1. Expressing set properties in Alloy
Roger L. Costello
March 10, 2018

2. Want to learn sets? Learn Alloy! Want to learn Alloy? Learn sets!
Certain properties of binary relations are so frequently encountered that they have been given names, including reflexive, symmetric, transitive, and connected.
All these apply only to relations in a set, i.e., in A x A, not to relations from A to B.

3. Reflexive property
Given a set A and a relation R in A, R is reflexive if and only if all the ordered pairs of the form (x, x) are in R for every x in A.
Example: Take the set A = { 1, 2, 3 } and the relation R1 = { (1,1), (1,2), (2,2), (2,3), (3,3), (3,1) } in A. R1 is reflexive because it contains the ordered pairs (1,1), (2,2), and (3,3). 1 2 3

4. Each person has the same birthday as themself
Take the set A = the set of human beings and the relation R2 = ‘has the same birthday’.
R2 is reflexive because it contains the ordered pairs:
Roger Costello has the same birthday as Stan Efferding.
Roger Costello has the same birthday as Roger Costello.
Stan Efferding has the same birthday as Roger Costello.
Stan Efferding has the same birthday as Stan Efferding. …
R2 = { (Roger Costello, Stan Efferding), (Roger Costello, Roger Costello), (Stan Efferding, Roger Costello), (Stan Efferding, Stan Efferding), … }

5. Alloy expression that constrains pairs of elem atoms to be reflexive
sig A {}
sig Test { R: A -> A } { (A <: iden) in R // Constrain R to be reflexive }
assert isReflexive { all x: A | (x -> x) in Test.R } check isReflexive

6. iden
sig A {}
sig Test { R: A -> A } { (A <: iden) in R // Constrain R to be reflexive }
assert isReflexive { all x: A | (x -> x) in Test.R } check isReflexive
Test0 A0 A1 A2

7. Reflexive Transitive Symmetric Connected Nonsymmetric Nonreflexive1 2 3 4 1 2 3 4 5 6 7 8 1 2 3 1 2 3
Nonsymmetric
Nonreflexive
Nontransitive
Nonconnected 1 2 3 1 2 3 4 1 2 3 1 2 3 4 5 6 7 8
Intransitive
Asymmetric
Irreflexive 1 2 3 4 1 2 3 1 2 3

8. Injective Functional Not Injective Not Functional 1 2 1 2 3 4 3 5 1 2

9. The ordering module
We have used the ordering module to order sets, e.g.,
open util/ordering[Snapshot]
// Order the set of Snapshots in the goat, cabbage, wolf model
open util/ordering[House]
// Order the set of Houses in the Einstein model
open util/ordering[Desktop]
// Order the set of Desktops in the Desktop model

10. The ordering module creates ordered pairs from your set
When the ordering module is called with a set then all these functions are suddenly available on the set: first, last, next.
first returns the first atom.
last returns the last atom.
next returns a set of pairs, such as this:
next
first returns the first Snapshot
first.next returns the second Snapshot
first.next.next returns the third Snapshot
And so forth
Snapshot0
Snapshot1
Snapshot1
Snapshot2
Snapshot2
Snapshot3

11. next: Snapshot Snapshot1 Snapshot3 Snapshot2 Snapshot0 Ordering Module
first
last
next:
Snapshot0
Snapshot1
Snapshot2
Snapshot3

12. Which of these properties do we want the relation (ordered pairs) to have?1 2 3
Reflexive
Transitive
Symmetric 4 5 6 7 8
Connected
Nonreflexive
Irreflexive
Nontransitive
Intransitive
Nonsymmetric
Asymmetric
Nonconnected

13. Which of these properties do we want the relation (ordered pairs) to have?1 2 3
Reflexive
Transitive
Symmetric 4 5 6 7 8
Connected
Nonreflexive
Irreflexive
Nontransitive
Intransitive
Nonsymmetric
Asymmetric
Nonconnected

14. Injective Functional Not Injective Not Functional 1 2 1 2 3 4 3 5 1 2

15. sig A {}
one sig Ord {
First: A,
Next: A -> A
} {
no x: A | (x -> First) in Next // First is the first atom in the list
irreflexive [A, Next] // Constrain Next to be irreflexive
intransitive [A, Next] // Constrain Next to be intransitive
asymmetric [A, Next] // Constrain Next to be asymmetric
nonconnected [A, Next] // Constrain Next to be non-connected
injective [A, Next] // Constrain Next to be injective
functional [A, Next] // Constrain Next to be functional
#Next = minus[#A, 1] // Constraint Next to contain all atoms in A }

16. Hold on! The ordering module also creates a prev function
The prev function allows you to traverse to the previous atom in the list.
The below graphic is not accurate, how should it be changed?
first
last
next:
Snapshot0
Snapshot1
Snapshot2
Snapshot3

17. Arrows must go both directions
first
last
next
Snapshot0
Snapshot1
Snapshot2
Snapshot3
prev

18. Now which of these properties do we want the relation (ordered pairs) to have?1 2 3
Reflexive
Transitive
Symmetric 4 5 6 7 8
Connected
Nonreflexive
Irreflexive
Nontransitive
Intransitive
Nonsymmetric
Asymmetric
Nonconnected

19. Which of these properties do we want the relation (ordered pairs) to have?1 2 3
Reflexive
Transitive
Symmetric 4 5 6 7 8
Connected
Nonreflexive
Irreflexive
Nontransitive
Intransitive
Nonsymmetric
Asymmetric
Nonconnected

20. Injective Functional Not Injective Not Functional 1 2 1 2 3 4 3 5 1 2

21. irreflexive [A, Next] // Constrain Next to be irreflexive
sig A {}
one sig Ord {
First: A,
Next: A -> A
} {
no x: A | (x -> First) in Next // First is the first atom in the list
irreflexive [A, Next] // Constrain Next to be irreflexive
intransitive [A, Next] // Constrain Next to be intransitive
symmetric [A, Next] // Constrain Next to be symmetric
nonconnected [A, Next] // Constrain Next to be non-connected
injective [A, Next] // Constrain Next to be injective
functional [A, Next] // Constrain Next to be functional
#Next = minus[#A, 1] // Constraint Next to contain all atoms in A }
Do Labs 5,6