# Approximation by $T$ means with respect to Vilenkin system in Lebesgue
spaces and Lipschitz classes

N. Anakidze, N. Areshidze, L.-E. Persson and G. Tephnadze  N. Anakidze, The
University of Georgia, School of Science and Technology, 77a Merab Kostava St,
Tbilisi, 0171, Georgia<EMAIL_ADDRESS>N. Areshidze, Tbilisi State
University, Faculty of Exact and Natural Sciences, Department of Mathematics,
Chavchavadze str. 1, tbilisi 0128, georgia<EMAIL_ADDRESS>L.-E.
Persson, UiT The Arctic University of Norway, P.O. Box 385, N-8505, Narvik,
Norway and department of Mathematics and Computer Science, Karlstad
University, 65188 Karlstad, Sweden<EMAIL_ADDRESS>G. Tephnadze, The
University of Georgia, School of Science and Technology, 77a Merab Kostava St,
Tbilisi, 0171, Georgia<EMAIL_ADDRESS>
###### Abstract.

In this paper we present and prove some new results concerning approximation
properties of $T$ means with respect to the Vilenkin system in Lebesgue spaces
and Lipschitz classes for any $1\leq p<\infty$. As applications, we obtain
extension of some known approximation inequalities.

The research was supported by Shota Rustaveli National Science Foundation
(SRNSF) grant no. FR-24-698.

2010 Mathematics Subject Classification: 42C10.

Key words and phrases: Vilenkin group, Vilenkin system, $T$ means, Nörlund
means, Fejér means, approximation, Lebesgue spaces, Lipschitz classes.

##  1\. Introduction

Concerning some definitions and notations used in this introduction we refer
to Section 2.

It is well-known (see e.g. [17], [29] and [40]) that, for any $1\leq
p\leq\infty$ and $f\in L_{p}(G_{m}),$ there exists an absolute constant
$C_{p},$ depending only on $p$ such that

$\left\|\sigma_{n}f\right\|_{p}\leq C_{p}\left\|f\right\|_{p}.$

Moreover, (see e.g. [29]) if $1\leq p\leq\infty$, $M_{N}\leq n<M_{N+1}$, $f\in
L^{p}(G_{m})$ and $n\in\mathbb{N},$ then

(1) $\displaystyle\left\|\sigma_{n}f-f\right\|_{p}\leq
2R^{5}\sum_{s=0}^{N}\frac{M_{s}}{M_{N}}\omega_{p}\left(1/M_{s},f\right),$

where $R:=\sup_{k\in\mathbb{N}}m_{k}$ and $\omega_{p}(\delta,f)$ is the
modulus of continuity of $L^{p}$ functions, $1\leq p<\infty$ functions defined
by

$\displaystyle\omega_{p}(\delta,f)=\sup_{|t|<\delta}\|f(x-t)-f(x)\|_{p},\quad\quad\delta>0.$

It follows that if $f\in Lip\left(\alpha,p\right),$ i.e.,

$\displaystyle Lip(\alpha,p):=\\{f\in
L^{p}:\omega_{p}(\delta,f)=O(\delta^{\alpha})\quad\textrm{as}\quad\delta\to
0\\},$

then

$\left\|\sigma_{n}f-f\right\|_{p}=\left\\{\begin{array}[]{c}O\left(1/M_{N}\right),\text{
\ \ \ if \ \ \ }\alpha>1,\\\ O\left(N/M_{N}\right),\text{ \ \ \ if \ \ \
}\alpha=1,\\\ O\left(1/M_{n}^{\alpha}\right),\text{ \ \ \ \ \ \ \ if
}\alpha<1.\end{array}\right.$

Moreover, (see e.g. [29]) if $1\leq p<\infty,$ $f\in L^{p}(G_{m})$ and

$\left\|\sigma_{M_{n}}f-f\right\|_{p}=o\left(1/M_{n}\right),\ \text{as}\
n\rightarrow\infty,$

then $f$ is a constant function.

For the maximal operators of Vilenkin-Fejer means $\sigma^{*}$, defined by

$\sigma^{*}f=\sup_{n\in\mathbb{N}}|\sigma_{n}f|$

the weak-$(1,1)$ type inequality

$\left\|\sigma^{*}f\right\|_{weak-L_{1}}\leq C\left\|f\right\|_{1},\text{
\quad}\left(f\in L^{1}(G_{m})\right)$

can be found in Schipp [30] for Walsh series and in Pál, Simon [28] and Weisz
[38] for bounded Vilenkin series. Boundedness of the maximal operators of
Vilenkin-Féjer means of the one- and two-dimensional cases can be found in
Fridli [12], Gát [14], Goginava [16], Nagy and Tephnadze [24, 25, 26, 27],
Simon [32, 33] and Weisz [39].

Convergence and summability of Nörlund means with respect to Vilenkin systems
were studied by Areshidze and Tephnadze [2], Blahota and Nagy [5], Blahota,
Persson and Tephnadze [9] (see also [3, 4, 7, 8]), Fridli, Manchanda and
Siddiqi [13], Goginava [15], Nagy [21, 22, 23] (see also [10] and [11]) and
Memic [18].

Móricz and Siddiqi [20] investigated the approximation properties of some
special Nörlund means of Walsh-Fourier series of $L^{p}$ functions in norm. In
particular, they proved that if $f\in L^{p}(G),$ $1\leq p\leq\infty,$
$n=2^{j}+k,$ $1\leq k\leq 2^{j}\ (n\in\mathbb{N}_{+})$ and
$(q_{k},k\in\mathbb{N})$ is a sequence of non-negative numbers, such that

$\frac{n^{\gamma-1}}{Q_{n}^{\gamma}}\sum_{k=0}^{n-1}q^{\gamma}_{k}=O(1),\ \
\text{for some}\ \ 1<\gamma\leq 2,$

then there exists an absolute constant $C_{p},$ depending only on $p$ such
that

$\|t_{n}f-f\|_{p}\leq\frac{C_{p}}{Q_{n}}\sum_{i=0}^{j-1}2^{i}q_{n-2^{i}}\omega_{p}\left(\frac{1}{2^{i}},f\right)+C_{p}\omega_{p}\left(\frac{1}{2^{j}},f\right),$

when the sequence $(q_{k},k\in\mathbb{N})$ is non-decreasing, while

$\|t_{n}f-f\|_{p}\leq\frac{C_{p}}{Q_{n}}\sum_{i=0}^{j-1}\left(Q_{n-2^{i}+1}-Q_{n-2^{i+1}+1}\right)\omega_{p}\left(\frac{1}{2^{i}},f\right)+C_{p}\omega_{p}\left(\frac{1}{2^{j}},f\right),$

when the sequence $(q_{k},k\in\mathbb{N})$ is non-increasing.

Tutberidze [34] (see also [29]) proved that if $T_{n}$ are $T$ means generated
by either a non-increasing sequence $\\{q_{k},k\in\mathbb{N}\\}$ or a non-
decreasing sequence $\\{q_{k},k\in\mathbb{N}\\}$ satisfying the condition

$\frac{q_{0}}{Q_{k}}=O\left(\frac{1}{k}\right),\text{ \ \ as \ \
}k\rightarrow\infty,$

then there exists an absolute constant $C,$ such that

$\left\|T^{*}f\right\|_{weak-L_{1}}\leq C\left\|f\right\|_{1},\text{
\quad}\left(f\in L^{1}(G_{m})\right)$

holds. From these results follows that if $f\in L^{p}(G_{m}),$ where $1\leq
p<\infty$ and either the sequence $\\{q_{k},k\in\mathbb{N}\\}$ is non-
increasing, or $\\{q_{k},k\in\mathbb{N}\\}$ is a sequence of non-decreasing
numbers, such that the condition

(2) $\frac{q_{n-1}}{Q_{n}}=O\left(\frac{1}{n}\right),\text{\ \ as \ \
}n\rightarrow\infty,$

is fulfilled, then

$\underset{n\rightarrow\infty}{\lim}\|{{T}_{n}}f-f\|_{p}\to 0,\ \ \ \text{as}\
\ \ n\to\infty.$

For the Walsh system in [19] Móricz and Rhoades proved that if $f\in L^{p},$
where $1\leq p<\infty,$ and ${{T}_{n}}$ are regular $T$ means generated by a
non-increasing sequence $\\{q_{k},\ k\in\mathbb{N}\\},$ then, for any
$2^{N}\leq n<2^{N+1},$ we have the following approximation inequality:

(3)
$\|T_{n}f-f\|_{p}\leq\frac{C_{p}}{Q_{n}}\sum_{s=0}^{N-1}2^{s}q_{2^{s}}\omega_{p}\left(1/2^{s},f\right)+C_{p}\omega_{p}\left(1/2^{N},f\right).$

In the case when the sequence $\\{q_{k},\ k\in\mathbb{N}\\}$ is non-decreasing
and satisfying the condition

(4) $\frac{q_{k-1}}{Q_{k}}=O\left(\frac{1}{k}\right),\text{ \ \ as \ \
}k\rightarrow\infty,$

then the following inequality holds:

(5) $\displaystyle\|T_{n}f-f\|_{p}\leq
C_{p}\sum_{j=0}^{N-1}2^{j-N}\omega_{p}\left(1/2^{j},f\right)+C_{p}\omega_{p}\left(1/2^{N},f\right).$

In this paper we use a new approach and generalize the inequalities in (3) and
(5) for $T$ means with respect to the Vilenkin system (see Theorems 1 and 2).
We also prove a new inequality for the subsequences $\\{T_{M_{n}}\\}$ means
when the sequence $\\{q_{k},\ k\in\mathbb{N}\\}$ is non-decreasing (see
Theorem 3).

The main results are presented in Section 3 together with 4 Corollaries, which
also generalize known results (see [19]). The detailed proofs are presented in
Section 4. In order not do disturb our presentations in these Sections we have
reserved Section 2 for some Preliminaries.

##  2\. Preliminaries

Let $\mathbb{N}_{+}$ denote the set of the positive integers,
$\mathbb{N}:=\mathbb{N}_{+}\cup\\{0\\}.$ Let $m=:(m_{0},m_{1},...)$ be a
sequence of positive integers not less than 2. Denote by

$Z_{m_{k}}:=\\{0,1,...,m_{k}-1\\}$

the additive group of integers modulo $m_{k}.$ Define the group $G_{m}$ as the
complete direct product of the group $Z_{m_{k}}$ with the product of the
discrete topologies of $Z_{m_{k}}\textrm{'s}$.

The direct product $\mu$ of the measures

$\mu_{k}\left(\\{j\\}\right):=1/m_{k}\text{ \ }(j\in Z_{m_{k}})$

is the Haar measure on $G_{m}$ with $\mu\left(G_{m}\right)=1.$

If $\sup_{k\in\mathbb{N}}m_{k}<+\infty$, then we call $G_{m}$ a bounded
Vilenkin group. If the sequence $\\{m_{k}\\}_{k\geq 0}$ is unbounded, then
$G_{m}$ is said to be unbounded Vilenkin group. In this paper we consider only
bounded Vilenkin groups.

The elements of $G_{m}$ are represented by the sequences

$x:=(x_{0},x_{1},\dots,x_{k},\dots)\qquad\left(\text{ }x_{k}\in
Z_{m_{k}}\right).$

It is easy to give a base for the neighborhood of $G_{m}$, namely

$\displaystyle I_{0}\left(x\right):=G_{m},\ \ \ I_{n}(x):=\\{y\in G_{m}\mid
y_{0}=x_{0},\dots,y_{n-1}=x_{n-1}\\}\text{ }(x\in G_{m},\text{
}n\in\mathbb{N}).$

For simplicity we also define $I_{n}:=I_{n}(0).$

Next, we define a generalized number system based on $m$ in the following way:

$M_{0}=:1,\quad M_{k+1}=:m_{k}M_{k}\quad(k\in\mathbb{N})$

Then every $n\in\mathbb{N}$ can be uniquely expressed as

$n=\sum_{k=0}^{\infty}n_{j}M_{j},\ \ \ \text{where }\ \ \ n_{j}\in Z_{m_{j}}\
\ \ (j\in\mathbb{N})$

and only a finite number of $n_{j}`$s differ from zero. Let

$|n|=:\max\\{j\in\mathbb{N},n_{j}\neq 0\\}.$

Moreover, Vilenkin (see [35, 36, 37]) investigated the group $G_{m}$ and
introduced the Vilenkin system $\\{{\psi}_{j}\\}_{j=0}^{\infty}$ as

$\psi_{n}\left(x\right):=\prod_{k=0}^{\infty}r_{k}^{n_{k}}\left(x\right)\text{
\quad}\left(n\in\mathbb{N}\right).$

where $r_{k}(x)$ are the generalized Rademacher functions defined by

$r_{k}(x):=\exp(2\pi ix_{k}/m_{k}),\text{ \ \ }\left(k\in\mathbb{N}\right).$

These systems include as a special case the Walsh system when $m_{k}=2$ for
any $k\in\mathbb{N}$.

The norms (or quasi-norms) $\left\|f\right\|_{p},$ $0<p<\infty,$ of the
Lebesgue spaces $L^{p}(G_{m})$ are defined by

$\left\|f\right\|_{p}^{p}:=\int_{G_{m}}\left|f\right|^{p}d\mu.$

The Vilenkin system is orthonormal and complete in $L^{2}\left(G_{m}\right)\,$
(see e.g. [1] and [31]).

If $f\in L^{1}\left(G_{m}\right)$, we can define the Fourier coefficients, the
partial sums of the Fourier series, the Fejér means, the Dirichlet and Fejér
kernels with respect to the Vilenkin system in the usual manner:

$\displaystyle\widehat{f}\left(k\right)$ $\displaystyle:$
$\displaystyle=\int_{G_{m}}f\overline{\psi}_{k}d\mu,\,\text{\quad}\left(k\in\mathbb{N}\right),$
$\displaystyle S_{n}f$ $\displaystyle:$
$\displaystyle=\sum_{k=0}^{n-1}\widehat{f}\left(k\right)\psi_{k},\text{
\quad}\left(n\in\mathbb{N}_{+},\text{ }S_{0}f:=0\right),$
$\displaystyle\sigma_{n}f$ $\displaystyle:$
$\displaystyle=\frac{1}{n}\sum_{k=1}^{n}S_{k}f,\text{ \quad\ \
}\left(n\in\mathbb{N}_{+}\right).$ $\displaystyle D_{n}$ $\displaystyle:$
$\displaystyle=\sum_{k=0}^{n-1}\psi_{k},\text{ \quad\ \
}\left(n\in\mathbb{N}_{+}\right).$ $\displaystyle K_{n}$ $\displaystyle:$
$\displaystyle=\frac{1}{n}\sum_{k=1}^{n}D_{k},\text{ \quad\ \
}\left(n\in\mathbb{N}_{+}\right).\ $

Recall that (see e.g. [1] and [29]),

(6) $\quad D_{M_{n}}\left(x\right)=\left\\{\begin{array}[]{l}\text{
}M_{n},\text{\thinspace\thinspace\thinspace if\thinspace\thinspace}x\in
I_{n},\\\ \text{
}0,\text{\thinspace\thinspace\thinspace\thinspace\thinspace\thinspace\thinspace\thinspace
if \thinspace\thinspace}x\notin I_{n},\end{array}\right.$

$\displaystyle D_{M_{n}-j}\left(x\right)$ $\displaystyle=$ $\displaystyle
D_{M_{n}}\left(x\right)-\overline{\psi}_{M_{n}-1}(-x)D_{j}(-x)$
$\displaystyle=$ $\displaystyle
D_{M_{n}}\left(x\right)-\psi_{M_{n}-1}(x)\overline{D_{j}}(x),\,\,\ 0\leq
j<M_{n}.$ (8) $n\left|K_{n}\right|\leq
2R^{2}\sum_{l=0}^{|n|}M_{l}\left|K_{M_{l}}\right|,$

and

(9) $\displaystyle\int_{G_{m}}K_{n}(x)d\mu(x)=1,\ \ \ \ \
\sup_{n\in\mathbb{N}}\int_{G_{m}}\left|K_{n}(x)\right|d\mu(x)\leq R^{5}.$

where $R:=\sup_{k\in\mathbb{N}}m_{k}.$ Moreover, if $n>t,$ $t,n\in\mathbb{N},$
then

(10)
$K_{M_{n}}\left(x\right)=\left\\{\begin{array}[]{ll}\frac{M_{t}}{1-r_{t}(x)},&x\in
I_{t}\backslash I_{t+1},\quad x-x_{t}e_{t}\in I_{n},\\\
\frac{M_{n}+1}{2},&x\in I_{n},\\\ 0,&\text{otherwise.}\end{array}\right.$

The $n$-th Nörlund mean $t_{n}$ and $T$ mean $T_{n}$ of $f\in L^{1}(G_{m})$
are defined by

$t_{n}f:=\frac{1}{Q_{n}}\overset{n}{\underset{k=1}{\sum}}q_{n-k}S_{k}f$

and

$T_{n}f:=\frac{1}{Q_{n}}\overset{n-1}{\underset{k=0}{\sum}}q_{k}S_{k}f,$

where

$Q_{n}:=\sum_{k=0}^{n-1}q_{k}.$

Here $\\{q_{k},\ k\geq 0\\}$ is a sequence of nonnegative numbers, where
$q_{0}>0$ and

(11) $\lim_{n\rightarrow\infty}Q_{n}=\infty.$

Then, a $T$ mean generated by $\\{q_{k},\ k\geq 0\\}$ is regular if and only
if the condition (11) is fulfilled (see [29]).

It is evident that

$T_{n}f\left(x\right)=\underset{G_{m}}{\int}f\left(t\right)F_{n}\left(x-y\right)d\mu\left(y\right)$

where

(12) $F_{n}:=\frac{1}{Q_{n}}\overset{n-1}{\underset{k=0}{\sum}}q_{k}D_{k},$

which are called the kernels of the $T$ means.

By applying Abel transformation we get the following two useful identities:

(13) $\displaystyle Q_{n}:=\overset{n-1}{\underset{k=0}{\sum}}q_{k}\cdot
1=\overset{n-2}{\underset{k=0}{\sum}}(q_{k}-q_{k+1})k+q_{n-1}(n-1)$

and

(14)
$T_{n}f=\frac{1}{Q_{n}}\left(\overset{n-2}{\underset{k=0}{\sum}}(q_{k}-q_{k+1})k\sigma_{k}f+q_{n-1}(n-1)\sigma_{n-1}f\right).$

##  3\. The Main Results

Our first main result reads:

######  Theorem 1.

Let $f\in L^{p}(G_{m}),$ where $1\leq p<\infty$ and ${{T}_{n}}$ are $T$ means
generated by a non-increasing sequence $\\{q_{k},\ k\in\mathbb{N}\\}.$ Then,
for any $n,N\in\mathbb{N},$ $M_{N}\leq n<M_{N+1},$ we have the following
inequality:

(15) $\displaystyle\ \ \
\|T_{n}f-f\|_{p}\leq\frac{6R^{6}}{Q_{n}}\sum_{j=0}^{N-1}M_{j}q_{{}_{M_{j}}}\omega_{p}\left(1/M_{j},f\right)+4R^{6}\omega_{p}\left(1/M_{N},f\right).$

Next we state and prove a similar inequality for non-decreasing sequences but
under some restrictions.

######  Theorem 2.

Let $f\in L^{p}(G_{m}),$ where $1\leq p<\infty$ and $T_{n}$ are regular $T$
means generated by a non-decreasing sequence $\\{q_{k},\ k\in\mathbb{N}\\}$.
Then, for any $n,N\in\mathbb{N},$ $M_{N}\leq n<M_{N+1},$ we have the following
inequality:

(16)
$\|T_{n}f-f\|_{p}\leq\frac{6R^{6}q_{n-1}}{Q_{n}}\sum_{j=0}^{N-1}M_{j}\omega_{p}\left(1/M_{j},f\right)+\frac{4R^{6}q_{n-1}M_{N}}{Q_{n}}\omega_{p}\left(1/M_{N},f\right).$

If, in addition, the sequence $\\{q_{k},\ k\in\mathbb{N}\\}$ satisfies the
condition (2), then the inequality

(17) $\displaystyle\|T_{n}f-f\|_{p}\leq
C_{p}\sum_{j=0}^{N}\frac{M_{j}}{M_{N}}\omega_{p}\left(1/2^{j},f\right)$

holds for some constant $C_{p}$ only depending on $p.$

Finally, we state and prove the third main result for the non-decreasing
sequences, where we prove a more precise result then that in (17) and without
the restriction (2) but only for subsequences.

######  Theorem 3.

Let $f\in L^{p}(G_{m}),$ where $1\leq p<\infty$ and ${{T}_{k}}$ are regular
$T$ means generated by a non-decreasing sequence $\\{q_{k},\
k\in\mathbb{N}\\}$. Then, for any $n\in\mathbb{N},$ the following inequality
holds:

$\displaystyle\|T_{M_{n}}f-f\|_{p}$ $\displaystyle\leq$ $\displaystyle
R^{2}\sum_{j=0}^{n-1}\frac{M_{j}}{M_{n}}\omega_{p}\left(1/M_{j},f\right)$
$\displaystyle+$
$\displaystyle\frac{2R^{4}}{q_{0}}\overset{n-1}{\underset{j=0}{\sum}}\frac{(n-j)(q_{{}_{M_{n}-M_{j}}})M_{j}}{M_{n}}\omega_{p}\left(1/M_{j},f\right)$
$\displaystyle+$
$\displaystyle\left(R^{2}+2\right)\omega_{p}\left(1/M_{n},f\right).$

We also point out the following generalizations of some results in [19] (in
this paper only the Walsh system was considered):

######  Corollary 1.

Let $\\{q_{k},\ k\geq 0\\}$ be a sequence of non-negative and non-increasing
numbers, while in case when the sequence is non-decreasing it is assumed that
also the condition (2) is satisfied. If $f\in Lip(\alpha,p)$ for some
$\alpha>0$ and $1\leq p<\infty,$ then

$\|T_{n}f-f\|_{p}=\left\\{\begin{array}[]{l}\text{
}O(n^{-\alpha}),\text{\qquad\qquad if\qquad}0<\alpha<1,\\\ \text{
}O(n^{-1}\log n),\text{\qquad if \quad\ }\alpha=1,\\\ \text{
}O(n^{-1}),\text{\qquad\qquad if \quad\ \ }\alpha>1,\end{array}\right.$

######  Corollary 2.

Let $\\{q_{k},\ k\geq 0\\}$ be a sequence of non-negative and non-increasing
numbers such that

$\displaystyle q_{k}\asymp k^{-\beta}\ \ \ \text{for some}\ \ \ 0<\beta\leq 1$

is satisfied.

If $f\in Lip(\alpha,p)$ for some $\alpha>0$ and $1\leq p<\infty,$ then

$\displaystyle\|T_{n}f-f\|_{p}=\left\\{\begin{array}[]{l}\text{
}O(n^{-\alpha}),\text{\qquad\qquad\qquad\ \ \ \ \ if \ \ \ \
}\alpha+\beta<1,\\\ \text{ }O(n^{-(1-\beta)}\log n+n^{-\alpha}),\text{\ \ \ if
\quad\ }\alpha+\beta=1,\\\ \text{ }O(n^{-(1-\beta)}),\text{\qquad\qquad\qquad
if \quad\ \ }\alpha+\beta>1,\ \beta>1,\\\ \text{ }O((\log
n)^{-1}),\text{\qquad\qquad\ \ \ \ if \quad\ }\ \beta=1.\\\
\end{array}\right.$

######  Corollary 3.

Let $\\{q_{k},\ k\geq 0\\}$ be a sequence of non-negative and non-increasing
numbers such that

$\displaystyle q_{k}\asymp{(\log k)}^{-\beta}\ \ \ \text{for some}\ \ \
\beta>0$

is satisfied.

If $f\in Lip(\alpha,p)$ for some $\alpha>0$ and $1\leq p<\infty,$ then

$\|T_{n}f-f\|_{p}=\left\\{\begin{array}[]{l}\text{
}O(n^{-\alpha}),\text{\qquad\qquad\qquad\ \ \ if \qquad}0<\alpha<1,\
\beta>0,\\\ \text{ }O(n^{-1}\log n),\text{\qquad\qquad\ \ \ if
\qquad}\alpha=1,\ 0<\beta<1,\\\ \text{ }O(n^{-1}\log n\log\log n),\text{\ \ \
\ if \quad\ \ }\ \alpha=\beta=1,\\\ \text{ }O(n^{-1}(\log
n)^{\beta}),\text{\qquad\qquad if \quad}\quad\alpha>1,\ \beta>0.\\\
\end{array}\right.$

######  Corollary 4.

Let $f\in L^{p}(G_{m}),$ where $1\leq p<\infty$ and $\\{q_{k},\ k\geq 0\\}$ is
a sequence of non-negative and non-increasing numbers, while in case when the
sequence is non-decreasing it is also assumed that the condition (2) is
satisfied. Then,

$\underset{n\rightarrow\infty}{\lim}\|{{T}_{n}}f-f\|_{p}\to 0,\ \ \ \text{as}\
\ \ n\to\infty.$

##  4\. Proofs

###### Proof of Theorem 1..

Let $M_{N}\leq n<M_{N+1}.$ Since $T_{n}$ are regular $T$ means generated by
the sequence of non-increasing numbers $\\{q_{k}:k\in\mathbb{N}\\},$ we can
combine (13) and (14) and conclude that

$\displaystyle\|T_{n}f-f\|_{p}$ $\displaystyle\leq$
$\displaystyle\frac{1}{Q_{n}}\left(\overset{n-2}{\underset{j=0}{\sum}}(q_{j}-q_{j+1})j\|\sigma_{j}f-f\|_{p}+q_{n-1}(n-1)\|\sigma_{n-1}f-f\|_{p}\right)$
$\displaystyle:=$ $\displaystyle I+II.$

Moreover,

$\displaystyle I$ $\displaystyle=$
$\displaystyle\frac{1}{Q_{n}}\overset{M_{N}-1}{\underset{j=1}{\sum}}\left(q_{j}-q_{j+1}\right)j\|\sigma_{j}f-f\|_{p}$
$\displaystyle+$
$\displaystyle\frac{1}{Q_{n}}\overset{n-1}{\underset{j=M_{N}}{\sum}}\left(q_{j}-q_{j+1}\right)j\|\sigma_{j}f-f\|_{p}$
$\displaystyle:=$ $\displaystyle I_{1}+I_{2}.$

Now we estimate all terms separately. By applying estimate (1) for $I_{1}$ we
obtain that

$\displaystyle I_{1}$ $\displaystyle\leq$
$\displaystyle\frac{2R^{5}}{Q_{n}}\overset{N-1}{\underset{k=0}{\sum}}\overset{M_{k+1}-1}{\underset{j=M_{k}}{\sum}}\left(q_{j}-q_{j+1}\right)j\sum_{s=0}^{k}\frac{M_{s}}{M_{k}}\omega_{p}\left(1/M_{s},f\right)$
$\displaystyle\leq$
$\displaystyle\frac{2R^{6}}{Q_{n}}\overset{N-1}{\underset{k=0}{\sum}}M_{k}\overset{M_{k+1}-1}{\underset{j=M_{k}}{\sum}}\left(q_{j}-q_{j+1}\right)\sum_{s=0}^{k}\frac{M_{s}}{M_{k}}\omega_{p}\left(1/M_{s},f\right)$
$\displaystyle\leq$
$\displaystyle\frac{2R^{6}}{Q_{n}}\overset{N-1}{\underset{k=0}{\sum}}\left(q_{{}_{M_{k}}}-q_{{}_{M_{k+1}}}\right)\sum_{s=0}^{k}M_{s}\omega_{p}\left(1/M_{s},f\right)$
$\displaystyle\leq$
$\displaystyle\frac{2R^{6}}{Q_{n}}\overset{N-1}{\underset{s=0}{\sum}}M_{s}\omega_{p}\left(1/M_{s},f\right)\sum_{k=s}^{N-1}\left(q_{{}_{{M_{k}}}}-q_{{}_{M_{k+1}}}\right)$
$\displaystyle\leq$
$\displaystyle\frac{2R^{6}}{Q_{n}}\overset{N-1}{\underset{s=0}{\sum}}M_{s}q_{{}_{M_{s}}}\omega_{p}\left(1/M_{s},f\right).$

Moreover,

$\displaystyle\quad\quad I_{2}$ $\displaystyle\leq$
$\displaystyle\frac{2R^{5}}{Q_{n}}\overset{n-1}{\underset{j=M_{N}}{\sum}}\left(q_{j}-q_{j+1}\right)j\sum_{s=0}^{N}\frac{M_{s}}{M_{N}}\omega_{p}\left(1/M_{s},f\right)$
$\displaystyle\leq$
$\displaystyle\frac{2R^{6}M_{N}}{Q_{n}}\overset{n-1}{\underset{j=M_{N}}{\sum}}\left(q_{j}-q_{j+1}\right)\sum_{s=0}^{N}\frac{M_{s}}{M_{N}}\omega_{p}\left(1/M_{s},f\right)$
$\displaystyle\leq$
$\displaystyle\frac{2R^{6}q_{{}_{M_{N}}}}{Q_{n}}\sum_{s=0}^{N}M_{s}\omega_{p}\left(1/M_{s},f\right)$
$\displaystyle\leq$
$\displaystyle\frac{2R^{6}}{Q_{n}}\overset{N}{\underset{s=0}{\sum}}M_{s}q_{{}_{M_{s}}}\omega_{p}\left(1/M_{s},f\right)$
$\displaystyle\leq$
$\displaystyle\frac{2R^{6}}{Q_{n}}\overset{N-1}{\underset{s=0}{\sum}}M_{s}q_{{}_{M_{s}}}\omega_{p}\left(1/M_{s},f\right)+2R^{6}\omega_{p}\left(1/M_{s},f\right).$

For $II$ we have that

$\displaystyle II$ $\displaystyle\leq$
$\displaystyle\frac{2R^{5}M_{N+1}q_{n-1}}{Q_{n}}\sum_{s=0}^{N}\frac{M_{s}}{M_{N}}\omega_{p}\left(1/M_{s},f\right)$
$\displaystyle\leq$
$\displaystyle\frac{2R^{6}}{Q_{n}}\sum_{s=0}^{N-1}M_{s}q_{{}_{M_{s}}}\omega_{p}\left(1/M_{s},f\right)+2R^{6}\omega_{p}\left(1/M_{N},f\right).$

The proof of (15) is complete by just combining (4)-(4).

∎

###### Proof of Theorem 2..

Let $M_{N}\leq n<M_{N+1}.$ Since $T_{n}$ are regular $T$ means, generated by a
sequence of non-decreasing numbers $\\{q_{k}:k\in\mathbb{N}\\},$ by combining
(13) and (14), we find that

$\displaystyle\|T_{n}f-f\|_{p}$ $\displaystyle\leq$
$\displaystyle\frac{1}{Q_{n}}\left(\overset{n-1}{\underset{j=1}{\sum}}\left(q_{j+1}-q_{j}\right)j\|\sigma_{j}f-f\|_{p}+q_{n-1}(n-1)\|\sigma_{n}f-f\|_{p}\right)$
$\displaystyle:=$ $\displaystyle I+II.$

Furthermore,

$\displaystyle I$ $\displaystyle=$
$\displaystyle\frac{1}{Q_{n}}\overset{M_{N}-1}{\underset{j=1}{\sum}}\left(q_{j+1}-q_{j}\right)j\|\sigma_{j}f-f\|_{p}$
$\displaystyle+$
$\displaystyle\frac{1}{Q_{n}}\overset{n-1}{\underset{j=M_{N}}{\sum}}\left(q_{j+1}-q_{j}\right)j\|\sigma_{j}f-f\|_{p}$
$\displaystyle=$ $\displaystyle I_{1}+I_{2}.$

Analogously to (4) we get that

$\displaystyle I_{1}$ $\displaystyle\leq$
$\displaystyle\frac{2R^{6}}{Q_{n}}\overset{N-1}{\underset{k=0}{\sum}}\left(q_{M_{k+1}}-q_{M_{k}}\right)\sum_{s=0}^{k}M_{s}\omega_{p}\left(1/M_{s},f\right)$
$\displaystyle\leq$
$\displaystyle\frac{2R^{6}}{Q_{n}}\sum_{s=0}^{N-1}M_{s}\omega_{p}\left(1/M_{s},f\right)\overset{N-1}{\underset{k=s}{\sum}}\left(q_{{}_{M_{k+1}}}-q_{{}_{M_{k}}}\right)$
$\displaystyle=$
$\displaystyle\frac{2R^{6}}{Q_{n}}\sum_{s=0}^{N-1}M_{s}\omega_{p}\left(1/M_{s},f\right)(q_{{}_{M_{N}}}-q_{{}_{M_{s}}})$
$\displaystyle\leq$
$\displaystyle\frac{2R^{6}q_{M_{N}}}{Q_{n}}\sum_{s=0}^{N-1}M_{s}\omega_{p}\left(1/M_{s},f\right)$
$\displaystyle\leq$
$\displaystyle\frac{2R^{6}q_{n-1}}{Q_{n}}\sum_{s=0}^{N-1}M_{s}\omega_{p}\left(1/M_{s},f\right).$

In the similar way as in (4) we find that

$\displaystyle I_{2}$ $\displaystyle\leq$
$\displaystyle\frac{2R^{5}}{Q_{n}}\overset{n-1}{\underset{j=1}{\sum}}\left(q_{j+1}-q_{j}\right)j\sum_{s=0}^{N}\frac{M_{s}}{M_{N}}\omega_{p}\left(1/M_{s},f\right)$
$\displaystyle=$
$\displaystyle\frac{2R^{5}}{Q_{n}}\left((n-1)q_{n-1}-Q_{n}\right)\sum_{s=0}^{N}\frac{M_{s}}{M_{N}}\omega_{p}\left(1/M_{s},f\right)$
$\displaystyle\leq$
$\displaystyle\frac{2R^{5}M_{N+1}q_{n-1}}{Q_{n}M_{N}}\sum_{s=0}^{N}M_{s}\omega_{p}\left(1/M_{s},f\right)$
$\displaystyle\leq$
$\displaystyle\frac{2R^{6}q_{n-1}}{Q_{n}}\sum_{s=0}^{N}M_{s}\omega_{p}\left(1/M_{s},f\right)$
$\displaystyle\leq$
$\displaystyle\frac{2R^{6}q_{n-1}}{Q_{n}}\sum_{s=0}^{N}M_{s}\omega_{p}\left(1/M_{s},f\right)+\frac{2R^{6}q_{n-1}M_{N}}{Q_{n}}\left(1/M_{N},f\right).$

For $II$ we have that

$\displaystyle II$ $\displaystyle\leq$
$\displaystyle\frac{2R^{5}q_{n-1}M_{N+1}}{Q_{n}}\sum_{s=0}^{N}\frac{M_{s}}{M_{N}}\omega_{p}\left(1/M_{s},f\right)$
$\displaystyle\leq$
$\displaystyle\frac{2R^{6}q_{n-1}}{Q_{n}}\sum_{s=0}^{N}M_{s}\omega_{p}\left(1/M_{s},f\right)$
$\displaystyle=$
$\displaystyle\frac{2R^{6}q_{n-1}}{Q_{n}}\sum_{s=0}^{N-1}M_{s}\omega_{p}\left(1/M_{s},f\right)+\frac{2R^{6}q_{n-1}M_{N}}{Q_{n}}\left(1/M_{N},f\right).$

By combining (4)-(4) we find that (16) holds. Moreover, by using condition (2)
we obtain the estimate (17) so the proof is complete. ∎

###### Proof of Theorem 3..

According to (2) we find that

$\displaystyle T_{M_{n}}f=D_{M_{n}}\ast
f-\frac{1}{Q_{M_{n}}}\overset{M_{n}-1}{\underset{k=0}{\sum}}q_{k}\left(\left(\psi_{M_{n}-1}\overline{D_{k}}\right)\ast
f\right).$

Hence, by using Abel transformation we get that

$\displaystyle\ \ \ \ \ T_{M_{n}}f$ $\displaystyle=$ $\displaystyle
D_{M_{n}}\ast f$ $\displaystyle-$
$\displaystyle\frac{1}{Q_{M_{n}}}\overset{M_{n}-2}{\underset{j=0}{\sum}}\left(q_{{}_{M_{n}-j}}-q_{{}_{M_{n}-j-1}}\right)j((\psi_{{}_{M_{n}-1}}\overline{K_{j}})\ast
f)$ $\displaystyle-$
$\displaystyle\frac{1}{Q_{M_{n}}}q_{{}_{M_{n}-1}}(M_{n}-1)(\psi_{{}_{M_{n}-1}}\overline{K}_{M_{n}-1}\ast
f)$ $\displaystyle=$ $\displaystyle D_{M_{n}}\ast f$ $\displaystyle-$
$\displaystyle\frac{1}{Q_{M_{n}}}\overset{M_{n}-2}{\underset{j=0}{\sum}}\left(q_{{}_{M_{n}-j}}-q_{{}_{M_{n}-j-1}}\right)j((\psi_{{}_{M_{n}-1}}\overline{K_{j}})\ast
f)$ $\displaystyle-$
$\displaystyle\frac{1}{Q_{M_{n}}}q_{{}_{M_{n}-1}}M_{n}(\psi_{{}_{M_{n}-1}}\overline{K}_{M_{n}}\ast
f)$ $\displaystyle+$
$\displaystyle\frac{q_{{}_{M_{n}-1}}}{Q_{M_{n}}}(\psi_{{}_{M_{n}-1}}\overline{D}_{M_{n}}\ast
f)$

so that

$\displaystyle T_{M_{n}}f(x)-f(x)$ $\displaystyle=$
$\displaystyle\int_{G_{m}}(f(x-t)-f(x))D_{M_{n}}(t)dt$ $\displaystyle-$
$\displaystyle\frac{1}{Q_{M_{n}}}\overset{M_{n}-2}{\underset{j=0}{\sum}}\left(q_{{}_{M_{n}-j}}-q_{{}_{M_{n}-j-1}}\right)j\int_{G_{m}}\left(f(x-t)-f(x)\right)\psi_{{}_{M_{n}-1}}(t)\overline{K}_{j}(t)dt$
$\displaystyle-$
$\displaystyle\frac{1}{Q_{M_{n}}}q_{{}_{M_{n}-1}}M_{n}\int_{G_{m}}(f(x-t)-f(x))\psi_{{}_{M_{n}-1}}(t)\overline{K}_{M_{n}}(t)dt$
$\displaystyle+$
$\displaystyle\frac{q_{{}_{M_{n}-1}}}{Q_{M_{n}}}\int_{G_{m}}(f(x-t)-f(x))\psi_{{}_{M_{n}-1}}(t)\overline{D}_{M_{n}}(t)dt$
$\displaystyle=:$ $\displaystyle I+II+III+IV.$

By combining generalized Minkowski’s inequality and (6) we find that

(31)
$\displaystyle\|I\|_{p}\leq\int_{I_{n}}\|f(x-t)-f(x))\|_{p}D_{M_{n}}(t)dt\leq\omega_{p}\left(1/M_{n},f\right).$

and

(32)
$\displaystyle\|IV\|_{p}\leq\int_{I_{n}}\|f(x-t)-f(x))\|_{p}{D}_{M_{n}}(t)dt\leq\omega_{p}\left(1/M_{n},f\right).$

Moreover, since

$M_{n}q_{M_{n}-1}\leq Q_{M_{n}},\ \ \text{ for any }\ \ \ n\in\mathbb{N},$

we can use (10) and generalized Minkowski’s inequality to find that

$\displaystyle\|III\|_{p}$ $\displaystyle\leq$
$\displaystyle\int_{G_{m}}\left\|f\left(x-t\right)-f\left(x\right)\right\|_{p}\left|\overline{K}_{M_{n}}\left(t\right)\right|d\mu(t)$
$\displaystyle=$
$\displaystyle\int_{I_{n}}\left\|f\left(x-t\right)-f\left(x\right)\right\|_{p}\left|\overline{K}_{M_{n}}\left(t\right)\right|d\mu(t)$
$\displaystyle+$
$\displaystyle\sum_{s=0}^{n-1}\sum_{n_{s}=1}^{m_{s}-1}\int_{I_{n}\left(n_{s}e_{s}\right)}\left\|f\left(x-t\right)-f\left(x\right)\right\|_{p}\left|\overline{K}_{M_{n}}\left(t\right)\right|d\mu(t)$
$\displaystyle\leq$
$\displaystyle\int_{I_{n}}\left\|f\left(x-t\right)-f\left(x\right)\right\|_{p}\frac{M_{n}+1}{2}d\mu(t)$
$\displaystyle+$
$\displaystyle\sum_{s=0}^{n-1}M_{s+1}\sum_{n_{s}=1}^{m_{s}-1}\int_{I_{n}\left(n_{s}e_{s}\right)}\left\|f\left(x-t\right)-f\left(x\right)\right\|_{p}d\mu(t)$
$\displaystyle\leq$
$\displaystyle\omega_{p}\left(1/M_{n},f\right)\int_{I_{n}}\frac{M_{n}+1}{2}d\mu(t)$
$\displaystyle+$
$\displaystyle\sum_{s=0}^{n-1}M_{s+1}\sum_{n_{s}=1}^{m_{s}-1}\int_{I_{n}\left(n_{s}e_{s}\right)}\omega_{p}\left(1/M_{s},f\right)d\mu(t)$
$\displaystyle\leq$
$\displaystyle\omega_{p}\left(1/M_{n},f\right)+R^{2}\sum_{s=0}^{n-1}\frac{M_{s}}{M_{n}}\omega_{p}\left(1/M_{s},f\right)$
$\displaystyle\leq$ $\displaystyle
R^{2}\sum_{s=0}^{n}\frac{M_{s}}{M_{n}}\omega_{p}\left(1/M_{s},f\right).$

From this inequality and the estimates in (4) it follows also that

$\displaystyle
M_{n}\int_{G_{m}}\left\|f\left(x-t\right)-f\left(x\right)\right\|_{p}|\overline{K}_{M_{n}}(t)|\,d\mu(t)$
$\displaystyle\leq$ $\displaystyle
R^{2}\sum_{s=0}^{n}M_{s}\omega_{p}\left(1/M_{s},f\right).$

Let $M_{k}\leq j<M_{k+1}$ By applying (8) and the last estimate we find that

$\displaystyle
j\int_{G_{m}}\left\|f\left(x-t\right)-f\left(x\right)\right\|_{p}|\overline{K}_{j}(t)|d\mu(t)$
$\displaystyle\leq$ $\displaystyle
2R^{4}\sum_{l=0}^{k}\sum_{s=0}^{l}M_{s}\omega_{p}\left(1/M_{s},f\right).$

Hence, by also using (8) we obtain that

(34) $\displaystyle\|II\|_{p}$ $\displaystyle\leq$
$\displaystyle\frac{1}{Q_{M_{n}}}\overset{M_{n}-1}{\underset{j=0}{\sum}}\left(q_{{}_{M_{n}-j}}-q_{{}_{M_{n}-j-1}}\right)j\int_{G_{m}}\|f(x-t)-f(x)\|_{p}|\overline{K}_{j}(t)|d\mu(t)$
$\displaystyle\leq$
$\displaystyle\frac{1}{Q_{M_{n}}}\overset{n-1}{\underset{k=0}{\sum}}\overset{M_{k+1}-1}{\underset{j=M_{k}}{\sum}}\left(q_{{}_{M_{n}-j}}-q_{{}_{M_{n}-j-1}}\right)j\int_{G_{m}}\|f(x-t)-f(x)\|_{p}|\overline{K}_{j}(t)|d\mu(t)$
$\displaystyle\leq$
$\displaystyle\frac{2R^{4}}{Q_{M_{n}}}\overset{n-1}{\underset{k=0}{\sum}}\overset{M_{k+1}-1}{\underset{j=M_{k}}{\sum}}\left(q_{{}_{M_{n}-j}}-q_{{}_{M_{n}-j-1}}\right)\sum_{l=0}^{k}\sum_{s=0}^{l}M_{s}\omega_{p}\left(1/M_{s},f\right)$
$\displaystyle\leq$
$\displaystyle\frac{2R^{4}}{Q_{M_{n}}}\overset{n-1}{\underset{k=0}{\sum}}\left(q_{{}_{M_{n}-M_{k}}}-q_{{}_{M_{n}-M_{k+1}}}\right)\sum_{l=0}^{k}\sum_{s=0}^{l}M_{s}\omega_{p}\left(1/M_{s},f\right)$
$\displaystyle\leq$
$\displaystyle\frac{2R^{4}}{Q_{M_{n}}}\overset{n-1}{\underset{l=0}{\sum}}\sum_{k=l}^{n-1}\left(q_{{}_{M_{n}-M_{k}}}-q_{{}_{M_{n}-M_{k+1}}}\right)\sum_{s=0}^{l}M_{s}\omega_{p}\left(1/M_{s},f\right)$
$\displaystyle\leq$
$\displaystyle\frac{2R^{4}}{Q_{M_{n}}}\overset{n-1}{\underset{l=0}{\sum}}q_{{}_{M_{n}-M_{l}}}\sum_{s=0}^{l}M_{s}\omega_{p}\left(1/M_{s},f\right)$
$\displaystyle\leq$
$\displaystyle\frac{2R^{4}}{Q_{M_{n}}}\overset{n-1}{\underset{s=0}{\sum}}M_{s}\omega_{p}\left(1/M_{s},f\right)\sum_{l=s}^{n-1}q_{M_{n}-M_{l}}$
$\displaystyle\leq$
$\displaystyle\frac{2R^{4}}{Q_{M_{n}}}\overset{n-1}{\underset{s=0}{\sum}}M_{s}\omega_{p}\left(1/M_{s},f\right)q_{{}_{M_{n}-M_{s}}}(n-s)$
$\displaystyle\leq$ $\displaystyle
2R^{4}\overset{n-1}{\underset{s=0}{\sum}}\frac{(n-s)M_{s}}{M_{n}}\frac{q_{{}_{M_{n}-M_{s}}}}{q_{0}}\omega_{p}\left(1/M_{s},f\right).$

Finally, by combining (31)-(34) we obtain (3) so the proof is complete. ∎

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# ASP-Based Declarative Process Mining

Francesco Chiariello1, Fabrizio Maria Maggi2, Fabio Patrizi1

###### Abstract

We put forward Answer Set Programming (ASP) as a solution approach for three
classical problems in Declarative Process Mining: Log Generation, Query
Checking, and Conformance Checking. These problems correspond to different
ways of analyzing business processes under execution, starting from sequences
of recorded events, a.k.a.  _event logs_. We tackle them in their _data-aware_
variant, i.e., by considering events that carry a payload (set of attribute-
value pairs), in addition to the performed activity, specifying processes
declaratively with an extension of linear-time temporal logic over finite
traces (ltlf). The data-aware setting is significantly more challenging than
the _control-flow_ one: Query Checking is still open, while the existing
approaches for the other two problems do not scale well. The contributions of
the work include an ASP encoding schema for the three problems, their
solution, and experiments showing the feasibility of the approach.

## Introduction

Process Mining (PM) for Business Process Management (BPM) is a research area
aimed at discovering common patters in Business Processes (BP) (van der Aalst
2016). The analysis starts from _event logs_ , i.e., sets of traces that
record the events associated with process instance executions, and typically
assumes a _process model_ , which may be taken as input, manipulated as an
intermediate structure, or produced in output. Events describe process
activities at different levels of details. In the simplest perspective, here
referred to as _control-flow-only_ , events model atomic _activity_ performed
by a process instance at some time point; in the most complex scenario,
typically referred to as _multi-perspective_ , events also carry a payload
including a timestamp and activity data.

Processes can be specified _prescriptively_ , i.e., as models, such as Petri
Nets, that generate traces, or _declaratively_ , i.e., through logical
formulas representing the constraints that traces must satisfy in order to
comply with the process. This is the approach we adopt here. Specifically, we
take a (untimed) _data-aware_ perspective where events include the activity
and set of attribute-value pairs, the _payload_.

In Declarative PM, the de-facto standard for expressing process properties is
Declare (van der Aalst, Pesic, and Schonenberg 2009), a temporal logic
consisting in a set of template formulas of the Linear-time Temporal Logic
over finite traces (ltlf) (De Giacomo and Vardi 2013); here, we use a strictly
more expressive extension, which we call _local_ ltlf, i.e., l-ltlf. This
logic features a simple automata-based machinery that facilitates its
manipulation, while retaining the ability to express virtually all the
properties of interest in declarative PM. Specifically, l-ltlf subsumes
Declare and even its multi-perspective variant MP-Declare (Burattin, Maggi,
and Sperduti 2016) without timestamps and _correlation conditions_ (i.e.,
conditions that relate the attributes of some event to those of other events),
but does not subsume full MP-Declare. Observe that since MP-Declare does not
subsume l-ltlf either, the two logics are incomparable.

Our goal is to devise techniques for three classical problems in Declarative
PM: _Event Log Generation_ (Skydanienko et al. 2018), i.e., generating an
event log consistent with a declarative model; _Query Checking_ (Räim et al.
2014), i.e., discovering hidden temporal properties in an event log; and
_Conformance Checking_ (Burattin, Maggi, and Sperduti 2016), i.e., checking
whether the traces of an event log conform to a process model. The main
challenge is dealing with data: in the data-aware framework, Query Checking is
still open, while existing tools for the other two problems do not scale well.

We put forward Answer Set Programming (ASP (Niemelä 1999)) as an effective
solution approach. ASP natively provides data-manipulation functionalities
which allow for formalizing data-aware constraints and has experienced over
the last years a dramatic improvement in solution performance, thus results in
a natural and promising candidate approach for addressing the problems of our
interest. We show how such problems can be conveniently modeled as ASP
programs, thus solved with any solver. Using the state-of-the-art solver
Clingo111potassco.org (Gebser et al. 2019), we experimentally compare our
approach against existing ones for Log Generation and Conformance Checking,
and show effectiveness of the approach for Query Checking in a data-aware
setting. Besides providing an actual solution technique, ASP facilitates reuse
of specifications: the ASP encodings we propose here, indeed, differ in very
few, although important, details.

Previous related work include (Wieczorek, Jastrzab, and Unold 2020), where ASP
is used to infer a finite-state automaton that accepts (resp. rejects) traces
from a positive (negative) input set. This can be seen as a form of
Declarative Process Discovery, i.e., the problem of obtaining a (declarative)
process specification, which is complementary to the problems we address here.
Our approach is similar, in that we use automata to model temporal properties.
However, we propose a different automata encoding and show the effectiveness
of the approach on three different problems. Another related paper is
(Heljanko and Niemelä 2003), which shows how ASP can be used to check a Petri
Net against an LTL specification, up to a bounded time horizon. Differently
from our work, it:  (i) deals with LTL over infinite, as opposed to finite,
runs;  (ii) adopts a prescriptive, as opposed to declarative, approach; and
(iii) does not deal with data in events.

From a broader perspective, we finally observe that while we deal with a set
of specific problems, the work paves the way for ASP to become a general
effective approach to Declarative PM.

## The Framework

An _activity (signature)_ is an expression of the form
$A(a_{1},\ldots,a_{n_{A}})$, where $A$ is the _activity name_ and each $a_{i}$
is an _attribute name_. We call $n_{A}$ the _arity_ of $A$. The attribute
names of an activity are all distinct, but different activities may contain
attributes with matching names. We assume a finite set $Act$ of activities,
all with distinct names; thus, activities can be identified by their name,
instead of by the whole tuple. Every attribute (name) $a$ of an activity $A$
is associated with a _type_ $D_{A}(a)$, i.e., the set of values that can be
assigned to $a$ when the activity is executed. For simplicity, we assume that
all domains are equipped with the standard relations $<,\leq,=,\geq,>$. All
results can be immediately adapted if some relations are absent in some
domain.

An _event_ is the execution of an activity (at some time) and is formally
captured by an expression of the form $e=A(v_{1},\ldots,v_{n_{A}})$, where
$A\in Act$ is an activity name and $v_{i}\in D_{A}(a_{i})$. The properties of
interest in this work concern _(log) traces_ , formally defined as finite
sequences of events $\tau=e_{1}\cdots e_{n}$, with
$e_{i}=A_{i}(v_{1},\ldots,v_{n_{A_{i}}})$. Traces model process executions,
i.e., the sequences of activities performed by a process instance. A finite
collection of executions into a set $L$ of traces is called an _event log_.

###  l-ltlf

We adopt a declarative approach to process modeling, meaning that processes
are specified through a set of constraints over their executions, i.e., over
the traces they produce. The formal language we use to express properties of
traces is a variant of the Linear-time Logic over finite traces, ltlf (De
Giacomo and Vardi 2013), adapted to deal with the data attributes of
activities. We call such variant _Linear-time Logic with local conditions over
finite traces_ , or _local_ ltlf for short, and denote it as l-ltlf.

Given a finite set of activities $Act$, the formulas $\varphi$ of l-ltlf over
$Act$ are inductively defined as follows:

$\varphi=\mathbf{true}\mid A\mid a\odot a^{\prime}\mid a\odot
v\mid\lnot\varphi\mid\varphi\land\varphi\mid\mathbf{X}\varphi\mid\varphi\mathbf{U}\varphi,$

where: $a$ and $a^{\prime}$ are attribute names from some activity in $Act$,
$v\in D_{A}(a)$, for some $A\in Act$, $\odot$ is an operator from
$\\{<,\leq,=,\geq,>\\}$, and $A$ is an activity name from $Act$. Formulas of
the form $\mathbf{true}$, $A$, $a\odot a^{\prime}$, and $a\odot v$ are called
_atomic_ ; formulas not containing the operators $\mathbf{X}$ and $\mathbf{U}$
are called _event formulas_.

The logic is interpreted over positions of finite traces. Formula
$\mathbf{true}$ holds at every position; $A$ checks whether activity $A$
occurs in the trace at the given position; $a\odot a^{\prime}$ (resp. $a\odot
v$) compares the value assigned to attribute $a$ with that of attribute
$a^{\prime}$ (resp. with that of value $v$), at the given position; boolean
operators combine formulas as usual; the _next_ operator $\mathbf{X}\varphi$
checks whether $\varphi$ holds in the suffix starting at the next position;
finally, the until operator $\varphi\mathbf{U}\varphi^{\prime}$ checks whether
$\varphi^{\prime}$ is satisfied at some position $k$, and whether $\varphi$
holds in all positions that precede $k$, up to the given position.

Formally, we define by induction when a trace $\tau=e_{1}\cdots e_{n}$
_satisfies_ an l-ltlf formula $\varphi$ at position $1\leq i\leq n$, written
$\tau,i\models\varphi$, as follows:

  * •

$\tau,i\models\mathbf{true}$;

  * •

$\tau,i\models A$ iff $e_{i}=A(v_{1},\ldots,v_{n_{A}})$;

  * •

$\tau,i\models a\odot a^{\prime}$ iff for
$e_{i}=A_{i}(v_{1},\ldots,v_{n_{A_{i}}})$ and
$A_{i}(a_{1},\ldots,a_{n_{A_{i}}})$ the signature of $A_{i}$, it is the case
that, for some $j$ and $k$, $a=a_{j}$, $a^{\prime}=a_{k}$, and $v_{j}\odot
v_{k}$; 222Notice that this requires _compatibility_ between the domains
$D_{A_{i}}(a_{j})$ and $D_{A_{i}}(a_{k})$ wrt relation $\odot$.

  * •

$\tau,i\models a\odot v$ iff for $e_{i}=A_{i}(v_{1},\ldots,v_{n_{A_{i}}})$ and
$A_{i}(a_{1},\ldots,a_{n_{A_{i}}})$ the signature of $A_{i}$, it is the case
that, for some $j$, $a=a_{j}$ and $v_{j}\odot v$;

  * •

$\tau,i\models\varphi_{1}\land\varphi_{2}$ iff $\tau,i\models\varphi_{1}$ and
$\tau,i\models\varphi_{2}$;

  * •

$\tau,i\models\neg\varphi$ iff $\tau,i\not\models\varphi$;

  * •

$\tau,i\models\mathbf{X}\varphi$ iff $i<n$ and $\tau,i+1\models\varphi$;

  * •

$\tau,i\models\varphi_{1}\mathbf{U}\varphi_{2}$ iff there exists $j\leq n$
s.t. $\tau,j\models\varphi_{2}$ and for every $1\leq k\leq j-1$, it is the
case that $\tau,k\models\varphi_{1}$.

Notice that while, in general, the satisfaction of an l-ltlf formula $\varphi$
at some position $i$ of $\tau$ depends on the whole trace, and precisely on
the suffix of $\tau$ starting at position $i$, event formulas depend only on
the event at $i$.

As in standard ltlf, $\mathbf{X}$ denotes the _strong next_ operator (which
requires the existence of a next event where the inner formula is evaluated),
while $\mathbf{U}$ denotes the _strong until_ operator (which requires the
right-hand formula to eventually hold, forcing the left-hand formula to hold
in all intermediate events). The following are standard abbreviations:
$\varphi_{1}\lor\varphi_{2}\doteq\neg(\neg\varphi_{1}\land\neg\varphi_{2})$;
$\varphi_{1}{\rightarrow}\varphi_{2}\doteq\neg\varphi_{1}\lor\varphi_{2}$;
$\mathbf{F}\varphi=\mathbf{true}\mathbf{U}\varphi$ (_eventually_ $\varphi$);
and $\mathbf{G}\varphi=\neg\mathbf{F}\neg\varphi$ (_globally_ , or _always_ ,
$\varphi$).

Through l-ltlf one can express properties of process traces that involve not
only the process control-flow but also the manipulated data.

###### Example 1

The l-ltlf formula $\varphi=\mathbf{G}(a{\rightarrow}\mathbf{F}b)$, a so-
called _Response_ constraint, says that whenever activity $a$ occurs, it must
be eventually followed by activity $b$. A possible _data-aware_ variant of
$\varphi$ is the formula $\varphi^{\prime}=\mathbf{G}((a\land
n<5){\rightarrow}\mathbf{F}b)$, which says that whenever activity $a$ occurs
with attribute $n$ less than $5$, it must be followed by activity $b$.

Formulas of ltlf, thus l-ltlf, have the useful property of being fully
characterized by finite-state, possibly nondeterministic, automata.
Specifically, for every l-ltlf formula $\varphi$ there exists a finite-state
automaton (FSA) $\cal A_{\varphi}$ that accepts all and only the traces that
satisfy $\varphi$ (De Giacomo and Vardi 2013). Such automata are standard FSA
with transitions labelled by event formulas. For a fixed set of activities
$Act$, let $\Gamma_{Act}$ be the set of event formulas over $Act$. An FSA over
a set of activities $Act$ is a tuple ${\cal{A}}=\langle
Q,q_{0},\rho,F\rangle$, where:

  * •

$Q$ is a finite set of _states_ ;

  * •

$q_{0}\in Q$ is the automaton _initial state_ ;

  * •

$\delta\subseteq Q\times\Gamma_{Act}\times Q$ is the automaton _transition
relation_ ;

  * •

$F\subseteq Q$ is the set of automaton _final states_.

Without loss of generality, we assume that formulas labeling transitions are
conjunctions of literals. It is immediate to show that every FSA can be
rewritten in this way.

A _run_ of an FSA ${\cal{A}}$ on a trace $\tau=e_{1}\cdots e_{n}$ (over $Act$)
is a sequence of states $\rho=q_{0}\cdots q_{n}$ s.t. for all $0\leq i\leq
n-1$ there exists a transition $\langle q_{i},\gamma,q_{i+1}\rangle\in\delta$
s.t. $\tau,i+1\models\gamma$. A trace $\tau$ over $Act$ is _accepted_ by
${\cal{A}}$ iff it _induces_ a _run_ of ${\cal{A}}$ that ends in a final
state. Notice that satisfaction of $\gamma$, this being an event formula, can
be established by looking at each event $e_{i+1}$ at a time, while
disregarding the rest of the trace; thus, in order to construct the induced
run $\rho$, one can proceed in an online fashion, as the next event arrives,
by simply triggering, at every step, a transition outgoing from $q_{i}$ whose
label is satisfied by the event.

###### Example 2

Consider again the formulas $\varphi$ and $\varphi^{\prime}$ shown above, and
the (paramtric) automaton depicted in Fig 1.

$s_{0}$$s_{1}$$\varphi_{3}$$\varphi_{1}$$\varphi_{4}$$\varphi_{2}$ Figure 1:
Automaton for the _Response_ constraint.

It is easy to see that for $\varphi_{1}=a$, $\varphi_{2}=b$,
$\varphi_{3}=\lnot a$, $\varphi_{4}=\lnot b$, the resulting automaton accepts
all and only the traces that satisfy $\varphi$, as well as that for
$\varphi_{1}=a\land n<5$, $\varphi_{2}=b$, $\varphi_{3}=\lnot(a\land n<5)$,
$\varphi_{4}=\lnot b$, the obtained automaton accepts all and only the traces
that satisfy $\varphi^{\prime}$.

The details about the construction of ${\cal{A}}_{\varphi}$ from $\varphi$ are
not in the scope of this work, and we refer the interested reader to (De
Giacomo and Vardi 2013) for more information; we rely on the results therein.
We observe that while the automaton construction is time-exponential in the
worst-case, wrt the size of the input formula $\varphi$, tools exist, such as
Lydia333github.com/whitemech/lydia/releases/tag/v0.1.1 (De Giacomo and
Favorito 2021), which exhibit efficient performance in practice; this,
combined with the fact that the specifications of practical interest are
typically small, makes the approaches based on automata construction usually
feasible in practice. We can now formalize the problems addressed in this
paper.

#### Event Log Generation.

Given a set $\Phi$ of l-ltlf formulas over a set of activities $Act$ and a
positive integer $t$, return a trace $\tau$ over $Act$ of length $t$ s.t., for
every formula $\varphi\in\Phi$, it is the case that $\tau\models\varphi$. In
words, the problem amounts to producing a trace of length $t$ over $Act$ that
satisfies all the input constraints in $\Phi$. A more general version of the
problem requires to generate a log $L$ of $n$ traces of fixed length $t$
satisfying the constraints. For simplicity, we consider the first formulation.

#### Query Checking.

Query Checking takes as input formulas from the extension of l-ltlf _with
activity variables_ , defined as follows:

$\varphi=\mathbf{true}\mid?V\mid A\mid a\odot a^{\prime}\mid a\odot
v\mid\lnot\varphi\mid\varphi\land\varphi\mid\mathbf{X}\varphi\mid\varphi\mathbf{U}\varphi,$

where symbols starting with “?” are _activity variables_ and the other symbols
are as in l-ltlf (given $Act$).

Given an l-ltlf formula with activity variables, by assigning an activity
(from $Act$) to every variable, we obtain a “regular” l-ltlf formula.
Formally, for an l-ltlf formula $\varphi$ (over $Act$), containing a (possibly
empty) set of activity variables $Vars_{\varphi}$, an _assignment_ to
$Vars_{\varphi}$ is a total function $\nu:Vars_{\varphi}\rightarrow Act$.
Given $\varphi$ and an assignment $\nu$ to its activity variables,
$\varphi[\nu]$ denotes the (regular) l-ltlf formula obtained by replacing, in
$\varphi$, every variable symbol $?V$ with an activity name from $Act$.
Observe that if $Vars_{\varphi}=\emptyset$ there exists only one assignment
$\nu$ and $\varphi=\varphi[\nu]$. Given a trace $\tau$, since $\varphi[\nu]$
is a regular l-ltlf formula, we can check whether $\tau\models\varphi[\nu]$.

An instance of Query Checking consists in a log $L$ and an l-ltlf formula
$\varphi$ with activity variables $Vars_{\varphi}$; a solution is a set
$\Lambda$ of assignments to $Vars_{\varphi}$ s.t. for every assignment
$\nu\in\Lambda$ and every trace $\tau\in L$, it holds that
$\tau\models\varphi[\nu]$.

In words, query checking requires to find a set $\Lambda$ of assignments
$\nu$, each transforming the input formula $\varphi$ into an l-ltlf formula
$\varphi[\nu]$ satisfied by all the traces of the input log $L$. Observe that
$\varphi$ variables can only span over activities.

#### Conformance Checking.

Given a trace $\tau$ and a set $\Phi$ of l-ltlf formulas, both over the same
set of activities $Act$, check whether, for all formulas $\varphi\in\Phi$,
$\tau\models\varphi$. The problem can also be defined in a more general form,
where $\tau$ is replaced by a log $L$ of traces over $Act$ and the task
requires to check whether for all the traces $\tau$ of $L$ and all
$\varphi\in\Phi$, it holds that $\tau\models\varphi$.

## Answer Set Programming (ASP)

An ASP program consists in a set of _rules_ which define _predicates_ and
impose relationships among them. The task of an ASP _solver_ is that of
finding a finite _model_ of the program, i.e., an interpretation of the
predicates that satisfies the program rules. ASP rules are written in a
fragment of (function-free) First-order Logic (FOL) extended with a special
_negation-as-failure_ (NAF) operator (in addition to classical negation) which
allows for distinguishing facts that are false from facts that are unknown.
The presence of this operator, combined with the classical FOL negation, has a
huge impact on the programs one can write and the way models are found. Here,
we do not discuss these details, referring the interested reader to (Gelfond
and Lifschitz 1988; Niemelä 1999). For our purposes, it will be sufficient
restricting to the class of rules with the NAF operator as the only available
negation operator (that is, disallowing classical negation).

### Syntax

The basic constructs of ASP programs are:  1. _constants_ , identified by
strings starting with a lower-case letter;  2. _variables_ , identified by
strings starting with an upper-case letter;  3. _terms_ , i.e., constants or
variables;  4. _atoms_ , i.e., expressions of the form
$p(t_{1},\ldots,t_{n})$, where $p$ is a _predicate_ , identified by a string,
and each $t_{i}$ is a _term_.  A predicate $p$ is said to have arity $n$ if it
occurs in an expression of the form $p(t_{1},\ldots,t_{n})$. An atom
containing only constant terms is said to be _ground_.

ASP rules are obtained by combining the basic elements through boolean
operators and the NAF operator. In this work, we use rules of the following
form:

$a\leftarrow
l_{1},\ldots,l_{n},\mathtt{not}~{}{l}_{n+1},\ldots,\mathtt{not}~{}{l}_{m}$

where $a$ and each $l_{i}$ are atoms $p(t_{1},\ldots,t_{n})$, $\mathtt{not}$
denotes the NAF operator, and every variable occurring in the rule also occurs
in some atom $l_{1},\ldots,l_{n}$. The left-hand side is called the rule’s
_head_ and is optional. When the head is absent, the rule is called an
_integrity constraint_. The right-hand side is called the _body_ and can be
left empty, in which case the $\leftarrow$ symbol is omitted and the rule is
called a _fact_.

### Semantics

Intuitively, a _model_ of an ASP program ${\cal{P}}$ is a set of ground atoms
that satisfies all program rules. In general, many models exist. Among these,
only those that are minimal wrt set inclusion and that contain a ground atom
only “if needed”, i.e., if it occurs as the head of a ground rule, are taken
as solutions, called the _answer sets_ of ${\cal{P}}$ in the ASP terminology.
The task of an ASP solver is to compute such sets.

Given an ASP program ${\cal{P}}$ and a rule $r\in{\cal{P}}$, the set of $r$
_ground instantiations_ is the set ${\cal{G}}(r)$ of rules obtained by
replacing all the variables in $r$ with all the constants mentioned in
${\cal{P}}$ (the so-called _Herbrand universe_ of ${\cal{P}}$), in all
possible ways, so that all rules in ${\cal{G}}(r)$ contain only ground atoms.
Then, the _ground_ instantiation ${\cal{G}}({\cal{P}})$ of a program
${\cal{P}}$ is the union of all the ground instantiations of its rules, i.e.,
${\cal{G}}({\cal{P}})=\bigcup_{r\in{\cal{P}}}{\cal{G}}({\cal{P}})$.

An _interpretation_ $I$ of a program ${\cal{P}}$ is a set of ground atoms
$p(c_{1},\ldots,c_{n})$, where $p$ is a predicate of arity $n$ occurring in
${\cal{P}}$ and $c_{1},\ldots,c_{n}$ are constants from the Herbrand universe
of ${\cal{P}}$. Given a _positive_ (i.e., without occurrences of
$\mathtt{not}$) program ${\cal{P}}$, an interpretation $I$ is a _model_ of
${\cal{P}}$ if, for every ground rule, $a\leftarrow l_{1},\ldots,l_{n}$ in
${\cal{G}}({\cal{P}})$, whenever $l_{i}\in I$ for $i=1,\ldots,l_{n}$, it holds
that $a\in I$. An _answer set_ of ${\cal{P}}$ is a model $I$ that is minimal
wrt set inclusion.

The semantics of general programs is obtained as a reduction to positive
programs. Namely, the _reduct_ of a ground program ${\cal{G}}({\cal{P}})$ wrt
an interpretation $I$ is the positive ground program
${\cal{G}}({\cal{P}})^{I}$ obtained by:

  * •

deleting all the rules

$a\leftarrow
l_{1},\ldots,l_{n},\mathtt{not}~{}{l}_{n+1},\ldots,\mathtt{not}~{}{l}_{m}$

of ${\cal{G}}({\cal{P}})$ s.t. $l_{i}\in I$ for some $i\in\\{n+1,\ldots,m\\}$;

  * •

replacing all the remaining rules

$a\leftarrow
l_{1},\ldots,l_{n},\mathtt{not}~{}{l}_{n+1},\ldots,\mathtt{not}~{}{l}_{m}$

with $a\leftarrow l_{1},\ldots,l_{n}$.

Intuitively, the first transformation removes a rule, as already satisfied by
$I$; the second transformation removes the so-called _negative body_ of the
rule, because it is satisfied. As it can be easily seen, the resulting program
${\cal{G}}({\cal{P}})^{I}$ does not mention the $\mathtt{not}$ operator. The
interpretation $I$ is an _answer set_ of ${\cal{P}}$ if it is an answer set of
${\cal{G}}({\cal{P}})^{I}$.

In this work, we do not discuss the algorithms to compute the answer sets of a
program, but focus on how the problems of our interest can be encoded in ASP
and then solved by an ASP solver, in such a way that the returned Answer Sets
represent the solution to our problems. This is the focus of the next section.
For the experiments, we use the state-of-the-art solver Clingo.

## ASP for Declarative Process Mining

We encode Log Generation, Conformance Checking, and Query Checking into ASP
programs. For every l-ltlf formula $\varphi$ we deal with, we assume available
the corresponding automaton ${{\cal{A}}}_{\varphi}$. The three programs share
some common parts, such as the automata and the traces, which are modeled
through suitable predicates and ASP rules. Each encoding re-uses some of these
parts, possibly customized, together with additional fragments used to model
problem-specific features.

_Activities_ are captured by the unary predicate $act(A)$, where $A$ is the
activity name. In the presence of data, _activity signatures_ are modeled by
the binary predicate $has\\_attr(A,N)$, where $A$ is the activity name and $N$
is the attribute name. Attributes may be typed by stating the set of values
they can take, through predicate $val(N,V)$, where $N$ is the attribute name
and $V$ one of its possible values. A _trace_ is modeled by the binary
predicate $trace(A,T)$, where $A$ is the activity and $T$ the time point where
it occurs. Time points come from predicate $time(T)$, which contains the
values $\\{0,\ldots,\ell\\}$, for $\ell$ the trace length. The trace is
defined on time points from $0$ to $\ell-1$. In the presence of data, activity
attributes are paired with values through predicate $has\\_val(N,V,T)$, where
$N$ is the attribute name, $V$ the assigned value, and $T$ the time point .
Notice that the association is based on the time point (exactly one activity
is performed at one time point). Simple integrity constraints are used to
ensure that the mentioned attributes belong in fact to the activity and that
the sassigned value comes from the corresponding type.

_Automata_ are encoded with predicates $init(S)$, $acc(S)$,
$trans(S,F,S^{\prime})$, and $hold(F,T)$. The first and the second one model
the initial state and the accepting states of the automaton, the third one
models the existence of a transition from $S$ to $S^{\prime}$ under the event
formula represented by integer $F$, and the last one models satisfaction of
(event) formula $F$ at time point $T$. In the presence of multiple l-ltlf
formulas, each automaton is identified by a unique integer value and an
additional parameter is added to the above predicates to refer to the various
automata.

###### Example 3

The ASP encoding of the automaton for the ltlf formula
$\mathbf{G}(a\rightarrow\mathbf{F}b)$, shown in Fig. 1, for $\varphi_{1}=a$,
$\varphi_{2}=b$, $\varphi_{3}=\lnot a$, $\varphi_{4}=\lnot b$, is as follows:

$\begin{array}[]{l}init(1,s0).\\\ acc(1,s0).\\\ trans(1,s0,1,s1).\\\
hold(1,1,T)\leftarrow trace(a,T).\\\ trans(1,s1,2,s0).\\\
hold(1,2,T)\leftarrow trace(b,T).\\\ trans(1,s0,3,s0).\\\
hold(1,3,T)\leftarrow\mathtt{not}~{}{hold}(1,1,T),time(T).\\\
trans(1,s1,4,s1).\\\ hold(1,4,T)\leftarrow trace(A,T),activity(A),A\neq
b.\end{array}$

where $a$ and $b$ are activities and each formula $\varphi_{i}$
($i=1,\ldots,4$) is identified by index $i$ in the encoding.

In a data-aware setting, conditions on data can be simply added to the rules
for $holds$. For example the following rule:

$\begin{array}[]{l}hold(1,1,T)\leftarrow\\\ \ \ \ \
trace(a,T),has\\_val(number,V,T),V<5.\end{array}$

expresses the fact that the event formula $\varphi_{1}$ holds at time $T$ if
activity $a$ occurs at time $T$ in the trace, with a value less than $5$
assigned to its attribute $number$.

To capture satisfaction of an l-ltlf formula $\varphi$ by a trace $\tau$, we
model the execution of the automaton ${\cal{A}}_{\varphi}$ on $\tau$. To this
end, we introduce predicate $state(I,S,T)$, which expresses the fact that
automaton (with index) $I$ is in state $S$ at time $T$. Since the automaton is
nondeterministic in general, it can be in many states at time point $T$
(except for the initial one). The rules defining $state$ are the following:

$\begin{array}[]{l}state(I,S,0)\leftarrow init(I,S).\\\
state(I,S^{\prime},T)\leftarrow state(I,S,T-1),\\\ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ trans(I,S,F,S^{\prime}),hold(I,F,T-1).\end{array}$

The first one says that at time point 0 every automaton $I$ is in its
respective initial state. The second one says that the current state of
automaton $I$ at time point $T$ is $S^{\prime}$ whenever the automaton is in
state $S$ at previous time point $T-1$, the automaton contains a transition
from $S$ to $S^{\prime}$ under some event formula with index $F$ and the
formula holds at time $T-1$ in the trace.

Finally, the fact that a trace is accepted by all automata, i.e., that the
trace satisfies the corresponding formulas, is stated by requiring that, for
each automaton, at least one of the final states be accepting ($tlength(T)$
denotes the length $T$ of the trace):

$\begin{array}[]{l}\leftarrow\\{state(I,S,T):\mathtt{not}~{}{acc}(I,S),tlength(T)\\}=0.\end{array}$

Next, we use these fragments to describe the ASP encodings for the problems of
interest. For lack of space, we discuss only the main rules.

#### Event Log Generation

The encoding schema of Event Log Generation is as follows:

  1. 1.

Activities, attributes, attribute types, and trace length are provided as
input and formalized as discussed above.

  2. 2.

For each input l-ltlf constraint $\varphi$, the corresponding automaton
${\cal{A}}_{\varphi}$ is generated and modeled as discussed, using a unique
integer value to identify it.

  3. 3.

Suitable integrity constraints are defined to ensure that: each time point in
the trace has exactly one activity; every attribute is assigned exactly one
value; and the attributes assigned at a given time point actually belong to
the activity occurring at that time point.

  4. 4.

Finally, predicate $state$ is defined as above and it is required that every
automaton ends up in at least one final state at the last time point.

  5. 5.

Predicates $trace$ and $has\\_val$ contain the solution, i.e., they model a
sequence of activities whose attributes have an assigned value, which
satisfies all the input constraints.

#### Query Checking

The ASP specification of query checking is analogous to that of Log Generation
except for the following. Firstly, the problem takes as input a set of fully
specified traces. This is dealt with in a simple way, by adding a parameter to
predicate $trace$ representing the (unique) identifier of the trace and,
consequently, by adding such parameter to all the predicates that depend on
$trace$ (e.g., $has\\_val$, $hold$, $state$). Secondly, the input l-ltlf
formulas contain activity variables. To deal with them, additional predicates
$var(V)$ and $assgnmt(V,W)$ are introduced to account for, respectively,
variables $V$ and assignments of value $W$ to variable $V$. Besides this, the
automata ${\cal{A}}_{\varphi_{i}}$ associated with the formulas are obtained
by treating activity variables as if they were activity symbols (without
affecting the construction, which does not consider the semantics of such
objects), thus obtaining automata whose transitions are labelled by event
formulas, possibly containing activity variables instead of activity symbols.
Such formulas become regular event formulas once values are assigned to
variables and can thus be evaluated on the (events of the) input trace.
Formally, this requires a slightly different definition of predicate $hold$,
which must now take $assgnmt$ into account. To see how this is done, consider
the formula $\varphi=\mathbf{G}((?A1\land number<5)\rightarrow\mathbf{F}?A2).$
The corresponding automaton is the same as that of Fig. 1, where
$\varphi_{1}=?A1\land number<5$, $\varphi_{2}=?A2$,
$\varphi_{3}=\lnot\varphi_{1}$, and $\varphi_{4}=\lnot\varphi_{2}$. For
formula $\varphi_{1}$, we have the following definition of predicate $hold$:

$\begin{array}[]{l}hold(1,1,J,T)\leftarrow trace(J,A,T),assgnmt(varA1,A),\\\ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
has\\_val(J,integer,V,T),V<5.\end{array}$

The parameter $J$ stands for the trace identifier, as discussed above. The
above rule generalizes the corresponding one in Log Generation in the presence
of activity variable $?A1$. As it can be seen, in order to evaluate formula
$\varphi_{1}$ (second parameter in $hold$) of automaton 1 (first parameter),
such variable (modeled as $varA1$) must be instantiated first, through
predicate $assgnmt$. Observe that once all variables are assigned a value, the
whole formula $\varphi$ becomes variable-free, and the corresponding automaton
is a regular automaton. The returned extensions of $assgnmt$ and $has\\_val$
represent, together, the problem solution.

#### Conformance Checking

Conformance Checking can be seen as a special case of Query Checking with a
single input trace and where all input formulas are variable-free. In this
case, the problem amounts to simply checking whether the whole specification
is consistent, which is the case if and only if the input trace, together with
the assignments to the respective activity attributes, satisfy the input
formulas.

We close the section by observing how these problems provide a clear example
of how the declarative approach allows for specification reuse. All the
specifications, indeed, share the main rules (for trace, automaton, etc.) and
are easily obtained as slight variants of each other, possibly varying the
(guessed) predicates representing the solution.

## Experiments

In this section, we provide both a comparison with state-of-the-art tools for
Log Generation and Conformance Checking, based on multi-perspective
declarative models, and an estimate of scalability for our query checking
tool, for which, instead, no competitors exist. The state-of-the art tool used
for Log Generation is the one presented in (Skydanienko et al. 2018), which is
based on Alloy444https://alloytools.org/ and tailored for MP-Declare; our
results show that our ASP implementation for Log Generation scales much better
than that and, at the same time, supports a more expressive data-aware rule
language. As to Conformance Checking, we considered the state-of-the-art tool
Declare Analyzer (Burattin, Maggi, and Sperduti 2016); we obtained comparable
execution times but Declare Analyzer is specifically tailored for Declare and
optimized to check conformance wrt Declare rules only, while our tool is more
general in this respect. The experiments have been carried out on a standard
laptop Dell XPS 15 with an intel i7 processor and 16GB of RAM. All execution
times have been averaged over 3 runs. Source code, declarative models and
event logs used in the experiments are available at
https://github.com/fracchiariello/process-mining-ASP.

### Log Generation

# constr. $\rightarrow$ | 3 | 5 | 7 | 10
---|---|---|---|---
Trace len $\downarrow$ |  |  |  |
10 | 595 | 614 | 622 | 654
15 | 876 | 894 | 904 | 956
20 | 1132 | 1155 | 1178 | 1250
25 | 1364 | 1413 | 1444 | 1543
30 | 1642 | 1701 | 1746 | 1874
10 | 249 | 270 | 289 | 340
15 | 349 | 390 | 408 | 457
20 | 436 | 496 | 538 | 601
25 | 519 | 568 | 611 | 712
30 | 622 | 666 | 726 | 837
10 | 35975 | 35786 | 36464 | 37688
15 | 50649 | 51534 | 54402 | 54749
20 | 69608 | 70342 | 73122 | 73222
25 | 85127 | 85598 | 87065 | 89210
30 | 101518 | 101882 | 106062 | 107520
10 | 18733 | 18947 | 19539 | 20007
15 | 25700 | 25723 | 27344 | 26897
20 | 32047 | 33837 | 33107 | 33615
25 | 39114 | 38666 | 40556 | 41055
30 | 46207 | 46706 | 47613 | 49410
Table 1: Log Generation (times in ms) Model (80) $\rightarrow$ | BPI2012 | DD | ID | PL | PTC | RP | RT | Sepsis
---|---|---|---|---|---|---|---|---
Trace len $\downarrow$ |  |  |  |  |  |  |  |
10 | 656 | 100* | 726* | 3901 | 1183 | 119* | 319 | 460
15 | 817 | 887 | 2865 | 4538 | 1820 | 1069 | 353 | 564
20 | 846 | 832 | 3160 | 4102 | 2194 | 813 | 860 | 640
25 | 1061 | 930 | 4129 | 6169 | 2889 | 1063 | 483 | 780
30 | 1433 | 1026 | 5226 | 9231 | 2370 | 1220 | 630 | 923
10 | 31935 | 2364* | 30762* | 59468 | 65783 | 2703* | 24909 | 38241
15 | 43337 | 58572 | 152188 | 85942 | 97098 | 66641 | 34408 | 57178
20 | 57596 | 80665 | 237777 | 122511 | 146420 | 95005 | 44608 | 85808
25 | 72383 | 118975 | 359665 | 174596 | 221434 | 134851 | 54808 | 120110
30 | 86910 | 181027 | 563794 | 236697 | 330753 | 187972 | 63379 | 174838
Table 2: Log Generation, real life (times in ms)

For testing the Log Generation tools, we have used 8 synthetic models and 8
models derived from real life logs. The experiments with synthetic models
allowed us to test scalability of the tools in a controlled environment and
over models with specific characteristics. The experiments with real models
have been used to test the tools in real environments. For the experiments
with synthetic models, we built 8 reference models containing 3, 5, 7, and 10
constraints with and without data conditions. Each model was obtained from the
previous one by adding new constraints and preserving those already present.
Times are in ms.

The first and second blocks of Table 1 show the execution times for the ASP-
based log generator, respectively with and without data conditions; the third
and fourth blocks show the results obtained with the Alloy log generator, with
and without data. Times refer to the generation of logs with 10000 traces (of
length from 10 to 30). Consistent results are obtained also on additional
experiments for logs of size between 100 and 5000, not reported here for space
reasons.

The results obtained with models containing data conditions show that the ASP-
based tool scales very well, requiring less than 2 sec in the worst case. This
occurs when a model with 10 constraints is used to generate 10000 traces of
length 30. As expected, the execution time increases linearly when the length
of the traces in the generated logs increases. The number of constraints in
the declarative model also affects the tool performance but with a lower
impact.

Without data conditions the results are similar but, as expected, the
execution time is lower and increases less quickly when the complexity of the
model and of the generated log increases. In the worst case (a model with 10
constraints used to generate 10000 traces of length 30), the execution time is
lower than 1 sec.

The results obtained with the Alloy-based tool show similar trends but with
execution times almost 60 times higher than those obtained with the ASP-based
tool.

The real life logs used in the experiments are taken from the collection
available at https://data.4tu.nl/. We used the declarative process model
discovery tool presented in (Maggi et al. 2018) to extract a process model
using the default settings. The models in the real cases are much more complex
and include a number of constraints between 10 and 49 for a minimum support of
80. The execution times needed for the Log Generation task with the ASP-based
log generator and with the Alloy-based tool are shown, respectively, in the
first and second block of Table 2. An asterisk indicates that for the specific
model it was not possible to generate 10000 unique traces. The complexity of
real life models makes even more evident the significant advantage of using
the ASP-based tool with respect to the Alloy-based one. In particular, in the
worst case, the ASP-based tool requires around 9 sec (to generate 10000 traces
of length 30 for log PL) while the Alloy-based generator almost 4 mins.

### Conformance Checking

Tool $\rightarrow$ | ASP | Declare Analyzer
---|---|---
Trace Len $\downarrow$ | data | no data | data | no data
10 | 665 | 635 | 598 | 110
15 | 1100 | 1035 | 805 | 145
20 | 1456 | 1354 | 1092 | 155
25 | 2071 | 1896 | 1273 | 177
30 | 2407 | 2219 | 1337 | 215
Table 3: Conformance Checking (times in ms) | BPI2012 | DD | ID | PL | PTC | RP | RT | Sepsis
---|---|---|---|---|---|---|---|---
60 | 33426 | 13084 | 49969 | 78625 | 8412 | 9354 | 49501 | 7116
70 | 33242 | 13245 | 46388 | 55475 | 5596 | 9359 | 35537 | 3796
80 | 24482 | 10176 | 29969 | 33775 | 4699 | 6836 | 35483 | 1778
90 | 8445 | 4568 | 17576 | 26590 | 2787 | 5608 | 35483 | 731
60 | 2882 | 2771 | 9800 | 8521 | 1549 | 2122 | 15262 | 1194
70 | 2852 | 3249 | 7416 | 5358 | 959 | 2102 | 10351 | 705
80 | 2291 | 2103 | 3993 | 2677 | 755 | 1532 | 11285 | 318
90 | 1691 | 1525 | 1946 | 1595 | 404 | 1091 | 10628 | 250
Table 4: Conformance Checking (times in ms)

Also for Conformance Checking we used synthetic and real life datasets. The
former include the same declarative models as those used for Log Generation,
plus synthetic logs of 1000 traces of lengths from 10 to 30. Table 3 shows the
execution times for the ASP-based tool, with and without data conditions, and
for the Declare Analyzer tool _for synthetic datasets_ (times in ms). The
results show that in all cases the execution times increase when the model
becomes larger and the traces in the log become longer. The execution times
obtained with the ASP-based tool and the Declare Analyzer are comparable for
data-aware constraints, while, model constraints do not contain data
conditions, the Declare Analyzer is around 5 times faster. This might be due
to the use of the #max aggregate to compute a trace’s length, which yields
performance degradations. A possible solution could be computing the trace
length in advance and then provide it in the ASP encoding as a fact.

In the real life experiments, we tested the Conformance Checking tools using
models obtained with the discovery tool by varying the minimum support between
60 and 90. The minimum support indicates the minimum percentage of traces in
which a constraint should be fulfilled to be added to the discovered model.
Clearly, a higher minimum support implies that the discovered models contain
less constraints. As expected (see Table 4), the execution times decrease when
the minimum support used to discover the reference models increases in size.
Also in this case, the Declare Analyzer (second block in Table 4) is faster.
However, the ASP-based tool also scales well (first block in Table 4)
requiring in the worst case around 1min.

### Query Checking

Constraints $\rightarrow$ | Existence | Responded | Response | Chain | Absence | Not Resp. | Not Resp. | Not Chain
---|---|---|---|---|---|---|---|---
Trace len $\downarrow$ |  | Existence |  | Response |  | Existence |  | Response
10 | 521 | 736 | 534 | 503 | 566 | 783 | 602 | 385
15 | 704 | 1113 | 801 | 788 | 784 | 1180 | 879 | 606
20 | 1321 | 1675 | 1143 | 1128 | 1373 | 1821 | 1304 | 865
25 | 1397 | 3218 | 1528 | 1561 | 1562 | 2823 | 1807 | 1104
30 | 1674 | 2878 | 1824 | 1906 | 1905 | 2784 | 2028 | 1301
10 | 399 | 658 | 541 | 632 | 441 | 799 | 806 | 772
15 | 616 | 1183 | 824 | 1057 | 595 | 1319 | 1121 | 1182
20 | 903 | 1778 | 1339 | 1550 | 874 | 1887 | 2127 | 2062
25 | 1188 | 2381 | 1724 | 2036 | 1101 | 3246 | 3200 | 2486
30 | 1461 | 3278 | 2066 | 2632 | 1333 | 3391 | 2766 | 2846
Table 5: ASP Query Checking (times in ms)

Since for Query Checking no competitor exists in the PM literature, we ran a
set of controlled experiments to check how execution times vary under
different conditions. We used the same synthetic logs used for Conformance
Checking and tested 8 queries corresponding to 8 standard Declare templates,
with and without data conditions. The results are shown in Table 5 (with and
without data in the first and second block respectively). The execution times
are comparable for different types of queries and the presence of data does
not affect performance. In addition, as expected, the execution times increase
when the traces in the log become longer.

## Conclusions

We have devised an ASP-based approach to solve three classical problems from
Declarative PM, namely Log Generation, Query Checking and Conformance
Checking, in a data-aware setting. Our results include correct ASP-encoding
schemata and an experimental evaluation against other approaches. The
experimental results show that, for Log Generation, our approach drastically
outperforms the state-of-the-art tool from PM. Time performance are slightly
worse wrt to the existing ad-hoc Conformance Checker Declare Analyzer, which
is optimized for Declare. As to Query Checking, our approach provides the
first solution in a data-aware setting, a problem still open so far. We
believe that, by showing how the selected problems can be encoded and solved
in ASP, we are not only offering a solution technique but, more in general, we
are putting forward ASP an effective modeling paradigm for Declarative PM in a
data-aware setting. For future work, we plan to extend the approach to deal
with actual, non-integer, timestamps in events and to go beyond local ltlf by
investigating the introduction of _across-state_ quantification to relate the
values assigned to attributes at a given time point to those assigned at a
different time point.

## Acknowledgments

Work partly supported by the ERC Advanced Grant WhiteMech (No. 834228), the EU
ICT-48 2020 project TAILOR (No. 952215), the Sapienza Project DRAPE, and the
UNIBZ project CAT.

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