=Paper=
{{Paper
|id=Vol-3914/short73
|storemode=property
|title=A Probabilistic Semantics for Process Mining (Short paper) 
|pdfUrl=https://ceur-ws.org/Vol-3914/short73.pdf
|volume=Vol-3914
|authors=Michela Vespa
|dblpUrl=https://dblp.org/rec/conf/aiia/Vespa24
}}
==A Probabilistic Semantics for Process Mining (Short paper) ==
<pdf width="1500px">https://ceur-ws.org/Vol-3914/short73.pdf</pdf>
<pre>
                                A probabilistic semantics for process mining
                                Michela Vespa1
                                1
                                    Dipartimento di Ingegneria, Università di Ferrara, Via Saragat 1, Ferrara, Italy


                                               Abstract
                                               In declarative Process Mining (PM), accounting for uncertainty is essential to accurately model real-
                                               world business processes. Up to now, most traditional approaches have overlooked the possibility of
                                               integrating probability into process management. Starting from our previous works on this topic, we
                                               present an extension to our semantics that underlies a probabilistic declarative framework for PM, in
                                               such a way that we can manage uncertainty at multiple levels, from individual events to entire logs, by
                                               assigning probabilities reflecting a degree of belief or confidence in them. This framework is based on
                                               the Distribution Semantics of Probabilistic Logic Programming.

                                               Keywords
                                               Process Mining, Declarative language, Distribution Semantics, Probability theory




                                1. Introduction
                                Ongoing research in Process Mining (PM) is increasingly focusing on the role of uncertainty
                                in business process management. Uncertainty in PM can manifest in multiple aspects of a
                                process, ranging from process models to process data, i.e. events and event attributes, traces,
                                and logs. For instance, real-world event logs may contain incomplete or noisy data, where
                                some events/traces are missing or misrecorded. Various approaches have been explored to
                                address uncertainty in procedural PM settings, dealing with structured, sequential process
                                models, typically represented as flow-based notations like Petri nets or BPMN diagrams. In
                                recent years, significant research built on foundational work by Pegoraro and Van der Aalst has
                                been devoted to address this challenge with respect to event data [1]. This has been achieved
                                through a framework designed to represent the control-flow dimension of uncertain events
                                as Petri nets, involving stochastic process modeling techniques like stochastic Petri nets [2],
                                behavioral nets [3, 4], and trace alignment [5, 6]. This research highlighted the complexities
                                of managing uncertain event data within procedural models, focusing on strong uncertainty
                                (unknown probability distributions for attribute values) at the attribute level of events.
                                   However, a distinct approach can be taken when dealing with uncertainty in declarative PM,
                                which focuses only on the constraints between activity sequences, rather than outlining exact
                                workflows [7, 8]. For example, [9] introduced the notion of probabilistic process constraints, by
                                associating probabilities to Declare constraints.
                                   Starting from our previous work based on probabilistic declarative process specifications [10]

                                Doctoral Consortium at the 23rd International Conference of the Italian Association for Artificial Intelligence Bolzano,
                                Italy, November 25-28, 2024
                                $ michela.vespa@unife.it (M. Vespa)
                                 0009-0004-4350-8151 (M. Vespa)
                                             © 2024 Copyright for this paper by its authors. Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).




CEUR
                  ceur-ws.org
Workshop      ISSN 1613-0073
Proceedings
and probabilistic events [11], here we extend the underlying semantics in order to comprehen-
sively handle uncertainty at all levels of process data, from traces to entire logs. This semantics
is inspired by the Distribution Semantics (DS) [12] of Probabilistic Logic Programming (PLP)
and handles uncertainty by expressing probabilities as a degree of belief (taking inspiration
from [13],[14]) in traces and logs.
   In [11], we treat uncertain events in a process trace by annotating them with a probability
expressing the user’s confidence in that event(s) happening. Here, we complete this framework
by considering probabilities attached to traces as a whole, which results in probabilistic logs.
For instance, there might be cases where a maintenance task, composed of different phases
(the trace’s events), is not logged correctly due to human error or system issues. If a technician
recalls performing it but later finds no documentation of this in the system, he might estimate,
based on his memory, interactions with colleagues, and standard operating procedures, that
there is a 95% probability the inspection was completed as required. This would generate two
possible logs, one with the trace included, with 0.95 probability, and the other log without the
trace, but much less probable (0.05 probability). To the best of our knowledge, previous efforts
in procedural PM have addressed either event-based or traced-based uncertainty separately,
while our approach is new in handling probabilities from events to logs, offering an integrated
semantics to manage and interpret uncertainty in process data at all levels of granularity.


2. Background: Distribution Semantics
PLP, notably through the Distribution Semantics, handles uncertain information by allowing
probabilities in logic programs, which define probability distributions over a set of possible
normal logic programs called "worlds" . In the following, the DS will be described with reference
to the language of LPADs (Logic Programs with Annotated Disjunctions) [15], even if it underlies
many other languages. A detailed survey of the DS in PLP can be found in [16]. In LPADs each
program clause has a disjunction in the head with each atom annotated by a probability. When
the clause body holds true, only one head atom is selected together with its probability.
   An annotated disjunctive clause 𝐶𝑖 is of the form ℎ𝑖1 : 𝑝𝑖1 ; . . . ; ℎ𝑖𝑛𝑖 : 𝑝𝑖𝑛𝑖 : − 𝑏𝑖1 , . . . , 𝑏𝑖𝑚𝑖 ,
where ℎ𝑖1 ,∑︀  . . . , ℎ𝑖𝑛𝑖 are logical atoms and {𝑝𝑖1 , . . . , 𝑝𝑖𝑛𝑖 } are real numbers
                                                                                    ∑︀ 𝑖 in the interval [0, 1]
                  𝑛𝑖
such that 𝑘=1           𝑝𝑖𝑘 ≤ 1; 𝑏𝑖1 , . . . , 𝑏𝑖𝑚𝑖 is indicated with 𝑏𝑜𝑑𝑦(𝐶𝑖 ). If 𝑛𝑘=1   𝑝𝑖𝑘 < 1, the head
implicitly contains∑︀      an extra atom 𝑛𝑢𝑙𝑙 that does not appear in the body of any clause and whose
annotation is 1 − 𝑛𝑘=1        𝑖
                                 𝑝𝑖𝑘 . We denote by 𝑔𝑟𝑜𝑢𝑛𝑑(𝑇 ) the grounding of an LPAD 𝑇 .
   An atomic choice [17] is a triple (𝐶𝑖 , 𝜃𝑗 , 𝑘) where 𝐶𝑖 ∈ 𝑇 , 𝜃𝑗 is a substitution that grounds
𝐶𝑖 and 𝑘 ∈ {1, . . . , 𝑛𝑖 } identifies one of the head atoms. (𝐶𝑖 , 𝜃𝑗 , 𝑘) means that, for the ground
clause 𝐶𝑖 𝜃𝑗 , the head ℎ𝑖𝑘 was chosen. A set of atomic choices 𝜅 is consistent if only one head is
selected from the same ground clause; we assume independence between the different choices. A
composite choice 𝜅 is a consistent set of atomic choices [17]. The probability 𝑃 (𝜅) of a composite
choice    𝜅 is the product of the probabilities of the independent atomic choices, i.e. 𝑃 (𝜅) =
   (𝐶𝑖 ,𝜃𝑗 ,𝑘)∈𝜅 𝑝𝑖𝑘 . A selection 𝜎 is a composite choice that, for each clause 𝐶𝑖 𝜃𝑗 in 𝑔𝑟𝑜𝑢𝑛𝑑(𝑇 ),
∏︀

contains an atomic choice (𝐶𝑖 , 𝜃𝑗 , 𝑘). Let us indicate with 𝑆𝑇 the set of all selections. A selection
𝜎 identifies a normal logic program 𝑤𝜎 defined as 𝑤𝜎 = {(ℎ𝑖𝑘 ← 𝑏𝑜𝑑𝑦(𝐶𝑖 ))𝜃𝑗 |(𝐶𝑖 , 𝜃𝑗 , 𝑘) ∈ 𝜎}.
𝑤𝜎 is called a (possible) world of 𝑇 . Since selections are composite choices, we can assign a
probability to worlds: 𝑃 (𝑤𝜎 ) = 𝑃 (𝜎) = (𝐶𝑖 ,𝜃𝑗 ,𝑘)∈𝜎 𝑝𝑖𝑘 .
                                            ∏︀

   We∑︀denote the set of all worlds of 𝑇 by 𝑊𝑇 . 𝑃 (𝑊𝑇 ) is a probability distribution over worlds,
i.e., 𝑤∈𝑊𝑇 𝑃 (𝑤) = 1. A composite choice 𝜅 identifies a set of worlds 𝑤𝜅 = {𝑤𝜎 |𝜎 ⋃︀    ∈ 𝑆𝑇 , 𝜎 ⊇
𝜅}. The set of possible worlds associated to a set of composite choices 𝐾 is 𝑊𝐾 = 𝜅∈𝐾 𝑤𝜅 .
Example 1. Consider the following LPAD T encoding the outcome of tossing a coin, which may be
either fair or biased:
            (𝐶1 )   ℎ𝑒𝑎𝑑𝑠(𝐶𝑜𝑖𝑛) : 0.5; 𝑡𝑎𝑖𝑙𝑠(𝐶𝑜𝑖𝑛) : 0.5 : −𝑡𝑜𝑠𝑠(𝐶𝑜𝑖𝑛), \+𝑏𝑖𝑎𝑠𝑒𝑑(𝐶𝑜𝑖𝑛).
            (𝐶2 )   ℎ𝑒𝑎𝑑𝑠(𝐶𝑜𝑖𝑛) : 0.6; 𝑡𝑎𝑖𝑙𝑠(𝐶𝑜𝑖𝑛) : 0.4 : −𝑡𝑜𝑠𝑠(𝐶𝑜𝑖𝑛), 𝑏𝑖𝑎𝑠𝑒𝑑(𝐶𝑜𝑖𝑛).
            (𝐶3 )   𝑓 𝑎𝑖𝑟(𝐶𝑜𝑖𝑛) : 0.9; 𝑏𝑖𝑎𝑠𝑒𝑑(𝐶𝑜𝑖𝑛) : 0.1.
            (𝐶4 )   𝑡𝑜𝑠𝑠(𝑐𝑜𝑖𝑛).

 If a coin is tossed, the probability of it landing heads or tails is influenced by whether it is fair or
biased: if the coin is fair ( \+biased), then it has an equal chance of landing heads or tails (0.5). If
the coin is biased, then it is more likely to land heads with a probability of 0.6, and tails with a
probability of 0.4. 𝐶3 states that the coin is fair with a probability of 0.9 or biased with a probability
of 0.1. 𝐶4 asserts that a coin is indeed tossed. Since we’re only considering 1 coin, each rule has 1
grounding 𝜃1 = {𝐶𝑜𝑖𝑛/𝑐𝑜𝑖𝑛}. Here, 𝑇 would have 2 × 2 × 2 = 8 possible worlds.

   Given a goal G, its probability 𝑃 (𝐺) can∑︀    be defined by marginalizing       the joint proba-
bility of the goal and the worlds: 𝑃 (𝐺) =
                                                                          ∑︀
                                                    𝑤∈𝑊𝑇 𝑃 (𝐺, 𝑤) =          𝑤∈𝑊𝑇 𝑃 (𝐺|𝑤)𝑃 (𝑤) =
                      The  probability  of a goal    given  a world      is    (𝐺|𝑤) = 1 if 𝑤 |= 𝐺
∑︀
   𝑤∈𝑊𝑇 :𝑤|=𝐺  𝑃 (𝑤).                              𝐺                 𝑤       𝑃
and 0 otherwise. 𝑃 (𝑤) = 𝑃 (𝜎), i.e. is the product of the annotations 𝑝𝑖𝑘 of the head atoms
selected in 𝜎. Therefore, the probability of G can be computed by summing the probability of
the worlds where the goal is true. In practice, given a goal to solve, it is unfeasible to enumerate
all the worlds where 𝐺 is entailed. Inference algorithms, instead, find explanations for a goal: a
composite choice 𝜅 is an explanation for 𝐺 if 𝐺 is entailed by every world of 𝑤𝜅 .
Example 2. (Ex.1 cont.) To determine the overall probabilities of the coin landing on heads or
tails, we need to ask the probability of the 2 goals heads and tails. Each goal is true in 4 worlds out
of the 8:
  𝑃 (heads) = (0.5 × 0.6 × 0.9) + (0.5 × 0.6 × 0.1) + (0.5 × 0.4 × 0.9) + (0.5 × 0.6 × 0.1) = 0.51

𝑃 (tails) = 1−𝑃 (heads) = (0.5×0.4×0.1)+(0.5×0.6×0.9)+(0.5×0.4×0.1)+(0.5×0.4×0.9) = 0.49


3. Probabilistic events, traces and logs
In this Section, we present our semantic framework for addressing uncertainty across events,
traces, and logs, building upon the DS. This approach acknowledges that in certain domains,
complete observation of a process instance may not be feasible, leading to uncertainty related to
events, traces and even logs. We can assign a probability to events [11], obtaining probabilistic
traces, or to traces as a whole, obtaining probabilistic logs. Probability always reflects the degree
of belief or confidence of the user in the happening of the event or the trace.
   With a finite alphabet of symbols Σ, representing activity names or descriptors, we can
define:
Definition 1 (Trace and Log). A Trace is a finite, ordered sequence of symbols over Σ, denoted as
𝑡 ∈ Σ* , where Σ* represents the infinite set of all possible finite sequences (sentences) 𝑡. Syntactically,
a trace is expressed as 𝑡 = ⟨e1 , e2 , . . . , e𝑛 ⟩, e𝑖 ∈ Σ, where 𝑛 is the length of the trace, and e𝑖 (for
𝑖 ∈ 1 . . . 𝑛) represents the 𝑖-th event in the trace. A log ℒ consists of a finite set of such traces.
Definition 2 (Probabilistic Event [11]). A Probabilistic Event is a couple Prob:EventDescription,
where EventDescription is a symbol describing an event (EventDescription ∈ Σ), while Prob
∈ [0, 1] is the probability that the event happened. A probability value of 1 means the event
happened, and we will refer to it as "certain".
   For example, the probabilistic event 0.8:early_mobilization in a trace of a medical log describes
the event of a patient’s early mobilization after surgery with probability 0.8, reflecting our
degree of belief associated with the event’s occurrence. In [11], we defined a trace where at
least one event is probabilistic as a probabilistic trace.
   Now we extend our framework to probabilistic logs, driven by real-world scenarios where
traces may not be accurately captured due to factors like software or hardware malfunctions and
human error. As a consequence, there is no certainty of the happening of some process instance.
However, due to the domain’s characteristics, it may be the case that the whole instance (trace)
happened with a certain probability.
Definition 3 (Probabilistic Log). A probabilistic log ℒ𝑝 is a log where at least one trace 𝑡𝑖 is
annotated with a probability 𝑝𝑖 . A probability value of 1 means the trace certainly happened and
the value will be omitted.
  Instead of considering the happening of the single events in a trace, as in Def. 2, here we
are inquiring about the certainty of the process instance as a whole: it certainly happened or
maybe it happened with a degree of confidence.
Example 3. In hospitals, patients are first admitted to the emergency department following an
initial screening known as triage. In exceptional situations, such as during serious emergencies, the
triage process might be performed but not recorded in the log. The probabilistic log:

                             ℒ𝑝 = {       𝑡1 ,    0.9 : 𝑡2 ,     𝑡3 ,   0.7 : 𝑡4      }

describes a scenario in which the process instances 𝑡1 and 𝑡3 were observed and recorded, while
𝑡2 was not observed but there is a high probability (0.9) that it happened. Similarly, 𝑡4 was not
observed but there is a fair probability (0.7) that it happened.
  We propose a straightforward extension of Sato’s distribution semantics, as done in [11], to
the case of probabilistic logs.
Definition 4 (Selection 𝜎 over a probabilistic log ℒ𝑝 ). A Selection 𝜎(ℒ𝑝 ) is defined as a composite
choice containing an atomic choice (t𝑖 , 𝑘) for each trace 𝑡𝑖 ∈ ℒ𝑝 . A selection 𝜎(ℒ𝑝 ) identifies a
world 𝑤𝜎 in this way: 𝑤𝜎 = {𝑡𝑖 |(𝑡𝑖 , 1) ∈ 𝜎}.
Example 4. Given the probabilistic log ℒ𝑝 described in Example 3, four selections are possible,
generating four corresponding worlds:
           𝜎1 (ℒ𝑝 ) = {   (𝑡2 , 1),   (𝑡4 , 1)}                𝑤𝜎1 (ℒ𝑝 ) = {   𝑡1 ,       𝑡2 ,   𝑡3 ,   𝑡4   }
            𝜎2 (ℒ𝑝 ) = {    (𝑡2 , 1),   (𝑡4 , 0)}               𝑤𝜎2 (ℒ𝑝 ) = {      𝑡1 ,   𝑡2 ,   𝑡3   }
            𝜎3 (ℒ𝑝 ) = {    (𝑡2 , 0),   (𝑡4 , 1)}               𝑤𝜎3 (ℒ𝑝 ) = {      𝑡1 ,   𝑡3 ,   𝑡4   }
            𝜎4 (ℒ𝑝 ) = {    (𝑡2 , 0),   (𝑡4 , 0)}               𝑤𝜎4 (ℒ𝑝 ) = {      𝑡1 ,   𝑡3     }

   Note that traces 𝑡1 and 𝑡3 always appear in the generated worlds as they are certain. A
possible world 𝑤𝜎𝑖 (ℒ𝑝 ) represents a possible log, determined by the presence or absence of
individual uncertain traces. A selection over such a log determines which traces are considered
to be part of a possible realization of the log.
Definition 5 (Probability of a Selection 𝜎(ℒ𝑝 )). The probability of a selection 𝜎(ℒ𝑝 ) over a
probabilistic log ℒ𝑝 is defined as:
                                        ∏︁           ∏︁
                           𝑃 (𝜎(ℒ𝑝 )) =        𝑝𝑖           (1 − 𝑝𝑖 )
                                               (𝑡𝑖 ,1)∈𝜎(ℒ𝑝 )     (𝑡𝑖 ,0)∈𝜎(ℒ𝑝 )

The probability of a selection corresponds to the probability of a possible log (i.e., a possible world),
obtained by multiplying the probabilities associated to each alternative (presence or absence of a
trace) as these
              ∑︀are considered independent of each other. This gives a probability distribution over
the logs, i.e. 𝑖 𝑃 (𝜎𝑖 (ℒ𝑝 )) = 𝑃 (𝑤𝜎𝑖 (ℒ𝑝 )) = 1.
Example 5 (Ex. 4 cont.). The probabilities of the four selections 𝜎𝑖 (ℒ𝑝 ) are:
   𝑃 (𝜎1 (ℒ𝑝 )) = 𝑃 (𝑤𝜎1 (ℒ𝑝 )) = 0.9 × 0.7 = 0.63 𝑃 (𝜎3 (ℒ𝑝 )) = 𝑃 (𝑤𝜎3 (ℒ𝑝 )) = 0.1 × 0.7 = 0.07
   𝑃 (𝜎2 (ℒ𝑝 )) = 𝑃 (𝑤𝜎2 (ℒ𝑝 )) = 0.9 × 0.3 = 0.27 𝑃 (𝜎4 (ℒ𝑝 )) = 𝑃 (𝑤𝜎4 (ℒ𝑝 )) = 0.1 × 0.3 = 0.03
 Note that 0.63+0.27+0.07+0.03=1. The 4 realizations of the log, with very different probabilities
in this case, highlight the fact the very high (low) values of confidence in the happening of some
traces may generate logs with much higher (lower) confidence than others. This means that a user
can rank the probabilistic realizations of the logs from the one with highest confidence (the most
probable) to the one with the lowest confidence.


4. Conclusions and Future Work
In this work, we presented a unified framework inspired by the distribution semantics of
Probabilistic Logic Programming, which integrates our recently proposed probabilistic semantics
for process events to handle uncertainty at various granularity levels: not only events, but also
traces and logs. In the future, we plan to extend this framework to include proof procedures for
conformance checking for probabilistic logs.


5. Acknowledgments


Research funded by the Italian Ministerial grant PRIN 2022 “Probabilistic Declarative Process Mining (PRODE)”, n.
20224C9HXA - CUP F53D23004240006, funded by European Union – Next Generation EU. Research funded by the
Italian Ministry of University and Research through PNRR - M4C2 - Investimento 1.3 (Decreto Direttoriale MUR
n. 341 del 15/03/2022), Partenariato Esteso PE00000013 - "FAIR - Future Artificial Intelligence Research" - Spoke 8
"Pervasive AI" - CUP J33C22002830006, funded by the European Union under the NextGeneration EU programme".
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