Measurement of Rule-based LTLf Declarative Process Specifications
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The document describes a method for calculating the probability of an Reactive Constraint (RCon) specification being satisfied, violated, or unaffected given a trace or log of event data. An RCon consists of an activation formula (φα) and target formula (φτ). The probability of a single RCon is calculated by determining the maximum likelihood estimate of the target formula occurring given the activation occurred, based on event labellings in the traces. The probability of an RCon specification (a set of RCon rules) is also defined based on the probabilities of individual RCon rules. This allows interestingness measures to be applied to entire specifications rather than just individual rules.
![“If d, then e will follow”
“If c, then previously a”
“If d, then 𝐅(e)”
“If c, then 𝐎(a)”
Rule: Reactive Constraint (RCon) 𝜓
3
[1] De Giacomo and Vardi, “Linear temporal logic and linear dynamic logic on finite traces,” in IJCAI, 2013
[2] Pesic, Bosnacki, van der Aalst, “Enacting declarative languages using LTL: avoiding errors and improving performance,” in SPIN, 2010
[3] C., DC., De Giacomo, and Mendling, “Interestingness of traces in declarative process mining: The Janus LTLp f approach,” in BPM, 2018
“If a viral infection is detected, then an intravenous antiviral administration will follow”
“If antibiotics are administered, then an antibiogram must have been previously registered”
LTL𝑓 operator (finally) [1]
LTL𝑓 operator (once)
d → 𝐅(e)
c → 𝐎(a)
Activation
Activation
Target
Target
𝜑𝛼1
𝜑𝛼2
𝜑𝜏1
𝜑𝜏2
LTL𝑓
fomulae
LTL𝑓
fomulae
Any
DECLARE
rule [2]
can be
encoded
as an
RCon [3]](https://image.slidesharecdn.com/cdcs-icpm2022-export-221028080459-8a4717e1/85/Measurement-of-Rule-based-LTLf-Declarative-Process-Specifications-3-320.jpg)
![Interestingess measures
• Based on probabilities [4]
– We needed to define the probability of single rules (done [5])
• We want to apply them also to entire specifications!
→ We need to define probabilities of specifications first (bear with us)
9
Cecconi, Di Ciccio, Senderovich: Interestingness of Specifications
[4] Geng and Hamilton, “Interestingness measures for data mining: A survey,” ACM Comput. Surv., 2006.
[5] C., De Giacomo, DC., Maggi, Mendling, “Measuring the interestingness of temporal logic behavioral specifications in process mining,” Inf. Syst., 2021](https://image.slidesharecdn.com/cdcs-icpm2022-export-221028080459-8a4717e1/85/Measurement-of-Rule-based-LTLf-Declarative-Process-Specifications-9-320.jpg)
![Probability: from LTL𝑓 rules to RCon specifications
𝜑
𝜓1 ≜ 𝜑𝛼1
→ 𝜑𝜏1
𝑆 ≜ {𝜓1, 𝜓2, … , 𝜓𝑠}
↦ 𝑃(𝜑) [5]
↦ 𝑃(𝜓) [5]
↦ 𝑃 𝑆 = ?
10
Cecconi, Di Ciccio, Senderovich: Interestingness of Specifications
In traces and logs!
[5] C., De Giacomo, DC., Maggi, Mendling, “Measuring the interestingness of temporal logic behavioral specifications in process mining,” Inf. Syst., 2021](https://image.slidesharecdn.com/cdcs-icpm2022-export-221028080459-8a4717e1/85/Measurement-of-Rule-based-LTLf-Declarative-Process-Specifications-10-320.jpg)
![Trace
Probability of an event 𝑒
in a trace 𝑡 of length 𝑛
to satisfy an LTL𝑓 formula 𝜑
• Bernoulli MLE [6]
𝑃 𝜑 𝑡 =
𝑖=1
𝑛
𝛺 𝑒𝑖, 𝜑
𝑛
,
𝛺 𝑒𝑖, 𝜑 ∈ 0,1
Log
Probability of a trace 𝑡
in a log 𝐿 of cardinality 𝑚
to satisfy an LTL𝑓 formula 𝜑
• MLE
𝑃(𝜑 𝐿 ) =
𝑖=1
𝑚
𝑃(𝑇 = 𝑡𝑖) 𝑃(𝜑 𝑡𝑖 )
Probability of an LTL𝑓 formula 𝜑
11
Cecconi, Di Ciccio, Senderovich: Interestingness of Specifications
Labelling
[6] Bickel and Doksum, Mathematical statistics: basic ideas and selected topics, volumes I-II package. CRC Press, 2015.](https://image.slidesharecdn.com/cdcs-icpm2022-export-221028080459-8a4717e1/85/Measurement-of-Rule-based-LTLf-Declarative-Process-Specifications-11-320.jpg)
![Trace
Conditional probability of an event
in a trace t to satisfy the target
given the satisfaction of the
activator
• Bivariate Bernoulli MLE [1]
𝑃 𝜓 𝑡 = 𝑃(𝜑τ(𝑡)|𝜑𝛼(𝑡)) =
𝑝11
𝑝01 + 𝑝11
where
𝑝11 = 𝑃(𝜑τ(𝑡) ∩ 𝜑𝛼(𝑡)) =
𝑖=1
𝑛 𝛺(𝑒𝑖,𝜑𝛼)𝛺(𝑒𝑖,𝜑τ)
𝑛
Log
Probability of a trace t
in a log 𝐿
to satisfy an RCon 𝜓
• MLE
𝑃 𝜓 𝐿 =
𝑡∈𝐿
𝑃(𝑇 = 𝑡)𝑃(𝜓(𝑡))
Probability of an RCon 𝝍 (activation 𝝋𝜶, target 𝝋𝝉)
17
Cecconi, Di Ciccio, Senderovich: Interestingness of Specifications
Labelling mechanism
𝛺𝑅 𝑒, 𝜓 =
0, if 𝛺 𝑒, 𝜑𝛼 = 1 and 𝛺 𝑒, 𝜑τ = 0
1, if 𝛺 𝑒, 𝜑𝛼 = 1 and 𝛺 𝑒, 𝜑τ = 1
⨯, otherwise
[1] Dai, Ding, Wahba, “Multivariate bernoulli distribution,” Bernoulli, 2013.](https://image.slidesharecdn.com/cdcs-icpm2022-export-221028080459-8a4717e1/85/Measurement-of-Rule-based-LTLf-Declarative-Process-Specifications-17-320.jpg)
Measurement of Rule-based LTLf Declarative Process Specifications
- 1. 𝐿 = {⟨a, b, c, d, b, c, e, c, b⟩17, ⟨b, d, a, b, b, d, e, d, c⟩6, ⟨c, d, a, b, c, e, b, c, b, c⟩5 , ⟨b, c, a, c, e, a⟩12, ⟨b, b, b⟩5} Log R1: if ‘c’ occurs, then ‘a’ must have previously occurred R2: if ‘d’ occurs, then ‘e’ must eventually occur Rules Confidence of R1: 82% Confidence of R2: 93% Measures If you consider a model consisting of R1 and R2 together, what is its confidence? Question 79 % 82 % 87.5 % 100 %
- 2. Measurement of Rule-based LTL𝑓 Declarative Process Specifications 4th Int. Conference on Process Mining, ICPM 2022, Bolzano (Italy) Alessio Cecconi Claudio Di Ciccio Arik Senderovich alessio.cecconi@wu.ac.at claudio.diciccio@uniroma1.it sariks@yorku.ca
- 3. “If d, then e will follow” “If c, then previously a” “If d, then 𝐅(e)” “If c, then 𝐎(a)” Rule: Reactive Constraint (RCon) 𝜓 3 [1] De Giacomo and Vardi, “Linear temporal logic and linear dynamic logic on finite traces,” in IJCAI, 2013 [2] Pesic, Bosnacki, van der Aalst, “Enacting declarative languages using LTL: avoiding errors and improving performance,” in SPIN, 2010 [3] C., DC., De Giacomo, and Mendling, “Interestingness of traces in declarative process mining: The Janus LTLp f approach,” in BPM, 2018 “If a viral infection is detected, then an intravenous antiviral administration will follow” “If antibiotics are administered, then an antibiogram must have been previously registered” LTL𝑓 operator (finally) [1] LTL𝑓 operator (once) d → 𝐅(e) c → 𝐎(a) Activation Activation Target Target 𝜑𝛼1 𝜑𝛼2 𝜑𝜏1 𝜑𝜏2 LTL𝑓 fomulae LTL𝑓 fomulae Any DECLARE rule [2] can be encoded as an RCon [3]
- 4. Rule: Reactive Constraint (RCon) 𝜓 4 Cecconi, Di Ciccio, Senderovich: Interestingness of Specifications Rule semantics (Interestingly) satisfied Violated Unaffected 𝜑𝜏 𝜑𝛼 𝜓 Trace 𝑡𝟐 ⟨ b, d, a, b, b, d, e, d, c ⟩ d → 𝐅(e) ● ● ● ● ● ● ● ● ● d → 𝐅(e) c → 𝐎(a) Activation Activation Target Target 𝜑𝛼1 𝜑𝛼2 𝜑𝜏1 𝜑𝜏2 LTL𝑓 fomulae LTL𝑓 fomulae
- 5. Rule: Reactive Constraint (RCon) 𝜓 5 Cecconi, Di Ciccio, Senderovich: Interestingness of Specifications Rule semantics (Interestingly) satisfied Violated Unaffected 𝜑𝜏 𝜑𝛼 𝜓 Trace 𝑡𝟐 ⟨ b, d, a, b, b, d, e, d, c ⟩ d → 𝐅(e) ● ● ● ● ● ● ● ● ● d → 𝐅(e) c → 𝐎(a) Activation Activation Target Target 𝜑𝛼1 𝜑𝛼2 𝜑𝜏1 𝜑𝜏2 LTL𝑓 fomulae LTL𝑓 fomulae
- 6. RCon Specification 𝑆 ≜ {𝜓1, 𝜓2, … 𝜓𝑛} • Semantics: a specification is • Satisfied iff an RCon is satisfied, but no violations occur • Violated iff an RCon is violated • Unaffected otherwise • A specification is like a single RCon 𝑆 = 𝑆𝛼 → 𝑆τ 𝑆𝛼 = 𝑗=1 𝑆 𝜑𝛼𝑗 ; 𝑆τ = 𝑗=1 𝑆 ¬(𝜑𝛼𝑗⋀¬𝜑𝜏𝑗 ) 6 Cecconi, Di Ciccio, Senderovich: Interestingness of Specifications d → 𝐅(e) c → 𝐎(a) Activation Activation Target Target 𝜑𝛼1 𝜑𝛼2 𝜑𝜏1 𝜑𝜏2 LTL𝑓 fomulae LTL𝑓 fomulae
- 7. d → 𝐅(e) c → 𝐎(a) Activation Activation Target Target 𝜑𝛼2 𝜑𝛼1 𝜑𝜏1 𝜑𝜏2 LTL𝑓 fomulae LTL𝑓 fomulae RCon Specification 𝑆 ≜ {𝜓1, 𝜓2, … 𝜓𝑛} • Semantics: a specification is • Satisfied iff an RCon is satisfied, but no violations occur • Violated iff an RCon is violated • Unaffected otherwise • A specification is like a single RCon 𝑆 = 𝑆𝛼 → 𝑆τ 𝑆𝛼 = 𝑗=1 𝑆 𝜑𝛼𝑗 ; 𝑆τ = 𝑗=1 𝑆 ¬(𝜑𝛼𝑗⋀¬𝜑𝜏𝑗 ) 7 Cecconi, Di Ciccio, Senderovich: Interestingness of Specifications d (¬ d ∧ ¬𝐅(e)) ∨ c ∧ (¬ c ∧ ¬𝐎(a)) 𝑆𝛼 𝑆τ LTL𝑓 fomula LTL𝑓 fomula
- 8. RCon specification measurement? 𝑡𝟏 𝑡𝟐 𝑡𝟑 𝑡𝟒 … 𝑡𝒏 𝜓1 ● ● ● ● … ● 𝜓2 ● ● ● ● … ● 𝜓3 ● ● ● ● … ● 𝑆 ● ● ● ● … ● 8 Cecconi, Di Ciccio, Senderovich: Interestingness of Specifications 𝒎(𝑺) ? ? ? ? … ?
- 9. Interestingess measures • Based on probabilities [4] – We needed to define the probability of single rules (done [5]) • We want to apply them also to entire specifications! → We need to define probabilities of specifications first (bear with us) 9 Cecconi, Di Ciccio, Senderovich: Interestingness of Specifications [4] Geng and Hamilton, “Interestingness measures for data mining: A survey,” ACM Comput. Surv., 2006. [5] C., De Giacomo, DC., Maggi, Mendling, “Measuring the interestingness of temporal logic behavioral specifications in process mining,” Inf. Syst., 2021
- 10. Probability: from LTL𝑓 rules to RCon specifications 𝜑 𝜓1 ≜ 𝜑𝛼1 → 𝜑𝜏1 𝑆 ≜ {𝜓1, 𝜓2, … , 𝜓𝑠} ↦ 𝑃(𝜑) [5] ↦ 𝑃(𝜓) [5] ↦ 𝑃 𝑆 = ? 10 Cecconi, Di Ciccio, Senderovich: Interestingness of Specifications In traces and logs! [5] C., De Giacomo, DC., Maggi, Mendling, “Measuring the interestingness of temporal logic behavioral specifications in process mining,” Inf. Syst., 2021
- 11. Trace Probability of an event 𝑒 in a trace 𝑡 of length 𝑛 to satisfy an LTL𝑓 formula 𝜑 • Bernoulli MLE [6] 𝑃 𝜑 𝑡 = 𝑖=1 𝑛 𝛺 𝑒𝑖, 𝜑 𝑛 , 𝛺 𝑒𝑖, 𝜑 ∈ 0,1 Log Probability of a trace 𝑡 in a log 𝐿 of cardinality 𝑚 to satisfy an LTL𝑓 formula 𝜑 • MLE 𝑃(𝜑 𝐿 ) = 𝑖=1 𝑚 𝑃(𝑇 = 𝑡𝑖) 𝑃(𝜑 𝑡𝑖 ) Probability of an LTL𝑓 formula 𝜑 11 Cecconi, Di Ciccio, Senderovich: Interestingness of Specifications Labelling [6] Bickel and Doksum, Mathematical statistics: basic ideas and selected topics, volumes I-II package. CRC Press, 2015.
- 12. Probability of an LTL𝑓 formula 𝜑 Example: 𝜑 ≐ 𝐅(e) Trace 𝑃 𝜑 𝑡2 = 7 9 = 0.78 12 Cecconi, Di Ciccio, Senderovich: Interestingness of Specifications Trace 𝑡𝟐 ⟨ b, d, a, b, b, d, e, d, c ⟩ 𝛺 𝑒, 𝜑 1 1 1 1 1 1 1 0 0 Trace Multiplicity 𝑃 𝜑 𝑡 𝑡1 17 0.78 𝑡2 6 0.78 𝑡3 5 0.60 𝑡4 12 0.83 𝑡5 5 0.00 Total 45 Labelling
- 13. Probability of an LTL𝑓 formula 𝜑 Example: 𝜑 ≐ 𝐅(e) Trace 𝑃 𝜑 𝑡2 = 7 9 = 0.78 Log 𝑃(𝜑 𝐿 ) = 17 45 × 0.78 + 6 45 × 0.78 + 5 45 × 0.60 + 12 45 × 0.83 + 5 45 × 0.0 = 0.69 13 Cecconi, Di Ciccio, Senderovich: Interestingness of Specifications Trace 𝑡𝟐 ⟨ b, d, a, b, b, d, e, d, c ⟩ 𝛺 𝑒, 𝜑 1 1 1 1 1 1 1 0 0 Trace Multiplicity 𝑃 𝜑 𝑡 𝑡1 17 0.78 𝑡2 6 0.78 𝑡3 5 0.60 𝑡4 12 0.83 𝑡5 5 0.00 Total 45 Labelling
- 14. Probability of an RCon 𝝍 (activation 𝝋𝜶, target 𝝋𝝉) Example: 𝜓 ≐ d → 𝐅(e) 14 Cecconi, Di Ciccio, Senderovich: Interestingness of Specifications Trace Multiplicity 𝑃 𝝍 𝑡 𝑡1 17 1.00 𝑡2 6 0.67 𝑡3 5 1.00 𝑡4 12 NaN 𝑡5 5 NaN Total 45 Trace 𝑡𝟐 ⟨ b, d, a, b, b, d, e, d, c ⟩ 𝛺 𝑒, 𝜑𝛼 0 1 0 0 0 1 0 1 0 𝛺 𝑒, 𝜑𝜏 1 1 1 1 1 1 1 0 0
- 15. Trace 𝑡𝟐 ⟨ b, d, a, b, b, d, e, d, c ⟩ 𝛺 𝑒, 𝜑𝛼 0 1 0 0 0 1 0 1 0 𝛺 𝑒, 𝜑𝜏 1 1 1 1 1 1 1 0 0 Probability of an RCon 𝝍 (activation 𝝋𝜶, target 𝝋𝝉) Example: 𝜓 ≐ d → 𝐅(e) Trace 𝑃 𝜓 𝑡 = 𝑃 𝜑𝜏 𝑡2 𝜑𝛼 𝑡2 = 2 2+1 = 0.67 15 Cecconi, Di Ciccio, Senderovich: Interestingness of Specifications 𝜑𝜏 ¬𝜑𝜏 𝜑𝛼 11 10 ¬𝜑𝛼 01 00 Trace Multiplicity 𝑃 𝝍 𝑡 𝑡1 17 1.00 𝑡2 6 0.67 𝑡3 5 1.00 𝑡4 12 NaN 𝑡5 5 NaN Total 45 Trace 𝑡𝟐 ⟨ b, d, a, b, b, d, e, d, c ⟩ 𝛺 𝑒, 𝜑𝛼 0 1 0 0 0 1 0 1 0 𝛺 𝑒, 𝜑𝜏 1 1 1 1 1 1 1 0 0 𝜴𝑹 ⅇ, 𝝍 ⨯ 1 ⨯ ⨯ ⨯ 1 ⨯ 0 ⨯ Division by 0 New labelling
- 16. Probability of an RCon 𝝍 (activation 𝝋𝜶, target 𝝋𝝉) Example: 𝜓 ≐ d → 𝐅(e) Trace 𝑃 𝜓 𝑡 = 𝑃 𝜑𝜏 𝑡2 𝜑𝛼 𝑡2 = 2 2+1 = 0.67 Log 𝑃(𝜓 𝐿 ) = 17 45 × 1.0 + 6 45 × 0. 67 + 5 45 × 1.0 = 0.58 16 Cecconi, Di Ciccio, Senderovich: Interestingness of Specifications 𝜑𝜏 ¬𝜑𝜏 𝜑𝛼 11 10 ¬𝜑𝛼 01 00 Trace Multiplicity 𝑃 𝝍 𝑡 𝑡1 17 1.00 𝑡2 6 0.67 𝑡3 5 1.00 𝑡4 12 NaN 𝑡5 5 NaN Total 45 Trace 𝑡𝟐 ⟨ b, d, a, b, b, d, e, d, c ⟩ 𝛺 𝑒, 𝜑𝛼 0 1 0 0 0 1 0 1 0 𝛺 𝑒, 𝜑𝜏 1 1 1 1 1 1 1 0 0 𝜴𝑹 ⅇ, 𝝍 ⨯ 1 ⨯ ⨯ ⨯ 1 ⨯ 0 ⨯ New labelling Division by 0
- 17. Trace Conditional probability of an event in a trace t to satisfy the target given the satisfaction of the activator • Bivariate Bernoulli MLE [1] 𝑃 𝜓 𝑡 = 𝑃(𝜑τ(𝑡)|𝜑𝛼(𝑡)) = 𝑝11 𝑝01 + 𝑝11 where 𝑝11 = 𝑃(𝜑τ(𝑡) ∩ 𝜑𝛼(𝑡)) = 𝑖=1 𝑛 𝛺(𝑒𝑖,𝜑𝛼)𝛺(𝑒𝑖,𝜑τ) 𝑛 Log Probability of a trace t in a log 𝐿 to satisfy an RCon 𝜓 • MLE 𝑃 𝜓 𝐿 = 𝑡∈𝐿 𝑃(𝑇 = 𝑡)𝑃(𝜓(𝑡)) Probability of an RCon 𝝍 (activation 𝝋𝜶, target 𝝋𝝉) 17 Cecconi, Di Ciccio, Senderovich: Interestingness of Specifications Labelling mechanism 𝛺𝑅 𝑒, 𝜓 = 0, if 𝛺 𝑒, 𝜑𝛼 = 1 and 𝛺 𝑒, 𝜑τ = 0 1, if 𝛺 𝑒, 𝜑𝛼 = 1 and 𝛺 𝑒, 𝜑τ = 1 ⨯, otherwise [1] Dai, Ding, Wahba, “Multivariate bernoulli distribution,” Bernoulli, 2013.
- 18. Probability of a specification 𝑺 18 Cecconi, Di Ciccio, Senderovich: Interestingness of Specifications Trace 𝑡𝟐 ⟨ b, d, a, b, b, d, e, d, c ⟩ 𝛺 𝑒, 𝜑𝛼1 0 0 0 0 0 0 0 0 1 𝛺 𝑒, 𝜑𝜏2 0 0 1 1 1 1 1 1 1 𝜴 ⅇ, 𝑺𝜶 0 1 0 0 0 1 0 1 1 𝜴 ⅇ, 𝑺𝝉 0 1 1 1 1 1 1 0 1 𝛺 𝑒, 𝜑𝛼2 0 1 0 0 0 1 0 1 0 𝛺 𝑒, 𝜑𝜏2 1 1 1 1 1 1 1 0 0 𝜓1 ≐ c → 𝐎(a) 𝜓2 ≐ d → 𝐅(e) 𝑆 ≐ {𝜓1, 𝜓2} MEMO 𝑆 ≜ 𝑆𝛼 → 𝑆τ where 𝑆𝛼 = 𝑗=1 𝑆 𝜑𝛼𝑗 ; 𝑆τ = 𝑗=1 𝑆 ¬(𝜑𝛼𝑗 ⋀¬𝜑𝜏𝑗 ) Trace Multiplicity 𝑃 𝑆 𝒕 𝑡1 17 1.00 𝑡2 6 0.75 𝑡3 5 0.80 𝑡4 12 0.50 𝑡5 5 NaN Total 45
- 19. Probability of a specification 𝑺 Trace: 𝑃 𝑆 𝑡 = 𝑃 𝑆τ 𝑆𝛼 = 3 3+1 = 0.75 Log: 𝑃(𝑆 𝐿 ) = 17 45 × 1.0 + 6 45 × 0.75 + 5 45 × 0.80 + 12 45 × 0.50 = 0.7 19 Cecconi, Di Ciccio, Senderovich: Interestingness of Specifications Trace 𝑡𝟐 ⟨ b, d, a, b, b, d, e, d, c ⟩ 𝛺 𝑒, 𝜑𝛼1 0 0 0 0 0 0 0 0 1 𝛺 𝑒, 𝜑𝜏2 0 0 1 1 1 1 1 1 1 𝜴 ⅇ, 𝑺𝜶 0 1 0 0 0 1 0 1 1 𝜴 ⅇ, 𝑺𝝉 0 1 1 1 1 1 1 0 1 𝛺 𝑒, 𝜑𝛼2 0 1 0 0 0 1 0 1 0 𝛺 𝑒, 𝜑𝜏2 1 1 1 1 1 1 1 0 0 𝜴 ⅇ, 𝑺𝜶 0 1 0 0 0 1 0 1 1 𝜴 ⅇ, 𝑺𝝉 0 1 1 1 1 1 1 0 1 𝜓1 ≐ c → 𝐎(a) 𝜓2 ≐ d → 𝐅(e) 𝑆 ≐ {𝜓1, 𝜓2} Trace Multiplicity 𝑃 𝑆 𝒕 𝑡1 17 1.00 𝑡2 6 0.75 𝑡3 5 0.80 𝑡4 12 0.50 𝑡5 5 NaN Total 45
- 20. Probability of a specification 𝑺 Trace: 𝑃 𝑆 𝑡 = 𝑃 𝑆τ 𝑆𝛼 = 3 3+1 = 0.75 Log: 𝑃(𝑆 𝐿 ) = 17 45 × 1.0 + 6 45 × 0.75 + 5 45 × 0.80 + 12 45 × 0.50 = 0.7 20 Cecconi, Di Ciccio, Senderovich: Interestingness of Specifications Trace 𝑡𝟐 ⟨ b, d, a, b, b, d, e, d, c ⟩ 𝛺 𝑒, 𝜑𝛼1 0 0 0 0 0 0 0 0 1 𝛺 𝑒, 𝜑𝜏2 0 0 1 1 1 1 1 1 1 𝜴 ⅇ, 𝑺𝜶 0 1 0 0 0 1 0 1 1 𝜴 ⅇ, 𝑺𝝉 0 1 1 1 1 1 1 0 1 𝛺 𝑒, 𝜑𝛼2 0 1 0 0 0 1 0 1 0 𝛺 𝑒, 𝜑𝜏2 1 1 1 1 1 1 1 0 0 𝜴 ⅇ, 𝑺𝜶 0 1 0 0 0 1 0 1 1 𝜴 ⅇ, 𝑺𝝉 0 1 1 1 1 1 1 0 1 𝜓1 ≐ c → 𝐎(a) 𝜓2 ≐ d → 𝐅(e) 𝑆 ≐ {𝜓1, 𝜓2} Trace Multiplicity 𝑃 𝑆 𝒕 𝑡1 17 1.00 𝑡2 6 0.75 𝑡3 5 0.80 𝑡4 12 0.50 𝑡5 5 NaN Total 45
- 21. Trace Conditional probability of an event in a trace t to satisfy the target given the satisfaction of the activator • MLE 𝑃 𝑆 𝑡 = 𝑃(𝑆τ(𝑡)|𝑆𝛼(𝑡)) = 𝑝11 𝑝01 + 𝑝11 where 𝑝11 = 𝑃(𝑆τ(𝑡) ∩ 𝑆𝛼(𝑡)) = 𝑖=1 𝑛 𝛺(𝑒𝑖,𝑆𝛼)𝛺(𝑒𝑖,𝑆τ) 𝑛 Log Probability of a trace t in a log 𝐿 to satisfy a specification 𝑆 • MLE 𝑃 𝑆 𝐿 = 𝑡∈𝐿 𝑃(𝑇 = 𝑡)𝑃(𝑆(𝑡)) Probability of a specification 𝑺 21 Cecconi, Di Ciccio, Senderovich: Interestingness of Specifications MEMO 𝑆 ≜ 𝑆𝛼 → 𝑆τ where 𝑆𝛼 = 𝑗=1 𝑆 𝜑𝛼𝑗 ; 𝑆τ = 𝑗=1 𝑆 ¬(𝜑𝛼𝑗 ⋀¬𝜑𝜏𝑗 )
- 22. Probability: From LTL𝑓 rules to RCon specifications 𝜑 𝜓1 ≜ 𝜑𝛼1 → 𝜑𝜏1 𝑆 ≜ {𝜓1, 𝜓2, … , 𝜓𝑠} ↦ 𝑃(𝜑) ↦ 𝑃(𝜓) = 𝑃 𝜑𝜏 𝜑𝛼 ↦ 𝑃 𝑆 = 𝑃(𝑆𝜏|𝑆𝛼) 22 Cecconi, Di Ciccio, Senderovich: Interestingness of Specifications Done! For traces and logs!
- 23. Specification measurements 23 Cecconi, Di Ciccio, Senderovich: Interestingness of Specifications Measure Formula for 𝐴 ⇒ 𝐵 Support 𝑃(𝐴𝐵) Recall 𝑃(𝐴|𝐵) Confidence 𝑃(𝐵|𝐴) Specificity 𝑃(¬𝐵|¬𝐴) Lift 𝑃(𝐴𝐵) 𝑃 𝐴 𝑃(𝐵) … … Measure Formula for 𝑆 (trace) Formula for 𝑆 (log) Support 𝑃(Sα ∩ Sτ, t) 𝑃(Sα ∩ Sτ, 𝐿) Recall 𝑃(Sα|Sτ, t) 𝑃(Sα|Sτ, 𝐿) Confidence 𝑃(Sτ|Sα, t) 𝑃(Sτ|Sα, 𝐿) Specificity 𝑃(¬Sτ|¬Sα, t) 𝑃(¬Sτ|¬Sα, 𝐿) Lift 𝑃(Sα ∩ Sτ, t) 𝑃 Sα, t 𝑃(Sτ, t) 𝑃(Sα ∩ Sτ, 𝐿) 𝑃 Sα, 𝐿 𝑃(Sτ, 𝐿) … … …
- 24. Evaluation: Discovery setup vs discovered quality 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1 0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 Discovred specification confidence Confidence threshold 24 Cecconi, Di Ciccio, Senderovich: Interestingness of Specifications
- 25. Evaluation: Discovery setup vs discovered quality 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1 0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 Discovred specification confidence Confidence threshold JANUS MINERful PERRACOTTA 25 Cecconi, Di Ciccio, Senderovich: Interestingness of Specifications
- 26. Recap and limitations • Computation of quality measures for declarative process specifications • Control-flow only • Assumption of probabilistic independence Future work • Multi-perspective approach • Upgrade to Bayesian networks • Logistic regression & statistical analysis • Applications of data featurization – E.g., for trace clustering Conclusions 26 Cecconi, Di Ciccio, Senderovich: Interestingness of Specifications
- 27. 𝐿 = {⟨a, b, c, d, b, c, e, c, b⟩17, ⟨b, d, a, b, b, d, e, d, c⟩6, ⟨c, d, a, b, c, e, b, c, b, c⟩5 , ⟨b, c, a, c, e, a⟩12, ⟨b, b, b⟩5} Log R1: if ‘c’ occurs, then ‘a’ must have previously occurred R2: if ‘d’ occurs, then ‘e’ must eventually occur Rules Confidence of R1: 82% Confidence of R2: 93% Measures If you consider a model consisting of R1 and R2 together, what is its confidence? Question 79 % 82 % 87.5 % 100 %
- 28. Measurement of Rule-based LTL𝑓 Declarative Process Specifications 4th Int. Conference on Process Mining, ICPM 2022, Bolzano (Italy) Alessio Cecconi Claudio Di Ciccio Arik Senderovich alessio.cecconi@wu.ac.at claudio.diciccio@uniroma1.it sariks@yorku.ca