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Reasoning on Labelled Petri Nets and Their Dynamics in a Stochastic Setting

Faculty of Computer Science - Free University of Bozen-Bolzano, profile picture
Faculty of Computer Science - Free University of Bozen-Bolzano

The document discusses labelled Petri nets and their application in process mining within a stochastic framework. It introduces concepts such as stochastic conformance checking, probabilistic trace alignment, and defines a labelled Petri net in detail, emphasizing its components and dynamics. The focus is on the structure of these nets and their role in modeling uncertain processes, particularly in an order-to-cash scenario.

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BPM 2022, Münster Reasoning on Labelled Petri Nets and Their Dynamics 
 in a Stochastic Setting Sander Leemans, Fabrizio Maggi, Marco Montali
analytics
analytics
Stochastic process mining Revived interest in stochastic processes Σ* Σ* trace process
Stochastic process mining Revived interest in stochastic processes Σ* Σ* trace process Σ* Σ* uncertain trace stochastic process
Stochastic process mining Revived interest in stochastic processes Σ* trace process Σ* stochastic conformance checking, probabilistic trace alignment, probabilistic declarative process mining uncertain trace stochastic process Σ* Σ*
Σ* Σ* Σ* stochastic conformance checking, probabilistic trace alignment, probabilistic declarative process mining uncertain trace Σ* Our focus stochastic process
Process control-flow with Petri nets Characteristics review claim claim received check claim info complete? obtain missing info check claim yes no …
Process control-flow with Petri nets Characteristics review claim claim received check claim info complete? obtain missing info check claim yes no … review claim check claim obtain missing info check claim …
Process control-flow with Petri nets Characteristics 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 silent repeated labels bounded control- fl ow
Unlogged tasks as silent transitions 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 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
Unlogged tasks as silent transitions 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 silent loop
Semantics via finite traces Start, end(s) 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 1 marking as initial state 1+ deadlocking markings as fi nal states
Semantics via finite traces Start, end(s) 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 1 marking as initial state 1+ deadlock markings as fi nal states initial state [q0] fi nal “paid” state [q6] fi nal “deleted” state [q8] fi nal “rejected” state [q7]
Semantics via finite traces Runs and traces 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 1 marking as initial state 1+ deadlock markings as fi nal states initial state [q0] Run: valid sequence of transitions from the initial state to some fi nal state Trace: projection of the run on labels of visible transitions How many runs for the same trace? Potentially in fi nitely many! fi nal “paid” state [q6] fi nal “deleted” state [q8] fi nal “rejected” state [q7]
Semantics via finite traces Runs and traces 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 1 marking as initial state 1+ deadlock markings as fi nal states initial state [q0] <open, fi nalize, accept, fi nalize, reject> fi nal “paid” state [q6] fi nal “deleted” state [q8] fi nal “rejected” state [q7]
Semantics via finite traces Runs and traces 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 1 marking as initial state 1+ deadlock markings as fi nal states initial state [q0] fi nal “paid” state [q6] fi nal “deleted” state [q8] <open, fi nalize, accept, fi nalize, reject> fi nal “rejected” state [q7]
From net to transition system 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 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 initial state [q0] fi nal “paid” state [q6] fi nal “deleted” state [q8] fi nal “rejected” state [q7] 6 Sander J.J. Leemans et al. 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
Stochastic Petri nets Every transition gets a weight • Immediate transition: relative likelihood to fi re • Timed transition: rate/decay of exponential distribution for waiting time 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 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
Stochastic Petri nets Every transition gets a weight • Immediate transition: relative likelihood to fi re • Timed transition: rate/decay of exponential distribution for waiting time Our interest: transition fi rings -> ordering without time 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 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
Stochastic Petri nets Focus: next transition fi ring/completion P( fi re t enabled in m | marking m) = weight of t ∑t′   enabled in m weight of t′  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 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 Works both for set of immediate OR of timed transitions
Stochastic Petri nets Focus: next transition fi ring/completion 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 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 P( fi re t enabled in m | marking m) = weight of t ∑t′   enabled in m weight of t′  fi re with probability r a + r fi re with probability a a + r
Stochastic Petri nets Focus: next transition fi ring/completion 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 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 Every fi ring is independent • Probability of a run = product of fi ring probabilities • Probability of a trace = (possibly in fi nite) sum of probabilities of its runs
Stochastic Petri nets Focus: next transition fi ring/completion 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 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 Every fi ring is independent • Probability of a run = product of fi ring probabilities • Probability of a trace = (possibly in fi nite) sum of probabilities of its runs So far: • approximate computation, or • exact computation for restricted nets (no fully silent loops)
Semantics via stochastic transition systems 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 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 initial state [q0] fi nal “paid” state [q6] fi nal “deleted” state [q8] fi nal “rejected” state [q7] 6 Sander J.J. Leemans et al. 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 Looks familiar?
Key problems 1. Probability of a trace 2. Probability of satisfying a qualitative property expressed in temporal logics over fi nite traces, or as a fi nite-state automaton 3. Conformance to a probabilistic Declare speci fi cation
 that de fi nes constraint scenarios, each coming with a di ff erent probability (or range of probabilities)
Observation 1. Probability of a trace 2. Probability of satisfying a qualitative property expressed in temporal logics over fi nite traces, or as a fi nite-state automaton Encode trace as an automaton <open, fi nalize, accept, fi nalize, reject> 0 1 2 3 4 5 open fi nalize accept fi nalize reject
Attack strategy Reasoning on states and probabilities Reasoning on tasks and transitions
Attack strategy Reasoning on states and probabilities Reasoning on tasks and transitions Markov chains Qualitative model checking Elegant trick to deal with silent transitions
Attack strategy Reasoning on states and probabilities Reasoning on tasks and transitions Markov chains Qualitative model checking Elegant trick to deal with silent transitions 1. Probability of a trace 2. Probability of satisfying a qualitative property 3. Conformance to a probabilistic Declare speci fi cation 0. Outcome probability
 Probability of completing the
 process in some fi nal states
Attack strategy Reasoning on states and probabilities Reasoning on tasks and transitions Markov chains Qualitative model checking Elegant trick to deal with silent transitions 1. Probability of a trace 2. Probability of satisfying a qualitative property 3. Conformance to a probabilistic Declare speci fi cation 0. Outcome probability
 Probability of completing the
 process in some fi nal states
Recall the good-old days of studying?
Recall the good-old days of studying?
Outcome probability What is the probability of ending with order paid? 6 Sander J.J. Leemans et al. 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.
Outcome probability What is the probability of ending with order paid? What matters: progression from state to state -> Markov chain 6 Sander J.J. Leemans et al. 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.
Outcome probability What is the probability of ending with order paid? Each state -> variable for probability of reaching the desired one from there 3 cases: • fi nal desired state (good deadlock) • fi nal non-desired state (bad deadlock) • other states… 6 Sander J.J. Leemans et al. 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.
Outcome probability What is the probability of ending with order paid? Each state -> variable for probability of reaching the desired one from there 3 cases: • fi nal desired state (good deadlock) -> 1 • fi nal non-desired state (bad deadlock) -> 0 • other states… -> recursive de fi nition via linear equations 6 Sander J.J. Leemans et al. 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. x6 = 1 x8 = 0 x1 = ρix2 + ρcx5
Outcome probability What is the probability of ending with order paid? Each state -> variable for probability of reaching the desired one from there 3 cases: • fi nal desired state (good deadlock) -> 1 • fi nal non-desired state (bad deadlock) -> 0 • other states… -> recursive de fi nition via linear equations 6 Sander J.J. Leemans et al. 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. } Solve for initial state variable! x6 = 1 x8 = 0 x1 = ρix2 + ρcx5
Outcome probability What is the probability of ending with order paid? 6 Sander J.J. Leemans et al. 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. 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
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 Outcome probability What is the probability of ending with order paid? 6 Sander J.J. Leemans et al. 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. } x0 = ρiρf ρaρp 1 − ρiρm − ρiρf ρaρb
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 Outcome probability What is the probability of ending with order paid? 6 Sander J.J. Leemans et al. 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. } x0 = ρiρf ρaρp 1 − ρiρm − ρiρf ρaρb
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 Outcome probability What is the probability of ending with order paid? 6 Sander J.J. Leemans et al. 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. } x0 = ρiρf ρaρp 1 − ρiρm − ρiρf ρaρb
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 Outcome probability What is the probability of ending with order paid? 6 Sander J.J. Leemans et al. 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. } x0 = ρiρf ρaρp 1 − ρiρm − ρiρf ρaρb
More care needed… 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 initial state [q0] fi nal state [q1]
More care needed… 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 initial state [q0] fi nal state [q1] livelock
More care needed… 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 initial state [q0] fi nal state [q1] livelock x3 = 0 x2 = 0
General case Exit/absorption probability computation from Markov chains 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) 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)
General case Exit/absorption probability computation from Markov chains 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) 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) 10 Sander J.J. Leemans et al. 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)
Attack strategy Reasoning on states and probabilities Reasoning on tasks and transitions Markov chains Qualitative model checking Elegant trick to deal with silent transitions 1. Probability of a trace 2. Probability of satisfying a qualitative property 3. Conformance to a probabilistic Declare speci fi cation 0. Outcome probability
 Probability of completing the
 process in some fi nal states
Model checking What are the traces of the net that satisfy my property? 6 Sander J.J. Leemans et al. 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. property: good, fi nite traces (via an automaton)
Model checking What are the traces of the net that satisfy my property? What matters: transitions and their labels -> transition system 6 Sander J.J. Leemans et al. 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. property: good, fi nite traces (via an automaton)
Automata-based product 6 Sander J.J. Leemans et al. 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 0 1 2 3 4 5 open fi nalize accept fi nalize reject
Automata-based product 6 Sander J.J. Leemans et al. 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 0 1 2 3 4 5 open fi nalize accept fi nalize reject X
Automata-based product 6 Sander J.J. Leemans et al. 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 1 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). 0 1 2 3 4 5 open fi nalize accept fi nalize reject X ???
Properties must “enjoy the silence” 6 Sander J.J. Leemans et al. 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 0 1 2 3 4 5 open fi nalize accept fi nalize reject
Properties must “enjoy the silence” 6 Sander J.J. Leemans et al. 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 0 1 2 3 4 5 open fi nalize accept fi nalize reject τ τ τ τ τ τ
Silence-preserving cross-product 6 Sander J.J. Leemans et al. 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 1 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). 0 1 2 3 4 5 open fi nalize accept fi nalize reject X τ τ τ τ τ τ
Silence-preserving cross-product 6 Sander J.J. Leemans et al. 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 1 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). 0 1 2 3 4 5 open fi nalize accept fi nalize reject X τ τ τ τ τ τ How to compute the probability that the speci fi cation is satis fi ed by a net trace?
Attack strategy Reasoning on states and probabilities Reasoning on tasks and transitions Markov chains Qualitative model checking Elegant trick to deal with silent transitions 1. Probability of a trace 2. Probability of satisfying a qualitative property 3. Conformance to a probabilistic Declare speci fi cation 0. Outcome probability
 Probability of completing the
 process in some fi nal states
1. Infuse the cross-product with net probabilities 6 Sander J.J. Leemans et al. 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 Reasoning on Labelled Petri Nets and their Dynamics in a Stochastic Setting 13 s1 s2 s3 s4 ⌧ ⌧ ⌧ ⌧ en fin acc fin ej 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
1. Infuse the cross-product with net probabilities 6 Sander J.J. Leemans et al. 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 Reasoning on Labelled Petri Nets and their Dynamics in a Stochastic Setting 13 s1 s2 s3 s4 ⌧ ⌧ ⌧ ⌧ en fin acc fin ej 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
2. Focus on states Reasoning on Labelled Petri Nets and their Dynamics in a Stochastic Setting 13 s1 s2 s3 s4 ⌧ ⌧ ⌧ ⌧ en fin acc fin ej 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
2. Focus on states Reasoning on Labelled Petri Nets and their Dynamics in a Stochastic Setting 13 s1 s2 s3 s4 ⌧ ⌧ ⌧ ⌧ en fin acc fin ej 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 An “incomplete” Markov chain
3. Solve the “outcome probability problem” Reasoning on Labelled Petri Nets and their Dynamics in a Stochastic Setting 13 s1 s2 s3 s4 ⌧ ⌧ ⌧ ⌧ en fin acc fin ej 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 An “incomplete” Markov chain
3. Solve the “outcome probability problem” Reasoning on Labelled Petri Nets and their Dynamics in a Stochastic Setting 13 s1 s2 s3 s4 ⌧ ⌧ ⌧ ⌧ en fin acc fin ej 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 An “incomplete” Markov chain x00 = ρiρf ρaρbρiρf ρr 1 − (ρiρm)2
Applications in stochastic process mining Probabilistic trace alignment
Compare with distributions over traces Induced distribution via qualitative veri fi cation Set of traces (behaviour)
Compare with distributions over traces Induced distribution via qualitative veri fi cation “Regular” scenarios (Partition of behaviour)
Compare with distributions over traces Induced distribution via qualitative veri fi cation “Regular” scenarios (Partition of behaviour) Speci fi cation distribution
Compare with distributions over traces Induced distribution via qualitative veri fi cation ProbDeclare speci fi cation “Regular” scenarios (Partition of behaviour) Speci fi cation distribution bounded stochastic Petri net
Compare with distributions over traces Induced distribution via qualitative veri fi cation ProbDeclare speci fi cation “Regular” scenarios (Partition of behaviour) Speci fi cation distribution Induced distribution bounded stochastic Petri net Using our approach
Compare with distributions over traces Induced distribution via qualitative veri fi cation ProbDeclare speci fi cation “Regular” scenarios (Partition of behaviour) Speci fi cation distribution Induced distribution (Earth mover’s) distance bounded stochastic Petri net
Conclusions Stochastic process mining calls for techniques to reason on stochastic process models Existing techniques do not readily apply due to the features of stochastic process models we are interested in (silent transitions) Analytic solution to key problems related to computing probabilities of behaviour • Combination and extension of techniques from Markov chain analysis and qualitative model checking of quantitative systems Just the beginning:
 timed analysis, discovery, more integration of mining&reasoning, …
Thank you! montali@inf.unibz.it

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Reasoning on Labelled Petri Nets and Their Dynamics in a Stochastic Setting

  • 1. BPM 2022, Münster Reasoning on Labelled Petri Nets and Their Dynamics 
 in a Stochastic Setting Sander Leemans, Fabrizio Maggi, Marco Montali
  • 4. Stochastic process mining Revived interest in stochastic processes Σ* Σ* trace process
  • 5. Stochastic process mining Revived interest in stochastic processes Σ* Σ* trace process Σ* Σ* uncertain trace stochastic process
  • 6. Stochastic process mining Revived interest in stochastic processes Σ* trace process Σ* stochastic conformance checking, probabilistic trace alignment, probabilistic declarative process mining uncertain trace stochastic process Σ* Σ*
  • 7. Σ* Σ* Σ* stochastic conformance checking, probabilistic trace alignment, probabilistic declarative process mining uncertain trace Σ* Our focus stochastic process
  • 8. Process control-flow with Petri nets Characteristics review claim claim received check claim info complete? obtain missing info check claim yes no …
  • 9. Process control-flow with Petri nets Characteristics review claim claim received check claim info complete? obtain missing info check claim yes no … review claim check claim obtain missing info check claim …
  • 10. Process control-flow with Petri nets Characteristics 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 silent repeated labels bounded control- fl ow
  • 11. Unlogged tasks as silent transitions 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
  • 12. Unlogged tasks as silent transitions 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
  • 13. Unlogged tasks as silent transitions 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 silent loop
  • 14. Semantics via finite traces Start, end(s) 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 1 marking as initial state 1+ deadlocking markings as fi nal states
  • 15. Semantics via finite traces Start, end(s) 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 1 marking as initial state 1+ deadlock markings as fi nal states initial state [q0] fi nal “paid” state [q6] fi nal “deleted” state [q8] fi nal “rejected” state [q7]
  • 16. Semantics via finite traces Runs and traces 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 1 marking as initial state 1+ deadlock markings as fi nal states initial state [q0] Run: valid sequence of transitions from the initial state to some fi nal state Trace: projection of the run on labels of visible transitions How many runs for the same trace? Potentially in fi nitely many! fi nal “paid” state [q6] fi nal “deleted” state [q8] fi nal “rejected” state [q7]
  • 17. Semantics via finite traces Runs and traces 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 1 marking as initial state 1+ deadlock markings as fi nal states initial state [q0] <open, fi nalize, accept, fi nalize, reject> fi nal “paid” state [q6] fi nal “deleted” state [q8] fi nal “rejected” state [q7]
  • 18. Semantics via finite traces Runs and traces 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 1 marking as initial state 1+ deadlock markings as fi nal states initial state [q0] fi nal “paid” state [q6] fi nal “deleted” state [q8] <open, fi nalize, accept, fi nalize, reject> fi nal “rejected” state [q7]
  • 19. From net to transition system 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 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 initial state [q0] fi nal “paid” state [q6] fi nal “deleted” state [q8] fi nal “rejected” state [q7] 6 Sander J.J. Leemans et al. 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
  • 20. Stochastic Petri nets Every transition gets a weight • Immediate transition: relative likelihood to fi re • Timed transition: rate/decay of exponential distribution for waiting time 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 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
  • 21. Stochastic Petri nets Every transition gets a weight • Immediate transition: relative likelihood to fi re • Timed transition: rate/decay of exponential distribution for waiting time Our interest: transition fi rings -> ordering without time 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 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
  • 22. Stochastic Petri nets Focus: next transition fi ring/completion P( fi re t enabled in m | marking m) = weight of t ∑t′   enabled in m weight of t′  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 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 Works both for set of immediate OR of timed transitions
  • 23. Stochastic Petri nets Focus: next transition fi ring/completion 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 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 P( fi re t enabled in m | marking m) = weight of t ∑t′   enabled in m weight of t′  fi re with probability r a + r fi re with probability a a + r
  • 24. Stochastic Petri nets Focus: next transition fi ring/completion 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 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 Every fi ring is independent • Probability of a run = product of fi ring probabilities • Probability of a trace = (possibly in fi nite) sum of probabilities of its runs
  • 25. Stochastic Petri nets Focus: next transition fi ring/completion 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 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 Every fi ring is independent • Probability of a run = product of fi ring probabilities • Probability of a trace = (possibly in fi nite) sum of probabilities of its runs So far: • approximate computation, or • exact computation for restricted nets (no fully silent loops)
  • 26. Semantics via stochastic transition systems 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 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 initial state [q0] fi nal “paid” state [q6] fi nal “deleted” state [q8] fi nal “rejected” state [q7] 6 Sander J.J. Leemans et al. 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 Looks familiar?
  • 27. Key problems 1. Probability of a trace 2. Probability of satisfying a qualitative property expressed in temporal logics over fi nite traces, or as a fi nite-state automaton 3. Conformance to a probabilistic Declare speci fi cation
 that de fi nes constraint scenarios, each coming with a di ff erent probability (or range of probabilities)
  • 28. Observation 1. Probability of a trace 2. Probability of satisfying a qualitative property expressed in temporal logics over fi nite traces, or as a fi nite-state automaton Encode trace as an automaton <open, fi nalize, accept, fi nalize, reject> 0 1 2 3 4 5 open fi nalize accept fi nalize reject
  • 29. Attack strategy Reasoning on states and probabilities Reasoning on tasks and transitions
  • 30. Attack strategy Reasoning on states and probabilities Reasoning on tasks and transitions Markov chains Qualitative model checking Elegant trick to deal with silent transitions
  • 31. Attack strategy Reasoning on states and probabilities Reasoning on tasks and transitions Markov chains Qualitative model checking Elegant trick to deal with silent transitions 1. Probability of a trace 2. Probability of satisfying a qualitative property 3. Conformance to a probabilistic Declare speci fi cation 0. Outcome probability
 Probability of completing the
 process in some fi nal states
  • 32. Attack strategy Reasoning on states and probabilities Reasoning on tasks and transitions Markov chains Qualitative model checking Elegant trick to deal with silent transitions 1. Probability of a trace 2. Probability of satisfying a qualitative property 3. Conformance to a probabilistic Declare speci fi cation 0. Outcome probability
 Probability of completing the
 process in some fi nal states
  • 33. Recall the good-old days of studying?
  • 34. Recall the good-old days of studying?
  • 35. Outcome probability What is the probability of ending with order paid? 6 Sander J.J. Leemans et al. 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.
  • 36. Outcome probability What is the probability of ending with order paid? What matters: progression from state to state -> Markov chain 6 Sander J.J. Leemans et al. 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.
  • 37. Outcome probability What is the probability of ending with order paid? Each state -> variable for probability of reaching the desired one from there 3 cases: • fi nal desired state (good deadlock) • fi nal non-desired state (bad deadlock) • other states… 6 Sander J.J. Leemans et al. 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.
  • 38. Outcome probability What is the probability of ending with order paid? Each state -> variable for probability of reaching the desired one from there 3 cases: • fi nal desired state (good deadlock) -> 1 • fi nal non-desired state (bad deadlock) -> 0 • other states… -> recursive de fi nition via linear equations 6 Sander J.J. Leemans et al. 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. x6 = 1 x8 = 0 x1 = ρix2 + ρcx5
  • 39. Outcome probability What is the probability of ending with order paid? Each state -> variable for probability of reaching the desired one from there 3 cases: • fi nal desired state (good deadlock) -> 1 • fi nal non-desired state (bad deadlock) -> 0 • other states… -> recursive de fi nition via linear equations 6 Sander J.J. Leemans et al. 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. } Solve for initial state variable! x6 = 1 x8 = 0 x1 = ρix2 + ρcx5
  • 40. Outcome probability What is the probability of ending with order paid? 6 Sander J.J. Leemans et al. 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. 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
  • 41. 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 Outcome probability What is the probability of ending with order paid? 6 Sander J.J. Leemans et al. 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. } x0 = ρiρf ρaρp 1 − ρiρm − ρiρf ρaρb
  • 42. 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 Outcome probability What is the probability of ending with order paid? 6 Sander J.J. Leemans et al. 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. } x0 = ρiρf ρaρp 1 − ρiρm − ρiρf ρaρb
  • 43. 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 Outcome probability What is the probability of ending with order paid? 6 Sander J.J. Leemans et al. 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. } x0 = ρiρf ρaρp 1 − ρiρm − ρiρf ρaρb
  • 44. 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 Outcome probability What is the probability of ending with order paid? 6 Sander J.J. Leemans et al. 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. } x0 = ρiρf ρaρp 1 − ρiρm − ρiρf ρaρb
  • 45. More care needed… 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 initial state [q0] fi nal state [q1]
  • 46. More care needed… 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 initial state [q0] fi nal state [q1] livelock
  • 47. More care needed… 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 initial state [q0] fi nal state [q1] livelock x3 = 0 x2 = 0
  • 48. General case Exit/absorption probability computation from Markov chains 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) 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)
  • 49. General case Exit/absorption probability computation from Markov chains 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) 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) 10 Sander J.J. Leemans et al. 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)
  • 50. Attack strategy Reasoning on states and probabilities Reasoning on tasks and transitions Markov chains Qualitative model checking Elegant trick to deal with silent transitions 1. Probability of a trace 2. Probability of satisfying a qualitative property 3. Conformance to a probabilistic Declare speci fi cation 0. Outcome probability
 Probability of completing the
 process in some fi nal states
  • 51. Model checking What are the traces of the net that satisfy my property? 6 Sander J.J. Leemans et al. 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. property: good, fi nite traces (via an automaton)
  • 52. Model checking What are the traces of the net that satisfy my property? What matters: transitions and their labels -> transition system 6 Sander J.J. Leemans et al. 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. property: good, fi nite traces (via an automaton)
  • 53. Automata-based product 6 Sander J.J. Leemans et al. 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 0 1 2 3 4 5 open fi nalize accept fi nalize reject
  • 54. Automata-based product 6 Sander J.J. Leemans et al. 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 0 1 2 3 4 5 open fi nalize accept fi nalize reject X
  • 55. Automata-based product 6 Sander J.J. Leemans et al. 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 1 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). 0 1 2 3 4 5 open fi nalize accept fi nalize reject X ???
  • 56. Properties must “enjoy the silence” 6 Sander J.J. Leemans et al. 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 0 1 2 3 4 5 open fi nalize accept fi nalize reject
  • 57. Properties must “enjoy the silence” 6 Sander J.J. Leemans et al. 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 0 1 2 3 4 5 open fi nalize accept fi nalize reject τ τ τ τ τ τ
  • 58. Silence-preserving cross-product 6 Sander J.J. Leemans et al. 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 1 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). 0 1 2 3 4 5 open fi nalize accept fi nalize reject X τ τ τ τ τ τ
  • 59. Silence-preserving cross-product 6 Sander J.J. Leemans et al. 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 1 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). 0 1 2 3 4 5 open fi nalize accept fi nalize reject X τ τ τ τ τ τ How to compute the probability that the speci fi cation is satis fi ed by a net trace?
  • 60. Attack strategy Reasoning on states and probabilities Reasoning on tasks and transitions Markov chains Qualitative model checking Elegant trick to deal with silent transitions 1. Probability of a trace 2. Probability of satisfying a qualitative property 3. Conformance to a probabilistic Declare speci fi cation 0. Outcome probability
 Probability of completing the
 process in some fi nal states
  • 61. 1. Infuse the cross-product with net probabilities 6 Sander J.J. Leemans et al. 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 Reasoning on Labelled Petri Nets and their Dynamics in a Stochastic Setting 13 s1 s2 s3 s4 ⌧ ⌧ ⌧ ⌧ en fin acc fin ej 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
  • 62. 1. Infuse the cross-product with net probabilities 6 Sander J.J. Leemans et al. 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 Reasoning on Labelled Petri Nets and their Dynamics in a Stochastic Setting 13 s1 s2 s3 s4 ⌧ ⌧ ⌧ ⌧ en fin acc fin ej 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
  • 63. 2. Focus on states Reasoning on Labelled Petri Nets and their Dynamics in a Stochastic Setting 13 s1 s2 s3 s4 ⌧ ⌧ ⌧ ⌧ en fin acc fin ej 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
  • 64. 2. Focus on states Reasoning on Labelled Petri Nets and their Dynamics in a Stochastic Setting 13 s1 s2 s3 s4 ⌧ ⌧ ⌧ ⌧ en fin acc fin ej 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 An “incomplete” Markov chain
  • 65. 3. Solve the “outcome probability problem” Reasoning on Labelled Petri Nets and their Dynamics in a Stochastic Setting 13 s1 s2 s3 s4 ⌧ ⌧ ⌧ ⌧ en fin acc fin ej 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 An “incomplete” Markov chain
  • 66. 3. Solve the “outcome probability problem” Reasoning on Labelled Petri Nets and their Dynamics in a Stochastic Setting 13 s1 s2 s3 s4 ⌧ ⌧ ⌧ ⌧ en fin acc fin ej 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 An “incomplete” Markov chain x00 = ρiρf ρaρbρiρf ρr 1 − (ρiρm)2
  • 67. Applications in stochastic process mining Probabilistic trace alignment
  • 68. Compare with distributions over traces Induced distribution via qualitative veri fi cation Set of traces (behaviour)
  • 69. Compare with distributions over traces Induced distribution via qualitative veri fi cation “Regular” scenarios (Partition of behaviour)
  • 70. Compare with distributions over traces Induced distribution via qualitative veri fi cation “Regular” scenarios (Partition of behaviour) Speci fi cation distribution
  • 71. Compare with distributions over traces Induced distribution via qualitative veri fi cation ProbDeclare speci fi cation “Regular” scenarios (Partition of behaviour) Speci fi cation distribution bounded stochastic Petri net
  • 72. Compare with distributions over traces Induced distribution via qualitative veri fi cation ProbDeclare speci fi cation “Regular” scenarios (Partition of behaviour) Speci fi cation distribution Induced distribution bounded stochastic Petri net Using our approach
  • 73. Compare with distributions over traces Induced distribution via qualitative veri fi cation ProbDeclare speci fi cation “Regular” scenarios (Partition of behaviour) Speci fi cation distribution Induced distribution (Earth mover’s) distance bounded stochastic Petri net
  • 74. Conclusions Stochastic process mining calls for techniques to reason on stochastic process models Existing techniques do not readily apply due to the features of stochastic process models we are interested in (silent transitions) Analytic solution to key problems related to computing probabilities of behaviour • Combination and extension of techniques from Markov chain analysis and qualitative model checking of quantitative systems Just the beginning:
 timed analysis, discovery, more integration of mining&reasoning, …