0 10pt 153mm 240mm 0.6 Postprint, monthyeardate[]

11institutetext: The University of Melbourne, VIC, 3010, Australia
11email<EMAIL_ADDRESS>11email:
<EMAIL_ADDRESS>The University of Adelaide,
Adelaide, Australia
22email<EMAIL_ADDRESS>York University,
Toronto, Ontario, Canada
33email<EMAIL_ADDRESS>
# Agent Miner: An Algorithm for Discovering Agent Systems from Event Data

Andrei Tour  11 Artem Polyvyanyy  11 Anna Kalenkova  22 Arik Senderovich  33

###### Abstract

Process discovery studies ways to use event data generated by business
processes and recorded by IT systems to construct models that describe the
processes. Existing discovery algorithms are predominantly concerned with
constructing process models that represent the control flow of the processes.
Agent system mining argues that business processes often emerge from
interactions of autonomous agents and uses event data to construct models of
the agents and their interactions. This paper presents and evaluates Agent
Miner, an algorithm for discovering models of agents and their interactions
from event data composing the system that has executed the processes which
generated the input data. The conducted evaluation using our open-source
implementation of Agent Miner and publicly available industrial datasets
confirms that our algorithm can provide insights into the process participants
and their interaction patterns and often discovers models that describe the
business processes more faithfully than process models discovered using
conventional process discovery algorithms.

##  1 Introduction

Process discovery is a subarea of process mining that studies ways to
construct models that faithfully describe processes of a system based on event
data the system has generated [26]. Constructed models assist analysts in
understanding the system and, consecutively, deciding how to improve it. The
state-of-the-art process discovery algorithms build models that describe the
control flow of the processes. This focus on control flow has at least two
limitations. Firstly, the resulting models are not well suited for analyzing
the behavior of individual process participants and their interactions, as
activities and interactions performed by a specific actor are often scattered
across a discovered control flow model. Secondly, control flow models
discovered from large data arrays are often too complex, the phenomenon known
as _spaghetti models_ [26]. Models of interacting agents, or _agent systems_ ,
where agents are the process participants, address the former limitation by
their very definition and, importantly, do _not_ necessarily grow in
complexity with the growth of the amount of data they represent [29].

Agent system mining is a type of process mining that studies ways to derive
and use knowledge about systems composed of interacting agents based on the
events these agents generate [24]. This paper presents and evaluates an agent
system discovery algorithm. Concretely, this paper makes these contributions:

  * •

It presents Agent Miner, a divide-and-conquer algorithm for discovering models
of agents and their interactions from event data. The algorithm “divides” the
input collection of events into several special subsets and “conquers” these
subsets using conventional discovery algorithms, like Inductive Miner [11] and
Split Miner [2], to construct an agent system that has generated the data.

  * •

It presents the results of an evaluation of Agent Miner based on our open-
source implementation of the algorithm that shows that Agent Miner can
discover agent and interaction models from publicly available industrial event
data that i) provide an additional perspective on the system that has
generated the data and ii) often describe the processes more faithfully than
the corresponding process models constructed using conventional control flow
discovery algorithms; thus, addressing the two stated limitations.

  * •

It demonstrates the value of agent system mining and invites the community to
more intensive explorations of the agent-based paradigm in process mining.

The remainder of this paper proceeds as follows. The next section presents a
motivating example. Then, Section 3 gives basic notions required for
understanding the subsequent discussions. Section 4 presents our new discovery
algorithm, while Section 5 discusses the results of its evaluation. Finally,
Section 6 surveys related work before Section 7 gives concluding remarks on
this work.

##  2 Motivating Example

Figure 1: A schematic visualization of the health surveillance process.

As a motivating example, we propose a hypothetical health surveillance process
sketched in Fig. 1. The process starts with a check (see the _check_ activity
in the figure) of the patient by doctor d1. If the analysis (_analyze_) of the
results of the check indicates a risk of developing ill-health, the doctor
prescribes (_prescribe_) medical tests and preventive therapy. The possible
tests are blood test (_B-test_), ultrasound (_U-sound_), and _X-ray_. The
therapy includes _yoga_ , _physio_ , _gym_ , and swimming (_swim_). The tests
and the therapy are performed independently; any subset of tests and exercises
can be prescribed. Doctors d2 and d4 perform the tests, while the therapy is
conducted by doctors d3 and d5. Once the tests and the therapy are completed,
doctor d1 rechecks (_check_) the patient. If the new check shows good results,
the patient is discharged (_discharge_), and the process terminates;
otherwise, further tests and therapy are prescribed. The information system
that supports the process recorded a _log_ of events that stem from managing 1
024 patients. Each recorded event has four attributes that specify the
_timestamp_ of the event occurrence, _activity_ that triggered the event, the
patient _case_ the event relates to, and the doctor, or _agent_ , that
performed the activity.

Figure 2 shows three directly-follows graphs (DFGs) [26] that describe control
flow dependencies between the activities of the health surveillance process
discovered from 2, 8, and 32 traces from the log, where a _trace_ is a
sequence of all events with the same case attribute ordered by their
timestamps. The complexity of the DFGs, defined as the number of nodes and
arcs, grows as they represent more data, where the model in Fig. 2(c) is an
example spaghetti model.

(a) 2 traces

(b) 8 traces

(c) 32 traces

Figure 2: DFGs that describe the health surveillance process (readability not
intended).

$t_{2}$ $p_{3}$ $p_{4}$ a3 $t_{3}$ a2 $t_{4}$ $p_{2}$ $t_{1}$ $p_{1}$ a1
$t_{5}$ $p_{5}$ $p_{6}$ $t_{6}$ $p_{7}$ $t_{7}$ $p_{8}$ (a)

_d_ $t_{5}$ _a_ $t_{6}$ $t_{7}$ $p_{6}$ $p_{3}$ $p_{4}$ $p_{5}$ _c_ $t_{2}$
_p_ $t_{3}$ $t_{4}$ $p_{2}$ $t_{1}$ $p_{1}$ (b)

Figure 3: Petri nets that describe: (a) interactions of the three agent types
a1 (doctor d1), a2 (d2 and d4), and a3 (d3 and d5) and (b) agent type a1 from
the health surveillance process with transition labels check (_c_), analyze
(_a_), prescribe (_p_), and discharge (_d_).

Figure 3(a) and Fig. 3(b) show the interaction net and one agent net that
describes one of the three _agent types_ discovered by Agent Miner from the
events of the 32 traces used to construct the DFG in Fig. 2(c) captured as
Petri nets [19]. Besides providing an alternative, modular perspective on the
process, that is, an explicit representation of process participants and their
interactions, these models, similar to the DFGs in Fig. 2, describe control
flow dependencies between the activities. These dependencies can be captured
explicitly in an integrated Petri net, called the Multi-Agent System (MAS)
net, obtained by refining labeled transitions of the interaction net with the
corresponding agent nets. Interestingly, this MAS net is smaller and
represents the traces encoded in the event data more faithfully than the Petri
net constructed from the same 32 traces using Inductive Miner (IM) [11]—a
conventional process discovery algorithm. The MAS net has 155 nodes and arcs,
while the IM net shown in Fig. 4 has 175 nodes and arcs. The precision of the
MAS net and the IM net is 0.37 and 0.16, respectively. To support model
comparison, both algorithms were configured to discover perfectly fitting
models (recall of 1.0). A model has good precision if it does not replay
traces not recorded in the log and a good recall if it replays many traces
from the log, where the values closer to 1.0 indicate better models. Precision
and recall were measured for the 32 traces used to construct the models using
the entropy-based approach [16, 17], the only existing measures that guarantee
that better discovered models result in better measurements [23].

A MAS net often does not require an increase in size to represent more data
well due to an ability of an agent system to simulate the non-decreasing
complexity of the behavior of a system [24, 29]. For instance, the same MAS
net we discovered from the 32 traces has precision of 0.43 and recall of 1.0
if measured for all 1 024 traces. This example confirms that Agent Miner can
construct modular, fitting, and precise models of process participants and
their interactions. Finally, while the interaction and agent nets enable
dedicated analyses of the respective artifacts, like identification of
repetitive work handovers and verification of safeness and liveness of
individual agent nets, the information about agents is scattered in the IM
net, see the _U-sound_ activity by agent a2 amidst activities _swim_ and
_yoga_ by agent a3 in Fig. 4, which hinders the analysis of individual agents
and their interactions. The input log, models, and measurements discussed in
this section are publicly available [25].

Figure 4: The IM net discovered from 32 traces of the example health
surveillance process.

##  3 Preliminaries

This section introduces Petri nets (Section 3.1) and event logs (Section 3.2).

###  3.1 Petri Nets and Workflow Nets

Petri nets formalism suits well for describing models of distributed systems
[19].

######  Definition 3.1 (Petri nets)

A _(labeled) Petri net_ , or a _net_ , $N$ is a quintuple $\left(P\mathchar
44\relax\penalty 0T\mathchar 44\relax\penalty 0F\mathchar 44\relax\penalty
0\Lambda\mathchar 44\relax\penalty 0\lambda\right)$, where $P$ is a finite set
of _places_ , $T$ is a finite set of _transitions_ , $F\subseteq(P\times
T)\cup(T\times P)$ is the _flow relation_ , $\Lambda$ is a set of _labels_ ,
such that $\tau\in\Lambda$ is the _silent label_ and sets $P$, $T$, and
$\Lambda$ are pairwise disjoint, and ${{\lambda}:{T}\rightarrow{\Lambda}}$ is
the _labeling function_. $\lrcorner$

If ${\mathit{\lambda}_{\\!}\left({t}\right)}=\tau$, $t\in T$, $t$ is _silent_
; otherwise $t$ is _observable_. Observable transitions represent activities
from the problem domain, and silent transitions encode internal actions of the
system. A _marking_ $M$ of a net encodes its state and is a multiset over its
places. Figure 3(a) shows a Petri net $N=\left(P\mathchar 44\relax\penalty
0T\mathchar 44\relax\penalty 0F\mathchar 44\relax\penalty 0\Lambda\mathchar
44\relax\penalty 0\lambda\right)$, with eight places
($P=\left\\{p_{1}\mathchar 44\relax\penalty 0\ldots\mathchar 44\relax\penalty
0p_{8}\right\\}$) and seven transitions ($T=\left\\{t_{1}\mathchar
44\relax\penalty 0\ldots\mathchar 44\relax\penalty 0t_{7}\right\\}$). In the
graphical notation, places are drawn as circles, while transitions as squares
or rectangles. Transitions $t_{1}$, $t_{2}$, $t_{6}$, and $t_{7}$ are silent,
shown as black rectangles. The labeling function assigns labels a1, a2, and a3
to transitions $t_{5}$, $t_{4}$, and $t_{3}$, respectively. The flow relation
is shown as directed arcs. A marking is denoted as an arrangement of black
dots, called _tokens_ , inside of the corresponding places. Marking $M$ shown
in Fig. 3(a) is the multiset $[p_{1}]$; see one black dot in place $p_{1}$.

Let $n\in P\,\cup\,T$ be a place or transition, then by $\bullet
n\\!=\\!\left\\{x\in(P\cup T)\;|\;(x,n)\in F\right\\}$ we denote its _preset_
and by $n\bullet\\!=\\!\left\\{x\in(P\cup T)\;|\;(n,x)\in F\right\\}$ we
denote its _postset_. Let $N=\left(P\mathchar 44\relax\penalty 0T\mathchar
44\relax\penalty 0F\mathchar 44\relax\penalty 0\Lambda\mathchar
44\relax\penalty 0\lambda\right)$ be a net. A transition $t\in T$ is _enabled_
in a marking $M$ of $N$, denoted by $\left(N\mathchar 44\relax\penalty
0M\right)\left[t\right\rangle$, if every _input place_ of $t$ contains at
least one token, i.e., $\forall\,p\in\bullet
t:{\mathit{M}_{\\!}\left({p}\right)}>0$; by
${\mathit{M}_{\\!}\left({p}\right)}$ we denote the multiplicity of $p$ in $M$.
An enabled transition $t\in T$ can occur. An _occurrence_ of $t$ leads to a
fresh marking $M^{\prime}=(M\setminus\bullet t)\uplus t\bullet$ of $N$,
denoted by $\left(N\mathchar 44\relax\penalty
0M\right)\left[t\right\rangle\left(N\mathchar 44\relax\penalty
0M^{\prime}\right)$.

Workflow nets are special nets used for modeling workflows [26]. A _workflow
net_ is a Petri net $\left(P\mathchar 44\relax\penalty 0T\mathchar
44\relax\penalty 0F\mathchar 44\relax\penalty 0\Lambda\mathchar
44\relax\penalty 0\lambda\right)$ with a dedicated _initial place_ $i\in P$,
$\bullet i=\emptyset$, a dedicated _final place_ $f\in P$,
$f\bullet=\emptyset$, and every place and transition on the vertex sequence of
some directed walk from $i$ to $f$ in graph $\left(P\cup T\mathchar
44\relax\penalty 0F\right)$. Marking $[i]$ is the initial marking of a
workflow net. A workflow net is _safe_ if every marking reachable from $[i]$
via a sequence of transition occurrences is a set. It is _sound_ if every
transition of the net can occur in some sequence of transition occurrences
that starts in $[i]$, every sequence of transition occurrences that starts in
$[i]$ can be extended to “put” a token in the final place, and once there is a
token in the final place no other places hold tokens [26]. Figure 3(a) and
Fig. 3(b) show safe and sound workflow nets with final markings $[p_{8}]$ and
$[p_{6}]$, respectively.

###  3.2 Events, Event Logs, and Traces

Table 1: An event selection. event | timestamp | case | activity |  agent
---|---|---|---|---
$e^{a}$ | 30 Mar 16:34 | case1 | _check_ |  d1/a1
$e^{b}$ | 30 Mar 16:35 | case1 | _analyze_ |  d1/a1
$e^{c}$ | 31 Mar 16:35 | case2 | _check_ |  d1/a1
$e^{d}$ | 01 Apr 08:22 | case2 | _analyze_ |  d1/a1
$e^{e}$ | 01 Apr 16:05 | case1 | _prescribe_ |  d1/a1
$e^{f}$ | 03 Apr 11:55 | case1 | _B-test_ |  d4/a2
$e^{g}$ | 03 Apr 16:59 | case1 | _X-ray_ |  d4/a2
$e^{h}$ | 06 Apr 10:02 | case1 | _physio_ |  d3/a3
$e^{i}$ | 06 Apr 11:01 | case1 | _swim_ |  d3/a3
$e^{j}$ | 07 Apr 15:55 | case1 | _yoga_ |  d3/a3
$e^{k}$ | 07 Apr 11:11 | case1 | _physio_ |  d3/a3
$e^{l}$ | 10 Apr 13:13 | case1 | _swim_ |  d3/a3
$e^{m}$ | 10 Apr 15:05 | case1 | _yoga_ |  d3/a3
$e^{n}$ | 11 Apr 09:12 | case1 | _physio_ |  d3/a3
$e^{o}$ | 11 Apr 10:05 | case1 | _swim_ |  d3/a3
$e^{p}$ | 13 Apr 11:03 | case1 | _yoga_ |  d3/a3
$e^{q}$ | 13 Apr 14:57 | case1 | _check_ |  d1/a1
$e^{r}$ | 16 Apr 12:11 | case1 | _analyze_ |  d1/a1
$e^{s}$ | 17 Apr 10:03 | case1 | _prescribe_ |  d1/a1
$e^{t}$ | 17 Apr 16:36 | case2 | _prescribe_ |  d1/a1

An _event log_ , or _log_ , represents real-world processes recorded by an
information system. In process mining, events are often organized into time-
ordered sequences, called _traces_. As explained in Section 2, in our work,
each event has at least four attributes: _timestamp_ , _activity_ , _case_ ,
and _agent_ (either instance or type). Let $\mathcal{A}$ be the universe of
attribute names, with $\left\\{\mathit{timestamp}\mathchar 44\relax\penalty
0\mathit{activity}\mathchar 44\relax\penalty 0\mathit{case}\mathchar
44\relax\penalty 0\mathit{agent}\right\\}\\!\\!\subseteq\\!\\!\mathcal{A}$.
Let $\mathcal{V}$ be the universe of attribute values. Then, an _event_ is an
attribute function ${{e}:{\mathcal{A}}\rightarrow{\mathcal{V}}}$ that maps
attribute names to attribute values. By $\mathcal{E}$ we denote the universe
of events. An _event selection_ $S\subseteq\mathcal{E}$ is a finite set of
events. Without loss of generality, we assume that events have unique
timestamps. Table 1 defines event selection $S=\left\\{{e^{a}}\mathchar
44\relax\penalty 0{\ldots}\mathchar 44\relax\penalty 0{e^{t}}\right\\}$. Each
row of the table describes one event. For example,
$e^{t}=\left\\{\left(\mathit{timestamp}\mathchar 44\relax\penalty
0\textnormal{17 Apr 16:36}\right)\mathchar 44\relax\penalty
0\left(\mathit{activity}\mathchar 44\relax\penalty
0\mathit{prescribe}\right)\mathchar 44\relax\penalty
0\left(\mathit{case}\mathchar 44\relax\penalty 0\textsl{case2}\right)\mathchar
44\relax\penalty 0\left(\mathit{agent}\mathchar 44\relax\penalty
0\textsl{a1}\right)\right\\}$ is the event from the last row of Table 1; the
agent attribute specifies agent type.

Let $X$ be a finite non-empty set. A _partition_ of $X$ is a set $\Pi$ of
disjoint subsets of $X$ such that the union of the subsets equals $X$; the
subsets are _parts_ of $\Pi$. Partition $\Pi$ can be defined by an equivalence
relation $\sim\,\subseteq X\times X$ such that if two events $e_{i},e_{j}\in
X$ are equivalent under $\sim$, that is, it holds that $\left(e_{i}\mathchar
44\relax\penalty 0e_{j}\right)\in\,\sim$, then $e_{i}$ and $e_{j}$ belong to
the same part of $\Pi$. We use notation $\Pi=X/\\!\sim$ to denote that
partition $\Pi$ is defined by the equivalence relation $\sim$ over $X$.
Against this background, we define a trace of an event selection as follows.

######  Definition 3.2 (Traces induced by partitions)

The _trace_ $\sigma$ of event selection $S$ induced by part $\pi$ of partition
$\Pi=S/\\!\sim$ is the ordered by timestamps sequence of all and only events
in $\pi$. $\lrcorner$

We refer to $\Pi$ and $\sim$ as the _trace partition_ and the _trace relation_
that induce $\sigma$. The _trace set_ of $S$ induced by $\Pi=S/\\!\sim$ is the
set of traces $\Sigma$ that for each part $\pi$ of $\Pi$ contains the trace of
$S$ induced by $\pi$, and contains no other traces.

######  Definition 3.3 (Event logs)

An _event log_ , or _log_ , of event selection $S$ induced by partition
$\Pi=S/\\!\sim$ is a triple $\left(S\mathchar 44\relax\penalty
0\Sigma\mathchar 44\relax\penalty 0\nu\right)$, where $\Sigma\subset S^{*}$ is
the trace set induced by $\Pi$, and ${{\nu}:{S}\rightarrow{\mathcal{V}}}$ is a
naming function that assigns names to events in $S$. $\lrcorner$

Multiple event logs induced by different partitions and naming functions can
be defined. For example, in classical process discovery, traces are induced by
case attributes. Let $\mathcal{C}$ be a universe of cases. The _case trace
set_ of event selection $S$ is the trace set of $S$ defined by relation
$\sim^{\textbf{c}}\,=\left\\{\left(e_{i}\mathchar 44\relax\penalty
0e_{j}\right)\in S\times
S\;|\;{\mathit{e_{i}}_{\\!}\left({\mathit{case}}\right)}={\mathit{e_{j}}_{\\!}\left({\mathit{case}}\right)}\right\\}$,
denoted by $\Sigma^{\textbf{c}}$; we refer to traces in $\Sigma^{\textbf{c}}$
as _case traces_. For instance, the case trace set of the event selection in
Table 1 consists of two case traces, namely $\langle e^{a}\mathchar
44\relax\penalty 0e^{b}\mathchar 44\relax\penalty 0e^{e}\mathchar
44\relax\penalty 0e^{f}\mathchar 44\relax\penalty 0e^{g}\mathchar
44\relax\penalty 0e^{h}\mathchar 44\relax\penalty 0e^{i}\mathchar
44\relax\penalty 0e^{j}\mathchar 44\relax\penalty 0e^{k}\mathchar
44\relax\penalty 0e^{l}\mathchar 44\relax\penalty 0e^{m}\mathchar
44\relax\penalty 0e^{n}\mathchar 44\relax\penalty 0e^{o}\mathchar
44\relax\penalty 0e^{p}\mathchar 44\relax\penalty 0e^{q}\mathchar
44\relax\penalty 0e^{r}\mathchar 44\relax\penalty 0e^{s}\rangle$ and $\langle
e^{c}\mathchar 44\relax\penalty 0e^{d}\mathchar 44\relax\penalty
0e^{t}\rangle$.

The naming function of an event log specifies the names of the events used by
the discovery algorithms to identify activity names in the constructed process
models. In process discovery, events are often identified by their activity
attributes, that is, $\nu=\left\\{\left(e\mathchar 44\relax\penalty
0{\mathit{e}_{\\!}\left({\mathit{activity}}\right)}\right)\;|\;e\in
S\right\\}$. In general, other naming functions can be used. We will use this
feature in the subsequent sections.

##  4 Agent Miner

In this section, we introduce the core notions required to define the Agent
Miner algorithm (Section 4.1) and present the algorithm (Section 4.2).

###  4.1 Agent Logs and Nets

A trace of an _agent trace set_ is a sequence of events that refer to the same
case, are performed by the same agent (identified by the agent attribute), and
are not interrupted by an event from the same case performed by a different
agent.

######  Definition 4.1 (Agent trace sets)

The _agent trace set_ of event selection $S$ is the trace set of $S$ induced
by the partition of $S$ defined by relation $\sim=\left\\{\left(e_{i}\mathchar
44\relax\penalty 0e_{j}\right)\in S\times
S\;|\;({\mathit{e_{i}}_{\\!}\left({\mathit{agent}}\right)}={\mathit{e_{j}}_{\\!}\left({\mathit{agent}}\right)})\right.$
$\left.\land\;\;({\mathit{e_{i}}_{\\!}\left({\mathit{case}}\right)}={\mathit{e_{j}}_{\\!}\left({\mathit{case}}\right)})\;\;\land\;\;(\neg\exists\,e_{k}\in
S:((({\mathit{e_{i}}_{\\!}\left({\mathit{timestamp}}\right)}<{\mathit{e_{k}}_{\\!}\left({\mathit{timestamp}}\right)}<\right.$
$\left.{\mathit{e_{j}}_{\\!}\left({\mathit{timestamp}}\right)})\;\;\lor\;\;({\mathit{e_{j}}_{\\!}\left({\mathit{timestamp}}\right)}\,\,<\,\,{\mathit{e_{k}}_{\\!}\left({\mathit{timestamp}}\right)}\,\,<\,\,{\mathit{e_{i}}_{\\!}\left({\mathit{timestamp}}\right)}))\;\;\land\,\right.$
$\left.({\mathit{e_{i}}_{\\!}\left({\mathit{agent}}\right)}\neq{\mathit{e_{k}}_{\\!}\left({\mathit{agent}}\right)}\,\land\,{\mathit{e_{i}}_{\\!}\left({\mathit{case}}\right)}={\mathit{e_{k}}_{\\!}\left({\mathit{case}}\right)})))\right\\}.$
$\lrcorner$

The set $\\{\langle e^{a}\mathchar 44\relax\penalty 0e^{b}\mathchar
44\relax\penalty 0e^{e}\rangle,\langle e^{c}\mathchar 44\relax\penalty
0e^{d}\mathchar 44\relax\penalty 0e^{t}\rangle,$ $\langle e^{f}\mathchar
44\relax\penalty 0e^{g}\rangle,\langle e^{h}\mathchar 44\relax\penalty
0e^{i}\mathchar 44\relax\penalty 0e^{j}\mathchar 44\relax\penalty
0e^{k}\mathchar 44\relax\penalty 0e^{l}\mathchar 44\relax\penalty
0e^{m}\mathchar 44\relax\penalty 0e^{n}\mathchar 44\relax\penalty
0e^{o}\mathchar 44\relax\penalty 0e^{p}\rangle,\langle e^{q}\mathchar
44\relax\penalty 0e^{r}\mathchar 44\relax\penalty 0e^{s}\rangle\\}$ is the
agent trace set of the event selection in Table 1. For instance, in trace
$\langle e^{a}\mathchar 44\relax\penalty 0e^{b}\mathchar 44\relax\penalty
0e^{e}\rangle$, all events are from case1, performed by agent a1, and, though
interrupted by events $e^{c}$ and $e^{d}$, the latter events are from a
different case. Note that, by definition, relation $\sim$ from 4.1 is an
equivalence relation.

Next, we define several useful logs. Traces of an interaction log are composed
of events that allow identifying all handovers of work between agents.

a3 $t_{2}$ $p_{2}$ $t_{1}$ $p_{1}$ $p_{3}$ a2 $t_{3}$ $p_{4}$ $t_{4}$ $p_{5}$
a1 $t_{5}$ Figure 5: An i-net.

######  Definition 4.2 (Interaction logs)

The _interaction log_ of event selection $S$ is the triple
$\left(\bar{S}\mathchar 44\relax\penalty 0\Sigma\mathchar 44\relax\penalty
0\nu\right)$, where $\bar{S}$ is the _agent event selection_ composed of the
first events of all the traces in the agent trace set $\Delta$ of $S$, that
is,
$\bar{S}=\left\\{{\mathit{\delta}_{\\!}\left({1}\right)}\;|\;\delta\in\Delta\right\\}$,
$\Sigma$ is the case trace set of $\bar{S}$, and $\nu=\left\\{\left(e\mathchar
44\relax\penalty
0{\mathit{e}_{\\!}\left({\mathit{agent}}\right)}\right)\;|\;e\in\bar{S}\right\\}$
names each event by the corresponding agent. $\lrcorner$

The interaction log of the event selection from Table 1 is, therefore, the
event log $\left(\bar{S}\mathchar 44\relax\penalty 0\Sigma\mathchar
44\relax\penalty 0\nu\right)$, where $\bar{S}=\left\\{e^{a}\mathchar
44\relax\penalty 0e^{c}\mathchar 44\relax\penalty 0e^{f}\mathchar
44\relax\penalty 0e^{h}\mathchar 44\relax\penalty 0e^{q}\right\\}$,
$\Sigma=\left\\{\langle e^{a}\mathchar 44\relax\penalty 0e^{f}\mathchar
44\relax\penalty 0e^{h}\mathchar 44\relax\penalty 0e^{q}\rangle\mathchar
44\relax\penalty 0\langle e^{c}\rangle\right\\}$, and
$\nu=\left\\{\left(e^{a}\mathchar 44\relax\penalty
0\textsl{a1}\right)\mathchar 44\relax\penalty 0\left(e^{c}\mathchar
44\relax\penalty 0\textsl{a1}\right)\mathchar 44\relax\penalty
0\left(e^{f}\mathchar 44\relax\penalty 0\textsl{a2}\right)\mathchar
44\relax\penalty 0\left(e^{h}\mathchar 44\relax\penalty
0\textsl{a3}\right)\mathchar 44\relax\penalty 0\left(e^{q}\mathchar
44\relax\penalty 0\textsl{a1}\right)\right\\}$. For instance, trace $\langle
e^{a}\mathchar 44\relax\penalty 0e^{f}\mathchar 44\relax\penalty
0e^{h}\mathchar 44\relax\penalty 0e^{q}\rangle$ in $\Sigma$ suggests that
agent a1 starts case1; note that
${\mathit{e^{a}}_{\\!}\left({agent}\right)}=\textsl{a1}$ and
${\mathit{e^{a}}_{\\!}\left({case}\right)}=\textsl{case1}$. Then, agent a2
takes over the work on case1. Agent a2 then passes the work on case1 to agent
a3, who later hands work back to agent a1.

Traces in the agent trace set done by the same agent compose its agent log.

######  Definition 4.3 (Agent logs)

The _agent log_ of agent a and event selection $S$ is the triple
$\left(S^{\textsl{a}}\mathchar 44\relax\penalty 0\Sigma\mathchar
44\relax\penalty 0\nu\right)$, where $S^{\textsl{a}}=\left\\{e\in
S\;|\;{\mathit{e}_{\\!}\left({\mathit{agent}}\right)}=\textsl{a}\right\\}$,
$\Sigma$ is the set of traces in the agent trace set $\Delta$ of $S$ performed
by a, that is,
$\Sigma=\left\\{\delta\in\Delta\;|\;\forall\,i\in{\left[1\,..\,\left|{\delta}\right|\right]}:{\mathit{{\mathit{\delta}_{\\!}\left({i}\right)}}_{\\!}\left({\mathit{agent}}\right)}=\textsl{a}\right\\}$,
and $\nu=\left\\{\left(e\mathchar 44\relax\penalty 0\left(\textsl{a}\mathchar
44\relax\penalty
0{\mathit{e}_{\\!}\left({\mathit{activity}}\right)}\right)\right)\;|\;e\in
S^{\textsl{a}}\right\\}$ is the naming function that names an event by the
pair comprising its agent and activity attributes. $\lrcorner$

The agent log of agent a1 and the event selection from Table 1 is defined by
the triple $\left(S^{\textsl{a1}}\mathchar 44\relax\penalty 0\Sigma\mathchar
44\relax\penalty 0\nu\right)$, where $S^{\textsl{a1}}=\left\\{e^{a}\mathchar
44\relax\penalty 0e^{b}\mathchar 44\relax\penalty 0e^{c}\mathchar
44\relax\penalty 0e^{d}\mathchar 44\relax\penalty 0e^{e}\mathchar
44\relax\penalty 0e^{q}\mathchar 44\relax\penalty 0e^{r}\mathchar
44\relax\penalty 0e^{s}\mathchar 44\relax\penalty 0e^{t}\right\\}$,
$\Sigma=\\{{\langle e^{a}\mathchar 44\relax\penalty 0e^{b}\mathchar
44\relax\penalty 0e^{e}\rangle},\\\ {\langle e^{c}\mathchar 44\relax\penalty
0e^{d}\mathchar 44\relax\penalty 0e^{t}\rangle},{\langle e^{q}\mathchar
44\relax\penalty 0e^{r}\mathchar 44\relax\penalty 0e^{s}\rangle}\\}$, and
$\nu$ maps events in $S^{\textsl{a1}}$ to their names and contains, for
instance, it holds that $\left\\{\left(e^{a}\mathchar 44\relax\penalty
0\left(\textsl{a1}\mathchar 44\relax\penalty
0\mathit{check}\right)\right)\mathchar 44\relax\penalty 0\left(e^{d}\mathchar
44\relax\penalty 0\left(\textsl{a1}\mathchar 44\relax\penalty
0\mathit{analyze}\right)\right)\right\\}\subset\nu$.

Next, we discuss several classes of workflow nets used by Agent Miner. An
interaction net describes the structure of interactions between agents.

######  Definition 4.4 (Interaction nets)

An _interaction net_ , or an _i-net_ , of event selection $S$ is a workflow
net $\left(P\mathchar 44\relax\penalty 0T\mathchar 44\relax\penalty
0F\mathchar 44\relax\penalty 0\Lambda\mathchar 44\relax\penalty
0\lambda\right)$, where
$\Lambda=\left\\{\textsl{a}\in\mathcal{V}\;|\;\exists\,e\in
S:{\mathit{e}_{\\!}\left({agent}\right)}=\textsl{a}\right\\}\cup\left\\{\tau\right\\}$.
$\lrcorner$

Figures 3(a) and 5 show two i-nets of the event selection from Table 1. They
describe alternative ways the agents could have interacted to generate the
event data. For example, the i-net in Figure 5 suggests that agent a1 starts
the interaction, and then any number of sequences of interactions of a1 with
agent a2, then of a2 with agent a3, and finally of a3 again with agent a1 can
occur.

A MAS net describes how agents perform activities in a collaborative setting.

######  Definition 4.5 (Multi-Agent System nets)

A  _Multi-Agent System (MAS) net_ of event selection $S$ is a workflow net
$(P,T,F,\\\ \Lambda,\lambda)$, with
$\Lambda=\left\\{(\textsl{a},b)\\!\in\\!\mathcal{V}\\!\times\\!\mathcal{V}\;|\;\exists\,e\\!\in\\!S\\!:\\!{\mathit{e}_{\\!}\left({\mathit{agent}}\right)}\\!=\\!\textsl{a}\land{\mathit{e}_{\\!}\left({\mathit{activity}}\right)}\\!=\\!b\right\\}\cup\left\\{\tau\right\\}$.
$\lrcorner$

A MAS net of events of a single agent is an agent net of this agent.

######  Definition 4.6 (Agent nets)

A MAS net of event selection $S$ such that the agent attribute of every event
in $S$ is equal to a, that is, $\forall\,e\in
S:{\mathit{e}_{\\!}\left({\mathit{agent}}\right)}=\textsl{a}$, is an _agent
net_ of a. $\lrcorner$

###  4.2 Algorithm

$\mathit{MAS\,net\,(\lx@cref{creftype~refnum}{def:mas_net})}$
$\mathit{Discover\,MAS\,net}$
$\mathit{agent\,nets\,(\lx@cref{creftype~refnum}{def:agent_net})}$n$\mathit{interaction\,net\,(\lx@cref{creftype~refnum}{def:interaction_net})}$
$\mathit{Discover\,agent\,net\,(ANDA)}$
$\mathit{Discover\,interaction\,net\,(INDA)}$
$\mathit{agent\,logs\,(\lx@cref{creftype~refnum}{def:agent_log})}$$\mathit{interaction\,log\,(\lx@cref{creftype~refnum}{def:interaction_log})}$
$\mathit{Create\,agent\,logs}$ n $\mathit{Create\,interaction\,log}$
$\mathit{agent\,trace\,set\,(\lx@cref{creftype~refnum}{def:agent-
trace})}$$\mathit{agent\,trace\,set\,(\lx@cref{creftype~refnum}{def:agent-
trace})}$ $\mathit{Create\,agent\,trace\,set}$
$\mathit{event\,selection\,with\,agents}$ Figure 6: The Agent Miner algorithm.

Figure 6 defines the Agent Miner algorithm as a workflow net. It is
parameterized by two conventional control flow discovery algorithms: an Agent
Net Discovery Algorithm (ANDA) and an Interaction Net Discovery Algorithm
(INDA). Agent Miner takes an event selection as input and produces an
interaction net, agent nets, and a MAS net. Similar to standard process
discovery algorithms, the latter explains how the discovered interaction and
agent nets represent the traces identified by the case attribute. The
algorithm has six steps detailed below.

Create agent trace set. The agent trace set of the input event selection is
created by associating each event with an agent trace, see 4.1.

Create interaction log. The interaction log is created by selecting the first
events of all traces in the agent trace set and constructing the case trace
set of this event selection, see 4.2.

Discover interaction net. In this step, INDA is used to discover an i-net, see
4.4, from the interaction log. For example, the i-net in Fig. 5 was discovered
from the interaction log of the event selection in Table 1.

Create agent logs. Multiple ($n$) agent logs are created, one for each agent
encountered in the input event selection, by extracting the corresponding
agent traces from the agent trace set, see 4.3.

Discover agent net. In this step, ANDA is used to discover $n$ agent nets (one
net is discovered from each agent log), see 4.6. Figure 7 shows three agent
nets discovered from the agent logs of the event selection in Table 1.

$p_{1}$ (a1,$\mathit{check}$)  $t_{1}$ $p_{2}$ (a1,$\mathit{analyze}$)
$t_{2}$ $p_{3}$ (a1,$\mathit{prescribe}$)  $t_{3}$ $p_{4}$ (a) Agent net a1

$p_{1}$ (a2,$\mathit{B}$-$\mathit{test}$)  $t_{1}$ $p_{2}$
(a2,$\mathit{X}$-$\mathit{ray}$)  $t_{2}$ $p_{3}$ (b) Agent net a2

$p_{1}$ $t_{1}$ $p_{2}$ (a3,$\mathit{physio}$)  $t_{2}$ $p_{3}$
(a3,$\mathit{swim}$)  $t_{3}$ $p_{4}$ (a3,$\mathit{yoga}$)  $t_{4}$ $p_{5}$
$t_{5}$ $p_{6}$ $t_{6}$ (c) Agent net a3

Figure 7: Three agent nets. $p_{1}$ $t_{1}$ $p_{2}$ (a1,$\mathit{check}$)
$t_{2}$ $p_{3}$ (a1,$\mathit{analyze}$)  $t_{3}$ $p_{4}$
(a1,$\mathit{prescribe}$)  $t_{4}$ $p_{5}$ (a2,$\mathit{B}$-$\mathit{test}$)
$t_{5}$ $p_{6}$ (a2,$\mathit{X}$-$\mathit{ray}$)  $t_{6}$ $p_{7}$ $t_{7}$
$p_{8}$ (a3,$\mathit{physio}$)  $t_{8}$ $p_{9}$ (a3,$\mathit{swim}$)  $t_{9}$
$p_{10}$ (a3,$\mathit{yoga}$)  $t_{10}$ $p_{11}$ $t_{11}$ $t_{12}$ $t_{13}$
$p_{12}$ Figure 8: A MAS net.

Discover MAS net. Finally, a MAS net, see 4.5, is constructed by “embedding”
the agent nets into the i-net and applying the Fusion of Series Places
reduction [14] to the refined i-net. Fig. 8 shows the embedding of the agent
nets in Fig. 7 into the i-net in Fig. 5. The embedding is performed by
refining each observable transition in the i-net with the corresponding agent
net. The resulting MAS net describes how agents (doctors) interact to support
the execution of cases (treatment of patients) in the health surveillance
process introduced in Section 2 discovered based on the event selection from
Table 1.

Thus, Agent Miner is a divide-and-conquer algorithm that “divides” the input
event selection into interaction and agent logs and “conquers” these logs
using ANDA and INDA. In this work, as ANDA, we used DFG translation to Petri
nets (DFG-PN) [12], while as INDA, we used Inductive Miner (IM) [11] and Split
Miner (SM) [2] and removed iterations of observable transitions in the
obtained i-nets. This approach follows the agent system paradigm, where agents
are sequential machines, and parallelism emerges through collaborations of
agents. If the algorithms guarantee that the constructed models are safe and
sound workflow nets, which holds for DFG-PN and IM, every constructed MAS net
is guaranteed to be a safe and sound workflow net (cf. Theorem 2 in [18]).

##  5 Evaluation

The Agent Miner algorithm (Figure 6) and its evaluation pipeline (Figure 9),
including the code, the discovered models, and detailed results, are publicly
available [25]. This section presents and discusses the design (Section 5.1)
and results (Section 5.2) of an evaluation of Agent Miner using real-world
datasets.

###  5.1 Design

Figure 9 presents our five-step evaluation pipeline. We executed this pipeline
for each real-world dataset used in this evaluation. The steps are explained
below.

_dataset_ _Select events_ _events_ _Identify agent types_ _events with agent
types_ _events with agent types_ _Discover AM models_ _AM models_ _Compare
models_ _precision, recall, size_ _CM models_ _Discover CM models_ Figure 9:
Evaluation pipeline.

Select events. In this step, we select events from the dataset to ensure
events have required attributes and remove infrequent case traces.
Specifically, every event must have a _case_ , _timestamp_ , _activity_ , and
_agent_ attribute. Moreover, each selected event must be part of a frequent
case variant (a trace within vff% of the most common case traces, where vff is
the variant frequency filter parameter).

Identify agent types. For the event selection, we identify agent instances
using agent attribute values of events and group them into agent types by
clustering them based on distances between agent instance DFGs, where an agent
instance DFG is constructed from the corresponding agent log (4.3). For a pair
of agents $\left(a_{1}\mathchar 44\relax\penalty 0a_{2}\right)$, we calculate
the distance between the DFGs of $a_{1}$ and $a_{2}$ as:

$1-\max(\nicefrac{{\left|{\mathit{DF}_{a_{1}}\cap\mathit{DF}_{a_{2}}}\right|}}{{\left|{\mathit{DF}_{a_{1}}}\right|}},\nicefrac{{\left|{\mathit{DF}_{a_{1}}\cap\mathit{DF}_{a_{2}}}\right|}}{{\left|{\mathit{DF}_{a_{2}}}\right|}}),$

where $\mathit{DF}_{a_{1}}$ and $\mathit{DF}_{a_{2}}$ are the directly-follows
relations (sets of edges) in the corresponding DFGs. Thus, the more one DFG
subsumes the directly-follows relations of the other DFG, the more likely the
corresponding agent instances belong to the same agent type. Once the agent
types are identified, the agent attribute values of all the events in the
input event selection are updated from instances to types. For example, this
step leads to the identification of three agent types a1 (doctor d1), a2
(doctors d2 and d4), and a3 (doctors d3 and d5) in the motivating example from
Section 2.

Discover CM models. We refer to a discovery algorithm based on the
conventional process mining paradigm as a Conventional Miner (CM) and a model
discovered by a CM algorithm as a CM model. We use two state-of-the-art CM
algorithms, Inductive Miner IMf variant (IM) [11] and Split Miner (SM) [2], to
construct CM models from the case traces induced by the event selection, that
is, traces identified by the case attributes, see 3.3. Given an event
selection $S$, to discover CM models, we use two naming functions: i)
activity-only labeling (AOL) defined by
$\nu_{\mathit{aol}}=\left\\{\left(e\mathchar 44\relax\penalty
0{\mathit{e}_{\\!}\left({\mathit{activity}}\right)}\right)\;|\;e\in
S\right\\}$ and ii) agent and activity labeling (AAL) defined by
$\nu_{\mathit{aal}}=\left\\{\left(e\mathchar 44\relax\penalty
0\left({\mathit{e}_{\\!}\left({\mathit{agent}}\right)}\mathchar
44\relax\penalty
0{\mathit{e}_{\\!}\left({\mathit{activity}}\right)}\right)\right)\;|\;e\in
S\right\\}$. Thus, for each event dataset, we construct four CM models. For
example, the model in Fig. 4 is the IM model constructed using the
$\nu_{\mathit{aol}}$ naming function. Hence, agent attributes of events were
not used to construct the model, and then observable transitions were
annotated with agent info. For each CM algorithm and naming function, we
construct ten CM models, one for each configuration of the _threshold_
parameter of the algorithm, _noise threshold_ for IM and _frequency threshold_
for SM (in the range from $0.0$ to $0.9$ in 0.1 increments). This approach
ensures that the discovered CM models show a range of quality characteristics.

Discover AM models. For the event selection over agent types, we use the Agent
Miner (AM) algorithm (refer to Section 4) to discover agent nets, the model of
agent interactions, and the MAS net that defines the semantics of the
resulting agent systems. DFG translation to Petri nets (DFG-PN) [12] algorithm
is used as an ANDA. Inductive Miner (IM) [11] and Split Miner (SM) [2] are
used as INDAs. For each configuration of ANDA and INDA, we run AM ten times
with ten different parameter pairs $(\mathit{ff}_{i},\mathit{th}_{i})$, with
$i\in{\left[1\,..\,10\right]}$, $\mathit{ff}_{i}=i\times 0.1$, and
$\mathit{th}_{i}=1-i\times 0.1$. Parameter $\mathit{ff}_{i}$ is the _activity
frequency filter_ parameter used by the ANDA. Parameter $\mathit{th}_{i}$ is
the _threshold_ parameter used by the INDA, _noise threshold_ for IM and
_frequency threshold_ for SM. The lower the value of $i$, the more filtered
the event selection is and the less the discovered models are fit to the input
data.

Compare models. To compare the discovered CM and AM models, we calculate their
_recall_ and _precision_ with respect to the case trace set of the original
event selection, and _size_ (as the number of nodes and arcs). To compute
precision and recall, we use the entropy-based measures [16, 17]. These
measures fulfill all the desired properties for these classes of measures
[23]. For example, the entropy-based precision measure guarantees that a model
that describes fewer traces not in the event log has a better precision score.
For CM models constructed using AOL and AAL naming functions of events, we
measure their precision and recall with respect to the event logs that
identify events using the corresponding naming functions. Note that labels of
observable transitions of AM models are composed of agent-activity pairs, see,
for example, the MAS net in Fig. 8. Hence, to compare MAS nets to CM models
constructed using the AOL naming function, we also measure precision and
recall of MAS nets after rewriting their transition labels to only mention
activity names.

###  5.2 Datasets and Results

To evaluate Agent Miner, we used publicly available real-world Business
Process Intelligence Challenge (BPIC) datasets and assumed they stem from
agent systems. These datasets are widely used to evaluate conventional process
discovery algorithms. We assumed that the resource attributes of events
specify agent instances that triggered them and selected all BPIC datasets
that specify events with resource attributes, leading to nine selected
datasets.111BPIC 2011
(https://doi.org/10.4121/uuid:d9769f3d-0ab0-4fb8-803b-0d1120ffcf54),
BPIC 2012 (https://doi.org/10.4121/uuid:3926db30-f712-4394-aebc-75976070e91f),
BPIC 2013 (https://doi.org/10.4121/uuid:a7ce5c55-03a7-4583-b855-98b86e1a2b07),
BPIC 2014 (https://doi.org/10.4121/uuid:3cfa2260-f5c5-44be-afe1-b70d35288d6d),
BPIC 2015 (https://doi.org/10.4121/uuid:31a308ef-c844-48da-948c-305d167a0ec1),
BPIC 2017 (https://doi.org/10.4121/uuid:5f3067df-f10b-45da-b98b-86ae4c7a310b),
BPIC 2018 (https://doi.org/10.4121/uuid:3301445f-95e8-4ff0-98a4-901f1f204972),
BPIC 2019 (https://doi.org/10.4121/uuid:d06aff4b-79f0-45e6-8ec8-e19730c248f1),
and
BPIC 2020 (https://doi.org/10.4121/uuid:52fb97d4-4588-43c9-9d04-3604d4613b51).

Initially, we used the vff% parameter of 80% and completed the evaluation
pipeline (cf. Fig. 9) for datasets BPIC 2012, 2013, 2015 (Municipality 1),
2017, and 2020 (Travel Permit Data). Due to performance reasons, to process
the other datasets, we lowered the vff% parameter to 10%. Table 2 summarizes
the quality measurements for workflow nets discovered using Inductive Miner
and MAS nets constructed by Agent Miner, using activity-only labeling (AOL) of
transitions in the discovered nets. For each dataset, the table lists size,
recall, and precision values for the models that scored the lowest size and
the greatest precision.

Table 2: Size, recall, and precision of IM nets and MAS nets discovered by
Agent Miner (Inductive Miner as INDA) that rely on activity-only labeling of
observable transitions.

BPIC dataset |  Variant frequency filter (vff%) | Inductive Miner | Agent Miner (MAS nets)
---|---|---|---
lowest size | greatest precision | lowest size | greatest precision
size | recall | prec. | size | recall | prec. | size | recall | prec. | size | recall | prec.
2011 | 10% | 592 | 0.72 | 0.41 | 592 | 0.72 | 0.41 | 466 | 0.92 | 0.35 | 466 | 0.92 | 0.35
2012 | 80% | 420 | 1.00 | 0.06 | 454 | 0.88 | 0.32 | 333 | 0.80 | 0.18 | 333 | 0.80 | 0.18
2013 | 80% | 54 | 0.70 | 0.53 | 105 | 0.78 | 0.62 | 69 | 0.62 | 0.64 | 69 | 0.62 | 0.64
2014 | 10% | 229 | 0.85 | 0.28 | 335 | 0.55 | 0.45 | 231 | 0.78 | 0.36 | 231 | 0.78 | 0.36
2015 | 80% | 199 | 0.95 | 0.35 | 199 | 0.95 | 0.35 | 122 | 0.95 | 0.34 | 122 | 0.95 | 0.34
2017 | 80% | 216 | 0.85 | 0.22 | 241 | 0.91 | 0.26 | 340 | 0.76 | 0.15 | 597 | 0.94 | 0.15
2018 | 10% | 350 | 0.88 | 0.13 | 350 | 0.88 | 0.13 | 535 | 0.85 | 0.09 | 535 | 0.86 | 0.09
2019 | 10% | 270 | 0.61 | 0.24 | 375 | 0.54 | 0.46 | 191 | 0.65 | 0.34 | 191 | 0.65 | 0.34
2020 | 80% | 204 | 0.72 | 0.18 | 220 | 0.69 | 0.25 | 200 | 0.75 | 0.19 | 200 | 0.75 | 0.19

These results confirm that the quality of the MAS nets discovered by Agent
Miner is comparable to the quality of CM models. This observation is
remarkable for at least two reasons. First, as stated above, we had no
background knowledge of whether the datasets stem from agent-driven business
processes. Still, for most datasets, we discovered interesting (in terms of
size, recall, and precision) MAS nets. Second, in addition to high-quality MAS
nets, Agent Miner constructs agent nets and i-nets that can be used as a
starting point for analysis and improvement of ways the process participants
work individually and together.

To support the above conclusions, Fig. 10 shows three types of trade-offs as
two-dimensional Pareto fronts for the nets discovered from the BPIC 2015
dataset. Each point in the plots denotes two quality measurements for one net
generated by the evaluation pipeline. The Pareto fronts indicate the nets with
better quality measurements. The MAS nets discovered by Agent Miner complement
workflow nets discovered by Inductive Miner to result in more saturated Pareto
fronts, that is, fronts with more models of interesting qualities. For the
BPIC15 dataset, MAS nets 1 to 8 discovered by Agent Miner belong to the
recall/precision Pareto front and demonstrate combinations of these quality
measurements better than most CM models discovered by Inductive Miner. MAS net
8 belongs to the size/precision Pareto front and is better in terms of
size/precision than most of the CM models. MAS nets 1 to 5, 7, and 8
demonstrate a better combination of size/recall measurements than almost all
CM models. Overall, the Pareto fronts contain more AM models than CM models,
except in the size/precision case, when the front is represented by one AM
model and one CM model. Similar to the BPIC 2015 results, the Pareto fronts
for the BPIC 2013 dataset, presented in Fig. 11, include points for the MAS
nets. The Pareto front plots for all the datasets are included in the
evaluation results [25].

(a) recall/prec. trade-off

(b) size/prec. trade-off

(c) size/recall trade-off

Figure 10: BPIC 2015 Pareto fronts: IM–AOL nets (“$\bullet$”) and AM–AOL MAS
nets (“$\circ$”).

(a) recall/prec. trade-off

(b) size/prec. trade-off

(c) size/recall trade-off

Figure 11: BPIC 2013 Pareto fronts: IM–AOL nets (“$\bullet$”) and AM–AOL MAS
nets (“$\circ$”).

Agent Miner uses an additional event attribute that specifies the _agent_ that
triggered the event. Hence, it is reasonable to expect it to construct
comparable or better models than conventional algorithms, which do not require
this attribute. To obtain generalizable conclusions, for all the datasets, we
analyzed Pareto fronts and performed paired samples t-tests to establish
whether CM and AM models are of the same qualities, for CM models discovered
using both activity-only labeling (AOL) and agent and activity labeling (AAL).
When measuring precision and recall of AOL and AAL models, both AM and CM
models, events in the logs were identified correspondingly. The null
hypotheses used in the t-tests are such that means of quality measurements for
AM and CM models are equal. Table 4 summarizes comparisons of Pareto fronts
for IM nets and AM nets (IM as INDA), while Table 4 shows the results of the
tests, where results are for size (s), recall (r) and precision (p)
measurements. In Table 4, AM, CM, and AM&CM entries stand for situations when
the Pareto front is composed of AM only, CM only, or AM and CM models,
respectively. In Table 4, for the significance level of 0.05, AM, CM, and
AM&CM entries stand for situations when AM models are significantly better, CM
models are significantly better, and the null hypothesis was not rejected,
respectively. We accept the AM and AM&CM entries as situations when Agent
Miner provides additional value to CM models and highlight them in bold (the
majority of the entries).

Table 3: Comparison of Pareto fronts. AM vs CM fronts (AOL) AM vs CM fronts
(ALL) BPIC data r/p s/p s/r r/p s/p s/r 2011 AM AM&CM AM AM&CM CM AM&CM 2012
AM&CM AM&CM AM&CM AM&CM AM&CM AM 2013 AM&CM AM&CM AM&CM AM&CM AM&CM AM&CM 2014
AM&CM AM&CM CM AM&CM AM&CM AM&CM 2015 AM&CM AM&CM AM&CM AM&CM CM CM 2017 AM&CM
CM CM AM&CM AM&CM CM 2018 CM CM CM CM CM CM 2019 AM&CM AM&CM AM&CM CM CM AM&CM
2020 AM&CM AM&CM AM&CM AM&CM AM&CM AM&CM Table 4: Results of t-tests. AM vs CM
tests (AOL) AM vs CM tests (ALL) BPIC data s r p s r p 2011 CM CM CM CM AM
AM&CM 2012 CM CM AM&CM AM&CM AM AM&CM 2013 CM AM&CM AM&CM AM&CM AM AM&CM 2014
CM CM AM&CM CM AM AM&CM 2015 CM AM&CM AM&CM CM AM AM&CM 2017 CM AM&CM AM&CM
AM&CM AM&CM AM&CM 2018 CM CM AM&CM AM&CM AM&CM AM&CM 2019 CM CM AM CM AM AM&CM
2020 CM CM AM&CM CM AM CM

The results confirm that Agent Miner discovers interesting models to
complement models constructed by conventional discovery algorithms that also
rely on event attributes that specify agents that triggered the events and
describe this information in the constructed models. The results for Split
Miner, CM models, and AM models (Split Miner as INDA), demonstrate similar
conclusions as for IM models presented above.

##  6 Related Work

This section reviews business process modeling with agents, process discovery
addressing agents, and agent system modeling in a broad context.

Agent Modeling in Business Process Management. The relationship between
business process management and agent-based modeling was first explored in the
‘90s by Jennings et al. [7]. In an extension of their work, dubbed ADEPT, the
authors propose to model a business process as a negotiation system between
agents, similar to our interaction nets [8]. However, their model of
negotiating agents is conceptual in its nature, while our model is formal and
executable. Several authors advocate for an agent-based perspective on
business process modeling [6, 22]. In these works, the process is seen as such
composed of interacting agents [7]. However, they do not provide automated
discovery of models from data.

Process Discovery. Traditional process discovery techniques assume a case
perspective when learning models from data while often ignoring additional
perspectives such as resources. Several process discovery techniques extended
traditional methods beyond the case perspective. Rozinat et al. [20] propose a
multi-perspective approach for mining simulation models from event data that
includes the resource perspective. The resulting models are executable and can
be used for performance analysis of the underlying system. However, resources
are considered static entities and not active agents. Van der Aalst et al. [1]
address the modeling of behavior and availability of resources. Yet, resources
play a secondary role, with cases being the dominant perspective that defines
business processes. Klijn et al. [9] develop a technique for querying event
logs to uncover interactions between process entities. Yet, they do not
provide a formal model that can be evaluated for its correctness or goodness-
of-fit. Discovering functional architecture models (FAM) that comprise
interacting modules that internally perform various activities was proposed in
[28]. Unlike the modules in FAMs, our agents are dynamic, decentralized, and
may interact not only with other agents, but also with the environment. Fettke
and Reisig presented an approach to system mining called Heraklit [5].
Heraklit proceeds by constructing distributed runs of participating agents and
then combining them into the overall system, whereas Agent Miner constructs
agent nets and an additional net that explicitly describes agents’
interactions.

Tour et al. [24] have shown that by shifting process mining paradigm from
case-based to agent-based, one can discover less complex models of business
processes. Within this agent paradigm, Nesterov et al. [15] have recently
proposed a process discovery solution. Their algorithm constructs sound
generalized workflow nets that capture agents’ behavior. In contrast to our
approach, Nesterov et al. assume to know the interaction patterns between
agents and aim to implement this given interaction pattern, while we discover
the interactions from the data. Moreover, their algorithm can only discover
pre-defined workflow patterns, while Agent Miner is generic and can model any
pattern its ANDA component can discover. Furthermore, we evaluated Agent Miner
over real-life logs, while Nesterov et al. tested their approach over
synthetically generated data.

Our approach can also be viewed as a log-decomposition process discovery
approach [27]. Such techniques propose to localize the event log to discover
different models tailored to the data in these local logs. In these
techniques, the log decomposition is usually driven by the case attribute,
whereas we propose an agent-driven log decomposition.

Multi-Agent System Modeling. Multi-agent systems (MASs) have been studied
extensively in the past; cf. [4] for a recent overview. Here, we focus on
approaches that model multi-agent systems using Petri nets, our formalism of
choice. A seminal paper by Moldt et al. [13] proposed to use colored Petri
nets (CPNs) to model agent systems. Their model captures three crucial
components in a MAS: communication, independence (between agents), and
intelligence. However, given the richness of CPNs, formal results on
correctness of constructed models were not obtain. Celaya et al. [3] model a
multi-agent system using elementary Petri nets to capture interactions between
agents and use Petri net analysis to ensure the models are deadlock-free. In
our work, we discover MAS models from event data rather than relying on human
expertise.

##  7 Conclusion

This paper presents and evaluates Agent Miner, a divide-and-conquer algorithm
for discovering models of agents and their interactions from event data. The
algorithm “divides” the input data into special parts and then “conquers” the
parts using conventional process discovery algorithms. The constructed agent
and interaction models provide a new, modular perspective on the data,
suitable for analyzing process participants and their interactions. These
artifacts can be integrated into a model that describes process control flow.
Such integrated models are often smaller and represent the event data more
faithfully than corresponding process models constructed using conventional
discovery algorithms. The configuration of Agent Miner used in the evaluation
reported in this paper ensures that the obtained integrated models are safe
and sound.

Agent Miner has several limitations representing areas of interest for future
work. First, the interaction logs are constructed by taking the first event in
each agent trace of each agent. This approach ignores the information on the
duration of agent activities and interactions. For instance, information on
the durations of activities between interactions can be obtained by
considering the first and the last event from agent traces between those
interactions. Consequently, one can apply lifecycle-aware process discovery
[10] or queue mining techniques [21] to infer agent interactions. Next, Agent
Miner associates each event with only one agent. One can relax this limitation
and study the effects of multiple agents sharing the same event. Finally, the
evaluation approach used in this article is limited to the traditional model
quality measures used in process mining. The use of new agent-specific quality
measures for discovered models may highlight additional benefits of Agent
Miner and other agent system mining algorithms in the context of agent-based
business process management.

Acknowledgments. Andrei Tour was supported via an “Australian Government
Research Training Program Scholarship.” Artem Polyvyanyy was in part supported
by the Australian Research Council project DP220101516.

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# How Mock Model Training Enhances
User Perceptions of AI Systems

Amama Mahmood, Gopika Ajaykumar, and Chien-Ming Huang
{amama.mahmood, gopika.ajaykumar<EMAIL_ADDRESS>
Department of Computer Science
Johns Hopkins University
Baltimore, MD 21218, USA

###### Abstract

Artificial Intelligence (AI) is an integral part of our daily technology use
and will likely be a critical component of emerging technologies. However,
negative user preconceptions may hinder adoption of AI-based decision making.
Prior work has highlighted the potential of factors such as transparency and
explainability in improving user perceptions of AI. We further contribute to
work on improving user perceptions of AI by demonstrating that bringing the
user in the loop through mock model training can improve their perceptions of
an AI agent’s capability and their comfort with the possibility of using
technology employing the AI agent.

##  1 Motivation and Background

Fueled by increasingly available user data, growing computing power, and
recent advances in machine learning, Artificial Intelligence (AI) technologies
are transforming our society and daily lives. However, users’ negative
preconceptions of AI may hinder adoption and continued use of AI technologies.
Negative user preconceptions can affect user trust, which is a key factor in
determining acceptance of technology [1]. Inadequate user trust can in turn
lead to misuse (i.e., inappropriate reliance on technology) and disuse (i.e.,
underutilization of technology due to rejection of its capability) [2, 3]. To
enhance user perceptions of AI systems, previous research has investigated AI
transparency, explainability, and interpretability (e.g., [4, 5]), as modern
machine learning methods are largely black boxes [6, 7]. For example, prior
work has explored how visualization may aid user understanding of how machine
learning models work (e.g., [8, 9]). Explanations of these models and
justifications for decisions made by intelligent machines help users
understand their inner workings once they begin interacting with the AI
technologies. In this work, we explore how to improve users’ existing
_preconceptions_ of AI agents prior to any interactions with the agents.

Simulated setups, such as mock trials, mock interviews, and drills, have been
used as low-cost, hands-on tools in early training phases to help people
become accustomed to unfamiliar practices and processes prior to engaging in
them. Similarly, we explore the potential of using mock interactions in which
users label training data for AI models in modulating users’ confidence in AI
agents’ capabilities and their comfort with the possibility of using
technologies employing the AI agents before engaging in real interactions with
the AI agents. We contextualize our exploration within the scenario of
training AI agents for use in autonomous vehicles—a safety-critical domain
that is likely to involve interactions with everyday users. Our findings
indicate that users’ perceptions of AI agents improved through participation
in mock model training, especially when they were able to precisely label
objects that they perceived to be important.

Figure 1: We explore how mock model training involving various data labeling
strategies may affect users’ perceptions of AI agents posed as driving
assistants.

##  2 Methods

###  2.1 Experimental Design, Task, and Conditions

We conducted a within-subjects study that consisted of four experimental
conditions (Figure 1). The study was contextualized within the scenario of
labeling images to train four AI agents to perform driving-related object
identification:

  * •

A1: To train this agent, the participant was presented with a grid of images
that included five positive examples for each of six item categories commonly
encountered during driving: stop sign, speed limit sign, traffic light, car,
bicyclist, and pedestrian. This labeling process is similar to image selection
tasks commonly used in web security checks. It represents low labeling
precision (i.e., the user did not localize the object within the image) and
passive labeling (i.e., the user only labeled items for the requested
categories). It is analogous to binary object _detection_ (i.e., indicating
whether or not a specified item is present).

  * •

A2: The participant followed a similar labeling process to train agent A2 as
done for A1, with the additional task of drawing bounding boxes around the
target item in all images. This process represents high labeling precision and
is analogous to binary object _recognition_..

  * •

A3: In training this agent, the participant was provided with a set of
individual images for labeling. For each image, the user was prompted to list
all items within the image that they considered to be relevant via text. The
user was free to specify as many item categories as they wanted. This method
is analogous to multiple object _detection_.

  * •

A4: Similarly to the training task for A3, the participant was prompted to
draw bounding boxes around all items that they considered to be relevant and
to specify the associated labels via text within each image in the set. This
process represents high labeling precision and is analogous to multiple object
_recognition_.

We also presented a baseline pre-trained agent to the participant at the
beginning of the study. The participant was able to review the images used to
the train the agent. We used this baseline condition as a reference to measure
users’ preconceptions of an AI agent without undergoing mock training.

###  2.2 Measures

We used a range of metrics to measure user perceptions that may affect user
trust in and adoption of AI technologies. For each trained agent, we computed
the _difference_ in comfort, projected capability, and task confidence
relative to the baseline, pre-trained agent (i.e., positive values indicate an
improvement in user perceptions relative to the baseline). We normalized the
data from all questionnaire responses to get values in a 0 – 1 range before
computing the difference.

  * •

Trustworthiness. Trust was measured through a single question asking which AI
agent the participant would trust the most if it was employed in an autonomous
vehicle.

  * •

Comfort. Comfort was measured through a custom scale consisting of six
statements (Cronbach’s $\alpha=0.90$) prompting users to rate how comfortable
they felt towards a self-driving car employing the trained agent (Appendix
A.1).

  * •

Projected Capability. Projected capability was measured through a custom scale
consisting of four statements (Cronbach’s $\alpha=0.87$) prompting
participants to rate how capable they felt the self-driving car employing the
trained agent to be (Appendix A.2).

  * •

Task Confidence. To quantify their perception of the AI agent’s performance,
we asked participants to rate their confidence (0–100%) in its ability to
identify specified items (e.g., stop sign) for a set of 14 images. This set
included two images for each of the six item categories (12 images) and two
images representing “unseen” items (e.g., no-left-turn sign and pedestrian-
crossing sign) that were not included in the object categories used for
training agents A1 and A2.

###  2.3 Procedure

The study consisted of five phases: (1) Introduction and consent. Upon opening
the website, participants were briefed about the study and were informed that
they would be training AI agents to become driving assistants by providing
examples of things (e.g., stop signs and pedestrians) that the agents may
encounter on the road. (2) Reference. The participants review the images used
to train the baseline, pre-trained agent and complete the confidence
assessment and perception survey. (3) Labeling training examples for AI agents
A1-A4. The participants labeled training data for for the four experimental
conditions, which were counterbalanced using a Latin square design. (4)
Confidence assessment and perception survey. Participants were asked to rate
task confidence and questions about trust. They then continued to the next
condition and repeated phase 3–4. (5) Post-study questionnaire. At the end,
participants filled out a post-study questionnaire, which asked which agent
they trusted the most and collected demographics information. The study was
approved by our institutional review board and took approximately 45 minutes
to complete. The participants were compensated with $10 USD upon completion of
the study.

Figure 2: One-way repeated measures ANOVAs were conducted to discover effects
of experimental condition on comfort, projected capability, and task
confidence for seen and unseen cases. Error bars represent 95% confidence
intervals; only significant comparisons ($p<.05$) are highlighted.

##  3 Results

A total of 35 participants (17 females, 17 males, 1 non-binary) were recruited
for this online study via convenience sampling. The participants were aged
between 18 to 35 ($M=25.91,SD=4.78$) and were from a variety of educational
backgrounds, including computer science, engineering and technology, social
work, healthcare, life sciences, business, law, media, public policy, and
education. The participants reported having minimal experience with self-
driving cars ($M=1.40,SD=0.85$), and moderate experience with AI products
($M=3.49,SD=1.72$) and with training AI or machine learning models
($M=3.09,SD=1.69$), using 6-point rating scales with 1 being no experience and
6 being lots of experience. Figure 2 summarizes our main findings. For all
statistical tests reported below, $p<.05$ was considered a significant effect.
We followed Cohen’s guidelines on effect size and considered
$\eta_{p}^{2}=0.01$ a small effect size, $\eta_{p}^{2}=0.06$ a medium effect
size, and $\eta_{p}^{2}=0.14$ a large effect size [10].

A chi-square goodness-of-fit test showed that users did not perceive AI
agents, including the baseline agent, as equally trustworthy,
$\chi^{2}(4,35)=43.60,p<.001,v=0.56$. In particular, A4 (active labeling with
high precision) was considered the most trustworthy agent by the most
participants (51%). A one-way repeated measures analysis of variance (ANOVA)
yielded a significant main effect of experimental condition on comfort,
$F(3,102)=3.75,p=.013,\eta_{p}^{2}=.099$. Post-hoc pairwise comparisons with a
Bonferroni correction revealed that comfort increased with active labeling
with precision, A4 ($M=0.10,SD=0.19$), more than with active labeling without
precision, A3 ($M=0.02,SD=0.22$), $p=.028$. Moreover, a one-way repeated
measures ANOVA yielded a significant main effect of the experimental condition
on projected capability, $F(3,102)=4.69,p=.004,\eta_{p}^{2}=.121$. Post-hoc
pairwise comparison with a Bonferroni correction revealed that active labeling
with precision, A4 ($M=0.09,SD=0.22$), had higher improvement in projected
capability than active labeling without precision, A3 ($M=-0.02,SD=0.22$),
$p=.009$.

A one-way repeated measures ANOVA yielded a significant main effect of the
experimental condition on task confidence for unseen cases,
$F(3,102)=6.76,p<.001,\eta_{p}^{2}=.166$. Post-hoc pairwise comparisons with a
Bonferroni adjustment revealed that active labeling with precision, A4
($M=18.00,SD=32.16$), had higher improvement in task confidence than passive
labeling with precision, A2 ($M=-2.57,SD=16.47$), $p=.004$. While a one-way
repeated measures ANOVA yielded a significant main effect of the experimental
condition on task confidence for seen cases,
$F(3,102)=4.44,p=.006,\eta_{p}^{2}=.115$, we did not observe any significant
differences in pairwise comparisons.

##  4 Discussion

In this study, we observed that users associated higher levels of comfort and
projected capability with the agents for which they labeled training data with
precision. Moreover, for unseen cases, users perceived the agent for which
they were able to freely label objects of interest to be more capable. Our
results suggest that everyday users can perceive the importance of high-
precision training data representative of diverse scenarios in determining AI
task performance. Therefore, involving users in mock training exercises where
they obtain hands-on experience with training data may help them in developing
accurate mental models of how an AI agent operates and in maintaining
appropriate trust levels in the AI agent’s performance before working with or
using the AI technology. Furthermore, our study suggests that greater levels
of user involvement (e.g., precise labeling using bounding boxes) may help
users feel more comfortable with using an AI agent, even in a more safety-
critical scenario. Overall, our study suggests that mock training setups can
serve to help set up appropriate user understanding and improved
preconceptions of how an AI agent will operate prior to real interaction with
the AI agent.

One of the limitations of this study is that we focused on user trust in AI
through a questionnaire item, rather than relying on behavioral (e.g., [11])
or physiological (e.g., [12]) measures. As a result, we may have failed to
accurately or fully capture actual user trust in AI systems. In future
studies, we would like to investigate alternate methods for measuring and
investigating trust so that we can better understand the range of factors that
contribute to user trust in human-AI interaction. We would also like to expand
our study of mock model training in AI systems to encompass new types of
interactions in different domains. In this work, we chose to contextualize our
study within the scenario of training AI agents for self-driving cars, which
is a safety-critical domain that many users may not have direct experience
with. Therefore, we would like to further investigate how our findings would
apply to more general, commonplace scenarios that may involve lower stakes,
such as speech-based interactions with AI agents in smart-speakers.
Furthermore, we investigated the effects of mock model training as explicit
participation in this work, but users may participate in different forms and
in other phases of machine learning, such as algorithm design or error
correction. Furthermore, modern machine learning systems may involve users
without their knowledge or explicit consent, such as recommender systems used
in online services. Future work should investigate if user participation still
positively influences perceptions of AI in cases where users are engaged
outside of AI training or implicitly without their awareness.

## Acknowledgements

This work was supported by the National Science Foundation award #1840088, the
National Science Foundation Graduate Research Fellowship Program under Grant
No. DGE-1746891, and the Nursing/Engineering joint fellowship from the Johns
Hopkins University.

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##  Appendix A Appendix

###  A.1 Comfort – Cronbach’s $\alpha=0.90$

Please rate the following regarding the self-driving car that has employed
Driving Assistant X:

  * •

I would be wary of the self-driving car111Reverse scale items

  * •

I would be afraid that the self-driving car would be harmful††footnotemark:

  * •

I would be confident riding in self-driving car

  * •

I would be comfortable riding in the self-driving car

  * •

I would be relaxed while riding in the self-driving car

  * •

I would be agitated while riding in the self-driving car††footnotemark:

###  A.2 Projected capability – Cronbach’s $\alpha=0.87$

Please rate following regarding the self-driving car that has employed Driving
Assistant X:

  * •

I believe that the self-driving car would NOT be dependable††footnotemark:

  * •

I believe that the self-driving car would be reliable

  * •

I would trust the self-driving car to identify pedestrians, signs and signals,
and obstacles correctly

  * •

I am confident that the self-driving car would comply with traffic rules

###  A.3 Image Sources

The images that we used in the study included public domain images from the
web and images from various datasets, including the Penn-Fudan Database for
Pedestrian Detection and Segmentation [13], the MIO-TCD Dataset [14], the LISA
Traffic Sign Dataset [15], and ImageNet [16]. Figure 1 shows examples of
images used for our study tasks.


On the divisibility of $q$-trinomial coefficients

Ji-Cai Liu

Department of Mathematics, Wenzhou University, Wenzhou 325035, PR China
<EMAIL_ADDRESS>

Abstract. We establish a congruence on sums of central $q$-binomial
coefficients. From this $q$-congruence, we derive the divisibility of the
$q$-trinomial coefficients introduced by Andrews and Baxter.

Keywords: $q$-trinomial coefficients; $q$-congruences; cyclotomic polynomials

MR Subject Classifications: 05A10, 11A07, 11B65, 13A05

##  1 Introduction

We consider the coefficients of the expanded form of this expression:

$\displaystyle(1+x+x^{2})^{n}=\sum_{j=-n}^{n}\left(n\choose j\right)x^{j+n}.$
(1.1)

The coefficients in (1.1) are called trinomial coefficients. Two simple
formulas for the trinomial coefficients (see [18, page 43]) are

$\displaystyle\left(n\choose j\right)=\sum_{k=0}^{n}{n\choose k}{n-k\choose
k+j},$

and

$\displaystyle\left(n\choose j\right)=\sum_{k=0}^{n}(-1)^{k}{n\choose
k}{2n-2k\choose n-j-k}.$

In 1987, Andrews and Baxter [4] introduced six apparently distinct
$q$-analogues of the trinomial coefficients, such as

$\displaystyle\left({n\choose j}\right)_{q}=\sum_{k=0}^{n}q^{k(k+j)}{n\brack
k}{n-k\brack k+j},$ (1.2)

where the $q$-binomial coefficients are defined as

${n\brack k}={n\brack
k}_{q}=\begin{cases}\displaystyle\frac{(1-q^{n})(1-q^{n-1})\cdots(1-q^{n-k+1})}{(1-q)(1-q^{2})\cdots(1-q^{k})},&\text{if
$0\leqslant k\leqslant n$},\\\\[10.0pt] 0,&\text{otherwise.}\end{cases}$

These $q$-trinomial coefficients play an important role in the solution of a
model in statistical mechanics (see [4]). In 1990, Andrews [2] investigated
some interesting properties for the $q$-trinomial coefficients, which leads
him to finite versions of dissections of the Rogers-Ramanujan identities into
even and odd parts.

In 1999, Andrews [3] showed that Babbage’s congruence [6]:

$\displaystyle{2p-1\choose p-1}\equiv 1\pmod{p^{2}}$ (1.3)

possesses the following nice $q$-analogue:

$\displaystyle{2p-1\brack p-1}\equiv q^{\frac{p(p-1)}{2}}\pmod{[p]_{q}^{2}},$
(1.4)

for any odd prime $p$. Notice that Wolstenholme [26] proved that (1.3) holds
modulo $p^{3}$ for any prime $p\geq 5$, which is known as the famous
Wolstenholme’s theorem.

In order to understand (1.4), we recall some necessary notation. For
polynomials $A_{1}(q),A_{2}(q),P(q)\in\mathbb{Z}[q]$, the $q$-congruence

$A_{1}(q)/A_{2}(q)\equiv 0\pmod{P(q)}$

is understood as $A_{1}(q)$ is divisible by $P(q)$ and $A_{2}(q)$ is coprime
with $P(q)$. In general, for rational functions $A(q),B(q)\in\mathbb{Z}(q)$,

$\displaystyle A(q)\equiv B(q)\pmod{P(q)}\Longleftrightarrow A(q)-B(q)\equiv
0\pmod{P(q)}.$

The $q$-integers are defined as $[n]_{q}=(1-q^{n})/(1-q)$ for $n\geq 1$, and
the $n$th cyclotomic polynomial is given by

$\displaystyle\Phi_{n}(q)=\prod_{\begin{subarray}{c}1\leq k\leq n\\\\[3.0pt]
(n,k)=1\end{subarray}}(q-\zeta^{k}),$

where $\zeta$ denotes an $n$th primitive root of unity.

Recently, Straub [19, Theorem 2.2] established a $q$-analogue of
Wolstenholme–Ljunggren congruence:

$\displaystyle{ap\choose bp}\equiv{a\choose b}\pmod{p^{3}},$

for any prime $p\geq 5$, i.e.

$\displaystyle{an\brack bn}\equiv{a\brack b}_{q^{n^{2}}}-(a-b)b{a\choose
b}\frac{n^{2}-1}{24}(q^{n}-1)^{2}\pmod{\Phi_{n}(q)^{3}}.$ (1.5)

The simple case $a=2$ and $b=1$ in (1.5) reads

$\displaystyle{2n\brack n}\equiv
1-q^{2n^{2}}-\frac{n^{2}-1}{12}(q^{n}-1)^{2}\pmod{\Phi_{n}(q)^{3}},$ (1.6)

which is an extension of (1.4) (notice that ${2n-1\brack n}={2n\brack
n}/(1+q^{n})$). We remark that Pan [17, Lemma 3.1] obtained another
interesting $q$-analogue of Wolstenholme–Ljunggren congruence and Zudilin [27]
extended both Straub and Pan’s $q$-congruences.

In the past few years, congruences for $q$-binomial coefficients as well as
basic hypergeometric series attracted many experts’ attention (see, for
example, [7, 8, 9, 10, 11, 12, 14, 15, 22, 24, 25]). Nowdays, there is also an
extensive literature on congruences for trinomial coefficients (see, for
instance, [5, 16, 23, 21]). It is worth mentioning that Gorodetsky [8]
investigated congruences for the $q$-trinomial coefficients $\left({n\choose
0}\right)_{q}$. However, the literature is still sparse on congruences for
$q$-trinomial coefficients.

The first aim of the paper is to prove the following congruence for the
$q$-trinomial coefficients $\left({n\choose 0}\right)_{q}$.

###### Theorem 1.1

For any positive integer $n$, the following congruence holds modulo
$\Phi_{n}(q)^{2}$:

$\displaystyle\left({n\choose
0}\right)_{q}\equiv{\begin{cases}\displaystyle(-1)^{m}(1+q^{m})q^{m(3m-1)/2},&\text{if}~{}~{}n=3m,\\\\[10.0pt]
(-1)^{m}q^{m(3m+1)/2},&\text{if}~{}~{}n=3m+1,\\\\[10.0pt]
(-1)^{m}q^{m(3m-1)/2},&\text{if}~{}~{}n=3m-1.\end{cases}}$

The elegant $q$-congruence (1.6) motivates us to establish the following
congruence for the $q$-trinomial coefficients $({2n\choose n})_{q}$, which
generalizes a conjectural $q$-congruence due to Apagodu and the author [5,
Conjecture 1].

###### Theorem 1.2

For any positive integer $n$, the following congruence holds modulo
$\Phi_{n}(q)^{2}$:

$\displaystyle\left(2n\choose n\right)_{q}\equiv{\begin{cases}\displaystyle
2(-1)^{m}(1+q^{m})q^{m(3m-1)/2}-3m(1-q^{3m}),&\text{if}~{}~{}n=3m,\\\\[10.0pt]
2(-1)^{m}q^{m(3m+1)/2}-(3m+1)(1-q^{3m+1}),&\text{if}~{}~{}n=3m+1,\\\\[10.0pt]
2(-1)^{m}q^{m(3m-1)/2}-(3m-1)(1-q^{3m-1}),&\text{if}~{}~{}n=3m-1.\end{cases}}$

The rest of the paper is organized as follows. In Section 2, we shall mainly
establish an auxiliary congruence on sums of central $q$-binomial
coefficients, which is interesting by itself. The proofs of Theorems 1.1 and
1.2 will be presented in Sections 3 and 4, respectively.

##  2 An auxiliary result

In order to prove Theorems 1.1 and 1.2, we first recall the following two
lemmas.

###### Lemma 2.1

(See [13, Lemma 2.3].) For any non-negative integer $n$, we have

$\displaystyle(1-q^{n})\sum_{k=0}^{\lfloor
n/2\rfloor}\frac{(-1)^{k}q^{k(k-1)/2}}{1-q^{n-k}}{n-k\brack k}$
$\displaystyle={\begin{cases}\displaystyle(-1)^{m}(1+q^{m})q^{m(3m-1)/2},&\text{if}~{}~{}n=3m,\\\\[10.0pt]
(-1)^{m}q^{m(3m+1)/2},&\text{if}~{}~{}n=3m+1,\\\\[10.0pt]
(-1)^{m}q^{m(3m-1)/2},&\text{if}~{}~{}n=3m-1,\end{cases}}$ (2.1)

where $\lfloor x\rfloor$ denotes the integral part of real $x$.

###### Lemma 2.2

(See [22, Lemma 3.3].) For $k=1,\cdots,n-1$, we have

$\displaystyle{2k-1\brack k}\equiv(-1)^{k}q^{k(3k-1)/2}{n-k\brack
k}\pmod{\Phi_{n}(q)}.$ (2.2)

We also require the following congruence on sums of the central $q$-binomial
coefficients ${2k\brack k}$, which is interesting by itself.

###### Proposition 2.3

For any positive integer $n$, we have

$\displaystyle\sum_{k=1}^{\lfloor
n/2\rfloor}\frac{q^{-k(k-1)}}{[2k]_{q}}{2k\brack
k}\equiv\frac{(1-q)(1-\mathcal{R}_{n}(q))}{1-q^{n}}\pmod{\Phi_{n}(q)},$ (2.3)

where $\mathcal{R}_{n}(q)$ denotes the right-hand side of (2.1).

Remark. Sun [20, Theorem 1.1] proved that for any prime $p\geq 5$,

$\displaystyle\sum_{k=1}^{(p-1)/2}\frac{{2k\choose
k}}{k}\equiv(-1)^{(p+1)/2}\frac{8p}{3}E_{p-3}\pmod{p^{2}},$ (2.4)

where $E_{0},E_{1},E_{2},\cdots$ are Euler numbers. Notice that (2.3) is a
$q$-analogue of (2.4) (the modulo $p$ version).

Proof of (2.3). By (2.2) and $q^{n}\equiv 1\pmod{\Phi_{n}(q)}$, we have

$\displaystyle\sum_{k=1}^{\lfloor
n/2\rfloor}\frac{q^{-k(k-1)}}{[2k]_{q}}{2k\brack k}$
$\displaystyle=(1-q)\sum_{k=1}^{\lfloor
n/2\rfloor}\frac{q^{-k(k-1)}}{1-q^{2k}}{2k\brack k}$
$\displaystyle=(1-q)\sum_{k=1}^{\lfloor
n/2\rfloor}\frac{q^{-k(k-1)}}{1-q^{k}}{2k-1\brack k}$
$\displaystyle\equiv(1-q)\sum_{k=1}^{\lfloor
n/2\rfloor}\frac{(-1)^{k}q^{k(k+1)/2}}{1-q^{k}}{n-k\brack k}$
$\displaystyle\equiv(q-1)\sum_{k=1}^{\lfloor
n/2\rfloor}\frac{(-1)^{k}q^{k(k-1)/2}}{1-q^{n-k}}{n-k\brack
k}\pmod{\Phi_{n}(q)}.$ (2.5)

We can rewrite (2.1) as

$\displaystyle\sum_{k=1}^{\lfloor
n/2\rfloor}\frac{(-1)^{k}q^{k(k-1)/2}}{1-q^{n-k}}{n-k\brack
k}=\frac{\mathcal{R}_{n}(q)-1}{1-q^{n}}.$ (2.6)

Substituting (2.6) into the right-hand side of (2.5) gives

$\displaystyle\sum_{k=1}^{\lfloor
n/2\rfloor}\frac{q^{-k(k-1)}}{[2k]_{q}}{2k\brack
k}\equiv\frac{(1-q)(1-\mathcal{R}_{n}(q))}{1-q^{n}}\pmod{\Phi_{n}(q)},$ (2.7)

as desired. $\square$

##  3 Proof of Theorem 1.1

For $1\leq k\leq\lfloor n/2\rfloor$, we have

$\displaystyle{n\brack k}$
$\displaystyle=\frac{(1-q^{n})(1-q^{n-1})\dots(1-q^{n-k+1})}{(1-q)\cdots(1-q^{k})}$
$\displaystyle\equiv\frac{(1-q^{n})(1-q^{-1})\dots(1-q^{-k+1})}{(1-q)\cdots(1-q^{k})}$
$\displaystyle=\frac{(-1)^{k-1}q^{-k(k-1)/2}(1-q^{n})}{1-q^{k}}$
$\displaystyle\equiv\frac{(-1)^{k}q^{-k(k+1)/2}(1-q^{n})}{1-q^{n-k}}\pmod{\Phi_{n}(q)^{2}},$

where we have used the fact that $1-q^{n}\equiv 0\pmod{\Phi_{n}(q)}$. It
follows from (1.2), (2.6) and the above that

$\displaystyle\left({n\choose 0}\right)_{q}$
$\displaystyle=\sum_{k=0}^{\lfloor n/2\rfloor}q^{k^{2}}{n\brack k}{n-k\brack
k}$ $\displaystyle=1+\sum_{k=1}^{\lfloor n/2\rfloor}q^{k^{2}}{n\brack
k}{n-k\brack k}$ $\displaystyle\equiv 1+(1-q^{n})\sum_{k=1}^{\lfloor
n/2\rfloor}\frac{(-1)^{k}q^{k(k-1)/2}}{1-q^{n-k}}{n-k\brack
k}\pmod{\Phi_{n}(q)^{2}}$ $\displaystyle=\mathcal{R}_{n}(q).$

This completes the proof of Theorem 1.1.

##  4 Proof of Theorem 1.2

Note that for $1\leq k\leq\lfloor n/2\rfloor$, we have

$\displaystyle{2n\brack k}$
$\displaystyle=\frac{(1-q^{2n})(1-q^{2n-1})\cdots(1-q^{2n+1-k})}{(1-q)(1-q^{2})\cdots(1-q^{k})}$
$\displaystyle\equiv\frac{2(1-q^{n})(1-q^{-1})\cdots(1-q^{-k+1})}{(1-q)(1-q^{2})\cdots(1-q^{k})}\pmod{\Phi_{n}(q)^{2}}$
$\displaystyle=\frac{2(-1)^{k-1}(1-q^{n})q^{-k(k-1)/2}}{1-q^{k}},$ (4.1)

and

$\displaystyle{2n-k\brack n+k}$ $\displaystyle={2n\brack
n}\frac{(1-q^{n-2k+1})\cdots(1-q^{n})}{(1-q^{n+1})\cdots(1-q^{n+k})(1-q^{2n-k+1})\cdots(1-q^{2n})}$
$\displaystyle\equiv\frac{(-1)^{k}q^{-k(3k-1)/2}}{2}{2n\brack n}{2k-1\brack
k}\pmod{\Phi_{n}(q)},$ (4.2)

where we have utilized the fact $q^{n}\equiv 1\pmod{\Phi_{n}(q)}$.

It follows from (1.2), (4.1) and (4.2) that

$\displaystyle\left(2n\choose n\right)_{q}$ $\displaystyle=\sum_{k=0}^{\lfloor
n/2\rfloor}q^{k(k+n)}{2n\brack k}{2n-k\brack n+k}$ $\displaystyle={2n\brack
n}+\sum_{k=1}^{\lfloor n/2\rfloor}q^{k(k+n)}{2n\brack k}{2n-k\brack n+k}$
$\displaystyle\equiv{2n\brack n}-(1-q^{n}){2n\brack n}\sum_{k=1}^{\lfloor
n/2\rfloor}\frac{q^{-k(k-1)}}{1-q^{k}}{2k-1\brack k}\pmod{\Phi_{n}(q)^{2}}$
$\displaystyle={2n\brack n}-\frac{1-q^{n}}{1-q}{2n\brack n}\sum_{k=1}^{\lfloor
n/2\rfloor}\frac{q^{-k(k-1)}}{[2k]_{q}}{2k\brack k}.$ (4.3)

From (1.6), we deduce that

$\displaystyle{2n\brack n}$ $\displaystyle\equiv 1+q^{n^{2}}$
$\displaystyle=2-(1-q^{n})(1+q^{n}+q^{2n}+\cdots+q^{(n-1)n})$
$\displaystyle\equiv 2-n(1-q^{n})\pmod{\Phi_{n}(q)^{2}}.$ (4.4)

Finally, substituting (2.3) and (4.4) into the right-hand side of (4.3) gives

$\displaystyle\left(2n\choose n\right)_{q}$ $\displaystyle\equiv
2\mathcal{R}_{n}(q)-n(1-q^{n})\pmod{\Phi_{n}(q)^{2}},$

as desired.

Acknowledgments. The author would like to thank Ofir Gorodetsky for
discussions on $q$-trinomial coefficients and useful suggestions regarding the
paper. This work was supported by the National Natural Science Foundation of
China (grant 12171370).

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