Equivalences in Alloy

We say that two non-temporal Alloy expressions E1 and E2 of the same type are equivalent, written E1 ≡ E2, if they evaluate to the same value in every Alloy instance. Below are some noteworthy equivalences in Alloy.

Set Operators

For all set/relations A, B, and C, the following equivalences hold in Alloy whenever their left- and right-hand sides are well typed:

• A & univ   ≡   A
• A & none   ≡   none
• A & A   ≡   A
• A + A   ≡   A
• A + univ   ≡   univ
• A + none   ≡   A
• A & B   ≡   B & A
• A + B   ≡   B + A
• A & (B & C)   ≡   (A & B) & C
• A + (B + C)   ≡   (A + B) + C
• A & (B + C)   ≡   (A & B) + (A & C)
• A + (B & C)   ≡   (A + B) & (A + C)
• A . (B . C)   ≡   (A . B) . C
• A . (B + C)   ≡   (A . B) + (A . C)
• (A + B) . C   ≡   (A . C) + (B . C)
• ~~A   ≡   A
• ~(A & B)   ≡   ~A & ~B
• ~(A + B)   ≡   ~A + ~B
• ~(A . B)   ≡   ~B . ~A
• ~^A   ≡   ^~A
• ~*A   ≡   *~A
• ^A   ≡   A    whenever is A is transitive
• *A   ≡   A    whenever is A is reflexive and transitive
• ^^A   ≡   ^A
• **A   ≡   *A

Boolean Operators

Let ⊤ denote any formula that is alway true (e.g., none = none) and let ⊥ denote any formula that is alway false (e.g., (none != none)).

For all set/relations A, B, C, and x where x is a singleton, the following equivalences hold in Alloy whenever their left- and right-hand sides are well typed:

• none in A   ≡   ⊤
• some A and A in none   ≡   ⊥
• A in A   ≡   ⊤
• A in B   ≡   (A & B) = A
• A in B   ≡   (A + B) = B
• A = B   ≡   (A in B) and (B in A)
• x in (A & B)   ≡   (x in A) and (x in B)
• x in (A + B)   ≡   (x in A) or (x in B)
• x in (A - B)   ≡   (x in A) and (x !in B)
• no A   ≡   A = none
• no A   ≡   all x: A | ⊥
• some A   ≡   A != none
• some A   ≡   some x: A | ⊤
• lone A   ≡   all x, y: A | x = y
• one A   ≡   some x: A | x = A
• one A   ≡   (some A) and all x, y: A | x = y

Boolean Connectives

For all formulas F, F1, F2, G, G1, and G2, the following equivalences hold in Alloy:

• F and ⊤   ≡   F
• F or ⊤   ≡   ⊤
• F and ⊥   ≡   ⊥
• F or ⊥   ≡   F
• F and G   ≡   G and F
• F or G   ≡   G or F
• F and (G and H)   ≡   (F and G) and H
• F or (G or H)   ≡   (F or G) or H
• F and (G or H)   ≡   (F and G) or (F and H)
• F or (G and H)   ≡   (F or G) and (F or H)
• not not F   ≡   F
• not (F and G)   ≡   (not F) or (not G)
• not (F or G)   ≡   (not F) and (not G)
• F implies G   ≡   (not F) or G
• F implies ⊤   ≡   ⊤
• F implies ⊥   ≡   not F
• ⊤ implies G   ≡   G
• ⊥ implies G   ≡   ⊤
• not (F implies G)   ≡   F and (not G)
• (F1 and F2) implies G   ≡   F1 implies (F2 implies G)
• F implies (G1 and G2)   ≡   (F implies G1) and (F implies G2)
• (F1 or F2) implies G   ≡   (F1 implies G) and (F2 implies G)
• F implies (G1 or G2)   ≡   (F implies G1) or (F implies G2)

Quantifiers

For all formulas F and all sets A and B, the following equivalences hold in Alloy:

• all x: A | ⊤   ≡   ⊤
• some x: A | ⊥   ≡   ⊥
• all x: none | F   ≡   ⊤
• some x: none | F   ≡   ⊥
• all x: A | all y: B | F   ≡   all y: B | all x: A | F
• some x: A | some y: B | F   ≡   some y: B | some x: A | F
• not all x: A | F   ≡   some x: A | not F
• not some x: A | F   ≡   all x: A | not F
• all x: A | (F and G)   ≡   (all x: A | F) and (all x: A | G)
• some x: A | (F or G)   ≡   (some x: A | F) or (some x: A | G)
• no x: A | F   ≡   all x: A | not F
• all x: A | F   ≡   { x: A | F } = A
• some x: A | F   ≡   some { x: A | F }
• no x: A | F   ≡   no { x: A | F }
• one x: A | F   ≡   one { x: A | F }
• lone x: A | F   ≡   (no x: A | F) or (one x: A | F)

Temporal operators

We say that two temporal Alloy formulas F and G are equivalent, written F ≡ G, if they are satisfied by exactly the same traces.

For all temporal formulas F and G, the following equivalences hold in Alloy 6.

Note: Keep in mind that unary temporal operators bind more strongly than any binary operators.
(E.g., always F and G parses as (always F) and G, not as always (F and G).)

Future operators

1. always F   ≡   F and after always F
2. eventually F   ≡   F or after eventually F
3. always always F   ≡   always F
4. eventually eventually F   ≡   eventually F
5. not always F   ≡   eventually not F
6. not eventually F   ≡   always not F
7. always (F and G)   ≡   (always F) and (always G)
8. eventually (F or G)   ≡   (eventually F) or (eventually G)
9. not after F   ≡   after not F
10. eventually F   ≡   ⊤ until F
11. F until G   ≡   G or (F and after (F until G))
12. after always before ⊤   ≡   ⊤

Past operators

1. historically F   ≡   F and (not before ⊤ or before historically F)
2. once F   ≡   F or before once F
3. historically historically F   ≡   historically F
4. once once F   ≡   once F
5. not historically F   ≡   once not F
6. not once F   ≡   historically not F
7. historically (F and G)   ≡   (historically F) and (historically G)
8. once (F or G)   ≡   (once F) or (once G)
9. once F   ≡   ⊤ since F
10. F since G   ≡   G or (F and before (F since G))