Process Reasoning and Mining with Uncertainty

Process Reasoning
and Mining
Marco Montali
Free University of Bozen Bolzano

montali@inf.unibz.it
with
Uncertainty
Credits to

Anti Alman, Sander J. J. Leemans,  
Fabrizio M. Maggi, Rafael Peñaloza
AdONE

16/05/2022

• Fix a
fi
nite alphabet of atomic tasks
• Process execution:
fi
nite sequence of events on a speci
fi
c case

• Event: atomic task in some position (data/time abstraction)

• Execution trace: element from
Σ
Σ*
Process executions
Setting the stage…
2

Process, trace
Setting the stage…
3
Σ*

Types of process representations
4

Σ* Σ*
Σ* Σ*
Where is uncertainty?
5
trace process

6
Plan today
1.Infusing declarative/imperative process
models with uncertainty
2.Show how we can reason on these
models and their traces
3.Use these techniques for stochastic
process mining (discovery, conformance
checking, monitoring)

7
Declarative process
specifications

Constraints declaratively predicating on the execution
of activities over time.

Examples: Declare [Pesic et al, EDOC07; _ et al,
TWEB11] and DCR Graphs [Slaats et al, BPM13].

Declare uses LTL over
fi
nite traces (LTLf) and
fi
nite-
state automata to provide support for the whole
lifecycle: consistency, enacment, monitoring, discovery.
Temporal process constraints
8

A front-end for linear temporal logic over
fi
nite traces
The Declare framework
9
Crisp semantics of constraints: an execution trace
conforms to the model if it satis
fi
es every constraint in
the model.
close
order
1..1
accept
refuse

• Best practices: constraints that must hold in the majority, but
not necessarily all, cases.

90% of the orders are shipped via truck.
• Outlier behaviors: constraints that only apply to very few, but
still conforming, cases.

Only 1% of the orders are canceled after being paid.
• Constraints involving external parties: contain uncontrollable
activities for which only partial guarantees can be given.

In 8 cases out of 10, the customer accepts the order and also
pays for it.
Some examples
Uncertainty is pervasive
10

Crisp and uncertain constraints
ProbDeclare
11
close
order
1..1
accept
refuse
{0.8}
{0.3}
{0.9}

ProbDeclare
12
close
order
1..1
accept
refuse
{0.8}
{0.3}
{0.9}

ProbDeclare
13
close
order
1..1
accept
refuse
Crisp!
Each trace in
the log
contains
exactly one
close order
{0.8}
{0.3}
{0.9}

ProbDeclare
14
close
order
1..1
accept
refuse
{0.8}
{0.3}
{0.9}
Uncertain!
90% traces
are so that an
order is not
accepted and
refused.

In 10% traces
the seller
changes their
mind

ProbDeclare
15
close
order
1..1
accept
refuse
Uncertain!
90% traces
are so that an
order is not
accepted and
refused.

In 10% traces
the seller
changes their
mind
{0.8}
{0.3}
{0.9}

• A stochastic language over is a function such
that

•
fi
nite if
fi
nitely many traces get a non-zero probability

• A log can be seen as a
fi
nite stochastic language
Σ ρ : Σ* → [0,1]
∑
τ∈Σ*
ρ(τ) = 1
ProbDeclare is interpreted over
fi
nite stochastic languages
Formally…
16

A ProbDeclare constraint over is a triple , where:

• the process condition is an LTLf formula over

• the probability operator is one of

• the probability reference value p is a rational value in [0,1]

A stochastic language satis
fi
es a ProbDeclare constraint
if

Σ ⟨φ, ⋈, p⟩
φ Σ
⋈ { = , ≠ , ≤ , ≥ , < , > }
ρ
⟨φ, ⋈, p⟩
∑
τ∈Σ*,τ⊧φ
ρ(τ) ⋈ p
Semantics of a ProbDeclare constraint
Formally…
17

A stochastic language satis
fi
es a ProbDeclare speci
fi
cation if:

• For every crisp constraint and every trace with non-zero
probability, we have that

• For every probabilistic constraint we have
ρ
φ τ ∈ Σ*
τ ⊧ φ
⟨φ, ⋈, p⟩
∑
τ∈Σ*,τ⊧φ
ρ(τ) ⋈ p
From one to many constraints
18
Key challenge: interplay of multiple constraints

A constraint scenario picks which probabilistic constraints must
hold, and which are violated (i.e., their negated version must hold).

It denotes a “process variant”.

All in all: up to 2n scenarios, denoting di
ff
erent “process variants”.
Constraint scenarios
Dealing with “n” probabilistic constraints
19
close
order
1..1
accept
refuse
{0.8}
{0.3}
{0.9}
1
2
3

A constraint scenario picks which probabilistic constraints must
hold, and which are violated (i.e., their negated version must hold).

It denotes a “process variant”.

All in all: up to 2n scenarios, denoting di
ff
erent “process variants”.
Constraint scenarios
Dealing with “n” probabilistic constraints
20
close
order
1..1
accept
refuse
{0.8}
{0.3}
{0.9}
1
2
3
8 scenarios
(1) (2) (3)
0 0 0
0 0 1
0 1 0
0 1 1
1 0 0
1 0 1
1 1 0
1 1 1
“1”: satis
fi
ed

“0”: violated

Interplay between logic and probabilities
Reasoning over scenarios is tricky
21
close
order
1..1
accept
refuse
{0.8}
{0.3}
{0.9}
Logically, there cannot be traces
satisfying the crips constraints and
also all the uncertain ones.

Hence, scenario 111 is inconsistent!

• A scenario indicates which constraints holds and which don’t

• A constraint holds in a trace if the trace satis
fi
es the constraint
formula

• A constraint does not hold in a trace if the trace violates the
constraint formula -> satis
fi
ed the negation of the constraint
formula

• Scenario characteristic formula:
LTLf to the rescue
Logical reasoning within scenarios
22
faction/violation of constraints as indicated by the scenario. Three questions
immediately arise: (i) how does one check to which scenario(s) a trace belongs?
(ii) Can a trace belong to multiple scenarios? (iii) Are all scenarios meaningful,
or should we discard some of them?
To answer such questions, we provide a logical characterization of scenarios.
First and foremost, we introduce a characteristic LTLf formula for a scenario:
a trace belongs to a scenario if and only if the trace satisfies the characteristic
formula of the scenario.
Definition 15. Let M = 〈Σ, C, 〈〈ϕ1, ⊲⊳1, p1〉, . . . , 〈ϕn, ⊲⊳n, pn〉〉〉 be ProbDe-
clare model. The characteristic formula induced by a scenario SM
b1···bn
over
M, compactly called SM
b1···bn
-formula, is the LTLf formula
Φ(SM
b1···bn
) =
'
ψ∈C
ψ ∧
'
i∈{1,...,n}
(
)
*
)
+
ϕi if bi = 1
¬ϕi if bi = 0
(8)
⊳
Definition 16. A trace τ belongs to scenario SM
b1···bn
if τ |= Φ(SM
b1···bn
). Sce-

Example
23
close
order
1..∗
accept
order
{0.8}
1
refuse
order
{0.3}
2
{0.9}
3
1 2 3 consistent?
S000 ✸(close ∧ ¬©✸acc) ✸(close ∧ ¬©✸ref) ✸acc ∧ ✸refuse no
S001 ✸(close ∧ ¬©✸acc) ✸(close ∧ ¬©✸ref) ¬(✸acc ∧ ✸refuse) yes
S010 ✸(close ∧ ¬©✸acc) ✷(close → ©✸ref) ✸acc ∧ ✸refuse no
S011 ✸(close ∧ ¬©✸acc) ✷(close → ©✸ref) ¬(✸acc ∧ ✸refuse) yes
S100 ✷(close → ©✸acc) ✸(close ∧ ¬©✸ref) ✸acc ∧ ✸refuse no
S101 ✷(close → ©✸acc) ✸(close ∧ ¬©✸ref) ¬(✸acc ∧ ✸refuse) yes
S110 ✷(close → ©✸acc) ✷(close → ©✸ref) ✸acc ∧ ✸refuse yes
S111 ✷(close → ©✸acc) ✷(close → ©✸ref) ¬(✸acc ∧ ✸refuse) no
Figure 1: A ProbDeclare model, with 8 constraint scenarios, out of which only 4 are consistent.
Recall that each scenario induces a formula that does not simply conjoin the positive/negated
variants of the probabilistic constraints, but includes also the conjunction of the formulae for
crisp constraints.

Interplay between logic and probabilities
Reasoning over scenarios is tricky
24
close
order
1..1
accept
refuse
{0.8}
{0.3}
{0.9}
0.8+0.3 > 1, hence there must be traces
where a closed order is accepted and
refused (and so the two activities coexist).
Hence, scenario 110 must have a non-zero probability!

• For n scenarios, let with be the probability
that an arbitrary trace belongs to scenario

• What are the legitimate probability distributions over scenarios?

• May be in
fi
nitely many

• No solution: inconsistent speci
fi
cation
xi i ∈ {0,…,2n−1
}
i
From probabilistic constraints to probability distributions over scenarios
Probabilities of scenarios
25
masses associated to consistent scenarios, we set up a system of inequalities
whose solutions constitute all the probability distributions that are compati-
ble with the logical and probabilistic characterization of the probabilistic con-
straints in the ProbDeclare model of interest. To do so, we associate each
scenario to a probability variable, keeping the same naming convention. For
example, the probability mass of scenario S001 is represented by variable x001.
For M = 〈Σ, C, 〈〈ϕ1, ⊲⊳1, p1〉, . . . , 〈ϕn, ⊲⊳n, pn〉〉〉, we construct the system LM of
inequalities using probability variables xi, with i ranging from 0 to 2n
− 1 (in
binary format):
xi ≥ 0 0 ≤ i < 2n
(9)
% 2n
−1
"
i=0
xi
&
= 1 (10)
% "
i∈{0,...,2n−1},
jth position of i is 1
xi
&
⊲⊳j pj 0 ≤ j < n (11)
xi = 0 0 ≤ i < 2n
, scenario Si is inconsistent (12)
The first two lines guarantee that variables xi indeed form a probability distri-
bution, being all non-negative and collectively summing up to 1. The schema of
inequalities captured in Equation (11) verifies the probability associated to each

Example
26
close
order
1..1
accept
refuse
{0.8}
{0.3}
{0.9}
1
2
3
sign
consent
close
order
1..∗
{0.8} 1
{0.1} 2
1 2 consistent?
S00 ¬sign U close ✸(close ∧ ¬©✸sign) yes
S01 ¬sign U close ✷(close → ©✸sign) yes
S10 ¬close W sign ✸(close ∧ ¬©✸sign) yes
S11 ¬close W sign ✷(close → ©✸sign) yes
Figure 2: A ProbDeclare model and its 4 constraint scenarios.
once the variables above are removed (being them all equal to 0):
x001 + x011 + x101 + x110 = 1
x101 + x110 = 0.8
x011 + x110 = 0.3
x001 + x011 + x101 = 0.9
It is easy to see that this system of equations admits only one solution: x001 = 0,
x = 0.2, x = 0.7, x = 0.1. This solution witnesses that scenario S

Reasoning in a scenario: standard LTLf reasoning:

• Inconsistent scenarios get probability 0 (no conforming trace).

Constraint probabilities induce probability distribution on scenarios:

• System of linear inequalities to compute scenario probabilities.
Which scenarios are possible? With which probability?
Reasoning over scenarios
27
close
order
accept
refuse
{0.8}
{0.3}
{0.9}
1
2
3
scenario
consistent? probability
(1) (2) (3)
0 0 0
0 0 1
0 1 0
0 1 1
1 0 0
1 0 1
1 1 0
1 1 1
1..1




28
close
order
accept
refuse
{0.8}
{0.3}
{0.9}
1
2
3
1..1
scenario
(1) (2) (3)
0 0 0 N
0 0 1 Y
0 1 0 N
0 1 1 Y
1 0 0 N
1 0 1 Y
1 1 0 Y
1 1 1 N




29
close
order
accept
refuse
{0.8}
{0.3}
{0.9}
1
2
3
1..1
scenario
(1) (2) (3)
0 0 0 N 0
0 0 1 Y
0 1 0 N 0
0 1 1 Y
1 0 0 N 0
1 0 1 Y
1 1 0 Y
1 1 1 N 0

scenario
(1) (2) (3)
0 0 0 N 0
0 0 1 Y 0
0 1 0 N 0
0 1 1 Y 0.2
1 0 0 N 0
1 0 1 Y 0.7
1 1 0 Y 0.1
1 1 1 N 0



30
close
order
accept
refuse
{0.8}
{0.3}
{0.9}
1
2
3
1..1




31
accept
refuse
{0.8}
{0.3}
{0.9}
1
2
3
scenario
(1) (2) (3)
0 0 0
0 0 1
0 1 0
0 1 1
1 0 0
1 0 1
1 1 0
1 1 1
scenario
(1) (2) (3)
0 0 0 N
0 0 1 Y
0 1 0 N
0 1 1 Y
1 0 0 N
1 0 1 Y
1 1 0 Y
1 1 1 N
scenario
(1) (2) (3)
0 0 0 N 0
0 0 1 Y
0 1 0 N 0
0 1 1 Y
1 0 0 N 0
1 0 1 Y
1 1 0 Y
1 1 1 N 0
scenario
(1) (2) (3)
0 0 0 N 0
0 0 1 Y 0
0 1 0 N 0
0 1 1 Y 0.2
1 0 0 N 0
1 0 1 Y 0.7
1 1 0 Y 0.1
1 1 1 N 0
scenario
(1) (2) (3)
0 0 0 N 0
0 0 1 Y 0
0 1 0 N 0
0 1 1 Y 0.2
1 0 0 N 0
1 0 1 Y 0.7
1 1 0 Y 0.1
1 1 1 N 0
1..1
close
order

Typical discovery procedure

1. Candidate constraints are generated by analysing the structure
of the log and the activities contained therein

2. Compute the support per constraint

3. Filter constraints based on support

4. Apply further
fi
lters based, redundancy, interestingness, vacuity,
…
Process discovery
Scenarios in action
32
Key challenge: consistency only guaranteed if support 100%
3. Candidate formulae are filtered, retaining only th
ceeds a given threshold.
4. Further filters are applied, for example considering
dancy, interestingness, and vacuity [7, 11, 27].
In this pipeline, the notion of support is typically fo
Definition 18. The support of an LTLf formula ϕ in a
suppL(ϕ) =
!
τ∈L,τ|=ϕ L(τ)
|L|
To obtain a meaningful Declare model in output, t
crucial catch: the formulae that pass all the steps of the
an overall inconsistent model. The reason is that formu
strictly less than 1 may actually conflict with each oth
recognized by the model, which does not keep nor use a
to support. Fixing these potential inconsistencies calls t
processing techniques [11].

Support naturally matches the semantics of probabilistic constraints

• Consistency by design if whenever we discover an interesting
constraint , we retain its support as probability:

• Two implications on “probabilistic interestingness” and
“probabilistic over
fi
tting”
φ
⟨φ, = , suppL
(φ)⟩
Process discovery
Scenarios in action
33

Scenarios in action
Probabilistic conformance checking
34
close
order
accept
<close order>
close
order
refuse
accept refuse
scenario
probability
(1) (2) (3)
0 0 0 0
0 0 1 0
0 1 0 0
0 1 1 0.2
1 0 0 0
1 0 1 0.7
1 1 0 0.1
1 1 1 0

Scenarios in action
35
scenario
probability
(1) (2) (3)
0 0 0 0
0 0 1 0
0 1 0 0
0 1 1 0.2
1 0 0 0
1 0 1 0.7
1 1 0 0.1
1 1 1 0
0
0
1
close
order
accept
<close order>
close
order
refuse
accept refuse

Scenarios in action
36
scenario
probability
(1) (2) (3)
0 0 0 0
0 0 1 0
0 1 0 0
0 1 1 0.2
1 0 0 0
1 0 1 0.7
1 1 0 0.1
1 1 1 0
0
0
1
Violation!
close
order
accept
<close order>
close
order
refuse
accept refuse

Scenarios in action
37
scenario
probability
(1) (2) (3)
0 0 0 0
0 0 1 0
0 1 0 0
0 1 1 0.2
1 0 0 0
1 0 1 0.7
1 1 0 0.1
1 1 1 0
close
order
accept
<close order, 
accept, 
refuse>
close
order
refuse
accept refuse

Scenarios in action
38
scenario
probability
(1) (2) (3)
0 0 0 0
0 0 1 0
0 1 0 0
0 1 1 0.2
1 0 0 0
1 0 1 0.7
1 1 0 0.1
1 1 1 0
close
order
accept
<close order, 
accept, 
refuse>
close
order
refuse
accept refuse
1
1
0

Scenarios in action
39
scenario
probability
(1) (2) (3)
0 0 0 0
0 0 1 0
0 1 0 0
0 1 1 0.2
1 0 0 0
1 0 1 0.7
1 1 0 0.1
1 1 1 0
1
1
0
Conforming! 
(but a rare  
case)
close
order
accept
<close order, 
accept, 
refuse>
close
order
refuse
accept refuse

Scenarios in action
Probabilistic monitoring
40
Whole scenarios have to be considered: one LTLf monitor per scenario.

Monitors used in parallel: if multiple return the same verdict, aggregate
probability values can be returned for sophisticated feedback.

Interpretability of these feedbacks is an interesting open question.

Scenarios in action
Probabilistic monitoring
41
Fully implemented!

From traces to logs
Stochastic conformance (granularity: scenario)
42
Log
ProbDeclare

speci
fi
cation

From traces to logs
43
Log
ProbDeclare

speci
fi
cation
Consistent scenarios

From traces to logs
44
Log
ProbDeclare

speci
fi
cation
Speci
fi
cation

distribution

From traces to logs
45
Log
ProbDeclare

speci
fi
cation
Speci
fi
cation

distribution
Log

distribution

From traces to logs
46
Log
ProbDeclare

speci
fi
cation
Speci
fi
cation

distribution
Log

distribution
(Earth mover’s) distance
Can be re
fi
ned
through trace
alignments

Process control-flow with Petri nets
Minimum requirements
48
review
claim
claim

received
check

claim
info

complete?
obtain
missing
info
check

claim
yes

no

Process control-flow with Petri nets
Minimum requirements
49
review
claim
claim

received
check

claim
info

complete?
obtain
missing
info
check

claim
yes

no

review
claim
check

claim
obtain
missing info
check

claim
labels repeated labels
silent transitions

Unlogged tasks as silent transitions
50
4 Sander J.J. Leemans et al.
q0
open
t01
o
q1
⌧
t12
i
(insert item)
q2
⌧
t21
m
finalize
t23
f
q3
reject
t37
r
q7
accept
t35
a
q4
⌧
t51
b
pay
t46
p
q6
⌧ t45
d
q5
cancel
t15
c
delete
t58
q8
Fig. 2: Stochastic net of an order-to-cash process. Weights are presented symbolically.
Transition t12 captures a task that cannot be logged, and so is modelled as silent.
Definition 1 (Labelled Petri net). A labelled Petri net N is a tuple hQ, T, F, `i, where:
(i) Q is a finite set of places; (ii) T is a finite set of transitions, disjoint from Q (i.e.,
Q T = ;); (iii) F ✓ (Q ⇥ T) [ (T ⇥ Q) is a flow relation connecting places to
𝗂
𝗇
𝗌
𝖾
𝗋
𝗍
𝗂
𝗍
𝖾
𝗆

Unlogged tasks as silent transitions
51
q0
open
t01
o
q1
⌧
t12
i
(insert item)
q2
⌧
t21
m
finalize
t23
f
q3
reject
t37
r
q7
accept
t35
a
q4
⌧
t51
b
pay
t46
p
q6
⌧ t45
d
q5
cancel
t15
c
delete
t58
q8

• Fix an initial marking (= initial state) and a set of deadlocking
fi
nal
markings (=
fi
nal states)

• Run: valid sequence of transitions from the initial state to some
fi
nal
state

• Trace: projection of the run on visible transitions

• How many runs for a trace? Potentially in
fi
nite!
Runs and traces
52
q0
open
t01
o
q1
⌧
t12
i
(insert item)
q2
⌧
t21
m
finalize
t23
f
q3
reject
t37
r
q7
accept
t35
a
q4
⌧
t51
b
pay
t46
p
q6
⌧ t45
d
q5
cancel
t15
c
delete
t58
q8

Via transition systems (interleaving semantics)
Execution semantics
53
q0
open
t01
o
q1
⌧
t12
i
(insert item)
q2
⌧
t21
m
finalize
t23
f
q3
reject
t37
r
q7
accept
t35
a
q4
⌧
t51
b
pay
t46
p
q6
⌧ t45
d
q5
cancel
t15
c
delete
t58
q8
transitions and transitions to places; (iv) ` : T ! ⌃ is a labelling function mapping
each transition t 2 T to a corresponding label `(t) that is either a task name from ⌃ of
the silent label ⌧. /
In the paper, we adopt a dot notation to extract the component of interest from a net, that
is, given a net N, its places are denoted by N.Q, etc. We will adopt the same notational
convention for the other definitions as well. Given a net N an element x 2 N.Q[N.T,
the preset and post-set of x are respectively defined by •
x = {y | hy, xi 2 F} and
•
initial state:  
[q0]
fi
nal states:  
[q6], [q7], [q8]
s0
[q0]
s1
[q1]
s2
[q2]
s3
[q3]
s4
[q4]
s7
[q7]
s5
[q5]
s6
[q6]
s8
[q8]
1
open
⇢i = i
i+c
⌧
⇢m = m
m+f
⌧
⇢f = f
m+f
fin ⇢a
=
a
a+r
acc
⇢r = r
a+r
rej
⇢b = b
b+p+d
⌧
⇢c = c
i+c can
⇢d = d
b+p+d
⌧
⇢p = d
b+p+d
pay
1
del

From nondeterminism to probability distribution over next markings
Stochastic decision making
54
q0
open
t01
o
q1
⌧
t12
i
(insert item)
q2
⌧
t21
m
finalize
t23
f
q3
reject
t37
r
q7
accept
t35
a
q4
⌧
t51
b
pay
t46
p
q6
⌧ t45
d
q5
cancel
t15
c
delete
t58
q8
• Every transition gets a weight

• Probability for an enabled transition to
fi
re: relative weight on all
enabled transitions

• Probability of a run: product of the
fi
ring probabilities

• Every transition gets a weight

• Probability for an enabled transition to
fi
re: relative weight on all
enabled transitions

• Probability of a run: product of the
fi
ring probabilities
From nondeterminism to probability distribution over next markings
Stochastic decision making
55
q0
open
t01
o
q1
⌧
t12
i
(insert item)
q2
⌧
t21
m
finalize
t23
f
q3
reject
t37
r
q7
accept
t35
a
q4
⌧
t51
b
pay
t46
p
q6
⌧ t45
d
q5
cancel
t15
c
delete
t58
q8
fi
re with probability
a
a + r
fi
re with probability
r
a + r

Via stochastic transition systems
Execution semantics
56
q0
open
t01
o
q1
⌧
t12
i
(insert item)
q2
⌧
t21
m
finalize
t23
f
q3
reject
t37
r
q7
accept
t35
a
q4
⌧
t51
b
pay
t46
p
q6
⌧ t45
d
q5
cancel
t15
c
delete
t58
q8
transitions and transitions to places; (iv) ` : T ! ⌃ is a labelling function mapping
each transition t 2 T to a corresponding label `(t) that is either a task name from ⌃ of
the silent label ⌧. /
In the paper, we adopt a dot notation to extract the component of interest from a net, that
is, given a net N, its places are denoted by N.Q, etc. We will adopt the same notational
convention for the other definitions as well. Given a net N an element x 2 N.Q[N.T,
the preset and post-set of x are respectively defined by •
x = {y | hy, xi 2 F} and
•
initial state:  
[q0]
fi
nal states:  
[q6], [q7], [q8]
s0
[q0]
s1
[q1]
s2
[q2]
s3
[q3]
s4
[q4]
s7
[q7]
s5
[q5]
s6
[q6]
s8
[q8]
1
open
⇢i = i
i+c
⌧
⇢m = m
m+f
⌧
⇢f = f
m+f
fin ⇢a
=
a
a+r
acc
⇢r = r
a+r
rej
⇢b = b
b+p+d
⌧
⇢c = c
i+c can
⇢d = d
b+p+d
⌧
⇢p = d
b+p+d
pay
1
del

Key questions
57
1. Probability of a trace?

2. Veri
fi
cation of qualitative properties: probability that the
net satis
fi
es a given declarative speci
fi
cation?

3. Conformance to a ProbDeclare speci
fi
cation?

1. Probability of a trace?

2. Veri
fi
cation of qualitative properties: probability that the
net satis
fi
es a given declarative speci
fi
cation?

3. Conformance to a ProbDeclare speci
fi
cation?
and answers
Key questions
58
A. Warm-up: outcome probability

B. Play with automata

C. Reduce all the three questions above to A.

Idea: labels are not important  
-> the stochastic transition system behaves like a Markov chain

 
Outcome probability ~ Markov chain exist distributions 
-> can be solved analytically
Probability of reaching a
fi
nal state
Outcome probability
59
s0
[q0]
s1
[q1]
s2
[q2]
s3
[q3]
s4
[q4]
s7
[q7]
s5
[q5]
s6
[q6]
s8
[q8]
1
open
⇢i = i
i+c
⌧
⇢m = m
m+f
⌧
⇢f = f
m+f
fin ⇢a
=
a
a+r
acc
⇢r = r
a+r
rej
⇢b = b
b+p+d
⌧
⇢c = c
i+c can
⇢d = d
b+p+d
⌧
⇢p = d
b+p+d
pay
1
del
Fig. 3: Stochastic reachability graph of the order-to-cash bounded stochastic PNP.
States are named. The initial state is shown with a small incoming edge. Final states
have a double countour.
Definition 7 (Labelled transition system). A labelled transition system is a tuple
hS, s0, Sf , %i where: (i) S is a (possibly infinite) set of states; (ii) s0 2 S is the ini-
tial state; (iii) Sf ✓ S is the set of accepting states; (iv) % ✓ S ⇥⌃ ⇥ S is a⌃-labelled
transition relation. A run is a finite sequence of transitions leading from s0 to one of the
q0
a
t01 q1
b
t02
b
q2
c
t23
c
q3
d
t32
e t33
(a) Stochastic net.
s0
[q0]
s1
[q1]
s2
[q2]
s3
[q3]
a
⇢a
=
a
a+b
b
⇢b = b
a+b c
d
⇢d = d
d+e e
⇢e = e
e+d
(b) Reachability graph.
Fig. 4: Reachability graph (b) of a bounded stochastic PNP with net shown in (a), initial
marking [q0] and final marking [q1]. States s2 and s3 are livelock markings.
By recalling that states of RG(N) are markings of N, the schema (1) of equations
deals with final (deadlock) states, that in (1) with non-final deadlock states, and that in
(1) with non-final, non-deadlock states.
EF
N has always at least a solution. However, it may be indeterminate and thus admit
infinitely many ones, requiring in that case to pick the least committing (i.e., minimal
non-negative) solution. The latter case happens when N contains livelock markings.
This is illustrated in the following examples.
Example 2. Consider bounded stochastic PNP Norder (Figure 2). We want to solve the
problem OUTCOME-PROB(Norder, [q6]), to compute the probability that a created order
eventually completes the process by being paid. To do so, we solve E
[q6]
Norder
by encoding
the reachability graph of Figure 3 into:
xs8
= 0 xs5
= xs8
xs2
= ⇢mxs1
+ ⇢f xs3
xs7
= 0 xs4
= ⇢bxs1
+ ⇢dxs5
+ ⇢pxs6
xs1
= ⇢ixs2
+ ⇢cxs5
xs6
= 1 xs3
= ⇢axs4
+ ⇢rxs7
xs0
= xs1
This yields xs0
=
⇢i⇢f ⇢a⇢pxs6
+⇢i⇢f ⇢rxs7
+(⇢i⇢f ⇢a⇢d+⇢c)xs8
1 ⇢i⇢m ⇢i⇢f ⇢a⇢b
=
⇢i⇢f ⇢a⇢p
1 ⇢i⇢m ⇢i⇢f ⇢a⇢b
, which

The issue of livelocks
60
Reasoning on Labelled Petri Nets and their Dynamics in a Stochastic Setting 9
q0
a
t01
a
q1
b
t02
b
q2
c
t23
c
q3
d
t32
d
e t33
e
(a) Stochastic net.
s0
[q0]
s1
[q1]
s2
[q2]
s3
[q3]
a
⇢a
=
a
a+b
b
⇢b = b
a+b c
d
⇢d = d
d+e e
⇢e = e
e+d
(b) Reachability graph.
Fig. 4: Reachability graph (b) of a bounded stochastic PNP with net shown in (a), initial
marking [q0] and final marking [q1]. States s2 and s3 are livelock markings.
By recalling that states of RG(N) are markings of N, the schema (1) of equations
deals with final (deadlock) states, that in (1) with non-final deadlock states, and that in
(1) with non-final, non-deadlock states.
initial state: [q0]
fi
nal state: [q1]

General solution
61
si
corresponds to a final marking, otherwise xsi = 0 (witnessing that F cannot be
reached);
Inductive case if s has at least one successor, then its variable is equal to sum of the
state variables of its successor states, weighted by the transition probability to
move to that successor.
Formally, OUTCOME-PROB(N, F) with RG(N) = hS, s0, Sf , %, pi gets encoded into
the following linear optimisation problem EF
N :
Return xs0 from the minimal non-negative solution of
xsi = 1 for each si 2 F (1)
xsj = 0 for each sj 2 S F s.t. |succRG(N )(sj)| = 0 (2)
xsk =
X
hsk,l,s0
k
i2succRG(N )(sk)
p(hsk, l, s0
ki) · xs0
k
for each sk 2 S s.t. |succRG(N )(sk)| > 0 (3)
Base case if s has no successor states (i.e., is a deadlock marking), then xsi = 1 if s
corresponds to a final marking, otherwise xsi = 0 (witnessing that F cannot be
reached);
Inductive case if s has at least one successor, then its variable is equal to sum of the
state variables of its successor states, weighted by the transition probability to
move to that successor.
Formally, OUTCOME-PROB(N, F) with RG(N) = hS, s0, Sf , %, pi gets encoded into
the following linear optimisation problem EF
N :
Return xs0 from the minimal non-negative solution of
xsi = 1 for each si 2 F (1
xsj = 0 for each sj 2 S F s.t. |succRG(N )(sj)| = 0 (2
xsk =
X
hsk,l,s0
k
i2succRG(N )(sk)
p(hsk, l, s0
ki) · xs0
k
for each sk 2 S s.t. |succRG(N )(sk)| > 0 (3
Example 3 illustrates how the technique implicitly gets rid of livelock markings,
associating to them a 0 probability. This captures the essential fact that, by definition,
a livelock marking can never reach any final marking. More in general, we can in fact
solve OUTCOME-PROB(N, F) by turning the linear optimisation problem EF
N into the
following system of equalities, which is guaranteed to have exactly one solution:
xsi
= 1 for each deadlock marking si 2 F (4)
xsj
= 0 for each deadlock marking sj 2 S F (5)
xsk
= 0 for each livelock marking sk 2 S (6)
xsh
=
X
hsh,l,s0
hi2succRG(N )(sh)
p(hsh, l, s0
hi) · xs0
h
for each remaining marking sh 2 S (7)
Recall that checking whether a marking s is livelock can be done over RG(N) by

Verification of qualitative properties
62
1. Express the property as a DFA  
(works for traces, LTLf, LDLf, regexp)

2. Extend the DFA with tau-loops, expressing that silent transitions
are always ok

3. Do the cross-product of the extended DFA and the input
stochastic transition system

4. Solve outcome probability of all
fi
nal states on the resulting TS
Key advantage: no ad-hoc techniques to get rid of
silent transitions!

Example: trace probability
Probability of: <open,
fi
n, acc,
fi
n, acc, pay>
63
s0
[q0]
s1
[q1]
s2
[q2]
s3
[q3]
s4
[q4]
s7
[q7]
s5
[q5]
s6
[q6]
s8
[q8]
1
open
⇢i = i
i+c
⌧
⇢m = m
m+f
⌧
⇢f = f
m+f
fin ⇢a
=
a
a+r
acc
⇢r = r
a+r
rej
⇢b = b
b+p+d
⌧
⇢c = c
i+c can
⇢d = d
b+p+d
⌧
⇢p = d
b+p+d
pay
1
del
Fig. 3: Stochastic reachability graph of the order-to-cash bounded stochastic PNP.
States are named. The initial state is shown with a small incoming edge. Final states
have a double countour.
Definition 7 (Labelled transition system). A labelled transition system is a tuple
hS, s0, Sf , %i where: (i) S is a (possibly infinite) set of states; (ii) s0 2 S is the ini-
tial state; (iii) Sf ✓ S is the set of accepting states; (iv) % ✓ S ⇥⌃ ⇥ S is a⌃-labelled
transition relation. A run is a finite sequence of transitions leading from s0 to one of the
states in Sf in agreement with %. /
Due to our requirement that all final markings are deadlock markings, accepting states
Reasoning on Labelled Petri Nets and their Dynamics in a Stochastic Setting
s0 s1 s2
s3
s4
s5
⌧ ⌧ ⌧
⌧
⌧
⌧
open fin
acc
fin
rej
0, 0 1, 1 2, 1 3, 2
2, 3
3, 4
7, 5
open
1 ⌧
⇢i
⌧
⇢m fin
⇢f
acc
⇢a
⇢
⌧
⇢i
⌧
⇢m
fin
⇢f
rej
⇢r
X
=
Reasoning on Labelled Petri Nets and their Dynamics in a Stochastic Setting 13
s0 s1 s2
s3
s4
s5
⌧ ⌧ ⌧
⌧
⌧
⌧
open fin
acc
fin
rej
(a) DFAs A and Ā .
0, 0 1, 1 2, 1 3, 2 4, 3
1, 3
2, 3
3, 4
7, 5
open
1 ⌧
⇢i
⌧
⇢m fin
⇢f
acc
⇢a
⌧
⇢b
⌧
⇢i
⌧
⇢m
fin
⇢f
rej
⇢r
(b) Product system between Ā and RG(Norder).

Applications in stochastic process mining
Probabilistic trace alignment
64

Conformance to a ProbDeclare specification
Induced distribution via qualitative veri
fi
cation
65
ProbDeclare

speci
fi
cation
Speci
fi
cation

distribution
Induced

distribution
(Earth mover’s) distance
Stochastic

Petri net

Important of incorporating uncertainty into declarative
and procedural process models
Combined reasoning via careful, loosely coupled
interplay of reasoning about time/dynamics and
reasoning about probabilities

Direct impact on a variety of stochastic process
mining tasks

Main techniques: all implemented!
“The future is uncertain, but the end is always near” (Jim Morrison)
Conclusion
66

Process Reasoning and Mining with Uncertainty

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