# Few remarks on the Poincaré metric on a singular holomorphic foliation

Sahil Gehlawat  Université de Lille, Laboratoire de Mathématiques Paul
Painlevé, CNRS U.M.R. 8524, 59655 Villeneuve d’Ascq Cedex, France.
<EMAIL_ADDRESS>
###### Abstract.

Let $\mathcal{F}$ be a Riemann surface foliation on $M\setminus E$, where $M$
is a complex manifold and $E\subset M$ is a closed set. Assume that $\cal{F}$
is hyperbolic, i.e., all leaves of the foliation $\cal{F}$ are hyperbolic
Riemann surface. Fix a hermitian metric $g$ on $M$. We will consider the
Verjovsky’s modulus of uniformization map $\eta$, which measures the largest
possible derivative in the class of holomorphic maps from the unit disk into
the leaves of $\cal{F}$. Various results are known to ensure the continuity of
the map $\eta$ along the transverse directions, with suitable conditions on
$M$, $\cal{F}$ and $E$. For a domain $U\subset M$, let $\cal{F}_{U}$ be the
holomorphic foliation given by the restriction of $\cal{F}$ to the domain $U$,
i.e., $\cal{F}|_{U}$. We will consider the modulus of uniformization map
$\eta_{U}$ corresponding to the foliation $\cal{F}_{U}$, and study its
variation when the corresponding domain $U$ varies in the Caratheodory kernel
sense, motivated by the work of Lins Neto–Martins.

###### Key words and phrases:

Poincaré metric, kernel convergence, singular Riemann surface foliation

###### 1991 Mathematics Subject Classification:

Primary: 32S65, 32M25 ; Secondary : 30F45

The author is supported by the Labex CEMPI (ANR-11-LABX-0007-01)

##  1\. Introduction

Let $M$ be a complex manifold of dimension $N$, and $E\subset M$ a closed
subset of codimension atleast 2. Recall that a singular holomorphic foliation
by Riemann surfaces $\cal{F}$ with singular set $E$ is defined by an atlas of
flow-boxes $(U_{\alpha},\phi_{\alpha})$ where
$\phi_{\alpha}:U_{\alpha}\rightarrow\mathbb{D}\times\mathbb{D}^{N-1}$
($\mathbb{D}\subset\mathbb{C}$ is the open unit disc) and the transition maps
$\phi_{\alpha\beta}$ are holomorphic in
$(x,y)\in\mathbb{D}\times\mathbb{D}^{N-1}$ and are of the form

$\phi_{\alpha\beta}=\phi_{\alpha\beta}(x,y)=\phi_{\alpha}\circ\phi^{-1}_{\beta}(x,y)=(P(x,y),Q(y)).$

The leaves of $\mathcal{F}$ are Riemann surfaces, which are locally given by
$\phi^{-1}_{\alpha}(\\{y=\mbox{constant}\\})$. A leaf passing through a given
$p\in M\setminus E$ will be denoted by $L_{p}$. A hyperbolic Riemann surface
foliation $(M,E,\mathcal{F})$ is a singular holomorphic foliation in which
every leaf $L_{p}$ of $\cal{F}$ is a hyperbolic Riemann surface. Fix a
hermitian metric $g$ on $M$ and denote the length of a tangent vector $v$ by
$|v|_{g}$. Consider the space $\mathcal{O}(\mathbb{D},M\setminus E)$, which is
the set of all holomorphic maps from the unit disc
$\mathbb{D}\subset\mathbb{C}$ into the complex manifold $M\setminus E$. Now
consider the subset
$\mathcal{O}(\mathbb{D},\cal{F})\subset\mathcal{O}(\mathbb{D},M\setminus E)$,
the set of holomorphic maps $f\in\mathcal{O}(\mathbb{D},M\setminus E)$ with
values in a leaf of the foliation $\cal{F}$, that is $f(\mathbb{D})\subset L$,
for some leaf $L$ of $\cal{F}$. Consider the map $\eta:M\setminus
E\rightarrow(0,\infty)$ given by

$\eta(z):=\sup\left\\{|f^{\prime}(0)|_{g}:f\in\mathcal{O}(\mathbb{D},\mathcal{F}),f(0)=z\right\\}.$

One can check that $\eta>0$, and that it achieves its extremal value with the
map $\pi_{z}$, where $\pi_{z}:\mathbb{D}\rightarrow L_{z}$ is the universal
covering map chosen such that $\pi_{z}(0)=z$ . Thus,
$\eta(z)=|\pi^{\prime}_{z}(0)|_{g}$. For a leaf $L\subset\cal{F}$ and a vector
$v$ tangent to $L$ at $z$, let $\lambda_{L}$ denote the Poincaré metric on $L$
and $|v|_{L}$ denote the Poincaré length of $v$. The extremal property of the
Kobayashi metric shows that $|v|_{g}=\eta(z)|v|_{L}$. Symbolically, this will
be written as $g=\eta\cdot\lambda_{L}$, and it can be understood that the
restriction of $g/\eta$ to a leaf $L$ induces the corresponding Poincaré
metric $\lambda_{L}$ on it.

Since the Poincaré metric $\lambda_{L}$ for a leaf $L$ is a positive real-
analytic function on $L$ and $g$ is smooth on $M$, therefore $\eta$ is also
smooth along leaves. However, the regularity of $\eta$ along transverse
directions is not a priori clear. In fact, the regularity of $\eta$ is related
to the variation of $\lambda_{L}$ along such directions. This question of the
regularity of $\eta$ corresponding to a hyperbolic singular foliation
$\cal{F}$ has been studied under various hypotheses on $M,E$ and
$\mathcal{F}$. One can have a look at Verjovsky [18], Lins Neto [13, 14],
Candel [4] in which the continuity of $\eta$ on $M\setminus E$ is established.
Recently, Dinh–Nguyên–Sibony [7, 8] gave a stronger estimate on the modulus of
continuity of $\eta$ under suitable hypotheses on $M,E$ and $\mathcal{F}$.
Related work on this theme can be found in [3], [6], [12], [9] and [10]. To
see few ramifications of studying the transverse regularity of the leafwise
Poincaré metric, one can look at [16], [17], where Nguyen highlights its
utility in studying the ergodic properties of Riemann surface laminations.

In [15], the variation of $\eta$ with respect to its domain of definition was
studied. To describe it, let $U\subset M$ be an open subset, denote the
restriction of the foliation $\mathcal{F}$ to $U$, that is $\cal{F}|_{U}$ by
$\mathcal{F}_{U}$. Let $\eta_{U}:U\rightarrow(0,\infty)$ be the positive
function corresponding to the foliation $\mathcal{F}_{U}$. Note that
$\eta_{U}$ is defined using holomorphic maps from $\mathbb{D}$ into $L_{p,U}$
as $p$ varies in $U$. The following facts were established in [15] by Lins
Neto–Martins. First, that $\eta_{U}$ is a monotone function of $U$, i.e., for
domains $U_{1}\subset U_{2}$, $\eta_{U_{1}}(p)\leq\eta_{U_{2}}(p)$ for all
$p\in U_{1}$. Also, if $\\{W_{n}\\}$ is an increasing exhaustion of $W$ and
the singular set of foliation $\cal{F}$ is discrete, then
$\eta_{W_{n}}\rightarrow\eta_{W}$ pointwise on $W\setminus E$. The convergence
is uniform on compact subsets of $W\setminus E$ if the functions
$\eta_{W_{n}}$ and $\eta_{W}$ are continuous. Similar observations about the
pointwise convergence of $\eta_{W_{n}}$ were made by the author (with co-
author Kaushal Verma) in [11], where the singular set $E$ is assumed to be
discrete, $W$ be a bounded taut domain and $W_{n}$ converges to $W$ in a
Hausdorff sense. The uniform convergence of $\eta_{W_{n}}$ to $\eta_{W}$ on
the compacts of $W\setminus E$ was also established in [11], with the
additional hypotheses that $\eta_{W}$ is continuous on $W\setminus E$ and that
there exist another bounded taut domain $V\subset M$ such that
$\overline{W}\subset V$. In [12], these observations were proved to be true
for the case when $E$ is not discrete. It was also proved that if the
foliation $\cal{F}$ is transversal type (see [12] for the definition), then
the uniform convergence will be on the compacts of $W$ instead of $W\setminus
E$.

The aim of this note is to study the variation of the function $\eta_{W_{n}}$
of the foliation $\cal{F}_{W_{n}}$ when the domains $W_{n}$ converges to the
domain $W$ in the kernel sense (see below for definitions). We will observe
that outside a defective set, the convergence results still holds true. We
will also see that these results are a generalization to the similar
observations made in [11, 12] for the case when domain $U$ varies in Hausdorff
sense, and in [15] where $U$ varies in monotonic increasing fashion. To keep
things simple, unless otherwise mentioned we will fix
$M=U\subset\mathbb{C}^{N}$, where $U$ is a connected open subset with the
metric $g$ to be the Euclidean metric throughout the article. Consider
$W_{n},W\subset U\subset\mathbb{C}^{N}$ be connected open subsets with $0\in
W_{n},W$ for all $n$. Let $\cal{F}$ a hyperbolic singular holomorphic
foliation on $U$ with singular set $E\subset U$. Throughout the article, we
will denote the foliation $\cal{F}_{W_{n}}$ by $\cal{F}_{n}$, the function
$\eta_{W_{n}}$ by $\eta_{n}$, and the leaf $L$ of $\cal{F}$ on $U$ passing
through $p\in U$ by $L_{p}$.

######  Definition 1.1.

Let $W_{n}\subset U$ be a sequence of connected open subsets of $U$ such that
$0\in W_{n}$ for all $n$. Let $\tilde{V}_{n}:=\cap_{k\geq n}W_{k}$ and $V_{n}$
be the connected component of the interior of $\tilde{V}_{n}$ containing $0\in
U$, that is $V_{n}=(\text{int}(\tilde{V}_{n}))_{0}$. The kernel of the
sequence $\\{W_{n}\\}_{n\in\mathbb{N}}$, denoted by $W$ is defined to be the
$\cup_{n\geq 1}V_{n}$ if it is non-empty, otherwise we say it is $\\{0\\}$.

We will say that $W_{n}$ converges to $W$, if every subsequence of
$\\{W_{n}\\}_{n\in\mathbb{N}}$ has the same kernel $W$.

Now we can define the kernel convergence of a sequence of domains.

######  Definition 1.2.

Let $W_{n}\subset U\subset\mathbb{C}^{N}$ be a sequence of connected open
subsets of $U$ such that $0\in W_{n}$ for all $n$, and $W\subset U$ be another
connected open set with $0\in W$. We say that $W_{n}$ converges to $W$ in the
kernel sense if $W$ is the kernel of the sequence
$\\{W_{n}\\}_{n\in\mathbb{N}}$ and $W_{n}$ converges to its kernel.

Note: We can check that if $W_{n}\to W$ in the kernel sense, then for any
compact set $K\subset W$, there exist $N_{K}\in\mathbb{N}$ such that $K\subset
W_{n}$ for all $n\geq N_{K}$.

The following example shows that even the pointwise convergence of $\eta_{n}$
to $\eta_{W}$ is not guaranteed on the whole set $W\setminus E$, when
$W_{n}\to W$ in the kernel sense.

######  Example 1.3.

Let $U=P(0,(5,5))\subset\mathbb{C}^{2}$ be the polydisc of polyradius
$(r_{1},r_{2})=(5,5)$ and centre $0\in\mathbb{C}^{2}$. Take
$W_{n}=P(0,(1,1))\cup P(0,(2,\frac{1}{n}))$ and $W=P(0,(1,1))$. Now consider
the vector field $X(z)=z$ on $U$ and let $\cal{F},\cal{F}_{n},\cal{F}_{W}$ be
the foliation induced by $X$ on the set $U,W_{n},W$ respectively. Here, the
singular set of the foliation is the singleton set $E=\\{0\\}$. Since $U$ is
bounded, the foliations $\cal{F},\cal{F}_{n},\cal{F}_{W}$ are all hyperbolic.

Observe that $P(0,(2,\frac{1}{n}))\searrow D(0,2)\times\\{0\\}$ as
$n\to+\infty$, that is

$\cap_{k\geq n}P(0,(2,\frac{1}{k}))=D(0,2)\times\\{0\\}.$

This tells us that the set $\tilde{V}_{n}=\cap_{k\geq n}W_{n}$ satisfies
$\text{int}(\tilde{V}_{n})=P(0,(1,1))=W$. Therefore, the kernel of the
sequence $W_{n}$ is $W$. One can easily check that this is true for all
subsequences of $\\{W_{n}\\}_{n\in\mathbb{N}}$. Thus, $W_{n}\to W$ in the
kernel sense.

Now take $p=(\frac{1}{2},0)\in U$, and observe that $p\in W_{n},W$ for all
$n\geq 1$. Denote the leaf of foliation $\cal{F}$ (or
$\cal{F}_{n},\cal{F}_{W}$) passing through point $p$ by $L_{p}$ (or
$L_{p,n},L_{p,W}$ respectively). One can easily check that
$L_{p,n}=D(0,2)\setminus\\{0\\}\times\\{0\\}\cong D(0,2)\setminus\\{0\\}$, and
$L_{p,W}=D(0,1)\setminus\\{0\\}\times\\{0\\}\cong\mathbb{D}^{*}$. Since the
Poincaré metric of the punctured disc $\mathbb{D}^{*}$ is given by

$ds^{2}=\frac{4}{|q|^{2}(\log{|q|^{2})^{2}}}|dq|^{2},$

and on $D(0,2)\setminus\\{0\\}$ by

$ds^{2}=\frac{4}{|q|^{2}(\log{\frac{|q|^{2}}{4})^{2}}}|dq|^{2}.$

Therefore

$\eta_{n}^{2}(p)=\frac{(\frac{1}{2})^{2}(\log{16})^{2}}{4},\;\eta_{W}^{2}(p)=\frac{(\frac{1}{2})^{2}(\log{4})^{2}}{4}.$

Thus, $\eta_{n}(p)$ doesn’t converge to $\eta_{W}(p)$, although $W_{n}\to W$
in the kernel sense.

But all is not lost though. We will see that the pointwise convergence will be
true away from a subset which we will call as defective set.

######  Definition 1.4.

Consider $(W_{n},W,U,\cal{F})$ be as in the above discussion. Now for a
subsequence $\sigma=\\{n_{k}\\}_{k\in\mathbb{N}}$ of the sequence
$\\{n\\}_{n\in\mathbb{N}}$, consider

$F_{\sigma}:=\\{p\in
U\setminus\overline{W}:\exists\\{p_{n_{k}}\\}_{k\in\mathbb{N}}\subset U\
\text{such that}\ p_{n_{k}}\in W_{n_{k}}\setminus\overline{W}\ \forall\ k,\
\text{and}\ p_{n_{k}}\to p\\}.$

Let $F:=\cup_{\sigma}F_{\sigma}$, where $\sigma$ varies over all possible
subsequence. The set $S:=\cup_{q\in F}(L_{q}\cap W)\subset W$ will be called
as the defective set corresponding to the foliation $\cal{F}$.

Note that $S\subset W$ is a $\cal{F}_{W}-$invariant set since it is union of
leaves in $W$.

######  Theorem 1.5.

Let $(W_{n},W,U,\cal{F})$ be as above with the additional hypothesis that $W$
is a bounded taut domain. If $W_{n}\to W$ in the kernel sense, then
$\eta_{n}(p)\to\eta_{W}(p)$ for all $p\in W\setminus(S\cup E)$.

The following proposition will be used in proving the above theorem.

######  Proposition 1.6.

Let $(W_{n},W,U,\cal{F})$ be as above. If $W$ is the kernel of the sequence
$\\{W_{n}\\}_{n\in\mathbb{N}}$, then

$\liminf_{n\to+\infty}{\eta_{n}(p)}\geq\eta_{W}(p),$

for all $p\in W\setminus E$.

###### Proof.

Let $p\in W\setminus E$, and $\alpha:\mathbb{D}\to L_{p,W}$ be the
uniformization of the leaf $L_{p,W}$ of the foliation $\cal{F}_{W}$. For
$\epsilon>0$, consider the map $\psi:\mathbb{D}\to\mathbb{D}$ given by
$\psi(z)=(1-\epsilon)z$. Now consider the map $\alpha\circ\psi:\mathbb{D}\to
L_{p,W}$ and observe that $\alpha\circ\psi(\mathbb{D})$ is relatively compact
in $W$. From the definition of kernel, we know $W=\cup_{n\geq 1}V_{n}$, where
$V_{n}$ satisfies $V_{k}\subset V_{k+1}$ for all $k$. This tells us that there
exist $N\in\mathbb{N}$ such that $\alpha\circ\psi(\mathbb{D})\subset V_{N}$,
which in turn gives us that $\alpha\circ\psi(\mathbb{D})\subset W_{k}$ for all
$k\geq N$. Thus $\alpha\circ\psi(\mathbb{D})\subset L_{p}\cap W_{k}$ and
$p\in\alpha\circ\psi(\mathbb{D})$. Since $L_{p,k}$ is the connected component
of $L_{p}\cap W_{k}$ containing $p$, therefore
$\alpha\circ\psi(\mathbb{D})\subset L_{p,k}$ for all $k\geq N$. By definition
of the map $\eta_{k}$, we get for $k\geq N$

$\eta_{k}(p)\geq|(\alpha\circ\psi)^{\prime}(0)|=(1-\epsilon)\eta_{W}(p).$

This gives us that $\liminf_{n\to+\infty}{\eta_{n}(p)}\geq\eta_{W}(p)$ for all
$p\in W\setminus E$. ∎

Now we shift our focus on the uniform convergence of $\eta_{n}$. First note
that we can extend the map $\eta$ on the singular set $E$ by defining
$\eta(p)=0$ for all $p\in E$. Before we state our next result, lets quickly
recall what we mean by a transversal type foliation. For a singular point
$p\in E$ of a holomorphic foliation $\cal{F}$ on a complex manifold $M$, we
consider the cone $C_{p}(\cal{F})\subset T_{p}{M}$ which is the collection of
all limit points of the vectors coming from the tangent line bundle of the
foliation $T\cal{F}=\cup_{q\in M\setminus E}T_{q}{\cal{F}}$ on $M\setminus E$.
Also, since $E\subset M$ is an analytic subset, we will denote the Whitney’s
$C_{4}$-tangent cone (see [5]) at $p\in E$ by $C_{p}E$. We say that $\cal{F}$
is transversal type at $p\in E$, if for some neighourhood $U_{p}\subset M$ of
$p$, $\overline{C_{q}{\cal{F}}}\cap C_{q}{E}=\\{0\\}$ holds for all $q\in
E\cap U_{p}$. The foliation $\cal{F}$ is transversal type if it is transversal
type at each $p\in E$. For a detailed definition, look at [12].

######  Theorem 1.7.

Let $(W_{n},W,U,\cal{F})$ be as before with the additional hypothesis that
$W,U$ are bounded taut domains. If $W_{n}\to W$ in the kernel sense and
$\eta_{W}$ is continuous on $W\setminus E$, then $\eta_{n}\to\eta_{W}$
uniformly on the compacts of $W\setminus(S\cup E)$. If in addition, $\cal{F}$
is transversal type, then $\eta_{n}\rightarrow\eta_{W}$ uniformly on the
compacts of $W\setminus S$.

It should be noted that not all points in the defective set $S$ of a foliation
$\cal{F}$ are $`$bad’. Given $(W_{n},W,U,\cal{F})$, we will say that a point
$p\in S$ is a removable defective point if $\eta_{n}(p)\to\eta_{W}(p)$. We
will denote the set of all such points by $\hat{S}$.

######  Theorem 1.8.

Suppose $W_{n},W\subset U$ be bounded connected open sets such that $0\in
W_{n},W$ for all $n\in\mathbb{N}$, $W$ is taut, and $W_{n}\to W$ in the kernel
sense. Let $\cal{F}$ be a singular hyperbolic Riemann surface foliation on
$U$, and $S\subset W$ be its defective set. If $p\in F$ satisfies

$\text{int}(L_{p}\cap F)=\emptyset,$

where $\text{int}(L_{p}\cap F)$ denotes the interior of the set $L_{p}\cap F$
when considered as a subset of the Riemann surface $L_{p}$, then

  1. (1)

$\eta_{n}(q)\to\eta_{W}(q)$ for all $q\in L_{p}\cap W$, that is $L_{p}\cap
W\subset\hat{S}$,

  2. (2)

given additional hypothesis of $U$ being taut, and $\eta_{W}$ being continuous
on $W\setminus E$, the uniform convergence $\eta_{n}\to\eta_{W}$ will be true
on the compacts of $(W\setminus(S\cup E))\cup(L_{p}\cap W)$.

What is left now is to show that the results proved in [11] (Theorem 1.1) and
[12] (Theorem 2.1) are special cases of Theorem 1.5 and Theorem 1.7. To do
that lets first recall the type of domain convergence considered in [11],
[12]. Let $(M,g)$ be a complex hermitian manifold and let $d$ be the distance
on $M$ induced by $g$. For $S\subset M$ and $\epsilon>0$, let $S_{\epsilon}$
be the $\epsilon$-thickening of $S$ with distances being measured using $d$.
Recall that the Hausdorff distance $\mathcal{H}(A,B)$ between compact sets
$A,B\subset M$ is the infimum of all $\epsilon>0$ such that $A\subset
B_{\epsilon}$ and $B\subset A_{\epsilon}$. For bounded domains $U,V\subset M$,
the prescription
$\rho(U,V)=\mathcal{H}(\overline{U},\overline{V})+\mathcal{H}(\partial
U,\partial V)$ defines a metric (see [1]) on the collection of all bounded
open subsets of $X$ with the property that if $\rho(U,U_{n})\rightarrow 0$,
then every compact subset of $U$ is eventually contained in $U_{n}$, and every
neighbourhood of $\overline{U}$ contains all the $U_{n}$’s eventually. We will
say that $U_{n}\rightarrow U$ in the Hausdorff sense if $\rho(U_{n},U)\to 0$.

In the following result $M=\mathbb{C}^{N}$, and $g$ is the Euclidean metric.

######  Theorem 1.9.

Let $W_{n},W$ be bounded connected domains in $\mathbb{C}^{N}$ such that $0\in
W_{n},W$ for all $n\in\mathbb{N}$. If $W_{n}\to W$ in the Hausdorff sense,
then $W=\text{ker}(W_{n})$ and $W_{n}\to W$ in the kernel sense with
$F=\emptyset$.

##  2\. Proof of Theorem 1.5

Having Proposition 1.5 at our disposal, it suffices to prove that for $p\in
W\setminus(S\cup E)$ and for some convergent subsequence
$\\{\eta_{n_{k}}(p)\\}_{k\in\mathbb{N}}$, it satisfies

$\lim_{k\to+\infty}{\eta_{n_{k}}(p)}\leq\eta_{W}(p).$

Fix $p\in W\setminus(S\cup E)$. Since $\eta_{n}(p)\leq\eta(p)<+\infty$ (here
$\eta$ corresponds to the set $U$), therefore we can choose a subsequence
$\\{\eta_{n_{k}}(p)\\}_{k\in\mathbb{N}}$ which converges to $M$. Consider the
corresponding uniformizers $\alpha_{n_{k}}:\mathbb{D}\to L_{p,n_{k}}$ with
$\alpha_{n_{k}}(0)=p$. Since $\alpha_{n_{k}}(\mathbb{D})\subset
L_{p,n_{k}}\subset L_{p}$, $\alpha_{n_{k}}(0)=p$ and $L_{p}$ being hyperbolic,
upto a further subsequence we can suppose that $\alpha_{n_{k}}$ converges to
some holomorphic map $\alpha:\mathbb{D}\to L_{p}$, with $\alpha(0)=p$.

Now we just need to show that $\alpha(\mathbb{D})\subset L_{p,W}$. Suppose
there exist a point $q\in\alpha(\mathbb{D})$ such that $q\in
U\setminus\overline{W}$. Let $\alpha(z_{0})=q$. Choose a neighourhood
$U_{q}\subset U$ of $q$ such that
$\overline{U}_{q}\cap\overline{W}=\emptyset$. Take
$q_{n_{k}}:=\alpha_{n_{k}}(z_{0})$, since $\alpha_{n_{k}}\to\alpha$, we have
$q_{n_{k}}\to q$. Therefore there exist $N\in\mathbb{N}$ such that
$q_{n_{k}}\in U_{q}$ for all $k\geq N$. Also
$q_{n_{k}}\in\alpha_{n_{k}}(\mathbb{D})=L_{p,n_{k}}\subset W_{n_{k}}$ for all
$k$. Therefore for $k\geq N$

$q_{n_{k}}\in W_{n_{k}}\setminus\overline{W},$

and $q_{n_{k}}\to q$. Thus $q\in F_{\sigma}\subset F$, and since $q\in L_{p}$,
therefore $p\in S$, which is a contradiction since $p\in W\setminus(S\cup E)$.

Hence $\alpha(\mathbb{D})\subset\overline{W}$, but $W$ is taut so
$\alpha(\mathbb{D})\subset W$. Since $\alpha(\mathbb{D})\subset L_{p}$ is
connected, therefore $\alpha(\mathbb{D})\subset L_{p,W}$. Now
$|\alpha^{\prime}(0)|=\lim_{k\to+\infty}{|\alpha_{n_{k}}^{\prime}(0)|}=\lim_{k\to+\infty}{\eta_{n_{k}}(p)}$,
and by definition of $\eta_{W}$,

$\eta_{W}(p)\geq|\alpha^{\prime}(0)|.$

So we have our result.

##  3\. Proof of Theorem 1.7

Suppose that there is a compact set $K\subset W\setminus(S\cup E)$ and a
sequence $\\{p_{n}\\}\in K$ such that

$|\eta_{n}(p_{n})-\eta_{W}(p_{n})|>\epsilon$

for some $\epsilon>0$. Assume that $p_{n}\rightarrow p\in W\setminus(S\cup E)$
after passing to a subsequence.

By the continuity of $\eta_{W}$, $|\eta_{W}(p_{n})-\eta_{W}(p)|<\epsilon/2$
and hence

(3.1) $|\eta_{n}(p_{n})-\eta_{W}(p)|>\epsilon/2$

for large $n$. Let $\alpha_{n}:\mathbb{D}\rightarrow L_{p_{n},W_{n}}\subset
W_{n}$ be a uniformization map with $\alpha_{n}(0)=p_{n}$. Since
$W_{n},W\subset U$ for all $n$ and $U$ is taut, hence the family
$\\{\alpha_{n}\\}$ is normal. By passing to a subsequence, let
$\tilde{\alpha}:\mathbb{D}\rightarrow\overline{U}$ be a holomorphic limit with
$\tilde{\alpha}(0)=p$. To complete the proof, it suffices to show that
$\tilde{\alpha}:\mathbb{D}\rightarrow L_{p,W}$ is a uniformization. Indeed, if
this were the case, then $\eta_{W}(p)=|\tilde{\alpha}^{\prime}(0)|$ which
would then imply that

$\eta_{n}(p_{n})=|\alpha^{\prime}_{n}(0)|\rightarrow|\tilde{\alpha}^{\prime}(0)|=\eta_{W}(p)$

and this contradicts (3.1). That $\tilde{\alpha}$ uniformizes the leaf
$L_{p,W}$ is a consequence of the following observations:

(a) $\tilde{\alpha}$ satisfies $\tilde{\alpha}(\mathbb{D})\subset W$.

(b) $\tilde{\alpha}$ satisfies $\tilde{\alpha}(\mathbb{D})\subset L_{p,W}\cup
E$.

(c) The map
$\tilde{\alpha}:\mathbb{D}\setminus\tilde{\alpha}^{-1}(E)\rightarrow L_{p,W}$
is a covering map.

(d) $\tilde{\alpha}^{-1}(E)=\emptyset$, and $\tilde{\alpha}(\mathbb{D})\subset
L_{p,W}$.

We will only prove the observations $(a)$ and $(b)$, since the proof of other
two are almost identical to the similar observations made in [11] (see proof
of Theorem 1.1).

(a) Since $U$ is taut and $\tilde{\alpha}(0)=p\in U$, it follows that
$\tilde{\alpha}(\mathbb{D})\subset U$. Now suppose that $\tilde{\alpha}(a)\in
U\setminus\overline{W}$ for some $a\in\mathbb{D}$. Then $\alpha_{n}(a)\in
U\setminus\overline{W}$ for all large $n$, and therefore $\alpha_{n}(a)\in
W_{n}\setminus\overline{W}$ which gives us that $\tilde{\alpha}(a)\in F$. Now
since $\tilde{\alpha}(a)\in L_{p}$, therefore $p\in S$ which is a
contradiction to the fact that $p\in W\setminus(S\cup E)$. Hence
$\tilde{\alpha}:\mathbb{D}\rightarrow\overline{W}$. Again, the tautness of $W$
implies that $\tilde{\alpha}:\mathbb{D}\rightarrow W$.

(b) To see that $\tilde{\alpha}(\mathbb{D})\subset L_{p,W}\cup E$, we will use
the following lemma.

######  Lemma 3.1.

Let $\mathcal{F}$ be a hyperbolic SHFC on a complex manifold $M$, with
singular set $E\subset M$. If $\\{\alpha_{n}\\}_{n\in\mathbb{N}}$ is a
sequence in $\mathcal{O}(\mathbb{D},\cal{F})$, which converges in the compact
parts of $\mathbb{D}$ to some $\alpha:\mathbb{D}\to M$, then
$\alpha(\mathbb{D})\subset L\cup E$, where either $L=\emptyset$ or $L$ is a
leaf of $\mathcal{F}$. In particular, if $L=\emptyset$,
$q\in\alpha(\mathbb{D})\cap E$, and $\mathcal{F}$ is transversal type at $q\in
E$, and then $\alpha\equiv q\in E$.

###### Proof.

If $\alpha$ is constant or $\alpha(\mathbb{D})\subset E$, then we are done.

Now suppose $\alpha$ is non-constant, and
$p\in\alpha(\mathbb{D})\cap(M\setminus E)$. Consider the open set
$N:=\\{z\in\mathbb{D}:\alpha(z)\notin E\\}\neq\emptyset$. Define for $w_{0}\in
N$, the set $N_{w_{0}}=\\{z\in N:L_{z}=L_{w_{0}}\\}$. To see that $N_{w_{0}}$
is open in $\mathbb{D}$, pick $w_{1}\in N_{w_{0}}$ and a chart
$\phi=(x,y):\tilde{W}\rightarrow\mathbb{D}\times\mathbb{D}^{n-1}$ around
$\tilde{\alpha}(w_{1})=q$ such that $(x,y)(q)=0\in\mathbb{C}^{n}$ and the
leaves of $\mathcal{F}_{\tilde{W}}$ are described by $y=\mbox{constant}$. Let
$V$ be the connected component of $\alpha^{-1}(\tilde{W})\subset\mathbb{D}$
containing $w_{1}$. Now, the claim is that $y(\alpha(V))=0$. If this is not
the case, then $y(\alpha(V))\not=\\{0\\}$. Since $y(\alpha(w_{1}))=0$, it
follows that $y\circ\alpha:V\rightarrow\mathbb{D}^{n-1}$ is a non-constant map
which implies that its derivative $d(y\circ\alpha)(w_{2})\not=0$ for some
$w_{2}\in V$ near $w_{1}$. Since $\alpha_{n}$ converges to $\alpha$,
$d(y\circ\alpha_{n})(w_{2})\not=0$ for $n$ large. This means that
$\alpha_{n}(\mathbb{D})$ is not contained in a leaf of $\mathcal{F}$. This
contradiction shows that $y(\alpha(V))=0$. Hence every $w\in V$ has the
property that the leaf of $\mathcal{F}_{\tilde{W}}$ containing $\alpha(w)$
coincides with $y=0$ which is also the leaf passing through $w_{1}$. This
shows that $N_{w_{0}}$ is open.

Now let $F=\alpha^{-1}(E)$. It suffices to show that if $F$ is non-empty, then
$F$ is a discrete set. Suppose $F$ is not discrete, and $z\in D$ is a limit
point of $F$. Since $F$ is closed, $z\in F$. Consider the point
$\alpha(z)=p\in E$. Since $E$ is an analytic subset of $M$, let
$f_{1},f_{2},\ldots,f_{m}$ be the local defining functions for $E$ near the
point $p$, i.e., there exist a neighourhood $V\subset M$ of $p$ such that
$V\cap E=\\{f_{1}=f_{2}=\ldots=f_{m}=0\\}$. Let $r>0$ be such that
$\alpha(D(z,r))\subset V$. Consider the holomorphic functions
$f_{1}\circ\alpha|_{D(z,r)},f_{2}\circ\alpha|_{D(z,r)},\ldots,f_{m}\circ\alpha|_{D(z,r)}$
defined on the disc $D(z,r)$. Since $z$ is an accumulation point of the zero
set $\mathcal{Z}(f_{i}\circ\alpha|_{D(z,r)})$ in $D(z,r)$, for all $1\leq
i\leq m$. Therefore by Identity principle, $f_{i}\circ\alpha|_{D(z,r)}\equiv
0$ on $D(z,r)$ for all $1\leq i\leq m$, which in turn gives us that
$D(z,r)\subset F$. Now if $F\neq\mathbb{D}$, then there exist a point
$\tilde{z}\in\partial(\mathbb{D}\setminus F)$ which is an accumulation point
of $F$ (since $F$ is not discrete). But by above argument $\tilde{z}$ belongs
to the interior of $F$, which is a contradiction. Thus $F=\mathbb{D}$, that is
$\alpha(\mathbb{D})\subset E$ which is a contradiction to the assumption that
$p\in\alpha(\mathbb{D})\cap(M\setminus E)$. Therefore $F$ is a discrete set.

Now if $L=\emptyset$, and consider $q\in\alpha(\mathbb{D})\cap E$ such that
$\mathcal{F}$ is transversal type at $q$. We need to show that $\alpha$ is a
constant map. Suppose not. Let $z_{0}\in\mathbb{D}$, $r>0$ be such that
$\alpha(z_{0})=q$ and $\alpha(D(z_{0},r))\subset U_{q}$. Since
$\alpha|_{D(z_{0},r)}:D(z_{0},r)\rightarrow U_{q}\cap E$ is non-constant, take
$z_{1}\in D(z_{0},r)$ be such that $\alpha^{\prime}(z_{1})\neq 0$, and since
$\alpha_{n}^{\prime}(z_{1})\rightarrow\alpha^{\prime}(z_{1})$, therefore
$\alpha^{\prime}(z_{1})\in C_{\alpha(z_{0})}\mathcal{F}$. Also since
$\alpha(\mathbb{D})\subset E$, therefore $\alpha^{\prime}(z_{1})\in
C_{\alpha(z_{1})}E$, which is a contradiction to the transversality property
of the foliation $\cal{F}$ in the neighourhood $U_{q}$. Therefore,
$\alpha\equiv q\in E$. ∎

Since each $\alpha_{n}\in\cal{O}(\mathbb{D},\cal{F})$, therefore by above
lemma $\tilde{\alpha}:\mathbb{D}\to L_{p}\cup E$. From $(a)$, we already saw
that $\tilde{\alpha}(\mathbb{D})\subset W$. Since
$p\in\tilde{\alpha}(\mathbb{D})$ and
$\mathbb{D}\setminus(\tilde{\alpha})^{-1}(E)$ is connected, therefore
$\tilde{\alpha}:\mathbb{D}\to L_{p,W}\cup E$.

Now if $\mathcal{F}$ is of transversal type, in addition to the above
hypothesis. To show that the convergence is uniform on compacts of $W\setminus
S$, it suffices to prove that for $p\in E$, and $p_{n}\in W\setminus(S\cup E)$
such that $p_{n}\to p$,

$\eta_{n}(p_{n})\to\eta_{W}(p)=0.$

Suppose not, then $|\eta_{n}(p_{n})|>\epsilon$. Consider the uniformization
map $\alpha_{n}$ of the leaf $L_{p_{n},W_{n}}$ with $\alpha_{n}(0)=p_{n}$.
Therefore after passing to a subsequence we get that
$\\{\alpha_{n}\\}_{n\in\mathbb{N}}$ converges uniformly on the compacts of
$\mathbb{D}$ to a map $\tilde{\alpha}:\mathbb{D}\rightarrow W$ with
$\tilde{\alpha}(0)=p\in E$. Since $\alpha_{n}\in\cal{O}(\mathbb{D},\cal{F})$
for all $n$, therefore by Lemma 3.1, $\tilde{\alpha}$ is constant and
$\tilde{\alpha}\equiv p$. Thus $\eta_{n}(p_{n})\to 0$ which is a
contradiction.

Thus, $\eta_{n}$ converges to $\eta_{W}$ uniformly on compact subsets of
$W\setminus S$.

##  4\. Proof of Theorem 1.8

(1) We need to show $\eta_{n}(q)\to\eta_{W}(q)$ holds true for all $q\in
L_{p}\cap W$. To see this, suppose there exist $q\in L_{p}\cap W$ such that
$\eta_{n}(q)\not\to\eta_{W}(q)$. Upto a subsequence, we can suppose that
$\eta_{n}(q)\to R\neq\eta_{W}(q)$, for some constant $R>0$. Let
$\alpha_{n}:\mathbb{D}\to L_{q,n}$ be the uniformizing map such that
$\alpha_{n}(0)=q$. Upto further subsequence, we can suppose that
$\alpha_{n_{k}}\to\alpha:\mathbb{D}\to L_{q}$ with $\alpha(0)=q$. Since $W$ is
a bounded taut domain, it suffices to show that
$\alpha(\mathbb{D})\subset\overline{W}$ to get a contradiction to the
assumption $\eta_{n_{k}}(q)\not\to\eta_{W}(q)$. Now suppose if possible there
exist $\tilde{q}=\alpha(z_{0})\in U\setminus\overline{W}$. Take a neighourhood
$U_{\tilde{q}}$ of $\tilde{q}$ such that $\overline{U_{\tilde{q}}}\subset
U\setminus\overline{W}$. Let $r>0$ be such that $\alpha(D(z_{0},r))\subset
U_{\tilde{q}}$. Observe that $\alpha(D(z_{0},r))$ is connected and since
$\alpha$ is not a constant map, therefore $\alpha(D(z_{0},r))$ is open in the
hyperbolic Riemann surface $L_{p}$. Choose $\epsilon>0$ such that

$\overline{(U_{\tilde{q}})_{\epsilon}}\subset U\setminus\overline{W},$

where $(U_{\tilde{q}})_{\epsilon}$ denotes the $\epsilon-$neighourhood of the
domain $U_{\tilde{q}}$. Since $\alpha_{n_{k}}\to\alpha$ uniformly on compacts
of $\mathbb{D}$, therefore there exist $N_{1}>0$ such that
$\alpha_{n_{k}}(D(z_{0},r))\subset(U_{\tilde{q}})_{\epsilon}$ for all $k\geq
N_{1}$. Now clearly $\alpha_{n_{k}}(z)\in W_{n_{k}}\setminus\overline{W}$ for
all $k\geq N_{1}$ and for all $z\in D(z_{0},r)$. This gives us that
$\alpha(z)\in F$ for all $z\in D(z_{0},r)$, that is $\alpha(D(z_{0},r))\subset
F$. So $\alpha(D(z_{0},r))\subset F\cap L_{q}=F\cap L_{p}$, which is a
contradiction since $L_{p}\cap F$ has empty interior in $L_{p}$. Thus
$\alpha(\mathbb{D})\subset\overline{W}$.

(2) Suppose on the contrary that $\eta_{n}$ doesn’t converge to $\eta_{W}$
uniformly on the compacts of $(W\setminus(S\cup E))\cup(L_{p}\cap W)$. Let
$K\subset W$ be a compact set on which $\eta_{n}\not\to\eta_{W}$ uniformly.
Then for $\epsilon>0$ and upto a subsequence, there exist a sequence $q_{n}\in
W$ such that for all $n\in\mathbb{N}$

$|\eta_{n}(q_{n})-\eta_{W}(q_{n})|\geq\epsilon.$

Upto another subsequence, suppose $q_{n}\to q$. Since $\eta_{W}$ is continuous
on $W\setminus E$, therefore
$|\eta_{W}(q_{n})-\eta_{W}(q)|<\frac{\epsilon}{4}$, for all $n\geq N$ for some
positive $N>0$. So, we get for all $n\geq N$

(4.1) $|\eta_{n}(q_{n})-\eta_{W}(q)|\geq\frac{\epsilon}{2}.$

If $q\in W\setminus(S\cup E)$, Theorem 1.7 gives us a contradiction to the
above inequality. So, it suffices to assume that $q\in(L_{p}\cup W)$ and get a
contradiction. Let $\alpha_{n}:\mathbb{D}\to L_{q_{n},n}$ be the
uniformization of the leaf $L_{q_{n},n}$ in $W_{n}$, such that
$\alpha_{n}(0)=q_{n}$. Since $\alpha_{n}(\mathbb{D})\subset U$ for all $n$,
and $U$ being taut, therefore $\alpha_{n}$ converges to a map
$\alpha:\mathbb{D}\to U$ uniformly on compacts of $\mathbb{D}$. Again
$\alpha_{n}\in\cal{O}(\mathbb{D},\cal{F})$ for all $n\in\mathbb{N}$, therefore
Lemma 3.1 tells us that $\alpha(\mathbb{D})\subset L_{q}\cup E$. If we prove
that $\alpha(\mathbb{D})\subset L_{q,W}\cup E$, then the exact arguments that
we made in the proof of Theorem 1.7 can be used to show that
$\alpha(\mathbb{D})\subset L_{q,W}$. In fact, $\alpha$ will be a
uniformization of the leaf $L_{q,W}$, which will give a contradiction to the
inequality (4.1), thus giving us the result.

Here, it suffices to show that $\alpha(\mathbb{D})\subset\overline{W}$. To see
this, note that using tautness of $W$, we have $\alpha(\mathbb{D})\subset W$,
and therefore

$\alpha(\mathbb{D})\subset W\cap(L_{q}\cup E).$

Now, since $\alpha(\mathbb{D})$ is connected and $\alpha(0)=q$, we get
$\alpha(\mathbb{D})\subset L_{q,W}\cup E$.

Suppose if possible that there exist $s=\alpha(z_{0})\in
U\setminus\overline{W}$ for some $z_{0}\in\mathbb{D}$. As we did in the proof
of $(1)$, we choose a neighourhood $U_{s}\subset U$ of $s$ and $\delta>0$
small enough such that $\overline{(U_{s})_{\delta}}\subset
U\setminus\overline{W}$. Now choose $r>0$ such that $\alpha(D(z_{0},r))\subset
U_{s}$. Since $\alpha_{n}\to\alpha$ uniformly on compacts of $\mathbb{D}$,
therefore there exist $N_{1}>0$ such that
$\alpha_{n}(D(z_{0},r))\subset(U_{s})_{\delta}$ for all $n\geq N_{1}$. Observe
that for all $z\in D(z_{0},r)$ and for all $n\geq N_{1}$, we have
$\alpha_{n}(z)\in W_{n}\setminus\overline{W}$, and therefore $\alpha(z)\in F$.
Thus we get

$\alpha(D(z_{0},r))\subset L_{p}\cap F\cap E.$

As $\alpha(0)=q\notin U_{s}$, therefore $\alpha:\mathbb{D}\to L_{p}\cup E$
cannot be a constant map. Again, Lemma 3.1 tells us that
$(\alpha)^{-1}(E)\subset\mathbb{D}$ is discrete, and so
$\mathbb{D}\setminus\alpha^{-1}(E)$ is connected. Therefore, the map
$\alpha|_{\mathbb{D}\setminus\alpha^{-1}(E)}:\mathbb{D}\setminus\alpha^{-1}(E)\to
L_{p}$ is open. We can choose $0<\tilde{r}<r$ such that
$\alpha(D(z_{0},\tilde{r})\setminus\\{z_{0}\\})\subset L_{p}$ and is open in
$L_{p}$, that is $\alpha(D(z_{0},\tilde{r})\setminus\\{z_{0}\\})\subset
L_{p}\cap F$ which is a contradiction to the assumption that $L_{p}\cap F$ has
empty interior. Thus $\alpha(\mathbb{D})\subset\overline{W}$ and we have our
result.

##  5\. Proof of Theorem 1.9

We will prove this in 3 steps.

$(1)$ The first step is to show that $W=\text{ker}(W_{n})$. Let
$W_{0}:=\text{ker}(W_{n})$ and let $p\in W$. Consider a connected neighourhood
$U_{p}$ of $p$ in $W$ such that $0\in U_{p}$ and $\overline{U}_{p}\subset W$
is compact. Therefore there exist $N_{1}>0$ such that

$\overline{U}_{p}\subset W_{n}$

for all $n\geq N_{1}$, which gives us that $\overline{U}_{p}\subset\cap_{n\geq
N_{1}}W_{n}=\tilde{V}_{N_{1}}$. Since $0\in U_{p}$ and $U_{p}$ is connected,
therefore $p\in V_{N_{1}}\subset W_{0}$. Thus $W\subset W_{0}$.

Now let $p\in W_{0}$. Therefore, $p\in V_{N_{2}}$ for some $N_{2}>0$. We can
choose a neighourhood $U_{p}$ of $p$ in $V_{N_{2}}$ such that $0\in V_{N_{2}}$
and $\overline{U}_{p}\subset V_{N_{2}}$ is compact. So, we get
$\overline{U}_{p}\subset W_{n}$ for all $n\geq N_{2}$. Suppose if possible
$p\notin\overline{W}$. Therefore there exist $N_{3}>0$ such that $p\notin
W_{n}$ for all $n\geq N_{3}$ which is a contradiction since
$p\in\overline{U}_{p}\subset W_{n}$ for all $n\geq N_{2}$. Thus
$p\in\overline{W}$. Now suppose if possible $p\in\partial W$. Since
$H(\partial W_{n},\partial W)\to 0$, therefore for any neighourhood
$\hat{U}_{p}$ of $p$, there exist $N_{4}>0$ such that

$\partial W_{n}\cap\hat{U}_{p}\neq\emptyset$

for all $n\geq N_{4}$, which is a contradiction since $U_{p}\cap\partial
W_{n}=\emptyset$ for all $n\geq N_{2}$. So $W_{0}\subset W$.

Thus $W=W_{0}=\text{ker}(W_{n})$.

$(2)$ In this step, we will show that $W_{n}\to W$ in the kernel sense. By
definition, we need to show that for any subsequence
$\\{W_{n_{j}}\\}_{j\in\mathbb{N}}$, we have $\text{ker}(W_{n_{j}})=W$. It is
easy to see that $\rho(W_{n},W)\to 0$ implies $\rho(W_{n_{j}},W)\to 0$ as
$j\to+\infty$. Now using Step $(1)$, we get $\text{ker}(W_{n_{j}})=W$. Thus
$W_{n}\to W$ in the kernel sense.

$(3)$ The final step is to show that $F=\emptyset$. Since
$F=\cup_{\sigma}F_{\sigma}$. Consider any subsequence
$\sigma=\\{W_{n_{j}}\\}_{j\in\mathbb{N}}$ of the sequence
$\\{W_{n}\\}_{n\in\mathbb{N}}$ such that $F_{\sigma}\neq\emptyset$. Let $p\in
F_{\sigma}$ which by definition tells us that $p\notin\overline{W}$.
Therefore, there exist a sequence of points $\\{p_{n_{j}}\\}_{j\in\mathbb{N}}$
such that $p_{n_{j}}\in W_{n_{j}}\setminus\overline{W}$ for all $j$, and
$p_{n_{j}}\to p$. We can choose a neighourhood $U_{p}$ of $p$ and $\epsilon>0$
small enough so that

$\overline{U}_{p}\cap(W)_{\epsilon}=\emptyset,$

where $(W)_{\epsilon}$ is the epsilon neighourhood of the domain $W$.
Therefore, there exist $N_{5}>0$ such that $\overline{U}_{p}\cap
W_{n}=\emptyset$, for all $n\geq N_{5}$. This is a contradiction since
$p_{n_{j}}\in\overline{U}_{p}\cap W_{n_{j}}$ for all $n_{j}\geq N_{6}$, for
some $N_{6}>0$.

Thus $F_{\sigma}=\emptyset$, which gives us that $F=\emptyset$.

##  6\. Remarks & Examples

Remark 1: It should be noted that this defective set $S$ is not always very
small. It is possible that $S$ could be dense in $W$. Suppose
$U=P(0,(5,5))\subset\mathbb{C}^{2}$. Consider a countable dense subset
$\\{q_{j}\\}_{j\in\mathbb{N}}\subset\partial W$, where $W=P(0,(1,1))$. Let
$L_{j}$ be the complex line passing through the point $q_{j}$ and the origin
in $U$. Consider a sequence of decreasing neighourhoods
$\\{R_{j,m}\\}_{m\in\mathbb{N}}$ of the line $L_{j}$ for each $j$, such that

$\cap_{m\in\mathbb{N}}R_{j,m}=L_{j},$

for all $j\in\mathbb{N}$. Now define $W_{j,m}:=P(0,(1,1))\cup R_{j,m}\subset
U$ for all $j,m\in\mathbb{N}$. Now we would like to define a sequence of
domains $\\{W_{n}\\}_{n\in\mathbb{N}}$ which contains $W_{j,m}$ for all
$j,m\in\mathbb{N}$. One way is to start with $W_{1}:=W_{1,1}$ and follow the
process shown in the Figure 1, with the first few terms would be
$W_{1,1},W_{1,2},W_{2,1},W_{3,1},W_{2,2},W_{1,3}$, and so on.

Figure 1.

It is easy to see that $W_{n}\to W$ in the kernel sense, and for each
$j\in\mathbb{N}$, $\\{W_{j,m}\\}_{m\in\mathbb{N}}$ is a subsequence of
$\\{W_{n}\\}_{n\in\mathbb{N}}$. Therefore, the defective set $S_{j}=L_{j}\cap
W$ corresponding to the subsequence $\\{W_{j,m}\\}_{m\in\mathbb{N}}$ is subset
of the defective set $S$ of the sequence of domains
$\\{W_{n}\\}_{n\in\mathbb{N}}$. Thus $L_{j}\cap W\subset S$ for all
$j\in\mathbb{N}$. But note that $\cup_{j\geq 1}(L_{j}\cap W)$ is a dense
subset of W. Therefore, the defective set $S$ is a dense subset of $W$.

Remark 2: Although we have proved the results in this note for connected
domains in $\mathbb{C}^{N}$, but one can check that everything still follows
along the same lines if we suppose $U$ to be an open subset of a complex
hermitian manifold $(M,g)$, where $g$ is a smooth hermitian metric on $M$.
Also, as we did in [11], we can just work with smooth Riemann surface
laminations $\cal{L}$ instead of holomorphic foliations, and all the results
still holds true. We proved the results for the case
$(M,g)=(\mathbb{C}^{N},|dz|^{2})$ and a hyperbolic Riemann surface foliation
$\cal{F}$ on $U\subset\mathbb{C}^{N}$ just to keep calculations simple.

######  Example 6.1.

Let $\cal{F}$ be the hyperbolic singular holomorphic foliation induced by the
vector field

$X=x\frac{\partial}{\partial x}+zy\frac{\partial}{\partial
y}+zy\frac{\partial}{\partial z}$

on the open polydisc of radius $r=(5,5,5)$ denoted by
$U=P(0,r)\subset\mathbb{C}^{3}$. The singular set of $\cal{F}$ is
$E=\\{(0,y,z)\in U:yz=0\\}=\\{y-\text{axis}\\}\cup\\{z-\text{axis}\\}$ (here
dim$(E)=1$). Let $W_{n}=P(0,(1,1,1))\cup P(0,(\frac{1}{n},2,\frac{1}{n}))$ and
$W=P(0,(1,1,1))$. We can check that $W_{n}\to W$ in the kernel sense with the
set $F=\\{(0,y,0)\in U:y\in D(0,2)\setminus\overline{D(0,1)}\\}$. Note that
$F\subset E$, therefore the defective set corresponding to the foliation
$\cal{F}$ is empty, that is $S=\emptyset$.

As shown in [12] (Example 1.17), using the fact that
$\Sigma_{1}=\\{x=0\\},\Sigma_{2}=\\{y=0\\}\;\&\;\Sigma_{3}=\\{z=0\\}$ are
$\cal{F}-$invariant to conclude that $\eta_{W}$ is continuous on $W\setminus
E$. It has also been proved that $\cal{F}$ is transversal type. Therefore,
$\eta_{W}$ can be extended continuously on whole of $W$.

Since $W,U$ are bounded taut domains in $\mathbb{C}^{3}$, therefore Theorem
1.7 tells us that $\eta_{n}\to\eta_{W}$ uniformly on the compacts of $W$.

######  Example 6.2.

Let $\cal{F}$ be the hyperbolic singular holomorphic foliation induced by the
vector field

$X=xy\frac{\partial}{\partial x}+zy\frac{\partial}{\partial
y}+zx\frac{\partial}{\partial z}$

on the open polydisc of radius $r=(5,5,5)$ denoted by
$U=P(0,r)\subset\mathbb{C}^{3}$. The singular set of $\cal{F}$ is given by
$E=\\{(x,y,z)\in
U:xy=yz=zx=0\\}=\\{x-\text{axis}\\}\cup\\{y-\text{axis}\\}\cup\\{z-\text{axis}\\}$
(here dim$(E)=1$). Let $W_{n}=P(0,(1,1,1))\cup
P(0,(\frac{1}{n},2,\frac{1}{n}))$ and $W=P(0,(1,1,1))$. Here $W_{n}\to W$ in
the kernel sense with the set $F=\\{(0,y,0)\in U:y\in
D(0,2)\setminus\overline{D(0,1)}\\}$. Note that $F\subset E$, therefore the
defective set corresponding to the foliation $\cal{F}$ is empty, that is
$S=\emptyset$.

As shown in [12] (Example 1.18), using the fact that
$\Sigma_{1}=\\{x=0\\},\Sigma_{2}=\\{y=0\\}\;\&\;\Sigma_{3}=\\{z=0\\}$ are
$\cal{F}-$invariant to conclude that $\eta_{W}$ is continuous on $W\setminus
E$.

Take $p=(x,0,0)\in E$ for $x\neq 0$. Observe that $C_{p}E=\langle
e_{1}\rangle$, and therefore $(1,0,0)=e_{1}\in C_{p}E$. Consider the sequence
$p_{n}=(x,\frac{1}{n},0)\in W\setminus E$ and note that $e_{1}\in
T_{p_{n}}\cal{F}$ for all $n\in\mathbb{N}$. Therefore
$e_{1}\in\overline{C_{p}\cal{F}}$. Thus $\cal{F}$ is not transversal type at
$p$. Since $p$ is arbitrary, $\cal{F}$ is not transversal type at any point of
$\\{(x,0,0):x\in\mathbb{D}^{\ast}\\}\subset E$.

Now on $\Sigma_{3}=\\{z=0\\}$, the foliation $\cal{F}_{W}$ is given by
$X=xy\frac{\partial}{\partial x}$. One can check that the leaf of
$\cal{F}_{W}$ passing through $p_{n}$ is given by
$L_{p_{n},W}=\\{(\xi,\frac{1}{n},0):\xi\in\mathbb{D}^{\ast}\\}\cong\mathbb{D}^{\ast}$
for all $n\in\mathbb{N}$. Therefore $\eta_{W}(p_{n})=\eta_{W}(p_{m})\neq 0$
for all $n,m\in\mathbb{N}$.

On $\Sigma_{2}=\\{y=0\\}$, the foliation $\cal{F}_{W}$ is given by
$X=xz\frac{\partial}{\partial z}$. Here the leaf of $\cal{F}_{W}$ passing
through $\tilde{p}=(x,0,z)$ is given by $L_{\tilde{p},W}=\\{(x,0,\xi):\xi\in
D(0,r_{3})^{\ast}\\}\cong\mathbb{D}^{\ast}$. Note that
$L_{\tilde{p},W}\cup\\{p\\}\cong\mathbb{D}$, i.e., $L_{\tilde{p},W}$ is a
separatrix through $p$. Take any sequence $\\{q_{n}\\}_{n\in\mathbb{N}}\subset
L_{\tilde{p},W}$ such that $q_{n}\to p$, we have $\eta_{W}(q_{n})\to 0$.

Therefore, we get two sequences $p_{n},q_{n}\to p$, but $\eta_{W}(p_{n})\to
k\neq 0$, and $\eta_{W}(q_{n})\to 0$. Thus, $\eta_{W}$ cannot be extended
continuously to $p$.

We use similar arguments to check that for every $p\in
E\setminus\\{0\\}=:\tilde{E}$, $\cal{F}$ is not transversal type at $p$ (in
fact $\cal{F}$ is not transversal type for any $p\in E$). Once again, by
looking at the structure of leaves in the invariant hyperplanes
$\Sigma_{i}$’s, we can conclude that $\eta_{W}$ can not be extended
continuously on any point $p\in\tilde{E}=E\setminus\\{0\\}$.

Since $W,U$ are bounded taut domains in $\mathbb{C}^{3}$, therefore Theorem
1.7 tells us that $\eta_{n}\to\eta_{W}$ uniformly on the compacts of
$W\setminus E$.

######  Example 6.3.

Let $\cal{F}$ be the hyperbolic singular holomorphic foliation induced by the
vector field

$X=x\frac{\partial}{\partial x}+2y\frac{\partial}{\partial y}$

on the open polydisc of radius $r=(5,5)$ denoted by
$U=P(0,r)\subset\mathbb{C}^{2}$. The singular set of the foliation $\cal{F}$
is $E=\\{(0,0)\\}$.

Consider the complex line segment $R=\\{(x,y)\in P(0,(3,3))\mid y=x\\}$. Let
$V_{n}\subset U$ be a sequence of decreasing open connected neighourhoods of
the segment $R$ such that $\cap_{n\geq 1}V_{n}=R$. Now let
$W_{n}=P(0,(1,1))\cup V_{n}$, and $W=P(0,(1,1))$. Note that $W_{n}\to W$ in
the kernel sense with the set $F=\\{(x,y)\in
P(0,(3,3))\setminus\overline{P(0,(1,1))}\mid y=x\\}$.

Let $p=(x_{0},y_{0})\in U\setminus E$, the leaf $L_{p}$ of $\cal{F}$ is given
by the solution of the differential equation

$\frac{dy}{dx}=\frac{2y}{x},$

with the initial condition $y(x_{0})=y_{0}$. One can check that

$L_{p}=\\{(x,y)\in U\setminus E\mid x_{0}^{2}y=y_{0}x^{2}\\}.$

Now if $q=(x,x)\in R\cap L_{p}$, then $x_{0}^{2}x=y_{0}x^{2}$. Since
$(0,0)\notin L_{p}$, therefore $R\cap L_{p}\neq\emptyset$ if and only if
$x_{0}y_{0}\neq 0$. If $p\in U\setminus E$ is such that $x_{0}y_{0}\neq 0$,
then $q=(\frac{x_{0}^{2}}{y_{0}},\frac{x_{0}^{2}}{y_{0}})$, that is

$R\cap L_{p}=\\{(\frac{x_{0}^{2}}{y_{0}},\frac{x_{0}^{2}}{y_{0}})\\}.$

So, $F\cap L_{p}\neq\emptyset$ if and only if $|y_{0}|<|x_{0}|^{2}$. Consider
the set

$T=\\{(x,y)\in W\setminus E:|y|<|x|^{2}\\}\subset W.$

Note that if $p\in T$, then we can see from above that $L_{p,W}\subset S$,
where $S$ is the defective set in $W$ corresponding to the foliation
$\cal{F}$. Since $p\in T$ is arbitrary, therefore $T\subset S$. Now one can
easily check from the description of leaves of $\cal{F}$ that for $p\in T$,
the leaf $L_{p,W}$ is contained in $T$, that is $L_{p,W}\subset T$. Therefore,
we get $S\subset T$. Thus, the defective set $S=T$ is an open set in $W$.

But as we already observed that for every $p\in S$, the intersection
$L_{p}\cap F$ is a singleton set. Therefore, for every $p\in S$,

$\text{int}(L_{p}\cap F)=\emptyset,$

where $\text{int}(L_{p}\cap F)$ denotes the interior of $L_{p}\cap F$ as a
subset of the leaf $L_{p}$. Thus, every $p\in S$ is a removable defective
point, that is $\hat{S}=S$.

Since $W,U$ are bounded taut domains in $\mathbb{C}^{2}$, [12] gives us the
continuity of $\eta_{W}$ on $W\setminus E$. Furthermore, $\cal{F}$ is
transversal type at $0\in E$, which gives us that $\eta_{W}$ is continuous on
whole of $W$.

Theorem 1.8 now tells us that $\eta_{n}\to\eta_{W}$ uniformly on the compacts
of $W$.

We will end the section with a few questions.

Question 1: In Theorem 1.8, we mentioned a sufficient condition for a
defective point $p\in S$ to be a removable defective point, that is
$p\in\hat{S}$. But we can easily construct an example of a sequence of domains
and foliation to show that the condition in Theorem 1.8 is not necessary. It
would be interesting to know the complete description of the set $\hat{S}$.

Question 2: Given $(W_{n},W,U,\cal{F})$, is it really necessary to assume the
continuity of $\eta_{W}$ on whole $W\setminus E$ in order to prove the uniform
convergence part in Theorem 1.7 and Theorem 1.8? For instance, if the
defective set $S\subset W$ is closed subset, then the uniform convergence of
$\eta_{n}$ to $\eta_{W}$ still holds on the compacts of $W\setminus(S\cup E)$
even if we only assume that the function $\eta_{W}$ is continuous on
$W\setminus(S\cup E)$. The proof will be exactly the same as we did for
Theorem 1.7 and Theorem 1.8.

## References

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# Data is Moody: Discovering Data Modification Rules from Process Event Logs

Marco Bjarne Schuster Airbus Operations GmbH, Bremen, Germany.
<EMAIL_ADDRESS>Boris Wiegand Stahl-Holding-Saar, Dillingen,
Germany.
<EMAIL_ADDRESS>Jilles Vreeken CISPA Helmholtz Center for
Information Security, Germany.
<EMAIL_ADDRESS>
###### Abstract

Although event logs are a powerful source to gain insight about the behavior
of the underlying business process, existing work primarily focuses on finding
patterns in the activity sequences of an event log, while ignoring event
attribute data. Event attribute data has mostly been used to predict event
occurrences and process outcome, but the state of the art neglects to mine
succinct and interpretable rules how event attribute data changes during
process execution. Subgroup discovery and rule-based classification approaches
lack the ability to capture the sequential dependencies present in event logs,
and thus lead to unsatisfactory results with limited insight into the process
behavior.

Given an event log, we are interested in finding accurate yet succinct and
interpretable if-then rules how the process modifies data. We formalize the
problem in terms of the Minimum Description Length (MDL) principle, by which
we choose the model with the best lossless description of the data.
Additionally, we propose the greedy Moody algorithm to efficiently search for
rules. By extensive experiments on both synthetic and real-world data, we show
Moody indeed finds compact and interpretable rules, needs little data for
accurate discovery, and is robust to noise.

##  1 Introduction

Given a process event log, process mining [29] provides a better understanding
of the underlying process and enables downstream tasks such as monitoring,
anomaly detection, simulation, and optimization. Existing work, however,
focuses on discovering patterns of event activities, but neglects how event
attribute data changes during the process. For instance, a textile company may
have an ordering process with the activities _Request_ , _Place_ , _Delay_ and
_Receive_. Process discovery algorithms [1, 27] only infer a graph of event
activities, where nodes refer to activities and edges visualize the flow of
execution. However, how data changes over time is crucial for understanding
the process as we show in Figure 1. In our textile example, we only know the
price and the delivery date of an order after placing the order. Delaying the
delivery of an order, e.g. due to a shortness of supplies, changes the
delivery date.

Surprisingly, data has only been used to predict the occurrences of event
activities or the outcome of processes [28, 31]. To the best of our knowledge,
none of the existing work mines interpretable rules for data modifications,
and related work does not satisfactorily transfer to our problem. Subgroup
discovery [19] and rule-based prediction methods [32] lack the ability to
model the sequential dependencies present in event logs, and thus lead to
unsatisfactory results with limited insight into the process behavior.

Given an event log, we are interested in finding accurate yet succinct and
interpretable if-then rules how the process modifies data. To this end, we
formalize the problem in terms of the Minimum Description Length (MDL)
principle, by which we choose the model with the best lossless description of
the data. Additionally, we propose our method Moody, which is short for
Modification rule Discovery, to efficiently search for rule models in
practice. Starting with an empty set, we greedily add the best compressing
rule to the model, until we no longer find a rule that improves our MDL score.
Through extensive experiments on both synthetic and real-world data, we show
Moody indeed finds succinct and interpretable rules, needs little data for
accurate discovery, and is robust to noise.

introduction-example RequestPlaceDelayRecieve product:
t-shirt amount:
100 product:
t-shirt amount:
100 price:
$1000 delivery:
May 1${}^{\text{st}}$ product:
t-shirt amount:
100 price:
$1000 delivery:
May 9${}^{\text{th}}$ product:
t-shirt amount:
100 price:
$1000 delivery:
May 9${}^{\text{th}}$

Figure 1: [Processes change data] Exemplary process for ordering textiles with
activities _Request_ , _Place_ , _Delay_ and _Receive_. Arrows indicate the
flow of events. Further, we show how the data of an exemplary order changes
throughout the process.

The contributions of our paper are as follows. We

  * (a)

formulate the problem of finding data modification rules from event logs,

  * (b)

formalize the problem using the Minimum Description Length (MDL) principle,

  * (c)

propose the Moody algorithm to efficiently find accurate yet succinct data
modification rules,

  * (d)

run extensive experiments on both synthetic and real-world data,

  * (e)

make code, data and appendix publicly
available.111https://eda.rg.cispa.io/prj/moody/

Our paper is structured as usual.

##  2 Related Work

While the problem of finding interpretable data modification rules from
process event logs has been neglected so far, related work on similar problems
exists. Krismayer[12] discovers if-then rules for data modification from
software execution logs. However, he only creates a large set of potentially
redundant candidates, which must be manually filtered by domain experts. Other
work [30, 8] infers extended finite state machines from software execution
logs. Since business process event logs in contrast to software event logs are
usually noisy and contain nondeterministic behavior, these methods are not
applicable to our problem.

While process mining [29] focuses on business process event logs, most of the
work only models the flow of process activities, and little work deals with
additional data. Mannhardt et al. [14] and Mozafari et al. [17] detect at
which event data changes, but do not model conditions or update values for
these changes. Schönig et al. [26] find rules which cover data modifications.
However, since they rely on support and confidence to filter a large set of
candidate rules, their method suffers from pattern explosion, i.e., it finds
many redundant rules.

Rule-based prediction is closely related to our problem. Classy [20] and its
successor Turs [32] find classification rules by minimizing an MDL score. Both
methods, however, require defining features and predicted variable beforehand,
whereas we are interested in finding relationships without any initial
knowledge of the data. Similarly, subgroup discovery algorithms such as SSD++
[19] find rules for differently behaving subgroups of a given dataset. None of
these methods are able to model sequential relationships present in event
logs.

In contrast to the above, Moody finds compact and interpretable rules for data
modifications from process event logs, needs little data for accurate
discovery, and is robust to noise.

##  3 Preliminaries

Before formalizing the problem, we introduce preliminary concepts and notation
we use in the paper.

###  3.1 Notation for Data Modification Rules

As input for finding data modification rules, we consider an event log or
dataset $D$ collecting traces of a single process. Each trace refers to an
instance of the process, such as a specific customer order, and consists of an
event sequence. We describe data attributes of events by a set of numerical
and categorical variables $V$.

To model how a process modifies these variables, we use different types of
update rules. For a categorical variable $v\in V$, we write
$v\in\\{\alpha,\beta,\dots\\}$, i.e., $v$ takes one of the values in the set.
For a numerical variable $v\in V$, we can set $v$ to a specific value,
$v=\alpha$, or to a range of values, $v\in[\alpha,\beta]$. We further denote
relative changes by $v=v+\alpha$, $v=v+[\alpha,\beta]$, and $v=\alpha\cdot v$.

Updates typically only occur in certain circumstances. For instance, the price
of an order may be dependent on the order volume, where a higher volume gives
discount. Therefore, we model conditions for update rules. In the simplest
case, we check for a specific value $v=\alpha$ or $v\neq\alpha$. Further, we
test lower and upper bounds of numerical values by $v\leq\alpha$ and
$v\geq\alpha$. Finally, we can condition on value transitions between the last
and the current event with $v:\alpha\rightarrow\beta$.

We combine a condition $c$ and an update rule $u$ into a data modification
rule IF $c$ THEN $u$. To join multiple rules into a model $M$ that covers the
full complexity of the process, we use an unordered set of rules, since this
allows for an independent interpretation of each rule [32]. Furthermore, since
we want to avoid contradictory predictions of our model, we only allow acyclic
models. For instance, if we condition on $v_{1}$ to update $v_{2}$, we are not
allowed to do the reverse in the same model.

###  3.2 Minimum Description Length

We use the Minimum Description Length (MDL) principle [21, 9] for model
selection. MDL defines the best model as the one with the shortest lossless
description of the given data. Formally, the best model minimizes
$L(M)+L(D\mid M)$, in which $L(M)$ is the length in bits of the description of
$M$, and $L(D\mid M)$ is the length of the data encoded with the model. This
form of MDL is known as two-part or crude MDL. Although one-part or refined
MDL has stronger theoretical guarantees, it is only computable in specific
cases [9]. Therefore, we use two-part MDL. In MDL, we only compute code
lengths, but are not concerned with actual code words. Next, we formalize our
problem in terms of MDL.

##  4 MDL for data modifications

From an event log $D$, we aim to find a model $M$ of data modification rules,
which accurately describe the data, yet are as succinct as possible, such that
domain experts gain insight into the process. Since real-world data is usually
noisy, we need a robust model selection criterion. Therefore, we formalize the
problem in terms of the MDL principle. To this end, we define length of the
data encoding $L(D\mid M)$, length of the model encoding $L(M)$, and finally
give a formal problem definition.

data-encoding-example  Model:
R1: IF amount $=$ 10 THEN vendor $=$ C R2: IF product $=$ bag THEN vendor
$\in$ {B, C} Code streams:
$C_{m}:$ ✓${}_{R\ref*{rule:cat-prediction-2}}$, ✓${}_{R\ref*{rule:cat-
prediction-1}}$, ✗${}_{R\ref*{rule:cat-prediction-1}}$ $C_{v}:$ C, A, A
$C_{r}:$ R2
Trace:
bag 20 C bag 10 C pants 10 A pants 20 A product: amount: vendor:
event:E1E2E3E4

Figure 2: [Data encoding] We use the model (top left), the model stream
$C_{m}$, value stream $C_{v}$ and rule selection stream $C_{r}$ (top right) to
decode the trace’s categorical variable _vendor_ (bottom).

###  4.1 Data encoding

To encode a given event log with a model of data modification rules, we encode
the variable values at each event. For each event, we check which rules in the
model fire, i.e., have satisfied conditions, and use the firing rules to
encode the data values. Whenever the model makes ambiguous predictions, we
encode the specific value among all possibilities. If no rule fires, we choose
and encode a value from the whole domain of the target variable. To ensure a
lossless encoding and to handle noisy data, we also encode any errors made by
the model.

Conceptually, we split the data encoding into three code streams: In the rule
selection stream $C_{r}$, we encode which of the rules with matching
conditions we choose to encode the current variable value. Then, we encode in
the model stream $C_{m}$ if the model predicts the correct value. If not, any
value of the target domain is possible. Whenever the model predicts multiple
values, we choose a value by a code in the value stream $C_{v}$.

We give a toy example of a data encoding in Figure 2, which we use to describe
how to decode the categorical variable _vendor_. First, at event E2, we see
that only rule R2 applies. Thus, we do not need to select a rule by reading
from $C_{r}$. Next, we find a checkmark as the first symbol in $C_{m}$, i.e.,
the model predicts correctly. However, the rule allows two values, B and C,
which we disambiguate by reading C from $C_{v}$. Next, at event E2, we observe
that both rules apply. Therefore, we check $C_{r}$ to find that we should use
rule R2 whose prediction is correct according to the second symbol in $C_{m}$.
Further, its prediction is not ambiguous and we get the value C in the trace.
Afterwards, at event E2, we find that only rule R2 applies but its prediction
is incorrect according to the last element in $C_{m}$. We obtain the correct
value by reading A from $C_{v}$. Finally, no rule applies at event E2, so we
neither read from $C_{m}$ nor from $C_{r}$. Instead, we read the last value
from $C_{v}$, A, by which we have successfully decoded the values for
_vendor_.

We compute the length of the data encoding by summing the code lengths in
$C_{r}$, $C_{m}$ and $C_{v}$. Whenever we have to disambiguate multiple firing
rules in $C_{r}$, we assume all rules in the model are equally important. We
compute the encoded length of $C_{r}$ by

(4.1) $L(C_{r})=\sum_{i=1}^{|C_{r}|}\log\left|{R_{i}}\right|\;,$

where $|R_{i}|$ denotes the set of firing rules at the $i$-event.

When we compute the encoded length of $C_{m}$, we do not know the
probabilities of codes for checkmarks and crosses in advance. Therefore, we
use a prequential plug-in code [5, 23] to compute $L(C_{m})$: We initialize
uniform counts for checkmarks and crosses and update counts after each event,
such that we have a valid probability distribution at each step in the
encoding. Asymptotically, this gives an optimal encoded length of $C_{m}$.
Formally, we have

(4.2)
$L(C_{m})=\sum_{i=1}^{|C_{m}|}\frac{\operatorname*{usg}_{i}C_{m}[i]+\epsilon}{\operatorname*{usg}_{i}\text{\char
51}+\operatorname*{usg}_{i}\text{\char 55}+2\epsilon}\;,$

where $\operatorname*{usg}_{i}C_{m}[i]$ denotes how often the $i$-th code in
$C_{m}$ has been used before, and $\epsilon$ with standard choice $0.5$ is for
additive smoothing.

To compute code lengths in $C_{v}$, we use the conditional empirical
probability of values. Let $u_{i}$ be the update rule we selected in $C_{r}$
to encode the $i$-th value in $C_{v}$. If no rule fires or we encoded an error
in $C_{m}$, $u_{i}$ falls back to all possible values in the domain of the
target variable. We formally define

(4.3)
$L(C_{v})=-\sum_{i=1}^{|C_{v}|}\log\frac{\operatorname{fr}(C_{v}[i])}{\sum_{j\in
u_{i}}\operatorname{fr}(j)}\;,$

where $\operatorname{fr}(C_{v}[i])$ denotes how often value $C_{v}[i]$ occurs
in the data, and we normalize this by the frequencies of all values $j$
possible according to the update rule $u_{i}$. To encode numerical variables,
we assume a histogram-based discretization, by which we can compute all
necessary probabilities and code lengths.

Altogether, this gives us a lossless data encoding.

###  4.2 Model encoding

To define the length of the model encoding $L(M)$, we need to encode the
number of rules in the model and all conditions $c$ and update rules $u$ in
the model. Since the number of rules is unbounded, we use the universal MDL
encoding for natural numbers $L_{\mathbb{N}}$ [22] that encodes any natural
number $x\geq 1$ as
$L_{\mathbb{N}}(x)=\log(2.865064)+\log(x)+\log(\log(x))+\dots,$ where we sum
only the positive terms and the first term ensures that Kraft’s inequality
holds, i.e., $L_{\mathbb{N}}$ is a lossless encoding. Denoting the number of
modification rules in the model as $|M|$, which can be zero, we then obtain
the encoded length of the model as

(4.4) $L(M)=L_{\mathbb{N}}(|M|+1)+\sum_{(c,u)\in M}L(c)+L(u)\;.$

To encode a condition $c$, we specify its type, which single variable $v\in V$
is tested by $c$, and all constants used in the condition. Formally, we define

(4.5)
$L(c)=\log\left|{\\{=,\neq,\leq,\geq,\rightarrow\\}}\right|+\log\left|{V}\right|+\sum_{\alpha\in
c}L(\alpha)\;.$

To encode the value $\alpha\in\operatorname{dom}{v}$ for a categorical
variable $v$, we have

$L(\alpha)=\log\left|{\operatorname{dom}{v}}\right|\;,$

and to encode a real-valued $\alpha$, we consider it in scientific notation
$\alpha=k\cdot 10^{s}$, and have

$L_{\mathbb{R}}(\alpha)=2+L_{\mathbb{N}}(|s|+1)+L_{\mathbb{N}}(\left\lceil|k|\cdot
10^{s}\right\rceil+1)\;,$

where we first encode the signs of $k$ and $s$ with 1 bit each, then the value
of $s$, and finally the value of $k$ up to a user-specified precision of $p$
significant digits [15].

To encode an update rule $u$ on a variable $v$, we first specify the type of
$u$, which is one of $v\in\\{\alpha,\beta,\ldots\\}$, $v=\alpha$,
$v\in[\alpha,\beta]$, $v=v+\alpha$, $v=v+[\alpha,\beta]$, or $v=\alpha\cdot
v$, for which we need $\log 6$ bits. Then, we encode which variable $v\in V$
is updated by $u$, and finally we encode the constants in $u$ the same way we
do for conditions. Formally, we have

(4.6) $L(u)=\log 6+\log\left|{V}\right|+\sum_{\alpha\in u}L(\alpha)\;.$

This gives us the encoded length of the model $L(M)$.

###  4.3 Formal problem definition

With this, we now have all the ingredients to formally define our problem.

Minimal Modification Rules Problem Given an event log $D$ with variables $V$,
find an acyclic model of data modification rules $M$ that minimizes the total
encoding cost $L(D,M)=L(M)+L(D\mid M)$.

In practice, it is infeasible to solve this problem optimally due to the
potentially large number of acyclic models. To approximate this number by a
lower bound, we consider the _rule dependency graph_ of a model, which is a
directed acyclic graph with variables as nodes and their dependencies induced
by modification rules as edges. We give a simple example of a rule dependency
graph in Figure 3. Since each edge requires at least one rule, there are at
least as many models as rule dependency graphs. According to Rodionov [24],
the number of acyclic graphs with $n$ nodes and up to $m$ edges is

$A(n,m)=\sum_{i=1}^{n}\sum_{j=0}^{m}(-1)^{i-1}\binom{n}{i}\binom{i(n-i)}{m-j}A(n-i,j)\;,$

with $A(1,\cdot):=1$. Because $A(n,m)$ grows exponentially in $n$ and $m$ [4,
p. 1186], the number of acyclic models grows exponentially in the number of
variables $|V|$ and the number of modification rules in the model $|M|$.

Furthermore, our search space has no trivial structure such as submodularity
or monotonicity, which we could exploit to find an optimal solution in
feasible time. We give counterexamples for both properties in the
supplementary material. Hence, we resort to heuristics.

IF product $=$ pants THEN vendor $=$ A IF product $=$ bag THEN amount $=$ 20
rule-dependency-graph amountproductvendor Figure 3: [Rule dependency graph] We
represent the variable dependencies of the model (top) as a graph (bottom)
where ovals represent variables and arrows show whether and in which direction
a modification rule induces a dependency between variables.

##  5 The Moody algorithm

To efficiently discover good sets of data modification rules in practice, we
prune the exponentially sized search space with a quickly computable estimate
of our score that avoids repetitively passing the whole event log, and we
introduce a greedy search.

###  5.1 Estimating the MDL score

Computing the MDL score $L(D,M)$ requires a pass over all events in the event
log, because we must check for each event, which of the rules in the model
fire. To avoid iterating over the event log each time we evaluate adding a new
rule to the model, we prune the large set of potential candidate rules by a
quickly computable estimate $\widehat{L}(D,M)$. We optimistically assume that
a newly generated rule is the only rule in the model which predicts its target
variable, such that we do not need to update $C_{r}$ or $C_{m}$.

Input: Rule $(c,u)$

Output: $\widehat{L}(C_{v}\mid c,u)$

1 $\widehat{L}(C_{v}\mid c,u)\leftarrow 0$;

2 $b\leftarrow\operatorname*{supp}(c)$;

3 forall $j\in u$ ordered by increasing $\operatorname{fr}(j)$ do

4 $\Delta b\leftarrow\min\\{b,\operatorname{fr}(j)\\}$;

5 $\widehat{L}(C_{v}\leavevmode\nobreak\ |\leavevmode\nobreak\
c,u)\leftarrow\widehat{L}(C_{v}\leavevmode\nobreak\ |\leavevmode\nobreak\
c,u)-\Delta b\cdot\log\frac{\operatorname{fr}(j)}{\sum_{i\in
u}\operatorname{fr}(i)}$;

6 $b\leftarrow b-\Delta b$;

7

8return $\widehat{L}(C_{v}\mid c,u)$;

Algorithm 1 Estimate $L(C_{v})$

By assuming independence between rules, we can independently estimate the
contribution of a single rule with condition $c$ and update rule $u$ to
$L(C_{v})$. We give the pseudocode for estimating $L(C_{v}\mid c,u)$ as
Algorithm 1. First, we compute the support $\operatorname*{supp}(c)$, i.e., at
how many events $c$ fires (ln. 1). For each value $j$ predicted by $u$, we
compute how many codes we must add to $C_{v}$, which is the minimum of the
remaining events to cover, $b$, and the frequency $\operatorname{fr}(j)$ of
$j$ (ln. 1). We compute the length of all these codes and add it to our
estimate (ln. 1). At the end of each iteration, we update how many codes we
still must add to $C_{v}$ (ln. 1).

While computing the support requires a pass over the dataset, we only need a
single pass when creating the candidate. By assuming independence between
rules, we do not need to update the estimated code lengths of all candidate
rules, every time we add another rule to the model. Using $\widehat{L}(D,M)$
we can prune out rules with high encoding costs, and thus avoid computing
$L(D,M)$ for those. Next, we use $\widehat{L}(D,M)$ and $L(D,M)$ to find good
modification rules for a given event log.

###  5.2 Finding Good Modification Rules

To reduce the search space, we let the domain experts control how much time
they want to invest in model search, and introduce two hyperparameters for a
greedy search. Instead of generating conditions with all possible combinations
of variables, signs and values, we only generate the $N_{c}$ most frequently
observed combinations of variables and values in the dataset for each sign of
the set $\\{=,\neq,\leq,\geq,\rightarrow\\}$. For a given condition, instead
of generating update rules with all possible combinations of variables and
values, we only generate the $N_{u}$ most frequent combinations of variables
and values in the dataset for each type of update rule.

Input: event log $D$ with variables $V$

Output: model of data modification rules $M$

1 $M\leftarrow\emptyset$;

2 do

3 forall $v\in V$ do

4 $Q\leftarrow$ priority queue of rules $r$ predicting $v$ ordered by
$\widehat{L}(D,M\cup\\{r\\})$;

5 $r^{*}\leftarrow\emptyset$;

6 while $Q\neq\emptyset\leavevmode\nobreak\
\textnormal{{and}}\leavevmode\nobreak\ \widehat{L}(D,M\cup\\{\text{top of
}Q\\})<L(D,M\cup\\{r^{*}\\})$ do

7 $r^{\prime}\leftarrow$ pop element from $Q$;

8
$r^{*}\leftarrow\operatorname*{arg\,min}_{r\in\\{r^{*},r^{\prime}\\}}L(D,M\cup\\{r\\})$

9 if $L(D,M\cup\\{r^{*}\\})<L(D,M)$ then

10 $M\leftarrow M\cup\\{r^{*}\\}$;

11

12

13

14while $M$ was extended in the last iteration;

15return $M$;

Algorithm 2 Moody

We provide the pseudocode of our greedy search Moody as Algorithm 2. Our
search starts with an empty model (ln. 2). We iteratively extend this model by
modification rules for all target variables (ln. 2). Since computing $L(D,M)$
needs a pass over the whole dataset, we manage the candidate rules in a
priority queue sorted by an estimate of our score $\widehat{L}(D,M)$ (ln. 2),
such that we check promising rules early. Next, we search the candidates from
most promising to least promising and compute their actual encoded length $L$
(ln. 2-2). To not waste computation time for computing $L$ on inferior rules,
we perform this search as long as the estimate $\widehat{L}$ is better than
the best actual code length $L$ that we have seen so far (ln. 2). After
evaluating the candidates, we only add the best candidate to our model if it
reduces the total encoded length (ln. 2-2). Finally, we end when no candidate
for any target variable could improve our score (ln. 2) and return the
resulting model (ln. 2).

In the worst case, all generated candidates improve our MDL score. Since the
number of candidates grows linearly with $N_{c}$ and $N_{u}$, the outer loop
of Moody grows linearly with $N_{c}$ and $N_{u}$. In the worst case, our
estimate does not prune any candidate, and we must compute our score for all
of the $O(N_{c}\cdot N_{u})$ candidates. Since we loop over all variables, and
computing our score requires a pass over the whole dataset, the runtime
complexity of Moody is $O\left((N_{c}\cdot N_{u})^{2}\cdot|V|\cdot|D|\right)$.

##  6 Experiments

In this section, we empirically evaluate Moody on both synthetic and real-
world data. When we defined our MDL score, we assumed discretization of
numerical variables. In our prototype implementation, we discretize numerical
values into variable-width histograms. For efficiency, we determine the
histogram boundaries by percentiles. We use 50 bins in all our experiments.

We run all our experiments in a Docker-based environment on a Linux server
with an Intel® Xeon® Gold 6244 CPU. In all experiments, we observe 16 GB of
RAM suffice. As a simple baseline, we consider the empty model $M=\emptyset$.
In addition, we learn if-then-else rules using the rule-based classifier Turs
[32], and using the subgroup discovery method SSD++ [19]. To ensure
reproducibility of all our results, we provide code, data, and further details
in the supplementary material.

moody_hyperparameter_tuning 2103050700.00.20.40.60.81.0Hyperparameter
$N_{c}$$\text{F}_{1}\leavevmode\nobreak\
\text{score}$210305070051015202530Hyperparameter $N_{c}$RMSE$10\%$ swap
noise$20\%$ swap noise

Figure 4: [Choosing $N_{c}$] Median $\text{F}_{1}\leavevmode\nobreak\
\text{score}$ on categorical variables (left) and median RMSE on numerical
variables (right) for different values of Moody’s hyperparameter $N_{c}$ for
$10\%$ and $20\%$ swap noise in the training set. Error bars show
interquartile ranges.

###  6.1 Synthetic event logs

To control data properties such as noise, we first experiment on synthetic
event logs, such that we know the generating ground-truth rules. We randomly
generate ten independent ground-truth models in our modeling language, where
each model contains five rules, two categorical variables and two numerical
variables. Since SSD++ cannot model sequential dependencies
$v:\alpha\rightarrow\beta$, we only create models with conditions
$v\leq\alpha$, $v\geq\alpha$ and $v=\alpha$. From these ground-truth models,
we generate event logs with $|D|=2000$ events. To test noise-robustness, we
add different amounts of noise, where we randomly swap values of variables.
10% swap noise means that for each variable in the dataset, we randomly swap
10% of its values.

Turs in contrast to SSD++ does only discover rules for categorical target
variables, and cannot predict numerical values. Therefore, we separately
evaluate on models that only predict categorical variables and on models that
only predict numerical variables. However, in both setups, we generate
conditions containing categorical and numerical variables.

#### Choosing Hyperparameters

Such that the user can control runtime by reducing the search space, we
introduced hyperparameters $N_{c}$ and $N_{u}$ when we proposed Moody. For
efficiency, we only search for the most compressing update rule given a
condition and set $N_{u}$ to 1. To evaluate the accuracy of a set of rules, we
compute its median $\text{F}_{1}\leavevmode\nobreak\ \text{score}$ on
predicting categorical variables, and its median root mean squared error
(RMSE) in predicting numerical variables. We report the influence of $N_{c}$
on prediction accuracy in Figure 4. We see that the prediction accuracy drops
for only very small $N_{c}$, and is relatively constant for larger values.
This means, the conditions with the highest support in the dataset tend to
give the best compression and accuracy. To have a safety margin on unknown
data, we set $N_{c}$ to 50.

moody_predictions_on_synthetic_data $0$$0.1$$0.2$$0.3$0.00.20.40.60.81.0Noise
proportionTest $\text{F}_{1}\leavevmode\nobreak\
\text{score}$$0$$0.1$$0.2$$0.3$$0$$10$$20$$30$51525Noise proportionTest
RMSE$0$$0.1$$0.2$$0.3$$0$$20$$40$$60$$80$$100$Noise proportionNo. of rule
termsMoodyTursSSD++empty modelground-truth model

Figure 5: [Moody predicts well under reasonable amounts of noise] Median
$\text{F}_{1}\leavevmode\nobreak\ \text{scores}$ on categorical variables
(left, higher is better), median root mean squared errors on numerical
variables (center, lower is better) and number of rule terms in the discovered
models (right, lower is less complex) at different noise levels for Moody,
SSD++ and Turs. Error bars indicate interquartile ranges.

#### Results on categorical variables

Next, we compare results on synthetic data with different amounts of swap
noise for Moody against all baselines in Figure 5. We see in the left plot
that Moody achieves by a wide margin the highest
$\text{F}_{1}\leavevmode\nobreak\ \text{score}$ on categorical variables for
data with 0% and 10% noise. Up to 20% noise, Moody shows good noise-robustness
and still has the highest median $\text{F}_{1}\leavevmode\nobreak\
\text{score}$. For 30% noise, the $\text{F}_{1}\leavevmode\nobreak\
\text{score}$ of Moody drops down to the $\text{F}_{1}\leavevmode\nobreak\
\text{score}$ of the empty model. We note that 30% noise may sound low, but
swapping 30% of the values for each variable in the dataset accumulates to a
much higher noise-ratio. Hence, Moody predicts well under reasonable amounts
of noise.

#### Results on numerical variables

We see similar results on predicting values of numerical variables in the
center plot of Figure 5. Since Turs cannot predict numerical variables, we
compare Moody to SSD++ and the empty model. Moody shows by far the smallest
test root mean squared error (RMSE) for training data with up to 20% noise.
For 30% noise, Moody’s RMSE converges to the RMSE of the empty model. We again
note swapping 30% of the values for each variable in the dataset accumulates
to a much higher noise-ratio than we expect in any real-world event log.
Hence, Moody predicts well under reasonable amounts of noise.

#### Model complexity

Not only does Moody give the best prediction results under reasonable amounts
of noise. It also discovers the rule sets with the lowest total number of rule
terms as we show in the right plot of Figure 5. Both Turs and SSD++ find rule
sets with a significantly higher number of rule terms. We see Moody’s models
have similar complexity compared to the ground-truth models for data with low
amounts of noise. If the noise level increases, Moody converges to the empty
model. This means, Moody is robust against finding spurious rules on noisy
data.

moody_sample_complexity 250500100015000.00.20.40.60.81.0No. of training
eventsTest $\text{F}_{1}\leavevmode\nobreak\ \text{score}$MoodyTursSSD++empty
model250500100015000102030405060No. of training eventsRuntime ${[s]}$

Figure 6: [Moody shows low sample complexity and scales well] Median test
$\text{F}_{1}\leavevmode\nobreak\ \text{score}$ on categorical variables
(left) and median runtime (right) dependent on the number of training events
for Moody, Turs and SSD++. Error bars show interquartile ranges.

#### Sample complexity

To empirically evaluate sample complexity, we compute
$\text{F}_{1}\leavevmode\nobreak\ \text{scores}$ on the test set dependent on
the number of events in the training set. For a realistic setup, we add 10%
swap noise to all training sets. We report results for Moody, Turs, SSD++ and
the empty model in the left plot of Figure 6. We see Moody predicts better the
more training data is available, while all baselines do not improve with more
training data. Already with 500 training events, Moody shows higher median
$\text{F}_{1}\leavevmode\nobreak\ \text{score}$ than the baselines.

#### Runtime

We report wall-clock runtime for single-threaded execution dependent on the
number of training events in the right plot of Figure 6. As we expect by our
theoretical runtime analysis, we see that Moody scales well and shows a growth
of runtime linear to the number of training events. While SSD++ shows a
constantly fast, almost zero runtime, Moody still finishes within reasonable
time and is significantly faster than Turs.

###  6.2 Real-world event logs

Data | $\left|{D}\right|$ | $||D||$ | $|\Omega|$ | $\left|{V_{\text{cat}}}\right|$ | $\left|{V_{\text{num}}}\right|$
---|---|---|---|---|---
Sepsis | 782 | 15214 | 18 | 25 | 7
Traffic Fines | 30074 | 112245 | 11 | 4 | 5
Table 1:  [Real-world event logs statistics] Number of traces
$\left|{D}\right|$, number of events $||D||$, number of different activities
$|\Omega|$, number of categorical variables $\left|{V_{\text{cat}}}\right|$,
and number of categorical variables $\left|{V_{\text{num}}}\right|$ for the
Sepsis and Traffic Fines real-world datasets.

Next, we evaluate on two publicly available real-world event logs, for which
we give the base statistics in Table 1. The first one, _Sepsis_ [13], contains
event traces from treating Sepsis patients in a Dutch hospital. The second
one, _Traffic Fines_ [6], [14, p. 20] is an event log of handling road-traffic
fines by the police of an Italian city. To reduce runtime, we randomly sample
20% of the original 150370 traces in the Traffic Fines event log. Furthermore,
we parallelize candidate generation and candidate evaluation of Moody on
twelve CPU cores.

First, we look at the insight we gain from some exemplary rules we find with
Moody on these event logs. Then, we evaluate how well the discovered rules
generalize to unseen test data.

IF group $=$ C THEN activity $=$ ER Triage IF group $=$ E THEN activity $=$
Release A IF group $=$ W THEN activity $=$ Admission IC IF group $=$ P THEN
activity $=$ Admission IC IF group $=$ F THEN activity $=$ Admission NC IF
group $=$ O THEN activity $=$ Admission NC Figure 7: [Responsibility rules for
the Sepsis event log] Different groups in the hospital are associated with
different activities for handling sepsis patients. IF Leukocytes $\leq$ 4.7
THEN Infusion $=$ True IF Leukocytes $\leq$ 2.8 THEN Diagnose $=$ GB IF
Leukocytes $=$ 7.8 THEN Diagnose $=$ AA Figure 8: [Leukocytes rules for the
Sepsis event log] Lower values for leukocytes (white blood cells) are
associated with certain diagnoses and infusions.

#### Sepsis rules

On the Sepsis dataset, Moody discovers in total 82 rules with an approximate
runtime of two hours. 23 of these rules express a correlation between the
_group_ attribute and the activity of an event, for which we show a subset in
Figure 7. The rules found by Moody indicate that certain groups in the
hospital are specialized in certain activities.

Furthermore, Moody finds rules, where leukocytes measurements imply the
diagnosis and the treatment of sepsis, as we show in Figure 8. These rules
enable to ask targeted questions to domain experts and thus can be a valuable
start to gain insight into this process.

IF amount $\leq$ 68.77 THEN points $=$ 0.0 IF 143.00 $\leq$ amount THEN points
$\in$ [0.0, 10.0] IF 131.00 $\leq$ amount THEN points $=$ 0.0 IF amount $=$
80.00 THEN points $\in$ [0.0, 2.0] Figure 9: [Traffic Fines rules] Moody finds
multiple relationships between the fine amount and the points deducted from
the offender’s driving license.

#### Traffic Fines rules

On the Traffic Fines dataset, Moody discovers in total 117 rules with an
approximate runtime of 4.5 hours. While we would expect that a higher fine
_amount_ correlates with a high number of _points_ deducted from the
offender’s driving license, the rules found by Moody contradict this
intuition, as we show in Figure 9. A closer look on the dataset indeed
confirms that there is no monotonic relationship between these two attributes.
Finding counter-intuitive but data-supported rules like these gives valuable
insight into the underlying process.

moody_generalization
$0$$5$$10$$15$$20$0.00.20.40.60.81.0Rule$\text{F}_{1}\leavevmode\nobreak\
\text{score}$$0$$20$$40$$60$$80$$100$$0$$100$$200$$300$$400$RuleRMSEtraintest

Figure 10: [Moody generalizes well] Train and test
$\text{F}_{1}\leavevmode\nobreak\ \text{score}$ on categorical variables
(left) and train and test root mean squared error on numerical variables
(right) for each rule found by Moody on the Traffic Fines log.

#### Generalization

Finally, we evaluate how well the rules discovered by Moody generalize to
unseen data. To this end, we split the Traffic Fines event log into a training
set and a test set with a distinct 20% of traces each. Then, we compare the
$\text{F}_{1}\leavevmode\nobreak\ \text{score}$ respectively RMSE on the
training set and the test set for each rule, which Moody discovers on the
training set. We show results in Figure 10. As we see, most of the rules have
a low prediction error on both sets. The gap between training and test
performance is small, which means that Moody finds well-generalizing rules.

##  7 Discussion

In our experiments, Moody does not only discover simple and thus interpretable
rules to unveil the data modifications within a process. It also finds rules
that accurately predicts the data values, and is more robust to sensible
amounts of noise than all baselines.

Yet, we see multiple interesting research directions to improve Moody. First,
extending the modeling language of Moody would allow understanding data
modifications of very complex processes. For example, this involves rules with
complex conditions joined by _(and)_ and _(or)_. We see adapting our MDL score
to such rules is relatively easy.

A more complex modeling language implies an even larger search space, and thus
we see the need to improve the runtime efficiency of Moody. As in many MDL-
based methods, the bottleneck of Moody is discrete combinatorial search.
Hence, we see an approach for mining data modification rules based on
differentiable pattern set mining [7] as a promising future direction.

Furthermore, we would like to put a stronger focus on causality of the
discovered rules. To this end, we may use well-defined measures for the causal
effect of a rule [3]. Alternatively, we would like to examine the link between
causality and two-part MDL codes in terms of algorithmic independence [16],
and how to use this during search for causal data modification rules.

Last but not least, we also see many interesting applications for Moody. As
the most compressing rules found by Moody define normal behavior, it would be
interesting to use them for anomaly detection [18]. Since the behavior of
real-world processes usually changes over time, we see Moody could help to
identify and understand concept drift [2, 25]. Finally, predicting data
attributes with Moody may be used in the simulation of process behavior for
process optimization [10].

##  8 Conclusion

We studied the hitherto largely neglected problem of discovering accurate yet
concise and interpretable rules how event attribute data changes in a business
process. We formalized the problem in terms of the Minimum Description Length
(MDL) principle, by which we choose the model with the best lossless
description of the data. To efficiently search for rule models in practice, we
proposed our greedy method Moody. Through extensive experiments on both
synthetic and real-world data, we showed Moody indeed discovers succinct and
interpretable if-then rules, needs little data for accurate discovery, is
robust to sensible amounts of noise, and thus gives valuable insight into data
modifications.

Besides applying Moody on downstream tasks such as anomaly detection, concept
drift detection and simulation, future work involves extending the rule
language of Moody to model more complex conditions for data changes, and
runtime optimizations to enable search for such complex rules in feasible
time.

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##  A Appendix

In this section, we give additional details of Moody and our empirical
evaluation, which we could not include into the main paper.

###  A.1 Submodularity of our MDL score

Rules:
R3: IF product $=$ shirt THEN vendor $=$ C R4: IF product $=$ bag THEN vendor
$\in$ {A, B}
Trace:
bag A shirt C shirt B pants C product: vendor: event:E5E6E7E8 Figure 11:
[Example rules and trace] We show two simple rules (top) and a simple trace
(bottom) to disprove submodularity and monotonicity for our score.

Submodularity [11, p. 15] requires our score to fulfill

$L(D,M_{a}\cap M_{b})+L(D,M_{a}\cup M_{b})\leq L(D,M_{a})+L(D,M_{b})$

for all valid models $M_{a},M_{b}$. As a counterexample for this, consider the
trace in Figure 11 as our event log $D$ and the models
$M_{1}=\\{R\ref*{rule:app-1}\\}$ and $M_{2}=\\{R\ref*{rule:app-2}\\}$. Here,
we get $L(D,M_{1}\cap M_{2})+L(D,M_{1}\cup M_{2})\approx 59.448$ and
$L(D,M_{1})+L(D,M_{2})\approx 59.199$ and thus

$L(D,M_{1}\cap M_{2})+L(D,M_{1}\cup
M_{2})>L(D,M_{1})+L(D,M_{2})\leavevmode\nobreak\ .$

###  A.2 Monotonicity of our MDL score

Monotonicity [11, p. 359] requires our score to be non-increasing or non-
decreasing as we add rules to our model. To disprove this for our score,
assume that our event log contains the trace in Figure 11 repeated 20 times.
Then, we get the scores $L(D,\emptyset)\approx 241.519$,
$L(D,\\{R\ref*{rule:app-1}\\})\approx 279.595$ and
$L(D,\\{R\ref*{rule:app-2}\\})\approx 239.595$. So, extending the empty model
with different rules can both increase and decrease the score, and thus it is
not monotone.

###  A.3 Supplementary material

In the supplementary material222https://eda.rg.cispa.io/prj/moody/ we make
code and data publicly available. This also includes the full Master’s Thesis
that the main paper is based on. In this thesis, we provide further details on
our encoding and the Moody algorithm. Further, we explain the setup of the
experiments on synthetic data in more detail.