$\displaystyle X_{1}$ $\displaystyle\equiv$
$\displaystyle\sqrt{\frac{\alpha^{\prime}p^{2}}{r_{h}^{4}}+8},$ $\displaystyle
X_{2}$ $\displaystyle\equiv$
$\displaystyle\sqrt{\frac{64\alpha^{\prime}p^{2}}{r_{h}^{4}}+2}.$ (4.9)

Therefore, the change of Helmholtz free energy can be expressed as,

$\Delta F=F(r_{h2})-F(r_{h1}).$ (4.10)

Now, the quantum work ($W_{Q}$) is expressed as [11],

$W_{Q}=e^{\frac{\Delta F}{T}},$ (4.11)

where $T$ is given by the equation (3.2). In this scenario, a dual depiction
of black hole geometry is articulated using superconformal field theory,
offering a potential resolution to the information loss paradox. The black
hole geometry of an $\alpha^{\prime}$ corrected Reissner-Nordstr”om AdS black
hole emits Hawking radiation, leading to the information loss paradox in
quantum thermodynamics. This paradox may find resolution through the quantum
behavior presented in the dual picture of a superconformal field theory.
Consequently, an $\alpha^{\prime}$ corrected Reissner-Nordstr”om AdS black
hole is described by a unitary superconformal field theory. This framework
enables the study of black hole geometry through quantum non-equilibrium
thermodynamics, achievable by computing the quantum work of an
$\alpha^{\prime}$ corrected Reissner-Nordstr”om AdS black hole. This quantum
work is symbolized by a unitary information-preserving process, serving as the
dual depiction of the black hole. The non-equilibrium quantum thermodynamics
on the alternate side facilitates the loss of mass from an $\alpha^{\prime}$
corrected Reissner-Nordstr”om AdS black hole through a unitary process.
Demonstrating that an $\alpha^{\prime}$ corrected Reissner-Nordstr”om AdS
black hole is unstable via a first-order phase transition reveals its eventual
evaporation. Consequently, information leakage from an $\alpha^{\prime}$
corrected Reissner-Nordstr”om AdS black hole during its final stages of
evaporation is conceivable through the unitary information-preserving process.
Thus, employing non-equilibrium quantum thermodynamics offers a pathway to
resolving the information loss paradox within the given system.

##  5 Conclusion

In this work, we have considered a 4-dimensional $\alpha^{\prime}$ corrected
Reissner-Nordström AdS black hole solution. Furthermore, we studied the
effects of $\alpha^{\prime}$ correction, charge parameter and cosmological
constant on the solution. Here, we have found that the decrease in the charge
parameter creates one coincidence horizon and two separated horizons. As the
charge of the black hole decreases, the outer horizons of the Reissner-
Nordström AdS black hole with/without $\alpha^{\prime}$ correction have more
overlap. When the cosmological constant decreases, the outer horizons
disappear but the inner horizon, which is constructed due to the existence of
$\alpha^{\prime}$ correction, exists for any cosmological constant.
Interestingly, the increasing $\alpha^{\prime}$ widens the separation of two
outer horizons, and the inner horizon radius increases.

Additionally, we have discussed the thermodynamics of the black hole. Here, we
have found that the existence of $\alpha^{\prime}$ prevents the outer
horizon’s final temperature from zero, but it doesn’t affect the critical mass
of the black hole. With increasing $\alpha^{\prime}$ correction, the unstable
phase is gradually corrected, and the heat capacity becomes continuously
positive. On the other hand, $\alpha^{\prime}$ correction causes instability
in the final evaporation stage when the black hole mass approaches the
critical mass. Finally, we have derived quantum work for this black hole. The
quantum work is represented by a unitary information preserving process, which
is the dual picture of an $\alpha^{\prime}$ corrected Reissner-Nordström AdS
black hole. We have observed that an $\alpha^{\prime}$ corrected Reissner-
Nordström AdS black hole is unstable through a first-order phase transition
and will evaporate at the final state.

## Authors’ Contributions

All authors have the same contribution.

## Data Availability

No data are available.

## Conflicts of Interest

The authors declare that there is no conflict of interest.

## Ethical Considerations

The authors have diligently addressed ethical concerns, such as informed
consent, plagiarism, data fabrication, misconduct, falsification, double
publication, redundancy, submission, and other related matters.

## Funding

This research did not receive any grant from funding agencies in the public,
commercial, or non-profit sectors.

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# Detecting Logical Relation In Contract Clauses

Alexandre Yukio Ichida
Felipe Meneguzzi
Pontifical Catholic University of Rio Grande do Sul
Porto Alegre, RS, Brazil
<EMAIL_ADDRESS>
<EMAIL_ADDRESS>
###### Abstract

Contracts underlie most modern commercial transactions defining define the
duties and obligations of the related parties in an agreement. Ensuring such
agreements are error free is crucial for modern society and their analysis of
a contract requires understanding the logical relations between clauses and
identifying potential contradictions. This analysis depends on error-prone
human effort to understand each contract clause. In this work, we develop an
approach to automate the extraction of logical relations between clauses in a
contract. We address this problem as a Natural Language Inference task to
detect the entailment type between two clauses in a contract. The resulting
approach should help contract authors detecting potential logical conflicts
between clauses.

## Introduction

Understanding existing logical relations between sentences is a difficult task
that requires an accurate understanding of natural language meaning. The
ambiguity and variability of linguistic expression in natural language
complicates the recognition of these relations such as entailment and
contradiction contained in texts. The ability to classify these logical
inferences among different text is a significant feature of an intelligent
system (?). Detecting these logical relations can help humans to interpret a
more complex text, where entailment and contradiction are crucial aspects to
fully understanding such as norms and contracts. Contracts are documents that
contain normative sentences formalizing agreements among the related parties,
which involve people and companies. The normative sentences describe the
duties that the related parties are subject to and the penalties in case of
rule violation. In a contract, the norms may be logically related, so that
some are entailed, or contradict each other (?).

For instance, in a contract that contains the following norms “All companies
must pay the Y tax” and “The company X must pay the Y tax”, it is not possible
the first norm is satisfied while the second norm is not satisfied. In the
case of company X not paying the tax Y, automatically violates both norms due
to the conditions of compliance. Considering that both norms are logically
linked and are in the same context, we have an entailment relation between
them. By contrast, conflicts in a contract may emerge through problems related
to a logical contradiction between norm clauses. Taking the example above, we
have a contradiction relation if we change the second norm to “The company X
must not pay the tax Y” due to their contradictory compliance condition.
Analyzing these conflicts in a contract demands a careful analysis from both
related parties. An automated way to detect a conflict between contract
clauses addresses these reviews of contract clauses, which is a long and
complex issue even for human experts.

Classifying the logical relation between norms is analogous to Natural
Language Inference (NLI), which is the task of determining whether a natural
language hypothesis h can be inferred from a natural language premise p (?).
In an entailment relation, if p is true then h cannot be false, otherwise
there is a contradiction. NLI is a broader task than conflict identification,
and thus, good models to classify logical relations will naturally be
applicable to detect contract conflicts. Importantly, since NLI has seen a
surge in research, including new machine learning models and dataset curation
(?; ?), it offers substantial labelled training data in much larger quantities
than contract conflict datasets (?).

We develop an automated approach to detecting logical relations between norms
in a contract as a natural language inference problem. First, we develop a
neural network that addresses the Natural Language Inference task to classify
the logical relation between two sentences written in natural language.
Second, we apply the trained neural network on conflicting norm pairs
reporting the logical relation that our model predicts concerning the conflict
types. The resulting model can help identify potential inconsistencies between
contract clauses by detecting logical relationships between natural language
sentences.

## Natural Language Inference

Automated reasoning and logical inference studied in natural logic are
important topics of artificial intelligence. Natural language inference (NLI)
is a widely-studied natural language processing task that is concerned with
determining the inferential relation between a premise p and a hypothesis h
(?). In NLI, the entailment relation inferred is formulated based on the
following representations: two-way classification and three-way classification
(?).

Two-way classification is the simplest representation of NLI, which describes
the task as a binary decision. The objective of this NLI task is to classify
whether the hypothesis follows the premise (entailment) or does not (non-
entailment). Alternatively, in three-way classification form, the relations
are divided into three categories: entailment, contradiction and neutral.
Given a pair of premise-hypothesis p and h, the entailment relation occurs
when h can be inferred from p (?). When h entails the negation of p, the pair
results in a contradiction. Otherwise, if none of these relations can be
inferred, the relation of p and h is neutral.

In NLI, both p and h are sentences written in natural language. The challenge
of this task differs of formal deduction from logic due to its focus in
informal reasoning (?). The emphasis of the NLI is on aspects of natural
language such as lexical semantic knowledge and the deal with the variability
of linguistic expression. Consider the following premise p and hypothesis h as
an instance of an NLI scenario (?):

  * •

p: Several airlines polled saw costs grow more than expected, even after
adjusting for inflation.

  * •

h: Some of the companies in the poll reported cost increases.

This example is considered a valid entailment inference in the NLI context
because any person that interprets p would likely accept that h implies the
information of p. Although this is a valid NLI classification, h is not a
strict logical consequence of p due to the fact that p informs that airline
companies saw the growth of the cost, not necessarily reporting the growth of
the cost. This example reflects the informal reasoning of the task definition
due to deal with ambiguity of natural language (?).

## Natural Language Inference Classifier

In this section, we explain our NLI model to predict over normative sentences.
First, we explain the Transformer neural network architecture that we use to
deal with NLI task. Second, we detail the attention mechanism we use to help
our classifier focus on key parts of normative sentences. Third, we describe
the feed-forward neural network that refines the internal representation of
words in sentences. Finally, we show how the model predicts a class given its
output representation.

### Transformer

Transformer is a type of neural network architecture that processes sequences
based solely on attention mechanisms instead of using recurrent connections in
the network. Vaswani et al. (?) developed this architecture to deal with the
machine translation task, achieving the state-of-the-art performance. This
architecture uses an Encoder-Decoder approach based on other machine
translation neural networks such as Sequence to Sequence learning (?).
Approaches that use Transformer variations recently achieve state-of-the-art
results on other natural language processing tasks, such as Question Answering
(?) and Sentiment Classification (?).

Instead of using the entire Transformer architecture from machine translation
tasks, we use only the Decoder part based on the work by Radford et al. (?)
that deals with the NLI classification task. The Transformer Decoder contains
blocks that consist of a layer with a Self Attention Mechanism and a feed-
forward neural network module. In the decoder block, we add a residual
connection applying a sum over the input of each layer with its output
followed by layer normalization. Figure 1 illustrates this overall
architecture with details of Decoder Transformer block of our model and how we
adapt to a classification task.

Figure 1: Diagram of a Transformer Decoder block that may contain more than
one stacked block in a single model.

#### Self-Attention

Self-attention is a neural network mechanism that relates different positions
of a single sequence to create a representation of the sequence. In natural
language processing models, the objective of self-attention mechanism is to
focus on relevant words rather than all words of the input sentence, giving an
attention score for each word relation of the sequence. For example, the word
”not” is more relevant to check whether exists some contradiction between two
sentences. The self-attention mechanism in a Transformer calculates the
attention of each word using a query that maps an output given a set of key-
value pairs (?). Initially, our model uses a fully connected layer on the
input representation and then split its output into three matrices, which
represent the initial query, key and value matrices of our attention model.

Our neural network uses the Scale Dot-Product attention model, which applies a
matrix multiplication between the query and key matrices followed by a softmax
function with the value matrix. To obtain more stable gradients, the attention
model scales the matrix multiplication dividing it by $\sqrt{d_{k}}$, where
$d_{k}$ represents the dimension of key matrix. Equation 1 computes the Scale
Dot-Product attention model given a query, key, and value matrices represented
by Q, K and V respectively.

$softmax(mask(\frac{Q.K}{\sqrt{d_{K}}})).V$ (1)

To prevent the self-attention mechanism from computing the attention score to
subsequent positions of a single word, we apply a mask that removes the score
of subsequent words. Given a certain word, we put a $-\infty$ value in its
subsequent words in input of softmax generating extremely low attention scores
in these illegal positions (?). This masking process ensures that a word can
depend only on already seen ones (?) ignoring its subsequent words. Table 1
describes an example of attention masking while processing the word “being” in
sentence pair.

Table 1: Example of sentence pair masked that contains very low values ($-\infty$) on subsequent positions. A | Car | is | being | driven | . | A | Car | is | stuck | [EOS]
---|---|---|---|---|---|---|---|---|---|---
0.5 | 0.78 | 0.1 | current | $-\infty$ | $-\infty$ | $-\infty$ | $-\infty$ | $-\infty$ | $-\infty$ | $-\infty$

To learn diverse representations of attention, we use the Multi-Head attention
approach (?) producing different attention scores for each word position. We
initialize each head randomly to represent different attention projections
with its respective $Q$, $K$ and $V$ matrices. Consequently, after the
training process, the Multi-Head attention approach produces a different
representation subspace for each head projecting distinct attention score of
each word. To propagate its results for the subsequent layers, we concatenate
all attention head results and apply a fully connected layer to reshape the
output to the original size. Figure 2 illustrates how the Multi-Head approach
computes each head in parallel receiving a word representation $x$ resulting
in a hidden representation $h$.

Figure 2: Diagram of attention model using Multi-Head approach using $n$ heads
given an word input representation $x$ and output representation $h$.

#### Position-Wise Feed Forward Network

After computing the self-attention layer, our decoder block uses a feed-
forward neural network to process each word of the sentence separately and
identically. Instead of using a ReLU activation function as the original
transformer work (?), we follow ? ? in using the gaussian error linear unit
(?) as the activation function computed in Equation 2. Equation 3 illustrates
feed-forward network (FFN) operations over each word $x$, which $W_{i}$ and
$b_{i}$ represent the weight and bias of the $i$-th layer respectively.

$GELU(x)=0.5x(1+tanh(\sqrt{2/\pi}(x+0.044715.x^{3})$ (2)
$FFN(x)=GELU(x.W_{1}+b_{1}).W_{2}+b_{2}$ (3)

#### Output Representation

The output of a Transformer decoder is a sequence of learned embeddings of all
tokens contained in the input sentence pair (i.e. the premise and hypothesis
pair). However, we need to convert the decoder output into a single
representation in order to predict the class of the whole sentence pair. The
Transformer decoder computes the embedding of this special token considering
all previous words due to the masking process that we defined in attention
model, which ignores the subsequent positions. We include a special token at
the end of the input pair to represent the whole sentence since the self-
attention mechanism computes its embedding considering all previous words.
With the special token embedding, we apply a fully connected layer to
represent the predicted class of sentence pair. Our model uses a softmax
activation function to generate a vector of probabilities containing a single
value for each class.

## Implementation Details

In this section, we describe how we develop our NLI classifier. First, we
detail the NLI corpus that we use to train our neural network. Second, we
describe how we represent a premise-hypothesis pair to feed our model. Third,
we detail how we implemented our model describing the sizes of each layer and
hyperparameters used. Finally, we report how we optimize our model and measure
its error reporting training results on NLI dataset.

### Stanford Natural Language Inference Corpus

The Stanford Natural language Inference (SNLI) corpus111The corpus is free
available on https://nlp.stanford.edu/projects/snli/ is a dataset that
contains 570 thousand sentence pairs that were written in English and manually
labeled by humans. The pair consists of a premise and a hypothesis sentence
that could follow the premise or not. Each sentence pair contains a label that
follows the NLI representation of three-way classification, which is
categorized as entailment, contradiction or neutral.

Bowman et al. (?) used Amazon Mechanical Turk for data collection by asking
each worker to supply a hypothesis text based on a scene description from a
pre-existing corpus (?). After this collection step, each worker received a
sentence pair and was asked to choose a single label (E for entailment, C for
contradiction or N for neutral) for each pair. This dataset contains five
judgments of different workers and a consensus judgment, which is described in
Table 2. Since the SNLI dataset contains descriptive phrases, all sentences
are in the present tense, which is a substantial limitation for the contract
conflict task, which contains multiple modal verbs.

Table 2: Examples of sentence pair and judgement resulted from the data collection process (?). Premise Text | Hypothesis text | Judgements
---|---|---
| A soccer game
---
with multiple males
playing a sport.
| Some men are
---
playing a sport.
| E E E E E
---
Entailment
| A black race car
---
starts up in front
of a crowd of
people.
| A man is driving
---
down a lonely
road.
| C C C N C
---
Contradiction

### Word Representation

Given a word vocabulary, our neural network receives as input the word indexes
followed by its relative position in the sentence. Since the Transformer
architecture does not have any recurrent network layer to obtain word order,
we use the information of relative position to provide order meaning for our
model. We use a numeric value that represents the word position to retrieve
positional embeddings as well as word embedding lookup process.

A positional embedding is a vector that contains the meaning of the position
of each item in a sequence. Instead of using fixed embeddings for each
position, our model learns positional embeddings via backpropagation. Finally,
we sum both embeddings resulting in a word representation that contains both
semantic and order information feeding the following layers.

Due to our model input being composed of two sentences (premise and
hypothesis), we concatenate the premise with its hypothesis followed by a
special token. This special token serves to retrieve the last hidden state,
which we use as a representation of the entire pair. Table 3 shows an example
of a premise-hypothesis contradiction pair followed by an end-of-sentence
token (EOS), which the first row is the word indexes, second is the position
indexes, third is the word embeddings, fourth is the position embeddings and
finally the last row is the sum of embeddings.

Table 3: Example of input representation of our model composed by the meaning of word itself and its position in sentence. Sentence pair: | A | Car | is | being | driven | . | A | Car | is | stuck | [EOS]
---|---|---|---|---|---|---|---|---|---|---|---
Word indexes: | 4 | 5 | 1 | 7 | 10 | 11 | 4 | 5 | 1 | 9 | 20
Positional indexes: | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11
Word embedding: | [1.3, …] | [2.5, …] | … | … | … | … | … | … | … | … | [6.7, …]
Positional embedding: | [1.5, …] | [3.7, …] | … | … | … | … | … | … | … | … | [0.2, …]
Input representation of pair: | [2.8, …] | [6.2, …] | … | … | … | … | … | … | … | … | [6.9, …]

### Neural Network Architecture Details

We implemented our neural network model using 12 stacked blocks of Transformer
decoders and included 12 heads in the attention layer of each decoder block
following Radford et al (?). Considering that the SNLI dataset contains a
vocabulary that holds 56220 distinct tokens, we include 56580 vectors with 240
dimensions in our embedding layer representing word embeddings including 360
vectors for position embeddings. Although our model supports sentences with
variable-length, we require to define a maximum length supported due to the
embedding layer has a finite size. We defined as maximum length supported 360
considering that the number of words in a contract clause is greater than SNLI
sentences.

### Training Details

We train our neural network with the Adam algorithm (?) using an initial
learning rate of 6.25e-5, which decays linearly with a scheduled warmup over
0.2% of training (?). We then apply gradient clipping during optimization to
avoid exploding gradients restricting the parameter’s values between -1 and 1
(?).

To measure network error, we apply the negative log-likelihood (NLL) loss
function in output probabilities computed by the output layer (?). Since we
deal with a multi-class classification task, our loss function accumulates the
log loss values of each class prediction. Given an output label $y_{c}$ for
class $c$ and a premise-hypothesis pair $(p,h)$, the goal of our model is to
minimize the function shown in Equation 4.

$NLL=-\sum_{c}y_{c}.\log P(c\mid p,h)$ (4)

During training execution, we define batches containing 16 randomly sampled
instances of SNLI dataset. We created batches with similar sizes to prevent an
excess of padding through ordering samples by the sum of premise and
hypothesis sentence lengths and apply early stopping to avoid overtraining our
neural network using the validation set as reference. In our training
procedure, we selected as metric the accuracy obtained by our model in the
validation set and 10 epochs of waiting. Our training stopped after 22 epochs
and chose the parameters with the highest accuracy obtained in epoch 12. We
validate our training results using accuracy and loss obtained in training and
validation sets throughout epochs. Figure 3 shows the value of the training
metrics obtained by our model up to epoch 12.

Figure 3: Accuracy and loss obtained on training and validation set throughout
the epochs.

Concerning the test set, our trained neural network achieved an accuracy of
82.1%. Although our work is similar to Radford et al(?), their work uses a
pre-trained model with unsupervised learning to improve language
understanding. Table 4 shows details about the state-of-the-art model and our
neural network concerning accuracy and number of parameters.

Table 4: Results comparison of our model with Radford et al model improved with pre-training. | Radford et al | Our work
---|---|---
Training Accuracy (%) | 96.6 | 90.5
Test Accuracy (%) | 89.9 | 82.1
Number of parameters | 80m | 20m

## Model Application on Conflicting Norms

In this section, we describe the application of our trained model in
conflicting norms stratified by conflict type. First, we introduce the norm
conflict dataset and explain the conflict types. Second, we report on the
results of norm pairs that contain conflicts based on conflicting modality.
Third, we report on the results of norm pairs that contain conflicts based on
norm structure. Finally, we discuss the limitations and issues of our model
that we found which concern in normative actions in conflicting norm pairs.

### Norm Conflict Classification Dataset

The Norm Conflict Dataset (?) consists of a corpus that contains clauses from
existing contracts labeled with different conflict types. The source of these
contract clauses is the Onecle222https://www.onecle.com/ site, which is a
repository of business contracts. Aires et al (?) labeled the normative
sentence pairs manually using a web-based tool that selects randomly a
contract clause requesting a human to create a second norm in such a way as to
create conflict with the selected norm. This dataset contains the following
conflict types: deontic-modality, deontic-structure, deontic-object and
object-conditional.

The deontic-modality conflict type indicates conflicts originated by the
deontic statement of each clause, i.e., prohibition $\times$ obligation,
obligation $\times$ permission, and permission $\times$ prohibition. Deontic-
structure conflict types involves different deontic meaning but with different
sentence structure. Deontic-object conflict occurs when norm actions of the
pair are conflicting, which represents the object of normative sentences. The
object-conditional conflict occurs when condition of norm actions are
conflicting. Table 5 shows examples of norm pairs contained in norm conflict
dataset with their respective conflict types.

Table 5: Examples of norm pairs with the respective conflict type. Norm Pair | Conflict Type
---|---
| \- The Specifications may be amended by the NCR design release process.
---
\- The Specifications shall not be amended by the NCR design release process.
deontic modality
| \- All inquiries that Seller receives on a worldwide basis relative to
Buyer’s air chamber ”Products” as
---
specified in Exhibit III, shall be directed to Buyer.
\- Seller may not redirect inquiries concerning Buyer’s air chamber
”Products”.
deontic structure
| \- Autotote shall make available to Sisal one (1) working prototype of the
Terminal by May 1, 1998.
---
\- Autotote shall make available to Sisal one (1) working prototype of the
Terminal by June 12, 1998.
deontic object
| \- The Facility shall meet all legal and administrative code standards
applicable to the conduct of the
---
Principal Activity thereat.
\- Only if previously agreed, the Facility ought to follow legal and
administrative code standards.
object conditional

In this work, we apply our trained model on conflicting pairs of this dataset
to find logical relations between contract clauses different conflict types.
We execute two times the model over pairs interleaving the roles of premise
and hypothesis.

### Deontic Modal Conflicts

In executions on deontic-modal conflict pairs, we note that our model could
detect the intensity of modal verbs. Given both sentences with similar
intensity, our model results in high scores of entailment and neutrality
considering the inference direction. For example, our model outputs the
entailment relation between a considerable number of pairs that have the modal
verbs shall and may in both directions. However, when the premise contains the
shall verb, the entailment score is greater than premises with may.

These scores show that our model increases entailment probability when the
premise contains an obligation and the hypothesis norm contains a permissible
action. On the other hand, our model shows that the opposite is not true,
which increases neutrality score when obligation comes from hypothesis norm.
However, our model fails to infer these relations when the hypothesis or
premise has the modal verb will. Since our training dataset contains only
sentences in the present tense, we suspect that our model does not recognize
the word will accurately. Table 6 shows examples of pairs with these modal
verbs with softmax score of our neural network output layer for neutral and
entailment classes.

Table 6: Examples of norm pairs with deontic-modal conflicts describing the model results given two norms with different modal verbs and deontic meaning. Norm Pair |  | E
---
(a,b)
| N
---
(a,b)
| E
---
(b,a)
| N
---
(b,a)
| (a) Purchaser shall also be responsible for all property taxes on the
equipment.
---
(b) Purchaser may also be responsible for all property taxes on the equipment.
0.96 | 0.03 | 0.88 | 0.08
| (a) CoPacker shall deliver all the Products that WWI purchases under this
Agreement to
---
WWI F.O.B.
(b) CoPacker may deliver all the Products that WWI purchases under this
Agreement to
WWI F.O.B.
0.71 | 0.20 | 0.30 | 0.53
| (a) CBSI will retain the originals in its archives.
---
(b) CBSI may retain the originals in its archives.
0.59 | 0.35 | 0.91 | 0.07

In norms that contains negation between deontic meanings, our model had
problems on bidirectional executions. The trained neural network does not
classify contradictions with reasonable accuracy when negations come from the
premise norm. We consider that this issue is related to our training dataset
(SNLI), which may be unbalanced concerning the negation side.

### Deontic Structure

In the deontic structure conflict type, our model has the same problems as in
deontic-modality conflicts. However, in this conflict case, we note that our
neural network could generalize contradictions and entailment regarding modal
verb in different sentence structures. This shows that our model can infer a
logical relation where norm pairs contain different words with similar
meanings. Table 7 shows unidirectional results of our model where the sentence
(a) is the premise and sentence (b) is the hypothesis.

Table 7: Norm pairs with distinct sentence structures and their softmax scores for entailment (E), contradiction (C) and neutral (N) classes generated by our model. Norm Pair |  | E
---
(a,b)
| C
---
(a,b)
| N
---
(a,b)
| (a) Autotote will own the Intellectual Property Rights to all said
prototypes.
---
(b) Autotote shall not own the Intellectual Property Rights to prototypes.
0.04 | 0.94 | 0.02
| (a) Medica shall also maintain records with respect to its costs,
obligations, and performance under
---
this agreement.
(b) With respect to its costs, obligations, and performance under this
agreement, Medica is not
obliged to maintain records.
0.11 | 0.81 | 0.08
| (a) Teknika will notify LSI that it considers that a Triggering Event has
occurred.
---
(b) Teknika shall not notify LSI of any regular event.
0.01 | 0.98 | 0.01
| (a) Customer will notify USF at least [***] days in advance of special
promotions that may cause
---
unusual or excessive demand on inventory.
(b) Customer should notify USF of special promotions that may cause unusual
and excessive
demand on inventory.
0.91 | 0.00 | 0.09

### Deontic-Object and Object-Conditional

Given conflicts that involve a difference between norm’s object, our model
fails to generate reasonable classifications. These results indicate that our
neural network does not detect object context in a normative sentence on
conflict context. Based on results in both conflicts type that concern norm
actions, our neural network output its prediction based on keywords such as
modal verb and negation words (not). Therefore, our model tends to disregard
norm actions when words of both norms are different or norm action structure
is composed of different words.

Table 8 shows examples of norm pairs that contain conflicts related to the
action of clauses. The first example shows that our model does not recognize
accurately the sentence action ignoring the associated objects. The second
example illustrates a contradiction instance that our model could capture due
to the similar structure between words of pair. Finally, the third example
describes an instance that involves conflicts that concerns the conditional
definition of norms.

Table 8: Example of norm pairs with conflicts that concerns in norm action with respective entailment (E), contradiction (C) and neutral (N) softmax score. Norm Pair |  | E
---
(a,b)
| C
---
(a,b)
| N
---
(a,b)
Type
| (a) The arbitration shall be conducted in Tampa, Florida.
---
(b) The arbitration shall be conducted in St. Petersburg, Florida.
0.92 | 0.02 | 0.06 | deontic-object
| (a) Hershey will cooperate in no shipping procedures.
---
(b) Hershey will cooperate in all shipping procedures.
0.05 | 0.88 | 0.07 | deontic-object
| (a) Where applicable, Taxes shall appear as separate items on Adaptec’s
---
invoice.
(b) If shipping products, the Taxes shall appear along the other items on
Adaptec’s invoice.
0.91 | 0.04 | 0.05 | object-conditional

## Related Work

In this section, we present related work that analyze normative sentences and
contract clauses. We describe the related works explaining the problem dealt,
their objectives and how they represent a normative sentence. We compare the
objective of this work with the related work and discuss the differences.

Aires et al (?) develop an approach that identifies potential conflicts
between norms in contracts. First, they focus on norm identification, which
results in a formal representation of a norm. Second, they use the formal
representation to detect and classify potential conflicts between norms using
techniques of the formal logic. This approach assumes that a norm follows a
well-defined 4-component structure: an indexing number or letter, one or more
named parties, a modal verb, and a behavior description. Given this structure,
they apply a regular expression to decide whether a sentence is a norm
sentence or not. After identifying norms, they create a formal representation
of the norm sentence extracting three components: party name, deontic meaning,
and the norm action. With the formal representation, they detect potential
conflicts following three relations between deontic meaning in norm pairs (?):

  * •

Permission and Prohibition

  * •

Permission and Obligation

  * •

Obligation and Prohibition

Instead of using a formal representation to use a strict logic approach, we
explored the use of techniques that deal with the informal reasoning of
natural language through neural network application. We use a neural network
considering SNLI dataset to deal with the challenges of NLI such as lexical
semantic knowledge and the variability of linguistic expression (?).

Aires et al (?) introduce a typology of conflicts in normative sentences and
present machine learning methods that classify these conflict types. These
learning methods rely on the semantic representation of norms using Sent2Vec
(?) to create embedding vectors to represent the norms. First, they describe
an extension of Aires Norm Dataset to include the conflict typology, which
introduces 228 new conflicting norms including the existing 111 from the
previous dataset. Second, they present an unsupervised learning method to
detect the presence or absence of norm conflicts. Finally, they present a
supervised learning method that deals with binary (i.e., conflicts and non-
conflict) and multi-class classification method to classify the conflict types
created. In this work, we use the dataset made by the authors to validate
logical relations. Instead of classifying conflicts, we use the conflict type
to illustrating how our model can help in contract analysis showing potential
conflicts that concern logical problems. These conflicts help us to detect
points that our model can improve regarding logical inference.

## Conclusion

In this work, we present an approach to identify a logical relation between
contract clauses. At this point, we have trained a neural network with SNLI
dataset to classify inference between a premise and a hypothesis. We use this
trained neural network on a corpus that contains a conflicting set of norms to
validate whether our model can help in contract analysis. The application of
our neural network on conflicting norms could help us to identify some issues
of our approach. Although our model has issues that involve the direction of
NLI inference, we show that our model can detect potential contradictions in
contract clauses regardless of their structures. Furthermore, we reported that
our model can identify deontic meaning between norms assigning an entailment
score based on modal verb intensity.

As future work, we intend to improve our neural network to detect the gaps
identified during contract clauses analysis. First, we intend to explore
others training datasets such as the Multi-Genre NLI Corpus (MNLI) (?), which
were modeled based no SNLI but differs in that covers a range of genres of
spoken and written text. Second, we aim to use pre-trained models such as BERT
(?), which are state-of-the-art in a wide variety of Natural Language Process
tasks such as Natural Language Inference.

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