aainstitutetext: Physics Department and Biruni Observatory, College of
Sciences, Shiraz University, Shiraz 71454, Iranbbinstitutetext: Research
Institute for Astrophysics and Astronomy of Maragha (RIAAM), P.O. Box
55134-441, Maragha, Iran

# Critical behaviour of Lifshitz dilaton black holes

Zeinab Dayyani a,b Ahmad Sheykhi<EMAIL_ADDRESS>
###### Abstract

Till now, critical behaviour of Lifshitz black holes, in an extended $P-v$
space, has not been studied, because it is impossible to find an analytical
equation of state, $P=P(v,T)$, for an arbitrary Lifshitz exponent $z$. In this
paper, we adopt a new approach toward thermodynamic phase space and
successfully explore the critical behaviour of $(n+1)$-dimensional Lifshitz
dilaton black holes. For this purpose, we write down the equation of state as
$Q^{s}=Q^{s}(T,\Psi)$ with $\Psi=\left({\partial M}/{\partial
Q^{s}}\right)_{S,P}$ is the conjugate of $Q^{s}$ and construct Smarr relation
based on this new phase space as $M=M(S,Q^{s},P)$, where $s=2p/(2p-1)$ with
$p$ is the power of the power-law Maxwell Lagrangian. We justify such a choice
mathematically and show that with this new phase space, the system admits the
critical behaviour and resembles the Van der Waals fluid system when the
cosmological constant (pressure) is treated as a fixed parameter, while the
charge of the system varies. We obtain Gibbs free energy of the system and
find swallow tail shape in Gibbs diagrams which represents the first order
phase transition. Finally, we calculate the critical exponents and show that
although thermodynamic quantities depend on the metric parameters such as $z$
, $p$ and $n$, the critical exponents are the same as Van der Walls fluid-gas
system. This alternative viewpoint toward phase space of lifshitz dilaton
black hole can be understood easily since one can imagine such a change for a
given single black hole i.e., acquiring charge which induces the phase
transition. Our results further support the viewpoint suggested in Dehy .

##  1 Introduction

Historically, Maldacena was the first who suggested, two decades ago, the
correspondence between gravity in an Anti-de Sitter (AdS) spacetime and the
Conformal Field Theory (CFT) living on the boundary of spacetime known as
AdS/CFT correspondence Mald . According to Maldacena’s conjecture the effects
of the string theory in a $d$-dimensional $AdS_{n+1}\times S^{d-n-1}$
spacetime can be appeared in the form of a field theory on an $n$-dimensional
$r$-constant brane which is the boundary of $AdS_{n+1}$ spacetime. This idea
has attracted a lot of enthusiasm and has been investigated from various point
of view AdS . The metric of the AdS spacetime is given by

$ds^{2}=-\frac{r^{2}}{l^{2}}dt^{2}+\frac{l^{2}}{r^{2}}dr^{2}+r^{2}\sum\limits_{i=1}^{n-1}dx_{i}^{2},$
(1)

which is invariant under an isotropic conformal transformation as follows:

$t\rightarrow\lambda t,\text{ \ \ \ \ }x_{i}\rightarrow\lambda x_{i},\text{ \
\ \ \ }r\rightarrow\lambda^{-1}r.$ (2)

On the other hand, the application of AdS/CFT is restricted to systems
respected isotropic scale invariance and quantum critical systems show scaling
symmetry as

$t\rightarrow\lambda^{z}t,\text{ \ \ \ \ }x_{i}\rightarrow\lambda x_{i},\text{
\ \ \ \ }r\rightarrow\lambda^{-1}r,$ (3)

where $z$ is a dynamical critical exponent and is restricted as $z>1$. This
parameter shows the degrees of anisotropy between space and time. The Lifshitz
spacetime was first introduced in Lif ; Mann  as

$ds^{2}=-\frac{r^{2z}}{l^{2z}}dt^{2}+{\frac{l^{2}dr^{2}}{r^{2}}}+r^{2}\sum\limits_{i=1}^{n-1}dx_{i}^{2}.$
(4)

The Lifshitz spacetime is not a vacuum solution of Einstein gravity and so
needs matter source. Usually a massive gauge field plays the role of this
matter source but it is nearly impossible to obtain an analytic solution for
arbitrary $z$ in such models. As shown in Ref. tarrio  considering a dilaton
field, instead of a massive gauge field, can lead to exact analytical
solutions in Lifshitz spacetime (see also Kord ). Another motivation is that
string theory in its low energy limit reduces to Einstein gravity with a
scalar dilaton field coupled to gravity and other fields dilaton .

On the other hand, the studies on the critical behavior of black holes have
got a lot of attentions in a wide range of gravity theories. For example,
critical behavior of charged AdS black holes has been studied in MannRN  and
the author completed the analogy between Reissner-Nordstrom-AdS black holes
with the Van der Walls liquid-gas system, with the same critical exponents.
The key assumption is to enlarge the thermodynamic phase space to include the
cosmological constant as a thermodynamic pressure and its conjugate quantity
as a thermodynamic volume Do1 ; Ka ; Do2 ; Do3 ; Ce1 ; Ur . When the gauge
field is the Born-Infeld nonlinear electrodynamics, one needs more extended
phase space to introduce a new thermodynamic quantity conjugate to the Born-
Infeld parameter which is necessary for consistency of both the first law of
thermodynamics and the corresponding Smarr relation MannBI . Treating the
cosmological constant as a thermodynamic pressure, thermodynamics and $P-v$
criticality of black holes in an extended phase space in the presence of
power-Maxwell HV  and exponential nonlinear electrodynamics Hendi1  have been
explored. The studies were also generalized to other gravity theories. In this
regards, the phase structure of asymptotically AdS black holes with higher
curvature corrections such as Gauss-Bonnet GB1 ; GB2  and Lovelock gravity
Lovelock  have also been investigated. The studies were also extended to the
rotating black holes, where phase transition and critical behavior of Myers-
Perry black holes have been investigated Sherkat . Other studies on the
critical behavior of black hole spacetimes in an extended phase space have
been carried out in Sherkat1 ; Rabin ; Zou ; John ; Amin2 .

Critical behavior of the Einstein-Maxwell-dilaton black holes has been studied
in Kamrani . When the gauge field is in the form of Born-Infeld Dayyani1  and
power-Maxwell Dayyani2  field, critical behavior of $(n+1)$-dimensional
dilaton black holes in an extended phase space have been investigated. Taking
into account the dilaton field in the presence of logarithmic and exponential
forms of nonlinear electrodynamics, and considering the cosmological constant
and nonlinear parameter as thermodynamic quantities which can vary, it was
shown that indeed there is a complete analogy between the nonlinear dilaton
black holes with Van der Waals liquid-gas system Dayyani3 . In all mentioned
above, one assumes the charge of the black hole as an external fixed parameter
and treats the cosmological constant as the pressure of the system which can
vary.

In the present work, we would like to investigate the critical behaviour of
Lifshitz black holes in Einstein-dilaton gravity in the presence of a power-
law Maxwell field. It is worthwhile to mention that the $(n+1)$-dimensional
Lagrangian in power Maxwell theory is conformally invariant provided
$p=(n+1)/4$ where $p$ the power of the Lagrangian. Let us first plot a
$3$-dimensional diagrams for equation of state of Lifshitz black hole to
understand the phase behaviour of this system and show its analogy with Van
der Walls liquid-gas system (see Fig. 1).

(a) Equation of state for Van der Walls liquid-gas system with $a=1$ and
$b=1$.

(b) Equation of state of dilaton lifshitz black hole with $Q=k=b=p=1$, $n=3$
and $z=1.1$.

Figure 1: The 3-d diagrams of equations of state for Van der Walls fluid
system and Lifshitz black holes. Comparing two diagrams indicate that these
systems have similar phase transition.

A close look at the temperature expression of Lifshitz-dilaton black holes
(see Kord5  and Eq. (LABEL:Temp) of the present paper), shows that it is
nearly impossible to solve this equation for $P$ (or more precisely for $l$).
Therefore, we cannot have an analytical equation of state, $P=P(v,T)$, to
investigate the critical behavior or calculate critical quantities of Lifshitz
black holes. Another way to investigate critical behavior of the black holes
is to use the method of Refs. Cha1 ; Cha2 , but as shown in Dehy , such a view
of thermodynamic conjugate variables ($Q$ and $\Phi=Q/r_{+}$) which are not
mathematically independent can lead to physically irrelevant quantities such
as $(\partial Q/\partial\Phi)_{T}$ which is supposed to be a thermodynamic
response function, but mathematically ill-defined.

To address this problem, an alternative viewpoint toward thermodynamic phase
space of black holes was developed in Dehy  by treating the cosmological
constant as a fixed parameter and considering the charge of the black hole as
a thermodynamic variable Amin1 ; Amin3 . It was argued that, with fixed
cosmological constant, the critical behavior indeed occurs in $Q^{2}$-$\Psi$
plane, where $\Psi=1/2r_{+}$ is conjugate of $Q^{2}$, and thus the equation of
state is written as $Q^{2}=Q^{2}(T,\Psi)$. We find out that in case of
Lifshitz dilaton black holes, the system admits a critical behaviour provided
we take the electrodynamics in the form of power-Maxwell field and considering
$Q^{s}$ as a thermodynamic variable with $\Psi=\left({\partial M}/{\partial
Q^{s}}\right)_{S,P}$ as its conjugate, where $s=2p/(2p-1)$. In this case we
can define a new response function which naturally leads to physically
relevant quantity. Thus, the equation of state is written in the form of
$Q^{s}=Q^{s}(T,\Psi)$ and Smarr relation based on this new phase space as
$M=M(S,Q^{s},P)$. Clearly, for $p=1$, the power Maxwell field reduces to
standard Maxwell field and $Q^{s}\rightarrow Q^{2}$. Following Dehy , in this
approach we keep the cosmological constant (pressure) as a fixed quantity,
while the charge of the system can vary.

This paper is outlined as follows. In the next section, we present the action,
basic field equations of of Lifshitz dilaton black holes and review
thermodynamic properties of this system. In section 3.1, we study the phase
structure of the solution and present the modified Smarr relation. In section
3.2, we obtain the equation of state and study the critical behavior of the
solutions and compare them with Van der Waals fluid system. We investigate the
Gibbs free energy and the critical exponents of the system in sections 3.3 and
3.4, respectively. The last section is devoted to summery and conclusion.

##  2 Thermodynamics of Lifshitz dilaton black holes

In this section we are going to review the solutions of charged Lifshitz black
holes with power Maxwell field Kord5 , with emphasizing on their thermodynamic
properties. The $(n+1)$-dimensional action of Einstein-dilaton gravity in the
presence of a power Maxwell electromagnetic and two linear Maxwell fields can
be written as

$\displaystyle S$ $\displaystyle=$
$\displaystyle-\frac{1}{16\pi}\int_{\mathcal{M}}d^{n+1}x\sqrt{-g}\left\\{\mathcal{R}-\frac{4}{n-1}(\nabla\Phi)^{2}\right.$
(5)
$\displaystyle\left.-2\Lambda+\left(-e^{-4/(n-1)\lambda_{1}\Phi}F\right)^{p}-\sum\limits_{i=2}^{3}e^{-4/(n-1)\lambda_{i}\Phi}H_{i}\right\\},$

where $\mathcal{R}$ is the Ricci scalar on manifold $\mathcal{M}$, $\Phi$ is
the dilaton field, $\lambda_{1}$ and $\lambda_{i}$ are constants. In Eq. (5)
$F=F_{\mu\nu}F^{\mu\nu}$ and $H_{i}$ are the Maxwell invariants of
electromagnetic fields, where $F_{\mu\nu}=\partial_{[\mu}A_{\nu]}$ and
$\left(H_{i}\right)_{\mu\nu}=\partial_{[\mu}\left(B_{i}\right)_{\nu]}$, with
$A_{\mu}$ and $\left(B_{i}\right)_{\mu}$ are the electromagnetic potentials.
Varying the action (5) with respect to the metric $g_{\mu\nu}$, the dilaton
field $\Phi$, electromagnetic potentials $A_{\mu}$ and
$\left(B_{i}\right)_{\mu}$, lead to the following field equations Kord5

$\displaystyle\mathcal{R}_{\mu\nu}=\frac{g_{\mu\nu}}{n-1}\left\\{2\Lambda+(2p-1)\left(-Fe^{-4\lambda_{1}\Phi/(n-1)}\right)^{p}\right.$
$\displaystyle\left.-\sum\limits_{i=2}^{3}H_{i}e^{-4\lambda_{i}\Phi/(n-1)}\right\\}+\frac{4}{n-1}\partial_{\mu}\Phi\partial_{\nu}\Phi$
$\displaystyle+2pe^{-4\lambda_{1}p\Phi/(n-1)}(-F)^{p-1}F_{\mu\lambda}F_{\nu}^{\text{
\ }\lambda}$
$\displaystyle+2\sum\limits_{i=2}^{3}e^{-4\lambda_{i}\Phi/(n-1)}\left(H_{i}\right)_{\mu\lambda}\left(H_{i}\right)_{\nu}^{\text{
\ }\lambda},$ (6)
$\displaystyle\nabla^{2}\Phi-\frac{p{\lambda_{1}}}{2}e^{-{4\lambda_{1}p\Phi}/({n-1})}(-F)^{p}$
$\displaystyle+\sum\limits_{i=2}^{3}\frac{{\lambda_{i}}}{2}e^{-{4\lambda_{i}\Phi}/({n-1})}H=0,$
(7)
$\displaystyle\triangledown_{\mu}\left(e^{-{4\lambda_{1}p\Phi}/({n-1})}(-F)^{p-1}F^{\mu\nu}\right)=0,$
(8)
$\displaystyle\triangledown_{\mu}\left(e^{-{4\lambda_{i}\Phi}/({n-1})}\left(H_{i}\right)^{\mu\nu}\right)=0.$
(9)

We assume the line element of the higher-dimensional asymptotic Lifshitz
spacetime has the following form Kord5

$ds^{2}=-\frac{r^{2z}f(r)}{l^{2z}}dt^{2}+{\frac{l^{2}dr^{2}}{r^{2}f(r)}}+r^{2}d\Omega_{n-1}^{2},$
(10)

where $d\Omega_{n-1}^{2}$ is an ($n-1$)-dimensional hypersurface with constant
curvature $(n-1)(n-2)k$ and volume $\omega_{n-1}$. Following the method of
Kord5 , one can find the solutions of the field equations (6)-(9) as

$\displaystyle f(r)$ $\displaystyle=$ $\displaystyle
1-{\frac{m}{{r}^{n-1+z}}}+{\frac{k{l}^{2}\left(n-2\right)^{2}}{\left(z+n-3\right)^{2}{r}^{2}}}+\frac{q^{2p}}{{r}^{\Gamma+z+n-1}},$
$\displaystyle{\Phi(r)}$ $\displaystyle{=}$
$\displaystyle\frac{(n-1)\sqrt{z-1}}{2}{\ln\left(\frac{r}{b}\right),}$ (12)
$\displaystyle\left(A_{1}\right)_{t}$ $\displaystyle=$
$\displaystyle-\frac{q_{1}{b}^{2(z-1)}}{\Gamma{r}^{\Gamma}},$ (13)
$\displaystyle\left(A_{2}\right)_{t}$ $\displaystyle=$
$\displaystyle\sqrt{\frac{z-1}{2(n+z-1)}}{\frac{{r}^{n+z-1}}{{l}^{z}{b}^{n-1}}},$
(14) $\displaystyle\left(A_{3}\right)_{t}$ $\displaystyle=$
$\displaystyle{\frac{\sqrt{{k\left(n-1\right)\left(n-2\right)\left(z-1\right)}}{r}^{z+n-3}\,}{\sqrt{2}(z+n-3)^{3/2}{l}^{z-1}{b}^{n-2}}},$
(15)

where

$\displaystyle\Gamma$ $\displaystyle=$ $\displaystyle z-2+(n-1)/(2p-1),$ (16)
$\displaystyle q^{2p}$ $\displaystyle=$
$\displaystyle\frac{\left(2\,p-1\right){b}^{2(z-1)}}{\left(n-1\right){l}^{-2\,p\left(z-1\right)-2}\Gamma}\left(2q_{1}^{2}\right)^{p},$
(17) $\displaystyle\Lambda$ $\displaystyle=$
$\displaystyle-\frac{(z+n-1)(z+n-2)}{2l^{2}}.$ (18)

It was argued in Kord5  that $p$ and $z$ are restricted as

$\text{for }p<1/2,$ | $z-1>(n-2p)/(1-2p),$
---|---
$\text{for }1/2<p\leq n/2,$ | $\text{all }z(\geq 1)\text{ values are allowed,}$
$\text{for }p>n/2,$ | $z-1>(2p-n)/(2p-1).$
(19)

Using the modified BY formalism BY , one can calculate the mass of the
solution per unit volume ${\omega_{n-1}}$ as Kord5

$M=\frac{(n-1)m}{16\pi l^{z+1}},$ (20)

where the mass parameter $m$ can be written in term of the horizon radius
$r_{+}$ by using the fact that $f(r_{+})=0$. We find

${m(r_{+})}={r}_{+}^{z+n-1}+{\frac{k{l}^{2}\left(n-2\right)^{2}{r}_{+}^{z+n-3}}{\left(z+n-3\right)^{2}}}+\frac{q^{2p}}{{r}_{+}^{\Gamma}}.$
(21)

One can also calculate the charge of the black hole by applying the Gauss law

$Q=\frac{\,{1}}{4\pi}\int
r^{n-1}e^{-{4\lambda_{1}p\Phi/(n-1)}}(-F)^{p-1}F_{\mu\nu}n^{\mu}u^{\nu}d{\Omega},$
(22)

where $n^{\mu}$ and $u^{\nu}$ are the unit spacelike and timelike normals to
the hypersurface of radius $r$ given as

$n^{\mu}=\frac{1}{\sqrt{-g_{tt}}}dt=\frac{l^{z}}{r^{z}\sqrt{f(r)}}dt,\text{ \
\ \ \ }u^{\nu}=\frac{1}{\sqrt{g_{rr}}}dr=\frac{r\sqrt{f(r)}}{l}dr.$

Using (22), we obtain the charge per unit volume $\omega_{n-1}$ as

$Q=\frac{2^{p-1}\left(q_{1}l^{z-1}\right)^{2p-1}}{4\pi}.$ (23)

The electric potential $U$, measured at infinity with respect to horizon is
defined by

$U=A_{\mu}\chi^{\mu}\left|{}_{r\rightarrow\infty}-A_{\mu}\chi^{\mu}\right|_{r=r_{+}},$
(24)

where $\chi=p\partial_{t}$ is the null generator of the horizon. Using (13),
we can obtain electric potential

$U=\frac{pq_{1}b^{2(z-1)}}{\Gamma r_{+}^{\Gamma}}.$ (25)

The entropy of the black holes can be calculated by using the area law of the
entropy which is applied to almost all kinds of black holes in Einstein
gravity including dilaton black holes. Thus, the entropy of our solutions per
unit volume $\omega_{n-1}$ is

$S=\frac{r_{+}^{n-1}}{4}.$ (26)

The Hawking temperature can also be obtained as

$\displaystyle T_{+}$ $\displaystyle=$
$\displaystyle\frac{r_{+}^{z+1}f^{\prime}\left(r_{+}\right)}{4\pi
l^{z+1}}=\frac{1}{4\pi}\left\\{\frac{{(n-1+z){r}_{+}^{z}}}{l^{z+1}}+{\frac{k\left(n-2\right)^{2}{r}_{+}^{z-2}}{l^{z-1}\left(z+n-3\right)}}-\frac{\Gamma
q^{2p}}{l^{z+1}{r}_{+}^{\Gamma+n-1}}\right\\}.$

As one can see from expression (LABEL:Temp) it is nearly impossible to solve
this equation for $P$ (or more precisely for $l$) and write an analytical
equation of state, $P=P(v,T)$ for an arbitrary Lifshitz exponent $z$. This
implies that, for the Lifshitz dialton black holes, one cannot investigate the
critical behavior of the system through an extended $P-v$ phase space by
treating the cosmological constant (pressure) as a thermodynamic variable.
However, as we shall see in the next section, it is quite possible to
investigate the critical behaviour of this system through a new $Q^{s}-\Psi$
phase space and show its similarity with Van der Waals fluid system.

##  3 Critical behaviour of Lifshitz dilaton black holes

###  3.1 Phase structure

It is now generally accepted that charged black holes in AdS spaces allow
critical behavior similar to the Van der Waals fluid system, provided one
treats the cosmological constant as a thermodynamic variable (pressure) in an
extended phase space MannRN . Also it has been shown in Dehy  that there is
deeper connection between charged AdS black holes and Van der Waals fluid
system. Indeed, it was argued that similar behavior can be found without
extending the phase space Dehy  by even keeping the cosmological constant as a
fixed parameter. The key assumption in this picture, is to treat the square of
the charge of black hole, $Q^{2}$, as a thermodynamic variable instead of
charge $Q$ Dehy . Besides, the equation of state has been written as
$Q^{2}=Q^{2}(T,\Psi)$ where $\Psi=1/v$ (conjugate of $Q^{2}$) is the inverse
of the specific volume. With this new picture, the authors completed analogy
of charged AdS black holes with Van der Waals fluid system with exactly the
same critical exponents. In this section, we would like to consider Lifshitz
dilaton black holes with power-law Maxwell field and investigate the critical
behavior as well as analogy with Van der Walls fluid for this system.

The usual first law of thermodynamics in an extended phase space is in the
from of

$dM=TdS+VdP+UdQ.$ (28)

From this point of view the usual Smarr relation which obtained from
thermodynamic variables (25)-(LABEL:Temp) and mass (20) can be written as

$M=\frac{n-1}{z+n-3}TS+\frac{-2}{z+n-3}VP+\frac{2p-1}{2p}\left(1+\frac{\Gamma}{z+n-3}\right)UQ.$
(29)

where

$P=\frac{n\left(n-1\right)r_{+}^{z-1}}{16\pi^{z+1}},\ \ \quad
V=\int{4Sdr_{+}}=\frac{r_{+}^{n}\omega_{n-1}}{n}.$ (30)

It was shown in Dehy  that by replacing term $UdQ$ in the first law with term
$\Psi dQ^{2}$, the system allows critical behavior similar to the Van der
Waals fluid system. First of all, let us review the motivation of this
selection. The well-known thermodynamic quantities of AdS black holes are
given Dehy ; MannRN

$M=\frac{r_{+}}{2}+\frac{Q^{2}}{2r_{+}}+\frac{r_{+}^{3}}{2l^{2}},$ (31)
$T=\frac{1}{4\pi
r_{+}}\left(1+\frac{3r_{+}^{2}}{l^{2}}-\frac{Q^{2}}{r_{+}^{2}}\right),$ (32)
$U=\frac{Q}{r_{+}}\Longrightarrow QU=\frac{Q^{2}}{r_{+}},$ (33)

and the usual Smarr formula is Dehy ; MannRN

$M=2\left(TS-VP\right)+QU.$ (34)

It is clear that $M$, $T$ and the term $(QU)$ in Smarr formula are
proportional to the square of the charge of black hole $Q^{2}$. Therefore,
when pressure ($\Lambda$) is a fixed parameter, $Q^{2}$ is the best choice as
a new variable Dehy  and one may replace $UdQ$ by $\Psi d^{2}Q$, where
$\Psi={1}/{2r_{+}}$ is the conjugate of $Q^{2}$ Dehy . Therefore, inspired by
the expression (32), one may write the equation of state as Dehy

$Q^{2}(T,\Psi)=r_{+}^{2}+\frac{3r_{+}^{4}}{l^{2}}-4\pi r_{+}^{3}T.$ (35)

As mentioned before, in case of constant $l$ (or $\Lambda$) the equation of
state (35) leads to a critical behavior similar to Van der Walls fluid system
Dehy . Now, we are going to employ this approach for charged Lifshitz black
holes with power Maxwell field. Let us write thermodynamic quantities in term
of $Q$ (the charge of black hole) instead of $q$ (charge parameter). It is a
matter of calculation to show by using Eqs. (17) and (23), that the mass (20),
temperature (LABEL:Temp) and electric potential (25) are the same as

$\displaystyle M=\frac{(n-1)\omega_{n-1}}{16\pi
l^{z+1}}\left\\{{r}_{+}^{z+n-1}+{\frac{k{l}^{2}\left(n-2\right)^{2}{r}_{+}^{z+n-3}}{\left(z+n-3\right)^{2}}}+\frac{(2p-1)b^{2z-2}l^{2}\pi^{s}Q^{s}}{(n-1)\Gamma
2^{\frac{-5s}{2}}\omega_{n-1}^{s}r_{+}^{\Gamma}}\right\\}$ (36) $\displaystyle
T_{+}=\frac{1}{4\pi}\left\\{\frac{{(n-1+z){r}_{+}^{z}}}{l^{z+1}}+{\frac{k\left(n-2\right)^{2}{r}_{+}^{z-2}}{l^{z-1}\left(z+n-3\right)}}-\frac{(2p-1)b^{2z-2}2^{\frac{5s}{2}}\pi^{s}Q^{s}}{(n-1)l^{z-1}\omega_{n-1}^{s}{r}_{+}^{\Gamma+n-1}}\right\\}$
(37) $U=\frac{pb^{2(z-1)}}{\Gamma l^{z-1}r_{+}^{\Gamma}}\left(\frac{\pi
Q}{2^{3-p}w}\right)^{\left(s-1\right)}\Longrightarrow
QU=\frac{pb^{2(z-1)}}{\Gamma
l^{z-1}r_{+}^{\Gamma}}\left(\frac{\pi}{2^{3-p}w}\right)^{\left(s-1\right)}Q^{s}.$
(38)

where

$s=2p/(2p-1).$ (39)

It is clear that the charge of black hole appears as $Q^{s}$ in the above
equations. This motivates us to choose the new thermodynamic variable as
$Q^{s}$ which can simplify all calculations. Besides, from Eq. (37) we see
that any other choice except $Q^{s}$, make the equation of state so
complicated which cannot be solved analytically to investigate it‘s critical
behavior and critical quantities. While selecting $Q^{s}$ as a new variable
causes to a simple and solvable equation of state. Fortunately, in the limit
of $p=z=1$ and $n=3$, $Q^{s}\longrightarrow Q^{2}$ i.e., our result reduces to
the one of Dehy . As we will show in the next sections, the system allows the
critical behavior similar to the Van der Walls fluid with fixed cosmological
constant by replacing $UdQ$ with $\Psi dQ^{s}$ . Thus, we write down the first
law in the form

$dM=TdS+VdP+\Psi dQ^{s},$ (40)

where $T=\left({\partial M}/{\partial S}\right)_{P,Q^{s}}$, $V=\left({\partial
M}/{\partial P}\right)_{S,Q^{s}}$ and the conjugate of $Q^{s}$ is

$\Psi=\left(\frac{\partial M}{\partial
Q^{s}}\right)_{S,P}=\frac{\left(2\,p-1\right){\omega}^{\frac{1}{2p-1}}{b}^{(2\,z-2)}{\pi}^{\frac{1}{2p-1}}}{16\Gamma
l^{z-1}r^{\frac{\eta}{2\,p-1}}}.$ (41)

while $\eta=2\,pz+n-4\,p-z+1$. When $p=z=1$ and in $3$-dimensions, the above
definition for $\Psi$ reduces to $\Psi={1}/{(2r_{+})}$ Dehy  . In this new
picture we can write the Smarr formula for the charged Lifshitz dilaton black
hole as

$M=\frac{n-1}{z+n-3}TS+\frac{-2}{z+n-3}VP+\frac{2p-1}{2p}\left(1+\frac{\Gamma}{z+n-3}\right)\Psi
Q^{s}.$ (42)

It is worth noting that we have replaced the usual $\Phi dQ$ term in the first
law with $\Psi dQ^{s}$. The extended phase space associated with
$P=-\Lambda/(8\pi)$ is still the same. Using (38) and (41), straightforward
calculations shows $QU=\Psi Q^{s}$ and so Smarr equation (29) and (42) are the
same.

###  3.2 Equation of state

Using Eq. (41) and treating pressure or more precisely $l$ as a fixed
parameter, Eq. (LABEL:Temp) can be written as

$\displaystyle
Q^{s}={\frac{\left(z+n-1\right){Y}^{2\,\beta}}{X{l}^{z+1}}{\Psi}^{{\frac{-2\left(2\,p-1\right)\beta}{\eta}}}}+{\frac{k\left(n-2\right)^{2}{Y}^{2\,\delta}}{\left(z+n-3\right)X{l}^{z-1}}{\Psi}^{{\frac{-2\left(2\,p-1\right)\delta}{\eta}}}}-{\frac{4\pi\,{\it
T}}{X}{Y}^{{\frac{\alpha}{2\,p-1}}}{\Psi}^{-{\frac{\alpha}{\eta}}}},$ (43)

where

$\displaystyle\alpha=2\,np+2\,pz-6\,p-z+2,\text{ \ \ \ \ \ \
}\beta={\frac{np+2\,pz-3\,p-z+1}{2\,p-1}},$
$\displaystyle\delta={\frac{np+2\,pz-5\,p-z+2}{2\,p-1}},\text{ \ \ \ \ \ \
}\eta=2\,pz+n-4\,p-z+1,$ (44)

and

$\displaystyle
X={\frac{\left(2\,p-1\right){b}^{2\,z-2}{\pi}^{s}{2}^{\frac{5s}{2}}}{{l}^{z-1}{w}^{s}\left(n-1\right)}},\text{
\ \ \ \ \ \
}Y=\left({\frac{\left(2\,p-1\right){b}^{2(z-1)}{2}^{\frac{5s}{2}}\pi^{\frac{s}{2p}}}{16{l}^{z-1}\Gamma
w^{\frac{s}{2p}}}}\right)^{{\frac{2\,p-1}{\eta}}},$ (45)

In order to compare the critical behavior of the system with Van der Waals
gas, one may plot isotherm diagrams $Q^{s}-\Psi$, which are displayed in Figs.
2-4. As we know second order phase transition occurs in the point with
following conditions:

Figure 2: $Q^{s}-\Psi$ diagram of Lifshitz black holes for $b=1$, $n=3$,
$q=1$, $l=1$, $p=1$ and $z=0.6$.

Figure 3: $Q^{s}-\Psi$ diagram of Lifshitz black holes for $b=1$, $n=3$,
$q=1$, $l=1$, $p=1.2$ and $z=1$.

Figure 4: $Q^{s}-\Psi$ diagram of Lifshitz black holes for $b=1$, $n=3$,
$q=1$, $l=1$, $p=1.2$ and $z=1.4$.

$\frac{\partial
Q^{s}}{\partial\Psi}\Big{|}_{T_{c}}=0,\quad\frac{\partial^{2}Q^{s}}{\partial\Psi^{2}}\Big{|}_{T_{c}}=0.$
(46)

Solving Eqs. (46) yield the coordinates of the critical point as

$\displaystyle\Psi_{c}=\left({\frac{k\left(n-2\right)^{2}\delta\,{l}^{2}\left(-z+2\right)}{{y}^{2}\left(z+n-3\right)\beta\,\left(z+n-1\right)z}}\right)^{{\frac{-\eta}{2(2\,p-1)}}},$
(47) $\displaystyle
T_{c}={\frac{\left(2\,p-1\right)\left(n-2\right)^{z}}{l\pi\,\alpha}\left({\frac{\left(z+n-1\right)\beta}{-z+2}}\right)^{1-\frac{z}{2}}\left({\frac{k\delta}{z\left(z+n-3\right)}}\right)^{\frac{z}{2}}},$
(48) $\displaystyle
Q^{s}_{c}={\frac{w^{s}\left(-z+2\right)\left(2\,p-1\right){b}^{-2\,z+2}\left(n-1\right)}{2^{\frac{5s}{2}}\alpha\pi^{s}}\left({\frac{{l}^{2}\left(-z+2\right)\left(2\,p-1\right)\delta}{z\left(z+n-1\right)}}\right)^{\delta}\left({\frac{k\left(n-2\right)^{2}}{\left(z+n-3\right)\left(2\,p-1\right)\beta}}\right)^{\beta}},$
(49)

Following the new definition $\rho_{c}=Q^{s}T_{c}\Psi_{c}$ Dehy , the energy
density of Lifshitz black hole at the critical point is

$\displaystyle\rho_{c}={\frac{\left(2p-1\right)^{2}w\left(n-1\right)\left(n-2\right)^{n+2\,z-1}}{{l}^{3-n}{\pi}^{2}\Gamma\,{\alpha}^{2}}\left(z+n-1\right)^{\frac{5-n+2z}{2}}\left({\frac{\left(-z+2\right)\delta}{z\beta}}\right)^{\frac{n-3+2z}{2}}\left({\frac{k}{z+n-3}}\right)^{\frac{n-1+2z}{2}}}.$
(50)

Since $\rho_{c}$ should be positive quantity, we can determine the range of
the parameters which satisfy this condition. In the general case it is
difficult to calculate it, but using diagram (5), show that $\rho_{c}$ is
positive provided Eq. (19) is satisfied. For $n=3$, $\rho_{c}$ is independent
of the value of $l$.

(a) $\rho_{c}-z$ for $b=1$, $n=3$, $q=1$ and $p=1$.

(b) $\rho_{c}-p$ for $b=1$, $n=3$, $q=1$ and $z=1$. As we see $\rho_{c}$ is
positive in the range $1/2<p<3/2$.

Figure 5: $\rho_{C}$ diagram of Lifshitz black holes.

As we expect when $n=3$, $z=1=p$, our results reduces to those of RN-AdS black
holes Dehy

$\Psi_{c}=\sqrt{\frac{3}{2l^{2}}},\ \quad Q^{2}_{c}=\frac{l^{2}}{36},\quad
T_{c}=\frac{1}{\pi l}\sqrt{\frac{2}{3}},\quad\rho=\frac{1}{36\pi}.$ (51)

To see how the critical quantities change with $p$ and $z$, one may plot the
Figs. 6-8. It is clear that in the limit of $z=2$, $\rho_{c}$ and $Q^{s}_{c}$
are equal to zero, while $\Psi_{c}$ and $T_{c}$ goes to infinity. Indeed
$T_{c}=0$ at $z=2$, while for $z>2$, $T_{c}$ may have an imaginary value.

(a) $T_{c}$ versus $z$ for $b=1$, $n=3$, $q=1$ and $p=1$

(b) $T_{c}$ versus $p$ for $b=1$, $n=3$, $q=1$ and $z=1$.

Figure 6: $T_{c}$ diagram of Lifshitz black holes.

(a) $\Psi_{c}$ versus $z$ for $b=1$, $n=3$, $q=1$ and $p=1$

(b) $\Psi_{c}$ versus $p$ for $b=1$, $n=3$, $q=1$ and $z=1$.

Figure 7: $\psi_{c}$ diagram of Lifshitz black holes.

(a) $Q^{s}_{c}$ versus $z$ for $b=1$, $n=3$, $q=1$ and $p=1$

(b) $Q^{s}_{c}$ versus $p$ for $b=1$, $n=3$, $q=1$ and $z=1$.

Figure 8: $Q^{s}_{c}$ diagram of Lifshitz black holes.

###  3.3 Gibbs free energy

Gibbs free energy is one of the most important item which can help us to study
phase transition of a thermodynamical system. As we know, there is no phase
transition when Gibbs free energy is a continuous function. Any discontinuity
in Gibbs free energy, known as a zero order phase transition. Also, first-
order phase transition occurs when the Gibbs free energy is continuous, but
its first derivative with respect to the temperature and pressure is
discontinuous. At first, we calculate the Gibbs free energy of Lifshitz
dilaton black hole. Then, we try to plot Gibbs diagrams to find out more
details about phase transition of the system. We associate the energy of the
system with the Gibbs free energy $G=M-TS$ Do1 . The Gibbs free energy can be
obtained as

$\displaystyle G$ $\displaystyle=$ $\displaystyle
G\left(Q^{s},T\right)=-\frac{k(n-2)^{2}w(z-2)l^{1-z}r^{n+z-3}}{16\pi(n+z-3)^{2}}-\frac{wl^{-z-1}(n+2z-1)r^{n+z-1}}{16\pi}$
(52)
$\displaystyle+\frac{2^{\frac{4-3p}{2p-1}}(2p-1)\pi^{\frac{1}{2p-1}}b^{2z-2}l^{1-z}Q^{s}w^{\frac{1}{1-2p}}(2np+n+2p(2z-5)-2z+3)r^{\frac{n+2p(z-2)-z+1}{1-2p}}}{(n-1)\eta}.$

In the limiting case where $p=z=1$ and $n=3$, the Gibbs free energy reduces to
Dehy

$G=G(T,Q^{2})=\frac{r_{+}}{4}+\frac{3Q^{2}}{4r_{+}}-\frac{r_{+}^{3}}{4l^{2}},$
(53)

where $r_{+}=r_{+}(T,Q^{2})$. We have plotted the Gibbs energy diagrams in
Figs. 9-11. These diagrams have been shifted for more clarity. The swallowtail
behavior of Figs. 9-11 show that a first order phase transition occurs in the
system.

Figure 9: $G-Q^{s}$ diagram of Lifshitz black holes for $b=1$, $n=3$, $q=1$,
$l=1$, $p=.8$ and $z=1.4$.

Figure 10: $G-Q^{s}$ diagram of Lifshitz black holes for $b=1$, $n=3$, $q=1$,
$l=1$, $p=1$ and $z=1.1$.

Figure 11: $G-Q^{s}$ diagram of Lifshitz black holes for $b=1$, $n=3$, $q=1$,
$l=1$, $p=1.2$ and $z=1.2$.

###  3.4 Critical exponents

The behavior of the physical quantities in the vicinity of critical point can
be characterized by the critical exponents. Following the approach of MannBI ,
one can calculate the critical exponents $\alpha^{\prime}$, $\beta^{\prime}$,
$\gamma^{\prime}$ and $\delta^{\prime}$ for the phase transition of charged
Lifshitz black holes in the presence of power-Maxwell field. To obtain the
critical exponents, we define the reduced thermodynamic variables as

$T_{r}=\frac{T}{T_{c}}\quad,\quad\psi_{r}=\frac{\psi}{\psi_{c}}\quad,\quad
Q^{s}_{r}=\frac{Q^{s}}{Q^{s}_{c}}.$ (54)

Since the critical exponents should be studied near the critical point, we
write the reduced variables in the form $T_{r}=1+t$ and $\psi_{r}=1+\phi$,
indicating deviation from the critical point. One may expand Eq. (43) near the
critical point as

$Q^{s}_{r}=1+At-Bt\phi-C\phi^{3}+O\left(t\phi^{2},\phi^{4}\right),$ (55)

where

$A={\frac{-4\delta\,\beta}{z\left(-z+2\right)}},\ \quad
B={\frac{-4\alpha\,\delta\,\beta}{z\left(-z+2\right)\eta}},\ \ \
C={\frac{2\left(2\,p-1\right)^{2}\alpha\,\beta\,\delta}{3{\eta}^{3}}}.$ (56)

To calculate the critical exponent $\alpha^{\prime}$, we consider the entropy
$S$ given in Eq. (26) as a function of $T$ and $\psi$. Using Eq. (41) we have

$S=S\left(T,\Psi\right)={\frac{{Y}^{n-1}}{4\pi}{\Psi}^{{{(-2\,p+1)}/{\eta}}}}.$

Obviously, this is independent of $T$ and therefore the specific heat
vanishes, $C_{\psi}=T\left(\partial S/\partial T\right)_{\psi}=0$. Since the
exponent $\alpha^{\prime}$ governs the behavior of the specific heat at fixed
$\psi$ $C_{v}\varpropto\left|t\right|^{\alpha^{\prime}}$, hence the exponent
$\alpha^{\prime}=0$.

Differentiating Eq. (55) at a fixed $t<0$ with respect to $\phi$, we get

$dQ^{s}_{r}=-\left(Bt+3C\phi^{2}\right)d\phi.$ (57)

Now, we apply the Maxwell’s equal area law callen . Denoting the variable
$\phi$ for small and large black holes with $\phi_{s}$ and $\phi_{l}$,
respectively, we obtain

$\displaystyle Q^{s}_{r}$ $\displaystyle=$ $\displaystyle 1+At-
Bt\phi_{l}-C\phi_{l}^{3}=1+At-Bt\phi_{s}-C\phi_{s}^{3},$ $\displaystyle 0$
$\displaystyle=$ $\displaystyle\int_{\phi_{l}}^{\phi_{s}}\phi dQ^{s}_{r}.$
(58)

Equation (58) leads to the unique non-trivial solution

$\phi_{l}=-\phi_{s}=\sqrt{-\frac{Bt}{C}},$ (59)

which gives the order parameter as

$\left|\phi_{s}-\phi_{l}\right|=2\phi_{s}=2\sqrt{-\frac{B}{C}}\ t^{1/2}.$ (60)

Thus, the exponent $\beta^{\prime}$ which describes the behaviour of the order
parameter near the critical point is $\beta^{\prime}=1/2.$ To calculate the
exponent $\gamma^{\prime}$, one may determine the behavior of the following
function near the critical point

$\chi_{T}=\frac{\partial\psi}{\partial Q^{s}}\Big{|}_{T}.$

Differentiating Eq. (55) with respect to $\phi$, near the critical point may
be written as

$\chi_{T}\propto-\frac{\psi_{c}}{BQ^{s}_{c}}\frac{1}{t}\quad\Longrightarrow\quad\gamma^{\prime}=1.$
(61)

Finally, the shape of the critical isotherm $t=0$ is given by Eq. (55). We
find

$Q^{s}_{r}-1=-C\phi^{3}\quad\Longrightarrow\quad\delta^{\prime}=3.$ (62)

##  4 Summery and conclusions

The critical behaviour of the Lifshitz dilaton black hole in an extended phase
space, where the cosmological constant is treated as the thermodynamical
pressure, cannot be studied due to the complicated form of the solution.
Indeed, it can be seen from Eq. (LABEL:Temp) that it is almost impossible to
solve this equation for $P$ (or more precisely for $l$). Therefore, we cannot
have an analytical equation of state, $P=P(v,T)$, to investigate $P-v$
critically of the system. Also, investigating the phase space of the system in
$U-Q$ plan leads to physically irrelevant quantities which are mathematically
ill-defined Dehy . Here, we address, for the first time, the critical behavior
of an $(n+1)$-dimensional dilaton Lifshitz black hole in the presence of a
power-law Maxwell field via an alternative phase space developed in Dehy . We
have treated the cosmological constant as a fixed parameter and the charge of
the system as thermodynamical variable. It was argued in Dehy  that without
extension the phase space and by keeping the cosmological constant (pressure)
as a fixed quantity instead of the charge of the system, it is quite possible
to have critical behaviour similar to those of Van der Walls system provided
one take the equation of state of the form $Q^{2}=Q^{2}(T,\psi)$ where
$\Psi=1/2r_{+}$ is conjugate of $Q^{2}$.

In this work, we disclosed that in order to investigate critical behaviour of
Lifshitz black holes with power Maxwell field, we should modify the method
developed in Dehy  by considering $Q^{s}$ as a thermodynamic variable and
write down the equation of state in the form of $Q^{s}=Q^{s}(T,\psi)$, where
$s=2p/(2p-1)$ with $p$ is the power of the power-Maxwell Lagrangian. In this
approach we keep the cosmological constant (pressure) as a fixed quantity and
treat the charge of the black hole as a thermodynamic variable. This is in
contrast to the extended phase space of MannRN , where the charge is fixed and
the cosmological constant is treated as a thermodynamic variable. The isotherm
diagrams $Q^{s}-\Psi$ show the complete analogy between our system and Van der
Walls liquid-gas system. Also the swallow tail behavior of the Gibbs free
energy represented a first order phase transition is occurring in the system.
Furthermore, we calculated the critical quantities such as $T_{c}$,
$\rho_{c}$, $P_{c}$ and $\Psi_{c}$, at the critical point which depend on the
metric parameters. Finally, we obtained the critical exponents of the system
and found out that they are universal and exactly the same as Van der Walls
fluid system. Indeed, our study shows that the approach here is powerful to
investigate the critical behaviour of Lifshitz black holes with power Maxwell
field. It could help to extract critical exponents of the system without
extending the phase space, which is useful in studying the thermodynamical
properties of the black holes. We expect to confirm that this approach is
viable and can be applied for other gravity theories such as Gauss-Bonnet and
Lovelock gravity.

###### Acknowledgements.

We thank Shiraz University Research Council. The work of AS has been supported
financially by Research Institute for Astronomy and Astrophysics of Maragha,
Iran.

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# Neural Morphology Dataset and Models for Multiple Languages, from the Large
to the Endangered

Mika Hämäläinen, Niko Partanen, Jack Rueter and Khalid Alnajjar
Faculty of Arts
University of Helsinki
<EMAIL_ADDRESS>

###### Abstract

We train neural models for morphological analysis, generation and
lemmatization for morphologically rich languages. We present a method for
automatically extracting substantially large amount of training data from FSTs
for 22 languages, out of which 17 are endangered. The neural models follow the
same tagset as the FSTs in order to make it possible to use them as fallback
systems together with the FSTs. The source
code111https://github.com/mikahama/uralicNLP/wiki/Neural-morphology,
models222http://doi.org/10.5281/zenodo.3926769 and
datasets333http://doi.org/10.5281/zenodo.3928628 have been released on Zenodo.

##  1 Introduction

Morphology is a powerful tool for languages to form new words out of existing
ones through inflection, derivation and compounding. It is also a compact way
of packing a whole lot of information into a single word such as in the case
of the Finnish word hatussanikinko (in my hat as well?). This complexity,
however, poses challenges for NLP systems, and in the work concerning
endangered languages, morphology is one of the first NLP problems people
address.

The GiellaLT infrastructure Moshagen et al. (2014) has HFST-based Lindén et
al. (2013) finite-state transducers (FSTs) for several morphologically rich
(and mostly Uralic) languages. These FSTs are capable of lemmatization,
morphological analysis and morphological generation of different words.

These transducers are at the core of this infrastructure, and they are in use
in many higher level NLP tasks, such as rule-based Trosterud (2004) and neural
disambiguation Ens et al. (2019), dependency parsing Antonsen et al. (2010)
and machine translation Pirinen et al. (2017). The transducers are also in
constant use in several real world applications such as online dictionaries
Rueter and Hämäläinen (2019), spell checkers Trosterud and Moshagen (2021),
online creative writing tools Hämäläinen (2018), automated news generation
Alnajjar et al. (2019), language learning tools Antonsen and Argese (2018) and
documentation of endangered languages Gerstenberger et al. (2017); Wilbur
(2018). As an additional important application we can mention the wide use of
FSTs in the creation of Universal Dependencies treebanks for low-resource
languages, at least with Erzya Rueter and Tyers (2018), Northern Saami Tyers
and Sheyanova (2017) Karelian Pirinen (2019a) and Komi-Zyrian Partanen et al.
(2018).

Especially in the context of endangered languages, accuracy is a virtue. Rule-
based methods not only serve as NLP tools but also as a way of documenting
languages in a machine-readable fashion. Members of language communities do
not benefit, for example, from a neural spell checker that works to a degree
in a closed test set, but fails miserably in real world usage. On the
contrary, a rule based description of morphology can only go so far. New words
appear and disappear all the time in a language, and keeping up with that pace
is a never ending job. This is where neural models come in as they can learn
to generalize rules for out-of-vocabulary words as well. Pirinen Pirinen
(2019b) also showed recently that at least with Finnish the neural models do
outperform the rule-based models. This said, Finnish is already a larger
language, so the experience doesn't necessarily translate into low-resource
scenario (see Hämäläinen 2021).

The purpose of this paper is to propose neural models for the three different
tasks the GiellaLT FSTs can handle: morphological analysis (i.e. given a form
such as kissan, produce the morphological reading +N+Sg+Gen), morphological
generation (i.e. given a lemma and a morphology, generate the desired form
such as kissa+N+Sg+Gen to kissan) and lemmatization (i.e. given a form,
produce the lemma such as kissan to kissa `a cat'). The goal is not to replace
the FSTs, but to produce neural fallback models that can be used for words an
FST does not cover. This way, the mistakes of the neural models can easily be
fixed by fixing the FST, while the overall coverage of the system increases by
the fact that a neural model can cover for an FST.

The main goal of this paper is not to propose a state of the art solution in
neural morphology. The goal is to first build the resources needed to train
such neural models so that they will follow the same morphological tags as the
GiellaLT FSTs, and secondly train models that can be used together with the
FSTs. All of the trained models will be made publicly available in a Python
library that supports the use of the neural models and the FSTs
simultaneously. The dataset built in this paper and the exact train,
validation and test splits used in this paper have been made publicly
available for others to use on the permanent archiving platform Zenodo.

##  2 Constructing the Dataset

We are well aware of the existence of the popular UniMorph dataset McCarthy et
al. (2020). However, it does not suit our needs of two reasons. One reason is
the incompatible morphological tagset. Our goal is to build models that can
directly be used side-by-side with the existing FSTs, which means that the
data has to follow the same formalism. Conversion is not a possibility, as the
main reason we are not interested in using the UniMorph data is its limited
scope; not only does it not cover all the languages we are dealing with in
this paper, but it does not cover any cases of complex morphology. For
example, the Finnish dataset does not cover possessive suffixes, question
markers, comparative, superlative etc. Such a data would not be on par with
the output produced by the FSTs.

We produce the data for the following languages: German (deu), Kven (fkv),
Komi-Zyrian (kpv), Mokhsa (mdf), Mansi (mns), Erzya (myv), Norwegian Bokmål
(nob), Russian (rus), South Sami (sma), Lule Sami (smj), Skolt Sami (sms),
Võro (vro), Finnish (fin), Komi-Permyak (koi), Latvian (lav), Eastern Mari
(mhr), Western Mari (mrj), Namonuito (nmt), Olonets-Karelian (olo), Pite Sami
(sje), Northern Sami (sme), Inari Sami (smn) and Udmurt (udm). A vast majority
of these languages are greatly endangered Moseley (2010).

| deu | fin | fkv | koi | kpv | lav | mdf | mhr | mns | mrj | myv | nob | olo | rus | sje | sma | sme | smj | smn | sms | udm | vro
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
N | 8741 | 51916 | 5936 | 558 | 20042 | 9738 | 17196 | 14079 | 2263 | 2529 | 10234 | 32009 | 5942 | 24691 | 2685 | 5946 | 37943 | 4331 | 13826 | 21158 | 10722 | 4703
Adv | 588 | 6036 | 652 | 89 | 2942 | 953 | 1771 | 2346 | - | 444 | 743 | 1743 | 14 | 2546 | - | 543 | 1314 | 343 | 1146 | 1729 | 985 | 122
V | 4021 | 27875 | 1445 | 532 | 12504 | 2601 | 11983 | 9954 | 4924 | 2456 | 3781 | 7432 | 2782 | 14348 | 1751 | 5208 | 7724 | 3130 | 5436 | 5033 | 3669 | 4129
A | 2768 | 13056 | 917 | 128 | 5218 | 1652 | 4407 | 5116 | - | 1031 | 2926 | 3236 | 2134 | 11054 | 185 | 645 | 2927 | 468 | 2295 | 3898 | 1550 | 1019
Table 1: The sizes of the GiellaLT dictionaries per part-of-speech

We use the FSTs and dictionaries from the GiellaLT with the UralicNLP
Hämäläinen (2019) library to build the datasets for training the models. We do
this in a clever way by taking all open class part-of-speech words from the
dictionaries for each language and use the FSTs to produce all morphological
readings for them. The number of words in the GiellaLT dictionaries is shown
in Table 1. The FSTs do not let us do this by default, so we build a regular
expression transducer that finds all possibilities for an input word and its
part-of-speech.In order to build the regular expression, we query all
alphabets in the transducer that contain one of the following strings for
exclusion: #, Der, Cmp or Err. This will remove compounds, erroneously spelled
forms and derivations. Derivations need to be excluded because otherwise the
transducers would produce derivations of derivations of derivations and so on.
Once the regular expression transducer is composed with the FST analyzer, we
can use HFST to extract the transducer paths to get a list of all the possible
morphological forms of the input word. From these, we filter out Clt and Foc
tags because these multiply the number of possible morphological forms,
especially since multiple different clitics can be appended after each other,
and some times even in multiple different orders. We also remove tags
indicating non-standard forms, Use and Dial, and Sem tags that are used in
language learning tools as well as contextual disambiguation to categorize
semantically similar words. Table 2 shows how many unique inflectional forms
each part-of-speech category has per language.

| deu | fin | fkv | koi | kpv | lav | mdf | mhr | mns | mrj | myv | nob | olo | rus | sje | sma | sme | smj | smn | sms | udm | vro
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
N | 24 | 850 | 50 | 788 | 183 | 24 | 83 | 208 | 151 | 162 | 19 | 17 | 98 | 75 | 16 | 50 | 727 | 297 | 496 | 339 | 744 | 26
Adv | 1 | 16 | 1 | 2 | 4 | - | 4 | 3 | - | 2 | 2 | 2 | - | 1 | - | 3 | 8 | 3 | 2 | 3 | 1 | 6
V | 254 | 6667 | 139 | 198 | 249 | 1245 | 894 | 59 | - | 40 | 10 | 21 | 726 | 693 | 38 | 58 | 302 | 144 | 382 | 177 | 156 | 119
A | 150 | 1244 | 77 | 4 | 244 | 44 | 127 | 4 | - | 2 | 5 | 15 | 217 | 39 | 52 | 75 | 1347 | 187 | 100 | 627 | 54 | 100
Table 2: Number of unique inflectional forms per part-of-speech category

We use the method described above to produce the data with all the open class
part-of-speech words in the GiellaLT dictionaries for each language. For
languages with bigger dictionaries, the maximum number of lemmas used per part
of speech is set to 2100, in which case the lemmas are also picked at random.
We use the typical split ratio and split 70% of the data for training, 15% for
validation and 15% for testing. The split is done on the lemma level and for
each part-of-speech separately. This means that the test and validation sets
will consist exclusively of out of vocabulary words that have not appeared in
the training in any inflectional from. This also means that the ratios are the
same for each part-of-speech, 70% of the adjectives are used in the training,
70% of the verbs and so on. The actual sizes can be seen in Table 3.

| deu | fin | fkv | koi | kpv | lav | mdf | mhr | mns | mrj | myv | nob | olo | rus | sje | sma | sme | smj | smn | sms | udm | vro
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
train | 394k | 14486k | 286k | 483k | 873k | 320k | 1267k | 666k | 283k | 232k | 45k | 37k | 1054k | 243k | 80k | 108k | 799k | 648k | 1167k | 2831k | 943k | 257k
val | 87k | 3061k | 62k | 105k | 186k | 68k | 276k | 142k | 60k | 50k | 9k | 8k | 229k | 51k | 17k | 22k | 177k | 145k | 249k | 628k | 202k | 54k
test | 84k | 3109k | 60k | 105k | 186k | 68k | 274k | 142k | 60k | 50k | 9k | 8k | 221k | 53k | 16k | 23k | 179k | 143k | 253k | 624k | 203k | 55k
Table 3: Sizes of the datasets for each language. The splits do not share
vocabulary.

The reason why we do the testing purely on out-of-vocabulary words is simply
to test the accuracy of the models in the scenario that is more close to the
one they are trained for, namely, in cases where the FSTs fail in their
coverage.

##  3 Experiments and Results

In this section, we cover the neural architecture for the three separate
morphological tasks: lemmatization, analysis and generation. We also show the
results of the models in these tasks for each language, and present an error
analysis on the Finnish and Komi-Zyrian by taking a closer look at the
results.

###  3.1 The Neural Model

Over recent years, there has been a growing body of work on different neural
approaches for low resourced languages in morphological analysis Moeller et
al. (2019); Schwartz et al. (2019), lemmatization Kondratyuk (2019);
Silfverberg and Tyers (2019) and generation Oseki et al. (2019); Yu et al.
(2020). Most notably the use of bi-directional LSTM architecture seems to be
supported by most of the recent related work for analysis, generation and
lemmatization.

It is important to note that we approach the lemmatization and analysis from
the same point of view as the FSTs. This means that it is a strictly
morphological process, and the question of disambiguation is left for another
part of the GiellaLT NLP pipeline, namely constraint grammar rules Bick and
Didriksen (2015). There is a plethora of work dealing with in-context
lemmatization Manjavacas et al. (2019); Malaviya et al. (2019), morphological
analysis Lim et al. (2018); Zalmout and Habash (2020) and part-of-speech
tagging Perl et al. (2020); Hoya Quecedo et al. (2020), but that is not what
we are aiming for. We are aiming for neural models that can be used to
complement the already existing systems relying on the GiellaLT
infrastructure.

For all three tasks, we train a character based bi-directional LSTM model
Hochreiter and Schmidhuber (1997) by using OpenNMT-py Klein et al. (2017) with
the default settings except for the encoder where we use a BRNN (bi-
directional recurrent neural network) Schuster and Paliwal (1997) instead of
the default RNN (recurrent neural network) as BRNN has been shown to provide a
performance gain in a variety of tasks. We use the default of two layers for
both the encoder and the decoder and the default attention model, which is the
general global attention presented by Luong et al. Luong et al. (2015).

| input | output
---|---|---
lemmatization | k a u n i i m p a n s a k o | k a u n i s
analysis | k a u n i i m p a n s a k o | A Comp Sg Gen PxSg3 Qst
generation | k a u n i s A Comp Sg Gen PxSg3 Qst | k a u n i i m p a n s a k o
Table 4: Example of the training data for each task

Table 4 shows an example of the input and output of the training data in each
of the three different tasks. Words are split into characters on both the
input and output side of the data. Different morphological tags are treated as
separate tokens, this means that FST morphologies consisting of multiple tags
such as N+Msc+Sg+Dat are simply split by the plus sign. We train a separate
model for each task, meaning that we train three different models for each
language: one for lemmatization, analysis and generation. All models have
shared the same random seed (3435), therefore training the models again with
this seed should result in the exact same results we are reporting in this
paper.

###  3.2 Results

We report the performance of the models in terms of accuracy, meaning how many
results were fully right (entirely correct lemma, entirely correctly generated
form and entirely correct morphological analysis). In addition, we report CER
(character error rate) for the lemmatizers and generators, and a MER
(morphological error rate) for the analyzers. These values indicate how close
the model got to the correct result even if some of the results were a bit
erroneous.

| deu | fin | fkv | koi | kpv | lav | mdf | mhr | mns | mrj | myv | nob | olo | rus | sje | sma | sme | smj | smn | sms | udm | vro
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
gen acc | 0,65 | 0,64 | 0,68 | 0,67 | 0,78 | 0,95 | 0,85 | 0,58 | 0,78 | 0,90 | 0,93 | 0,94 | 0,83 | 0,97 | 0,77 | 0,69 | 0,73 | 0,67 | 0,57 | 0,40 | 0,87 | 0,82
gen CER | 5,61 | 8,03 | 3,70 | 8,67 | 3,75 | 1,21 | 1,77 | 11,76 | 3,92 | 1,77 | 0,67 | 1,23 | 2,12 | 0,48 | 4,28 | 5,81 | 4,19 | 3,54 | 4,25 | 6,65 | 1,90 | 3,35
lem acc | 0,88 | 0,68 | 0,80 | 0,70 | 0,87 | 0,85 | 0,93 | 0,88 | 0,79 | 0,88 | 0,90 | 0,76 | 0,87 | 0,82 | 0,72 | 0,71 | 0,06 | 0,70 | 0,67 | 0,79 | 0,92 | 0,79
lem CER | 2,71 | 12,37 | 5,85 | 11,41 | 1,21 | 4,34 | 1,11 | 2,46 | 4,87 | 3,82 | 1,50 | 5,71 | 3,76 | 4,25 | 6,78 | 6,96 | 55,72 | 9,13 | 7,78 | 4,45 | 2,75 | 5,81
ana acc | 0,11 | 0,57 | 0,86 | 0,78 | 0,88 | 0,39 | 0,61 | 0,94 | 0,77 | 0,92 | 0,98 | 0,49 | 0,86 | 0,36 | 0,73 | 0,60 | 0,56 | 0,53 | 0,42 | 0,42 | 0,76 | 0,74
ana MER | 35,40 | 16,66 | 6,52 | 7,24 | 3,06 | 18,24 | 11,54 | 3,73 | 7,76 | 4,75 | 0,41 | 38,09 | 5,85 | 19,04 | 19,45 | 22,91 | 17,24 | 22,48 | 23,20 | 20,11 | 10,90 | 9,82
Table 5: Results of the models for different languages on out-of-vocabulary
data

The results can be seen in Table 5, the models reaching to an accuracy to over
80 % are highlighted in bold. The results indicate that lemmatization is the
easiest task for the model to learn, and after that generation. Morphological
analysis is the most difficult task as it receives the scores lower than the
generation or lemmatization. Needless to say, some results are exceptionally
good for specific languages such as for Erzya (myv) and Western Mari (mrj),
while they are not good for others like Finnish (fin) and German (deu). This
calls for more investigation of the results.

Figure 1: Accuracies based on morphological complexity

Figure 1 shows the accuracy of each model based on the morphological
complexity of the input. The complexity is measured by the number of
morphological tags in the FST produced data. The complexity axis of the plots
shows a relative complexity for each language, meaning that 1.0 has the
maximum number of tags, 0.8 shows results for input having 80% of the maximum
number of tags and so on. The maximum complexity is shown in brackets after
the language ISO-code. Analyzers seem to have a lower accuracy for most of the
languages when the complexity is small. This is probably due to the fact that
shorter word forms tend to have more ambiguity to begin with and might be
analyzed as a word different from the one in the gold standard. For many
languages, the accuracy increases towards the average complexity and drop
again for the most complex forms. It is to be remembered that these accuracies
are also affected by the peculiarities of the transducers themselves and their
tagging conventions.

Lemmatizers seem to follow the pattern of the analyzers but do so more
clearly. Lemmatization of morpholgically simple forms is not as easy as more
complex forms. However, as the complexity increases, the lemmatization
accuracy does not drop for most of the languages. This has probably something
to do with the fact that unlike morphological tags, the word forms follow
clearer patterns as they do not have such a large amount of subjectivity in
the tagging decisions the different linguists working on these transducers
have introduced.

Generators are very even for most of the languages in the sense that they
produce consistently around the same accuracy regardless of the morphological
complexity. Although, some of the languages follow a more analyzer like
pattern, generating wrong with small and large morphological complexity.

| Missing predictions | Wrong predictions
---|---|---
deu | Def, Pl, Acc, Dat, Neu, Gen, Msc, Fem, NoArt, Nom | NoArt, Fem, Msc, Indef, Sg, Gen, Acc, Nom, Def, Neu
fin | PxSg3, A, PxPl3, N, Sg, Pl, Nom, Gen, Par, Pss | V, Act, PxPl3, PrfPrc, PxSg3, Ind, PrsPrc, Pss, Prs, Sg
fkv | A, Act, N, Sg, V, Pl, Ind, Inf3, Nom, Pl3 | N, Act, A, V, Sg, Pl, Ind, Pass, Prs, Inf3
koi | IV, TV, AprIne, AprIll, Ill, Prs, V, So/CP, Apr, Ind | Apr, So/CP, TV, Ine, AprIne, Fut, Sg, N, IV, Nom
kpv | Ine, TV, IV, Fut, Prs, Ill, V, Pl3, Sg1, PxSg1 | Ill, IV, TV, Prs, Ine, Fut, Sg, Sg3, N, Pl1
lav | IV, Fem, Acc, Pl, Sg, Nom, TV, Voc, Def, Gen | TV, Gen, Msc, Sg, Pl, Indef, Loc, IV, Fem, Acc
mdf | IV, TV, V, Ind, Prt2, OcPl3, N, A, PxSg2, NomAct | TV, IV, N, Conj, OcSg3, Def, A, V, OcPl1, ScSg1
mhr | N, Sg, V, So/CP, Nom, Ill, Ind, So/PC, Gen, Ger | V, Ind, N, Sg, Adv, Nom, Prs, Sg3, Imprt, A
mns | PxPl2, Sg, Pl, PxDu2, Nom, PxSg2, Du, Lat, Abl, Loc | PxDu2, Pl, Sg, PxSg2, Nom, PxPl3, Du, Lat, Tra, PxPl2
mrj | Sg, N, Lat, Prs, V, Nom, Ind, Imprt, Ine, PxPl3 | Prt1, Ind, V, Ill, N, Sg, Nom, Sg3, Ine, Prs
myv | N, A, Tra, Ela, Abl, Ine, Interr, PxSg2 | A, N, Abl, Ine, Ela, Ill, Tra, NomAg, V, IV
nob | A, Pos, Indef, Sg, Pl, PrfPrc, Def, Fem, V, MF | Msc, Ind, V, Sg, Indef, Prt, Neu, Def, N, Pl
olo | Ins, N, A, ConNeg, V, Sg, Act, PrfPrc, Nom, Pl | Gen, ConNeg, Act, V, N, Ind, A, Sg, Prs, PrsPrc
rus | TV, Acc, Neu, Gen, Anim, Inan, Impf, Dat, Pass, IV | IV, Loc, Msc, AnIn, Nom, Perf, TV, Acc, Gen, Sg
sje | Sg, V, Com, Prs, N, Sg2, Pl, Gen, Ind, Nom | N, Sg, Pl, V, Ind, Nom, Prs, Prt, Gen, Ine
sma | IV, TV, N, Ind, V, Ess, Sg, Pl, A, Prs | Sg, IV, TV, N, Ind, Com, Prs, Pl, Nom, Ill
sme | Acc, Gen, TV, Sg, Pl, Loc, IV, N, Com, Nom | Gen, Sg, Acc, IV, TV, Nom, Pl, Com, Loc, A
smj | Com, Sg, N, Pl, PxDu2, PxDu1, Gen, Acc, IV, NomAg | Gen, TV, V, PxPl2, PxPl1, Pl, Sg, Nom, IV, Ine
smn | Acc, Sg, Gen, PxPl2, PxSg3, N, PxPl3, IV, PxPl1, TV | PxDu3, Gen, PxDu2, Nom, A, Ill, Sg, PxSg1, PxSg2, Pl
sms | N, Sg, V, Acc, Pl, Gen, Ill, Com, Ind, Nom | A, Pl, Nom, Sg, Gen, Loc, Acc, Com, Ill, Par
udm | Ill, N, Opt, Sg, ConNeg, Ind, PxSg3, Imprt, Sg2, Ine | ConNeg, Ind, Ine, Sg3, Fut, Nom, N, A, Det, Sg
vro | A, Pss, Act, V, Pl, Sg, Ind, N, Gen, ConNegII | Sg, Nom, Act, N, A, Pl, Sg1, Prs, Ind, Par
Table 6: The top 10 most difficult tags for the analyzers

Table 6 shows the most difficult tags for the analyzers. The missing
predictions column shows the most frequent tags the analyzer did not predict
even though they were in the gold data, and the wrong predictions column shows
the most frequent ones the analyzer predicted but were not in the gold data.
We can see that many of the most challenging tags are shared by different
languages. In various Uralic languages, for example, connegatives and
imperatives, or connegatives and infinitives, are homonymous, and cannot be
predicted correctly just from the surface form alone. Similarly cases such as
illative and inessive are in many complex forms homonymous in Permic
languages, which surfaces in missing predictions of all these languages. In
the languages where transitivity is a feature coded into FST, there are
regular problems in predicting these categories correctly. Similarly, in many
Indo-European languages gender is primarily a lexical category, and in many
instances the model cannot predict it correctly in cases where only the
surface form that doesn't show the gender is presented. In the Section 3.3 we
go through more in detail this kind of instances, for example, in relation to
purely lexically determined Komi-Zyrian stem consonants.

Table 7 shows the morphological constructions that were the most difficult
ones for the models to lemmatize and generate correctly in their respective
columns. For instance, the Erzya (myv) generation indicates the translative
with subsequent possessive-suffix marking is the most problematic. If it had
been lemmatization, the explanation would point to the extreme infrequency of
these translative forms and the fact that there is an ambiguity with genitive
and nominative forms of derivations in ks. Lemmatization for Erzya, however,
appears to have no issues with ambiguity at all. The same difficulties are not
shared by other languages, but seem to all be language specific. Eastern and
Meadow Mari (mhr), for example, appear to have difficulties with generation
and lemmatization of nearly the same tag set, namely, the illative plural with
a third person plural possessive suffix (ordered: possessive, plural and
finally case marker). Looking at the sibling language Western Mari (mrj), we
will note that there is a different tagging strategy in use, but here as well
there seems to be an intersection where the same forms present problems for
both generation and lemmatization.

This could be seen as a type of sanity test whereby simple flaws in the
transducers might be detected. The Latvian (lav) transducer is a blatant
example of inconsistencies in transducer development. The problem, which has
now been addressed and corrected, was in the multiple exponence of part-of-
speech tags, i.e. there are double +V and +N tags due to the introduction of
automated part-of-speech tagging in XML dictionary to FST formalism
transformation without removing the part-of-speech tagging in subsequent
continuation lexica of the rule-based transducer. Development of the Mari pair
might be greatly enhanced through the introduction of a segment-ordering tag
in Western or Hill Mari (mrj), which would bring it closer to the strategy
followed in the Eastern and Meadow Mari (mhr) use of +So/PNC. These questions
with tag and suffix ordering appear also as important factor in Komi-Zyrian
morphological generation, as discussed in Section 3.3.

| generator | lemmatizer
---|---|---
deu |  | V+PrfPrc+Pos+Pl+Nom+Indef, V+PrfPrc+Pos+Fem+Sg+Nom+Indef,
---
V+PrfPrc+Pos+Pred, V+PrfPrc+Pos+Fem+Sg+Acc+Indef,
V+PrfPrc+Pos+Neu+Sg+Acc+Def
| N+Msc+Pl+Dat, N+Msc+Pl+Gen, N+Msc+Pl+Nom,
---
N+Msc+Pl+Acc, N+Msc+Sg+Dat
fin |  | A+Sg+Ess+PxSg3, A+Sg+Ess+PxPl3,
---
A+Sg+Ess+PxPl3+Qst, A+Sg+Ess+PxSg3+Qst,
N+Pl+Par+PxPl3+Qst
| A+Sg+Ess+PxSg3, A+Sg+Ess+PxPl3, A+Sg+Ess+PxPl3+Qst,
---
A+Sg+Ess+PxSg3+Qst, N+Pl+Par+PxSg3
fkv |  | V+Act+Inf3+A+Pl+Superl+Par, A+Pl+Superl+Par, N+Pl+All,
---
V+Act+Inf3+A+Pl+Par, V+Act+Inf3+A+Pl+Gen
N+Pl+All, N+Pl+Par, N+Pl+Gen, N+Sg+Par, N+Pl+Abe
koi |  | V+Ind+Prt2+Pl3+Comp, V+Ind+Prt2+Pl3, V+IV+Ind+Prt2+Pl3+Comp,
---
V+IV+Ind+Prt2+Pl3, V+TV+Ind+Prt2+Pl3
| N+Sg+Ela+Comp+Cop+Pl, N+Sg+Ine+PxPl1+Comp+Cop+Pl,
---
N+Sg+Ine+PxPl2+Comp+Cop+Pl,
N+Sg+Ine+PxPl3+Comp+Cop+Pl, N+Sg+Ela+PxSg1+Comp+Cop+Pl
kpv |  | N+Sg+Com+PxSg2, N+Sg+Com+PxSg3, N+Sg+Egr+PxSg1+Comp,
---
N+Sg+Egr+PxSg1, N+Sg+Egr+PxSg1+Comp+Cop+Pl
| N+Sg+Acc, Adv, N+Sg+Prl+PxPl1, N+Sg+Com+PxSg2,
---
N+Sg+Com+PxSg3
lav |  | V+V+TV+PrsPrc+Act+Msc+Sg+Voc+Def,
---
V+V+TV+PrsPrc+Pss+Msc+Sg+Voc+Def,
V+V+TV+PrfPrc+Pss+Msc+Sg+Voc+Def,
N+N+Msc+Sg+Voc, V+V+IV+PrsPrc+Act+Msc+Sg+Voc+Def
| N+N+Msc+Sg+Voc, N+N+Fem+Sg+Voc, N+N+Fem+Pl+Gen,
---
N+N+Fem+Sg+Acc, N+N+Fem+Sg+Loc
mdf |  | V+IV+NomAg+Pl+Gen+PxSg3, V+IV+NomAg+Pl+Nom+PxSg3,
---
V+IV+NomAg+SP+Cau+Indef, V+IV+NomAg+Sg+Dat+PxSg1,
V+IV+NomAg+Pl+Dat+PxSg3
| N+Sg+Nom+Indef, N+SP+Gen+Indef, N+Sg+Dat+PxSg2,
---
N+Sg+Dat+PxSg1, N+SP+Tra+Indef
mhr |  | N+Pl+Ill+PxSg3+So/PNC, N+Pl+Ill+PxPl1+So/PNC,
---
N+Pl+Ill+PxPl2+So/PNC, N+Pl+Ill+PxPl3+So/PNC,
N+Pl+Ill+PxSg1+So/PNC
| Adv, N+Pl+Ill+PxSg3+So/PNC, N+Pl+Ill+PxSg2+So/PNC,
---
N+Sg+Ill+PxSg3+So/CP, N+Pl+Ill+PxSg1+So/PNC
mns |  | N+Du+PxDu1+Abl, N+Du+PxDu1+Ins,
---
N+Du+PxDu1+Nom, N+Du+PxDu1+Loc,
N+Du+PxDu1+Lat
| N+Pl+PxSg2+Ins, N+Pl+PxSg2+Loc, N+Pl+PxSg2+Abl,
---
N+Pl+PxSg2+Nom, N+Pl+PxSg2+Lat
mrj |  | N+Sg+Ine+PxSg3, N+Sg+Ill, N+PxSg2+Pl+Ill,
---
N+Sg+PxSg2+Ill, N+PxSg1+Pl+Lat
| N+Sg+Ill, N+Sg+Gen, N+Sg+Acc, N+Sg+Nom,
---
N+Sg+Ine+PxSg3
myv |  | N+SP+Tra+PxSg2, N+SP+Tra+PxPl1, N+SP+Tra+PxPl2,
---
N+SP+Tra+PxPl3, N+SP+Tra+PxSg3
| V+IV+Act+PrsPrc, V+IV+NomAg+SP+Ill+PxSg2,
---
V+IV+NomAg+SP+Ill+PxPl1, V+IV+NomAg+SP+Ill+PxSg3,
V+IV+NomAg+SP+Ine+PxPl1
nob |  | N+Neu+Pl+Def, V+Ind+Prt, N+Neu+Pl+Indef,
---
V+PrfPrc, A+Superl+Def
| V+Imp, A+Pos+Neu+Sg+Indef, A+Pos+Fem+Sg+Indef,
---
V+Ind+Prt, A+Pos+Msc+Sg+Indef
olo |  | V+Act+PrsPrc+Pl+Abe, V+Act+PrsPrc+Pl+Abe+Qst,
---
N+Pl+Abe, N+Pl+Abe+Qst, N+Sg+Abe+Qst
| N+Sg+Nom, N+Sg+Nom+Qst, N+Sg+Abe,
---
N+Pl+Abe+Qst, N+Sg+Abe+Qst
rus |  | V+Perf+IV+Imp+Pl2, V+Perf+IV+Imp+Sg2,
---
V+Perf+IV+Fut+Sg3, V+Perf+IV+Fut+Sg2, V+Perf+IV+Fut+Sg1
| Adv, A+Neu+Sg+Pred, A+Msc+Sg+Pred,
---
V+Perf+IV+Imp+Sg2, A+Msc+AnIn+Sg+Loc
sje | N+Pl+Ela, N+Pl+Com, N+Sg+Ela, N+Sg+Com, V+Pot+Sg3 | N+Pl+Com, N+Pl+Ela, N+Sg+Ela, N+Sg+Com, V+Ind+Prs+Sg3
sma | N+Pl+Gen, N+Pl+Com, N+Sg+Com, N+Pl+Ill, N+Ess | N+Pl+Gen, N+Sg+Gen, N+Ess, N+Pl+Ine, N+Pl+Nom
sme |  | A+Comp+Sg+Nom+Qst, A+Comp+Sg+Nom,
---
A+Comp+Attr, A+Comp+Attr+Qst, V+TV+VAbess+Qst
| A+Comp+Sg+Nom+Qst, A+Comp+Sg+Nom,
---
A+Comp+Attr, A+Comp+Attr+Qst, V+TV+VAbess
smj |  | N+Sg+Com+PxSg1, N+Sg+Com+PxSg2,
---
N+Sg+Abe, N+Pl+Abe, N+Pl+Gen+PxSg1
| N+Pl+Abe, N+Sg+Abe, N+Sg+Com+PxSg1,
---
N+Sg+Com+PxSg2, N+Pl+Com+PxSg2
smn |  | N+Pl+Com+Qst, A+Pl+Com+Qst, A+Comp+Pl+Com+Qst,
---
A+Superl+Pl+Com+Qst, V+PrsPrc+Qst
| N+Pl+Com+Qst, N+Pl+Gen+Qst, V+Ind+Prs+Sg3+Qst,
---
A+Pl+Com+Qst, V+Ind+Prs+ConNeg+Qst
sms |  | A+Superl+Sg+Abe, A+Superl+Sg+Abe+Qst/a,
---
A+Superl+Sg+Abe+Qst/ko, V+VAbess+Qst/a, V+VAbess+Qst/ko
| V+Ind+Prt+Pl1, V+VAbess+Qst/a, V+Ind+Prt+Pl1+Qst/ko,
---
V+VAbess+Qst/ko, V+VAbess
udm |  | N+Sg+Ela+PxPl1, N+Sg+Ela+PxSg3,
---
N+Sg+Ela+PxSg2, N+Sg+Ela+PxPl3+Qst, N+Sg+Ela+PxSg1
| V+Ind+Prs+Pl1, V+Ind+Prs+Pl1+Qst,
---
V+Ind+Fut+Pl1+Qst, V+Ind+Fut+Pl1, V+Imprt+Pl2
vro |  | V+Act+Sup+Ine, V+Act+Ind+Prt+Sg2, V+Pss+Ind+Prt+Sg2, V
---
+Pss+PrfPrc, V+Pss+PrfPrc+Sg+Nom
| V+Act+Ind+Prt+Sg2, N+Pl+Ill, V+Pss+PrfPrc,
---
V+Pss+PrfPrc+Sg+Nom, V+Pss+Ind+Prt+Sg2
Table 7: The top 5 morphological forms that were the most difficult to
lemmatize and generate

###  3.3 Error Analysis

In this section, we take a closer look at the result of the Finnish (fin) and
Komi-Zyrian (kpv) models in order to better understand their shortcomings.

####  3.3.1 Finnish

For lemmatization Finnish offered one of the worst results, which makes it an
interesting target for error analysis. Some of the obvious errors are related
to extremely common word formation patterns, which the model for some reason
is not able to generalize. One of these pattern belongs to adjectives and
nouns formed with suffix -inen, for example pienimuotoisissani `in my most
minor (things)' the correct lemmatization would be pienimuotoinen, but the
model returns pienimuotoida, which doesn't mean anything. Interestingly, it
gives very consistently similar forms to different variants of the same word,
so the model appears to believe this is the correct lemma. We can analyze that
out of all Finnish lemmatization errors -inen derivations are involved in 7.7%
of all mistakes. Thereby future work should investigate what can cause such a
gap in the models prediction abilities, as impact in this can lead into rapid
improvements. One phenomena we observed is that Finnish FST also produces
incorrect forms, such as pienimuotoisimmillean, which probably should end into
-een. We can also observe that in many Finnish lemmas that the model does
analyze correctly the forms are compounds. This leaves open the possibility
that the training data has contained either the second component independently
or within a comparable compound, which would had given the model some example.
One lemmatization issue that can be distinguished is that the model doesn't
lemmatize correctly proper names that are written with initial capital letter.
These include several words, for example Unkareinansako `as their Hungaries?'
should be lemmatized as Hungary, but the model returns nkareintaa. What this
shows is that the model struggles with uppercase characters, although those
would ideally be part of the correct lemmatization result.

The Finnish model has problems in generating forms for words ending in
-lainen, as it seems to inflect them as one would inflect the word laine
`wave', such as dominikaanilaineiltasi `$\approx$ from your Dominican waves'
instead of dominikaanilaisiltasi `from your Dominican people'. Also, other
adjectives ending in -inen are problematic such as keväneensä instead of
keväisensä `his spring-like'. In this case, the model has not learned the
typical inflectional category of adjectives ending in -inen. This issue has an
interesting parallel with the same problem being present in the lemmatization
task, described above. This shows that the problems the models encounter are
to some degree parallel to one another in different tasks, and either relate
to the complexity of the linguistic system, or somehow inadequately
represented input.

Interestingly, the generation model has problems with the plural forms of the
abessive and illative case, and often generates the singular form instead of
the plural such as in sähkömittariksesi `for your electricity meter' instead
of sähkömittareiksesi `for your electricity meters' or a completely erroneous
form such as sähkömittaritsiisi instead of sähkömittareihisi `to your
electricity meters'. In these erroneous cases, the model has tried to
pluralize the word, for example sähkömittarit is the correct plural form of
electricity meters in nominative, but it is no longer correct when inflected
in the illative case.

####  3.3.2 Komi-Zyrian

When we examine the lemmatization task, some particularities are obvious in
Komi-Zyrian. For example, many of word forms with interspersed white spaces in
them are not lemmatized correctly. We also see that some complex entries
borrowed from Russian are challenging to lemmatize, possibly due to their
rarity, for example: народно-освободительнӧйджыкъяснысланьджык `more in the
direction of their people who are more national-liberational' would correctly
result in народно-освободительнӧй, but the model predicts народнотильнӧй. In
this case the hyphen within the compound probably contributes to the rarity of
the form itself. Similarly, the model is also struggling when there are words
that follow orthographic conventions more typical to Russian than Komi, for
example областьсаас would be correctly lemmatized as областьса, but the model
predicts областььса. If this reflects the underlying code, model training like
this could be very useful for locating erroneously coded transducers. The
double soft sign would seem to allude to double exponence in the code. The
model also has challenges with rarer orthographical conventions in Komi
vocabulary. For example пипуа-кыддзаинӧйланьсянь `from the direction of my
aspen and birch grove' should be пипуа-кыддзаин `aspen and birch grove', but
we get пипуа-кыдзаин. These shortcomings, however, are relatively rare in the
Zyrian data, and the model learns to lemmatize at high accuracy. Much more so
than Finnish, which could be related to more concatenative morphology of Komi
where the word boundaries can be easier to detect.

In the case of Komi-Zyrian we can observe that a large portion of wrongly
recognized forms results from ambiguity that is inherent to the morphology of
this language. For example, it is not possible to distinguish some of the
cases, such as the inessive and illative, in all forms where they occur. As
the model inevitably returns only one reading, it is clear that the evaluation
accuracy cannot be perfect. This finding is consistent with analogous
ambiguity for other forms in the Skolt Sami (sms) model. There appears to be a
consistency in what is incorrectly predicted in Skolt Sami. When there is a
four-way ambiguity as in the Sg Gen, Sg Acc Sg Nom and Pl Nom, the tag Sg Gen
is consistently predicted to be Pl Nom, leaving the two readings Sg Acc and Sg
Nom out of the dichotomy. Komi models shows similar preferences into specific
categories when there are multiple homonymous possibilities.

In the analysis above it was already briefly discussed that some categories
are difficult to recognize correctly for Permic languages. Another example
like this is seen in the Komi-Zyrian and Komi-Permyak (koi) future tense
marking. As these languages have morphologically marked future in the third
person alone, every first and second person verb in the present tense also
gets a future reading, as both analyses can be seen as correct. One could also
argue, however, that if some analysis is not possible to resolve at this
level, some of the distinctions could be removed or merged at this level of
analysis.

What comes to morphological generation of Komi, the accuracy is rather high.
Some of the errors can be connected to the fact that some suffixes can occur
in varying orders. For example, with input кольквиж A Sg Egr PxPl1 Comp one
could assume the output кольквижнымсяньджык `more from the direction of our
yellows', but in this case the model outputs кольквижсяньнымджык. The only
difference is, however, in the order of markers for case Egr and possessive
suffix PxPl1. The model is actually giving a correct output, but the input
doesn't have all information about the suffix order that the model would need.

There are also instances of word generation where the correct prediction would
demand actual lexicographical knowledge, which the model cannot have. For
example, Komi displays with some nouns an additional stem consonant. It is not
possible to predict from the surface form whether this consonant exists and
what it is. So when the model is given input мек N Sg Ins, it doesn't predict
the correct мекйӧн `with a pelt', but offers the regular but incorrect form
мекӧн. This is a good example from construction where rule-formulated
linguistic knowledge may be necessary for optimal analysis. It also shows that
the model is capable to learn very well the regular structures of the language
and does predict them with high accuracy.

##  4 Conclusions

In this paper, we have presented a method for automatically extracting
inflectional forms from FST transducers by composing a regular expression
transducer for each word with an existing FST transducer. This way, we have
been able to gather very large morphological training data for analysis,
lemmatization and generation for 22 languages, 17 out of which are endangered
and fighting for their survival. We have used this dataset to train neural
models for each language. Because the data follows the tags and conventions
used in the GiellaLT infrastructure, these neural models can be used directly
side by side with the FST transducers in many of the applications that depend
on them.

The results look very good for some languages while being a bit more modest
for others. Analysis seems to be the hardest problem out of the three, and its
training also took the longest time. Despite this, many models reached to an
over 80% accuracy in the tasks. This is rather good given that the evaluation
was conducted entirely on out-of-vocabulary words.

The accuracies reported in this paper are a somewhat lower than what they
could be. This is due to the fact that we ran the evaluation by producing one
result only for each input with the neural models and compared that input
directly to the one in the test data. As we saw in our analysis, many of the
inputs in the test data were ambiguous, which caused the neural model to
produce an output that is correct, but not the one in the test data. However,
the right way to overcome this problem would be to research how to deal with
ambiguity. The neural models we trained can already now produce N best
candidates for each input.

It is probable that within those N best candidates, the models actually cater
for the ambiguity and produce other results that are correct as well. For
instance, the Finnish word noita, can be an accusative singular noun meaning
`witch' or a partitive of nuo meaning `them'. Knowing how to maximize the
number of forms the neural model produces while minimizing the number of
incorrect forms is a question for another paper. Although, some methods could
already be used with the models trained in this paper by introducing simple
modifications to how the results are predicted Silfverberg and Tyers (2019).

Even though we aimed for a real world scale morphological tag complexity by
querying all possibilities from the FSTs, there are still a couple of
morphological categories we did not tackle for practical reasons. One of them
is the use of clitics. The problem with these is that they can be attached to
almost any kind of word regardless of its part-of-speech and inflectional
form. On top of this, multiple clitics can be added one after another. To give
an idea of the scale, with clitics, Finnish has 9425 unique forms for nouns
(instead of 850), 216 for adverbs (instead of 16), 14794 for adjectives
(instead of 1244) and a whopping 88044 forms for verbs (instead of 6667). This
means that clitics need to be solved by taking a different approach than the
one we had. One could, for example, introduce some forms with different
combinations of clitics here and there in the training data, in which case the
question arises on how many forms need to appear with clitics in order for the
model to generalize their usage.

Compounds and derivations could not be included because of how the FSTs were
implemented. If you ask an FST for compounds and derivations, you will surely
get them! Even in such quantities that your computer will run out of RAM and
swap memory for the forms of a single word, as there is no limit to how many
words can be written together to form a compound or how many times one can
derive a new word from another. We people might have our cognitive limits for
that, but the FSTs will not444Take, for instance, a look at this derivational
Skolt Sami word produced by the FST Piân’njatõõvvõlltâs-
ttiatemesvuõt’tsážvuõðtõvvstõlškuät’tteškuättõõlstõlstââst- stõõstčâtttömâs
for piânnai+N+Der/Dimin+Der/N2A+Der/toovvyd
+Der/oollyd+Der/jed+V+Der/Caus+Der/Dimin+Der/NomAg+N
+Der/Dimin+Der/N2A+Der/teqm+A+Attr+Der/vuott+N
+Der/sazh+A+Err/Orth+Attr+Der/vuott+N+Der/toovvyd
+Err/Orth+Der/stoollyd+V+Der/shkueqtted+Der/jed
+V+Der/Caus+Der/shkueqtted+Der/oollyd+Der/stoollyd
+Der/Dimin+V+Der/Dimin+Der/Dimin+V+Der/Dimin+Der/ched
+Der/Caus+Der/t+A+Superl+Attr. The problem of compounds is probably best to
leave for a separate model to solve, as there are already methods out there
for predicting word boundaries Shao et al. (2018); Seeha et al. (2020). The
compound splits by such methods could then be fed into the neural models
trained in this paper. As for derivations, some of them could be included in
the training data, but the question of how many forms are needed would still
require further research.

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