This was a work-in progress presentation given in Nov 2010 (uploaded here for historic records). It was part of the ACSI EU project for process mining over processes with multiple objects. The slides discuss an example of an order process over multiple objects and propose how to model this data in a graph (event relation graph) that is stored in a database and supported by a relational algebra over behavior (e.g. to join event traces of multiple objects into a case). The final part of the presentation discusses the synchronous proclet model ultimately formalized in https://doi.org/10.1007/978-3-030-21571-2_1

1. Artifacts and Databases
Need for Event Relation Graphs and Synchronous Composition
Dirk Fahland, 2010-11-23

2.  abstract model of a database
• with events
• a normal form
• the Event Relation Graph
 What is a case?
• Some ideas
• a “case algebra”
 Synchronous artifact model
Overview
PAGE 1

3.  Database DB = { R1, R2, …, Rn}
• table is a relation Ri  D1  …  Dk
 events: defined by “time-stamp” columns in DB, so:
Ri  D1  …  Dk  Tk+1  …  Tl
 relations between different tables:
• key-foreign key relations defined in some DB schema
PAGE 2
Abstract model of a database
data in Ti time-stamps in Ri

4.  given table Ri  D1  …  Dk  Tk+1  …  Tl
 separate data Di from events Ti
 assume each Ri with events defines a key D1 = Ki
 Ri  Ki  D2  …  Dk  Tk+1  …  Tl
 data table event table
Ri  Ki  D2  …  Dk Ei  Ki  Type  Time
PAGE 3
Normal form
{k+1,…,l} distinguish events of the
different columns Tk+1  …  Tl

5.  DB in normal form {R1, R2, …, Rn , E1, E2, …, En}
Ri  Ki  D2  …  Dk, Ki – key of table Ri
Ei  Ki  Type  Time
key(e) := Ki iff e  Ei
 Event Relation Graph: G = (V, F)
 V = E1  … En, F  V  V
 F = { (e1,e2) | there is a key-foreign-key relation
from key(e1) to key(e2) or key(e1) = key(e2) }
PAGE 4
Event Relation Graph

6. PAGE 5
Example
Order
Customer : CustID
Amount : Euro
Created : DateTime
Paid : DateTime
Completed : DateTime
Orderline
Product : ProdID
NofItems : PosInt
TotalWeight : Weight
Entered : DateTime
BackOrdered : DateTime
Secured : DateTime
Delivery
DelAddress : Address
Contact : PhoneNo
Attempt
When : DateTime
Successful : Bool
1
1..*
0..* 1
0..1
1..*
OrderID : OrderID OrderID : OrderID
OrderLineID : OrderLineID
DelID : DelID
DelID : DelID
DelID : DelID
Order = [OrderID:OrderID, Customer:CustomerID, Amount:Euro]
EOrder = [OrderID:OrderID, Type:{Created,Paid,Completed}, Time:DateTime]

7. PAGE 6
Example
Order
Customer : CustID
Amount : Euro
Created : DateTime
Paid : DateTime
Completed : DateTime
Orderline
Product : ProdID
NofItems : PosInt
TotalWeight : Weight
Entered : DateTime
BackOrdered : DateTime
Secured : DateTime
Delivery
DelAddress : Address
Contact : PhoneNo
Attempt
When : DateTime
Successful : Bool
1
1..*
0..* 1
0..1
1..*
OrderID : OrderID OrderID : OrderID
OrderLineID : OrderLineID
DelID : DelID
DelID : DelID
DelID : DelID
OrderLine = [OrderLineID:OrderLineID, OrderID:OrderID,…]
EOrderLine = [OrderLineID:OrderLineID, Type:{Entered, …}, Time:DateTime]

8. PAGE 7
Example: events
Order
OrderID Amount Created Paid Completed
91245 100 € 28-11-10(8:12) 2-12-10(13:45) 5-12-10(11:33)
Customer
“John”
91561 530 € 28-11-10(12:22) 3-12-10(14:34) 5-12-10(9:32)
“Mike”
91812 234 € 29-11-10(9:45) 2-12-10(9:44) 4-12-10(13:33)
“Mary”
92233 110 € 29-11-10(10:12) null null
“Sue”
92345 195 € 29-11-10(14:45) 2-12-10(13:45) null
“Kirsten”
92355 320 € 29-11-10(16:32) null null
“Pete”
... ... ... ... ... ...
[91245, Created, 28-11-10(8:12)]
Eorder:
[91245, Created, 02-12-10(13:45)]
[91245, Created, 05-12-10(11:33)]
[92233, Created, 29-11-10(10:12)]
the other events
for 92233 do not exist
…
…

9. PAGE 8
Example: Event Relation Graph
Order
OrderID Amount Created Paid Completed
91245 100 € 28-11-10(8:12) 2-12-10(13:45) 5-12-10(11:33)
Customer
“John”
91561 530 € 28-11-10(12:22) 3-12-10(14:34) 5-12-10(9:32)
“Mike”
Orderline
Product NofItems TotalWeight Entered BackOrdered Secured
OrderID
OrderLineID DelID
91245
91245
112345
91561
112346
112448
“iPhone 3GS”
“iPhone 3GS”
“iPod nano”
1
2
1
0.250
0.300
0.250
28-11-10(8:13)
28-11-10(8:14)
28-11-10(12:23)
null 28-11-10(8:55) 882345
28-11-10(8:55) 30-11-10(9:06) 882346
null 28-11-10(12:59) 882345
245,Cr 245,P 245,Co 561,Cr 561,P 561,Co
345,E 345,Se 346,E 346,Bo 346,Se 448,E 448,Se

10. PAGE 9
Example: Event Relation Graph
Order
OrderID Amount Created Paid Completed
91245 100 € 28-11-10(8:12) 2-12-10(13:45) 5-12-10(11:33)
Customer
“John”
91561 530 € 28-11-10(12:22) 3-12-10(14:34) 5-12-10(9:32)
“Mike”
Orderline
Product NofItems TotalWeight Entered BackOrdered Secured
OrderID
OrderLineID DelID
91245
91245
112345
91561
112346
112448
“iPhone 3GS”
“iPhone 3GS”
“iPod nano”
1
2
1
0.250
0.300
0.250
28-11-10(8:13)
28-11-10(8:14)
28-11-10(12:23)
null 28-11-10(8:55) 882345
28-11-10(8:55) 30-11-10(9:06) 882346
null 28-11-10(12:59) 882345
245,Cr 245,P 245,Co 561,Cr 561,P 561,Co
345,E 345,Se 346,E 346,Bo 346,Se 448,E 448,Se
transitively
reduced

11. PAGE 10
Example: Event Relation Graph
245,Cr 245,P 245,Co 561,Cr 561,P 561,Co
345,E 345,Se 346,E 346,Bo 346,Se 448,E 448,Se
transitively
reduced
Attempt
When Successful
DelID
...
882345
882346
882345
882345
5-12-10(8:55)
6-12-10(9:12)
7-12-10(8:56)
5-12-10(8:43)
...
false
false
true
true
...
Orderline
OrderID
OrderLineID DelID
91245
91245
112345
91561
112346
112448
882345
882346
882345
882345,A,t1 882345,A,t2 882345,A,t3 882346,A,t1

12.  Abstract model of database
• with events
• a normal form
• the event relation graph
 What is a case?
• some ideas
• a “case algebra”
 Synchronous artifact model
Overview
PAGE 11

13.  describes which event relates by data to which
other events
 directed:
• event relations that can be retrieved by following the key-
foreign key relations forward
• directions indicate which events should be considered
together for a “case”
 suggests notion of a case: subgraph
• starting at a specific event (without predecessor?)
• containing all transitive predecessor
and/or successors (or only up to a certain point)
• “viewpoint” = set of all cases of a specific kind
PAGE 12
Properties of the Event Relation Graph

14. PAGE 13
Example: Cases
245,Cr 245,P 245,Co 561,Cr 561,P 561,Co
345,E 345,Se 346,E 346,Bo 346,Se 448,E 448,Se
882345,A,t1 882345,A,t2 882345,A,t3 882346,A,t1

15. PAGE 14
Example: Order Line, Case 1
245,Cr 245,P 245,Co 561,Cr 561,P 561,Co
345,E 345,Se 346,E 346,Bo 346,Se 448,E 448,Se
882345,A,t1 882345,A,t2 882345,A,t3 882346,A,t1

16. PAGE 15
Example: Order Line, Case 2
245,Cr 245,P 245,Co 561,Cr 561,P 561,Co
345,E 345,Se 346,E 346,Bo 346,Se 448,E 448,Se
882345,A,t1 882345,A,t2 882345,A,t3 882346,A,t1

17. PAGE 16
Example: Order Line, Case 3
245,Cr 245,P 245,Co 561,Cr 561,P 561,Co
345,E 345,Se 346,E 346,Bo 346,Se 448,E 448,Se
882345,A,t1 882345,A,t2 882345,A,t3 882346,A,t1

18. PAGE 17
Example: Complete Order, Case 1
245,Cr 245,P 245,Co 561,Cr 561,P 561,Co
345,E 345,Se 346,E 346,Bo 346,Se 448,E 448,Se
882345,A,t1 882345,A,t2 882345,A,t3 882346,A,t1

19. PAGE 18
Example: Complete Order, Case 2
245,Cr 245,P 245,Co 561,Cr 561,P 561,Co
345,E 345,Se 346,E 346,Bo 346,Se 448,E 448,Se
882345,A,t1 882345,A,t2 882345,A,t3 882346,A,t1

20. PAGE 19
Example: Complete Delivery, Case 1
245,Cr 245,P 245,Co 561,Cr 561,P 561,Co
345,E 345,Se 346,E 346,Bo 346,Se 448,E 448,Se
882345,A,t1 882345,A,t2 882345,A,t3 882346,A,t1
etc.

21.  a case is a subgraph of the event relation graph
 different notions of a case, cases can be combined,
e.g. complete order + complete delivery etc.
 define a case algebra:
• basic cases
• composition of cases (union)
• composition wrt. case properties (only completed cases)
• joins, etc.
 operations all should follow from relational algebra
of the underlying database
PAGE 20
Case Algebra for Artifact Models

22. PAGE 21
Example:
Complete Order 
 Complete Delivery, 1
245,Cr 245,P 245,Co 561,Cr 561,P 561,Co
345,E 345,Se 346,E 346,Bo 346,Se 448,E 448,Se
882345,A,t1 882345,A,t2 882345,A,t3 882346,A,t1

23. PAGE 22
Example:
Complete Order 
 Complete Delivery, 2
245,Cr 245,P 245,Co 561,Cr 561,P 561,Co
345,E 345,Se 346,E 346,Bo 346,Se 448,E 448,Se
882345,A,t1 882345,A,t2 882345,A,t3 882346,A,t1

24. PAGE 23
Example:
Complete Order 
 Complete Delivery, 3
245,Cr 245,P 245,Co 561,Cr 561,P 561,Co
345,E 345,Se 346,E 346,Bo 346,Se 448,E 448,Se
882345,A,t1 882345,A,t2 882345,A,t3 882346,A,t1

25.  Abstract model of database
• with events
• a normal form
• the event relation graph
 What is case?
• Some ideas
• a “case algebra”
 Synchronous artifact model
Overview
PAGE 24

26.  Proposition: Synchronous communication
describes more precisely how a set of artifacts
relates to a database.
PAGE 25
Aim: relate Database to Artifact model

27. Artifact
B C
Artifact C
Artifact B
Artifact A
Artifact
B A
PAGE 26
A first idea…
Table A
Table C
Table B
date1, date2
date3, date4
date5, date6
d1 d2 d1 d2
d3 d4
d5 d6 d5 d6
d3 d4
Event Relation Graph
Database schema

28. Artifact
B C
Artifact C
Artifact B
PAGE 27
A first idea…
Table A
Table C
Table B
date1, date2
date3, date4
date5, date6
d1 d2 d1 d2
d3 d4
d5 d6 d5 d6
d3 d4
“Artifact-Centric Traces”
Database schema
Artifact A
Artifact
B A
d1 d2
d3 d4 d3 d4
d5 d6 d5 d6
d3 d4 d3 d4
d1 d2

29. Artifact
B C
Artifact C
Artifact B
PAGE 28
A first idea…
d1 d2 d1 d2
d3 d4
d5 d6 d5 d6
d3 d4
“Artifact-Centric Traces”
Artifact A
Artifact
B A
d1 d2
d3 d4 d3 d4
d5 d6 d5 d6
d3 d4 d3 d4
d1 d2
describes occurrences of
events wrt. Table A
describes occurrences of
events the relation from B to A
describes occurrences of
events wrt. Table B

30. PAGE 29
Corresponding Petri net model
d1 d2 d1 d2 Artifact A
Artifact
B A
d1 d2
d3 d4 d3 d4
d1 d2
d1
d2
d1
d2
d3
d4
d3
d4
Artifact B
d3 d4 d3 d4
A B  A B
…

31. PAGE 30
Corresponding Artifact-Model
d1 d2 d1 d2 Artifact A
Artifact
B A
d1 d2
d3 d4 d3 d4
d1 d2
d1
d2
d1
d2
d3
d4
d3
d4
Artifact B
d3 d4 d3 d4
A B  A B
1,1
1,1
1,*
1,*
1,*
1,*
1,*
1,*

32. PAGE 31
Semantics of Synchronous Ports?
d1
d2
d1
d2
d3
d4
d3
d4
A B  A B
1,1
1,1
1,*
1,*
1,*
1,*
1,*
1,*
c,m c,m
X X
A B
multiplicity m: how often X occurs in
one instance of A
cardinality c: with how many
instances of B does an occurrence
of X in A synchronize?

33. PAGE 32
Semantics of Synchronous Ports?
d1
d2
d1
d2
A B
1,1
*,1
*,1
*,1
d1
d2
B1
d1
d2
A1
d1
d2
B2
d1
d2
A2
d1
d2
A3
N:M Proclet Model
Instances

34. PAGE 33
Semantics of Synchronous Ports?
d1
d2
B1
d1
d2
A1
d1
d2
B2
d1
d2
A2
d1
d2
A3
A2
d1
d2
A1
d1
d2
B1
d1
d2
A3
d1
d2
B2
d1
d2
the partially ordered runs of A1,…,B2
+ synchronization

35. PAGE 34
Semantics of Synchronous Ports?
d1
d2
B1
d1
d2
A1
d1
d2
B2
d1
d2
A2
d1
d2
A3
A2
d1
d2
A1
d1
d2
B1
d1
d2
A3
d1
d2
B2
d1
d2
the partially ordered runs of A1,…,B2
+ synchronization