1) The document discusses unification and the union-find algorithm for efficiently solving unification problems. 2) Unification involves resolving constraints between terms to find a substitution that makes the terms equal. This can have exponential time and space complexity. 3) The union-find algorithm uses set representatives and linking to maintain disjoint sets in near-constant time complexity, improving on naive recursive algorithms. 4) An example shows how union-find can be applied to fully unify a complex expression by progressively linking variables and subterms between the two sides.

1. Hendrik van Antwerpen
IN4303 Compiler Construction
TU Delft
September 2017
Declare Your Language
Chapter 8: Constraint Resolution II

2. Reading Material
2

3. Unification
3
Baader, F., and W. Snyder
"Unification Theory”
Ch. 8 of Handbook of Automated Deduction
Springer Verlag, Berlin (2001)
http://www.cs.bu.edu/~snyder/publications/UnifChapter.pdf

4. Efficient Unification
with Union-Find
4

5. Space complexity
- Exponential
- Representation of unifier
Time complexity
- Exponential
- Recursive calls on terms
Solution
- Union-Find algorithm
- Complexity growth can be
considered constant
Complexity of Unification (recap)
5
h(a1 , …,an , f(b0,b0), …, f(bn-1,bn-1), an) ==
h(f(a0,a0), …,f(an-1,an-1), b1, …, bn-1 , bn)
a1 -> f(a0,a0)
a2 -> f(f(a0,a0), f(a0,a0))
ai -> … 2i+1-1 subterms …
b1 -> f(a0,a0)
b2 -> f(f(a0,a0), f(a0,a0))
bi -> … 2i+1-1 subterms …
a1 -> f(a0,a0)
a2 -> f(a1,a1)
ai -> … 3 subterms …
b1 -> f(a0,a0)
b2 -> f(a1,a1)
bi -> … 3 subterms …
fully applied triangular

6. Set Representatives
6

7. Set Representatives
6
a == b
c == a

8. Set Representatives
6
a == b
c == a
a
b c

9. Set Representatives
6
a == b
c == a
u == w
v == u
x == v
a
b c

10. Set Representatives
6
a == b
c == a
u == w
v == u
x == v
a
b c
u
w v
x

11. Set Representatives
6
a == b
c == a
u == w
v == u
x == v
a
b c
u
w v
x
representative

12. Set Representatives
6
a == b
c == a
u == w
v == u
x == v
a
b c
u
w v
x
representative

13. Set Representatives
6
a == b
c == a
u == w
v == u
x == v
a
b c
u
w v
x

14. Set Representatives
6
a == b
c == a
u == w
v == u
x == v
x == c
a
b c
u
w v
x

15. Set Representatives
6
a == b
c == a
u == w
v == u
x == v
x == c
a
b c
u
w v
x

16. Set Representatives
6
a == b
c == a
u == w
v == u
x == v
x == c
a
b c
u
w v
x

17. Set Representatives
6
a == b
c == a
u == w
v == u
x == v
x == c
a
b c
u
w v
x

18. Set Representatives
6
FIND(a):
b := rep(a)
if b == a:
return a
else
return FIND(b)
UNION(a1,a2):
b1 := FIND(a1)
b2 := FIND(a2)
LINK(b1,b2)
LINK(a1,a2):
rep(a1) := a2
a == b
c == a
u == w
v == u
x == v
x == c
a
b c
u
w v
x

19. Path Compression
7
a
b c
u
w v
x

20. Path Compression
7
…
x == b
x == c
x == w
x == v
a
b c
u
w v
x

21. Path Compression
7
…
x == b
x == c
x == w
x == v
a
b c
u
w v
x

22. Path Compression
7
…
x == b
x == c
x == w
x == v
a
b c
u
w v
x

23. Path Compression
7
…
x == b
x == c
x == w
x == v
a
b c
u
w v
x

24. Path Compression
7
…
x == b
x == c
x == w
x == v
a
b c
u
w v
x

25. Path Compression
7
FIND(a):
b := rep(a)
if b == a:
return a
else
b := FIND(b)
rep(a) := b
return b
UNION(a1,a2):
b1 := FIND(a1)
b2 := FIND(a2)
LINK(b1,b2)
LINK(a1,a2):
rep(a1) := a2
…
x == b
x == c
x == w
x == v
a
b c
u
w v
x

26. Tree Balancing
8
…
x == c
a
b c
u
w v
x

27. Tree Balancing
8
…
x == c
a
b c
u
w v
x

28. Tree Balancing
8
…
x == c
a
b c
u
w v
x

29. Tree Balancing
8
…
x == c
a
b c
u
w v
x
3 steps

30. Tree Balancing
8
…
x == c
a
b c
u
w v
x

31. Tree Balancing
8
…
x == c
a
b c
u
w v
x
2 steps

32. Tree Balancing
8
…
x == c
a
b c
u
w v
x
?

33. Tree Balancing
8
…
x == c
a
b c
u
w v
x
1
111
?

34. Tree Balancing
8
…
x == c
a
b c
u
w v
x
1
2111
?

35. Tree Balancing
8
…
x == c
a
b c
u
w v
x
1
21
4
11
3
?

36. Tree Balancing
8
…
x == c
a
b c
u
w v
x
1
21
4
11
3

37. Tree Balancing
8
FIND(a):
b := rep(a)
if b == a:
return a
else
b := FIND(b)
rep(a) := b
return b
UNION(a1,a2):
b1 := FIND(a1)
b2 := FIND(a2)
LINK(b1,b2)
LINK(a1,a2):
if size(a2) > size(a1):
rep(a1) := a2
size(a2) += size(a1)
else:
rep(a2) := a1
size(a1) += size(a2)
…
x == c
a
b c
u
w v
x
1
21
4
11
3

38. The Complex Case
9
h(a1 , …,an , f(b0,b0), …, f(bn-1,bn-1), an) ==
h(f(a0,a0), …,f(an-1,an-1), b1, …, bn-1 , bn)

39. The Complex Case
9
h(a1 , …,an , f(b0,b0), …, f(bn-1,bn-1), an) ==
h(f(a0,a0), …,f(an-1,an-1), b1, …, bn-1 , bn)
a1
a0
an-1
an
b1
b0
bn-1
bn
f(a0,a0)
f(an-2,an-2)
f(an-1,an-1)
f(b0,b0)
f(bn-2,bn-2)
f(bn-1,bn-1)

40. The Complex Case
9
h(a1 , …,an , f(b0,b0), …, f(bn-1,bn-1), an) ==
h(f(a0,a0), …,f(an-1,an-1), b1, …, bn-1 , bn)
a1
a0
an-1
an
b1
b0
bn-1
bn
f(a0,a0)
f(an-2,an-2)
f(an-1,an-1)
f(b0,b0)
f(bn-2,bn-2)
f(bn-1,bn-1)
an == bn

41. The Complex Case
9
h(a1 , …,an , f(b0,b0), …, f(bn-1,bn-1), an) ==
h(f(a0,a0), …,f(an-1,an-1), b1, …, bn-1 , bn)
a1
a0
an-1
an
b1
b0
bn-1
bn
f(a0,a0)
f(an-2,an-2)
f(an-1,an-1)
f(b0,b0)
f(bn-2,bn-2)
f(bn-1,bn-1)
an == bn
f(an-1,an-1) == f(bn-1,bn-1)

42. The Complex Case
9
h(a1 , …,an , f(b0,b0), …, f(bn-1,bn-1), an) ==
h(f(a0,a0), …,f(an-1,an-1), b1, …, bn-1 , bn)
a1
a0
an-1
an
b1
b0
bn-1
bn
f(a0,a0)
f(an-2,an-2)
f(an-1,an-1)
f(b0,b0)
f(bn-2,bn-2)
f(bn-1,bn-1)
an == bn
f(an-1,an-1) == f(bn-1,bn-1)
an-1 == bn-1 an-1 == bn-1

43. The Complex Case
9
h(a1 , …,an , f(b0,b0), …, f(bn-1,bn-1), an) ==
h(f(a0,a0), …,f(an-1,an-1), b1, …, bn-1 , bn)
a1
a0
an-1
an
b1
b0
bn-1
bn
f(a0,a0)
f(an-2,an-2)
f(an-1,an-1)
f(b0,b0)
f(bn-2,bn-2)
f(bn-1,bn-1)
an == bn
f(an-1,an-1) == f(bn-1,bn-1)
an-1 == bn-1 an-1 == bn-1
f(an-2,an-2) == f(bn-2,bn-2)

44. The Complex Case
9
h(a1 , …,an , f(b0,b0), …, f(bn-1,bn-1), an) ==
h(f(a0,a0), …,f(an-1,an-1), b1, …, bn-1 , bn)
a1
a0
an-1
an
b1
b0
bn-1
bn
f(a0,a0)
f(an-2,an-2)
f(an-1,an-1)
f(b0,b0)
f(bn-2,bn-2)
f(bn-1,bn-1)
an == bn
f(an-1,an-1) == f(bn-1,bn-1)
an-1 == bn-1 an-1 == bn-1
f(an-2,an-2) == f(bn-2,bn-2)
⠸

45. The Complex Case
9
h(a1 , …,an , f(b0,b0), …, f(bn-1,bn-1), an) ==
h(f(a0,a0), …,f(an-1,an-1), b1, …, bn-1 , bn)
a1
a0
an-1
an
b1
b0
bn-1
bn
f(a0,a0)
f(an-2,an-2)
f(an-1,an-1)
f(b0,b0)
f(bn-2,bn-2)
f(bn-1,bn-1)
an == bn
f(an-1,an-1) == f(bn-1,bn-1)
an-1 == bn-1 an-1 == bn-1
f(an-2,an-2) == f(bn-2,bn-2)
⠸
a1 == b1 a1 == b1

46. The Complex Case
9
h(a1 , …,an , f(b0,b0), …, f(bn-1,bn-1), an) ==
h(f(a0,a0), …,f(an-1,an-1), b1, …, bn-1 , bn)
a1
a0
an-1
an
b1
b0
bn-1
bn
f(a0,a0)
f(an-2,an-2)
f(an-1,an-1)
f(b0,b0)
f(bn-2,bn-2)
f(bn-1,bn-1)
an == bn
f(an-1,an-1) == f(bn-1,bn-1)
an-1 == bn-1 an-1 == bn-1
f(an-2,an-2) == f(bn-2,bn-2)
⠸
a1 == b1 a1 == b1
f(a0,a0) == f(b0,b0)

47. The Complex Case
9
h(a1 , …,an , f(b0,b0), …, f(bn-1,bn-1), an) ==
h(f(a0,a0), …,f(an-1,an-1), b1, …, bn-1 , bn)
a1
a0
an-1
an
b1
b0
bn-1
bn
f(a0,a0)
f(an-2,an-2)
f(an-1,an-1)
f(b0,b0)
f(bn-2,bn-2)
f(bn-1,bn-1)
an == bn
f(an-1,an-1) == f(bn-1,bn-1)
an-1 == bn-1 an-1 == bn-1
f(an-2,an-2) == f(bn-2,bn-2)
⠸
a1 == b1 a1 == b1
f(a0,a0) == f(b0,b0)
a0 == b0 a0 == b0

48. The Complex Case
9
h(a1 , …,an , f(b0,b0), …, f(bn-1,bn-1), an) ==
h(f(a0,a0), …,f(an-1,an-1), b1, …, bn-1 , bn)
a1
a0
an-1
an
b1
b0
bn-1
bn
f(a0,a0)
f(an-2,an-2)
f(an-1,an-1)
f(b0,b0)
f(bn-2,bn-2)
f(bn-1,bn-1)
an == bn
f(an-1,an-1) == f(bn-1,bn-1)
an-1 == bn-1 an-1 == bn-1
f(an-2,an-2) == f(bn-2,bn-2)
⠸
a1 == b1 a1 == b1
f(a0,a0) == f(b0,b0)
a0 == b0 a0 == b0

49. The Complex Case
9
h(a1 , …,an , f(b0,b0), …, f(bn-1,bn-1), an) ==
h(f(a0,a0), …,f(an-1,an-1), b1, …, bn-1 , bn)
a1
a0
an-1
an
b1
b0
bn-1
bn
f(a0,a0)
f(an-2,an-2)
f(an-1,an-1)
f(b0,b0)
f(bn-2,bn-2)
f(bn-1,bn-1)
an == bn
f(an-1,an-1) == f(bn-1,bn-1)
an-1 == bn-1 an-1 == bn-1
f(an-2,an-2) == f(bn-2,bn-2)
⠸
a1 == b1 a1 == b1
f(a0,a0) == f(b0,b0)
a0 == b0 a0 == b0

50. The Complex Case
9
h(a1 , …,an , f(b0,b0), …, f(bn-1,bn-1), an) ==
h(f(a0,a0), …,f(an-1,an-1), b1, …, bn-1 , bn)
a1
a0
an-1
an
b1
b0
bn-1
bn
f(a0,a0)
f(an-2,an-2)
f(an-1,an-1)
f(b0,b0)
f(bn-2,bn-2)
f(bn-1,bn-1)
an == bn
f(an-1,an-1) == f(bn-1,bn-1)
an-1 == bn-1 an-1 == bn-1
f(an-2,an-2) == f(bn-2,bn-2)
⠸
a1 == b1 a1 == b1
f(a0,a0) == f(b0,b0)
a0 == b0 a0 == b0

51. The Complex Case
9
h(a1 , …,an , f(b0,b0), …, f(bn-1,bn-1), an) ==
h(f(a0,a0), …,f(an-1,an-1), b1, …, bn-1 , bn)
a1
a0
an-1
an
b1
b0
bn-1
bn
f(a0,a0)
f(an-2,an-2)
f(an-1,an-1)
f(b0,b0)
f(bn-2,bn-2)
f(bn-1,bn-1)
an == bn
f(an-1,an-1) == f(bn-1,bn-1)
an-1 == bn-1 an-1 == bn-1
f(an-2,an-2) == f(bn-2,bn-2)
⠸
a1 == b1 a1 == b1
f(a0,a0) == f(b0,b0)
a0 == b0 a0 == b0

52. The Complex Case
9
h(a1 , …,an , f(b0,b0), …, f(bn-1,bn-1), an) ==
h(f(a0,a0), …,f(an-1,an-1), b1, …, bn-1 , bn)
a1
a0
an-1
an
b1
b0
bn-1
bn
f(a0,a0)
f(an-2,an-2)
f(an-1,an-1)
f(b0,b0)
f(bn-2,bn-2)
f(bn-1,bn-1)
an == bn
f(an-1,an-1) == f(bn-1,bn-1)
an-1 == bn-1 an-1 == bn-1
f(an-2,an-2) == f(bn-2,bn-2)
⠸
a1 == b1 a1 == b1
f(a0,a0) == f(b0,b0)
a0 == b0 a0 == b0
How about occurrence checks?

53. The Complex Case
9
h(a1 , …,an , f(b0,b0), …, f(bn-1,bn-1), an) ==
h(f(a0,a0), …,f(an-1,an-1), b1, …, bn-1 , bn)
a1
a0
an-1
an
b1
b0
bn-1
bn
f(a0,a0)
f(an-2,an-2)
f(an-1,an-1)
f(b0,b0)
f(bn-2,bn-2)
f(bn-1,bn-1)
an == bn
f(an-1,an-1) == f(bn-1,bn-1)
an-1 == bn-1 an-1 == bn-1
f(an-2,an-2) == f(bn-2,bn-2)
⠸
a1 == b1 a1 == b1
f(a0,a0) == f(b0,b0)
a0 == b0 a0 == b0
How about occurrence checks? Postpone!

54. Main idea
- Represent unifier as graph
- One variable represent equivalence class
- Replace substitution by union & find operations
- Testing equality becomes testing node identity
Optimizations
- Path compression make recurring lookups fast
- Tree balancing keeps paths short
Complexity
- Linear in space and almost linear in time (technically inverse Ackermann)
- Easy to extract triangular unifier from graph
- Postpone occurrence checks to prevent traversing (potentially) large terms
Union-Find
10
Martelli, Montanari. An Efficient Unification Algorithm. TOPLAS, 1982

55. Error Reporting
11

56. Type errors
- Types are supposed to prevent us from making mistakes
- Error messages are supposed to help us fix mistakes
- This does not always work out so well
‣ Wrong error location
‣ Unclear error message
‣ Not always clear where inferred types come from
- Users expect concise, informative messages, at the right place
Type errors = Unsatisfiable constraints
- Constraint satisfaction is a binary property
- Language designers want to influence error messages
Reporting Type Errors
12

57. Error Example
13
1. let
2. function f(x) =
3. x + 1
4. in
5. (f "Hello!") * 2

58. Error Example
13
1. let
2. function f(x) =
3. x + 1
4. in
5. (f "Hello!") * 2
ty1 == FUN(ty2, ty3)
ty2 == INT()
ty3 == INT()
ty1 == FUN(ty4, ty5)
ty4 == STRING()
ty5 == INT()

59. Error Example
13
1. let
2. function f(x) =
3. x + 1
4. in
5. (f "Hello!") * 2
ty1 == FUN(ty2, ty3)
ty2 == INT()
ty3 == INT()
ty1 == FUN(ty4, ty5)
ty4 == STRING()
ty5 == INT()
[function f(x)]
[x]
[_ + _]
[f _]
[“Hello!”]
[_ * 2]

60. Error Example
13
1. let
2. function f(x) =
3. x + 1
4. in
5. (f "Hello!") * 2
ty1 == FUN(ty2, ty3)
ty2 == INT()
ty3 == INT()
ty1 == FUN(ty4, ty5)
ty4 == STRING()
ty5 == INT()
[function f(x)]
[x]
[_ + _]
[f _]
[“Hello!”]
[_ * 2]
ty1 == FUN(ty2, ty3)
ty1 == FUN(ty4, ty5)
ty2 == INT()
ty3 == INT()
ty5 == INT()
ty4 == STRING()

61. Error Example
13
1. let
2. function f(x) =
3. x + 1
4. in
5. (f "Hello!") * 2
ty1 == FUN(ty2, ty3)
ty2 == INT()
ty3 == INT()
ty1 == FUN(ty4, ty5)
ty4 == STRING()
ty5 == INT()
[function f(x)]
[x]
[_ + _]
[f _]
[“Hello!”]
[_ * 2]
ty1 == FUN(ty2, ty3)
ty1 == FUN(ty4, ty5)
ty2 == INT()
ty3 == INT()
ty5 == INT()
ty4 == STRING()
✔
✔
✔
✔
✔
❌

62. Error Example
13
1. let
2. function f(x) =
3. x + 1
4. in
5. (f "Hello!") * 2
ty1 == FUN(ty2, ty3)
ty2 == INT()
ty3 == INT()
ty1 == FUN(ty4, ty5)
ty4 == STRING()
ty5 == INT()
[function f(x)]
[x]
[_ + _]
[f _]
[“Hello!”]
[_ * 2]
ty1 == FUN(ty2, ty3)
ty1 == FUN(ty4, ty5)
ty2 == INT()
ty3 == INT()
ty5 == INT()
ty4 == STRING()
✔
✔
✔
✔
✔
❌

63. Error Example
13
1. let
2. function f(x) =
3. x + 1
4. in
5. (f "Hello!") * 2
ty1 == FUN(ty2, ty3)
ty2 == INT()
ty3 == INT()
ty1 == FUN(ty4, ty5)
ty4 == STRING()
ty5 == INT()
[function f(x)]
[x]
[_ + _]
[f _]
[“Hello!”]
[_ * 2]
ty1 == FUN(ty2, ty3)
ty1 == FUN(ty4, ty5)
ty2 == INT()
ty3 == INT()
ty5 == INT()
ty4 == STRING()
✔
✔
✔
✔
✔
❌
ty1 == FUN(ty2, ty3)
ty3 == INT()
ty4 == STRING()
ty5 == INT()
ty2 == INT()
ty1 == FUN(ty4, ty5)

64. Error Example
13
1. let
2. function f(x) =
3. x + 1
4. in
5. (f "Hello!") * 2
ty1 == FUN(ty2, ty3)
ty2 == INT()
ty3 == INT()
ty1 == FUN(ty4, ty5)
ty4 == STRING()
ty5 == INT()
[function f(x)]
[x]
[_ + _]
[f _]
[“Hello!”]
[_ * 2]
ty1 == FUN(ty2, ty3)
ty1 == FUN(ty4, ty5)
ty2 == INT()
ty3 == INT()
ty5 == INT()
ty4 == STRING()
✔
✔
✔
✔
✔
❌
ty1 == FUN(ty2, ty3)
ty3 == INT()
ty4 == STRING()
ty5 == INT()
ty2 == INT()
ty1 == FUN(ty4, ty5)
✔
✔
✔
✔
✔
❌

65. Error Example
13
1. let
2. function f(x) =
3. x + 1
4. in
5. (f "Hello!") * 2
ty1 == FUN(ty2, ty3)
ty2 == INT()
ty3 == INT()
ty1 == FUN(ty4, ty5)
ty4 == STRING()
ty5 == INT()
[function f(x)]
[x]
[_ + _]
[f _]
[“Hello!”]
[_ * 2]
ty1 == FUN(ty2, ty3)
ty1 == FUN(ty4, ty5)
ty2 == INT()
ty3 == INT()
ty5 == INT()
ty4 == STRING()
✔
✔
✔
✔
✔
❌
ty1 == FUN(ty2, ty3)
ty3 == INT()
ty4 == STRING()
ty5 == INT()
ty2 == INT()
ty1 == FUN(ty4, ty5)
✔
✔
✔
✔
✔
❌

66. Error Example
13
1. let
2. function f(x) =
3. x + 1
4. in
5. (f "Hello!") * 2
ty1 == FUN(ty2, ty3)
ty2 == INT()
ty3 == INT()
ty1 == FUN(ty4, ty5)
ty4 == STRING()
ty5 == INT()
[function f(x)]
[x]
[_ + _]
[f _]
[“Hello!”]
[_ * 2]
ty1 == FUN(ty4, ty5)
ty4 == STRING()
ty5 == INT()
ty2 == INT()
ty1 == FUN(ty2, ty3)
ty3 == INT()
ty1 == FUN(ty2, ty3)
ty1 == FUN(ty4, ty5)
ty2 == INT()
ty3 == INT()
ty5 == INT()
ty4 == STRING()
✔
✔
✔
✔
✔
❌
ty1 == FUN(ty2, ty3)
ty3 == INT()
ty4 == STRING()
ty5 == INT()
ty2 == INT()
ty1 == FUN(ty4, ty5)
✔
✔
✔
✔
✔
❌

67. Error Example
13
1. let
2. function f(x) =
3. x + 1
4. in
5. (f "Hello!") * 2
ty1 == FUN(ty2, ty3)
ty2 == INT()
ty3 == INT()
ty1 == FUN(ty4, ty5)
ty4 == STRING()
ty5 == INT()
[function f(x)]
[x]
[_ + _]
[f _]
[“Hello!”]
[_ * 2]
ty1 == FUN(ty4, ty5)
ty4 == STRING()
ty5 == INT()
ty2 == INT()
ty1 == FUN(ty2, ty3)
ty3 == INT()
ty1 == FUN(ty2, ty3)
ty1 == FUN(ty4, ty5)
ty2 == INT()
ty3 == INT()
ty5 == INT()
ty4 == STRING()
✔
✔
✔
✔
❌
✔
✔
✔
✔
✔
❌
ty1 == FUN(ty2, ty3)
ty3 == INT()
ty4 == STRING()
ty5 == INT()
ty2 == INT()
ty1 == FUN(ty4, ty5)
✔
✔
✔
✔
✔
❌

68. Error Example
13
1. let
2. function f(x) =
3. x + 1
4. in
5. (f "Hello!") * 2
ty1 == FUN(ty2, ty3)
ty2 == INT()
ty3 == INT()
ty1 == FUN(ty4, ty5)
ty4 == STRING()
ty5 == INT()
[function f(x)]
[x]
[_ + _]
[f _]
[“Hello!”]
[_ * 2]
ty1 == FUN(ty4, ty5)
ty4 == STRING()
ty5 == INT()
ty2 == INT()
ty1 == FUN(ty2, ty3)
ty3 == INT()
ty1 == FUN(ty2, ty3)
ty1 == FUN(ty4, ty5)
ty2 == INT()
ty3 == INT()
ty5 == INT()
ty4 == STRING()
✔
✔
✔
✔
❌
✔
✔
✔
✔
✔
❌
ty1 == FUN(ty2, ty3)
ty3 == INT()
ty4 == STRING()
ty5 == INT()
ty2 == INT()
ty1 == FUN(ty4, ty5)
✔
✔
✔
✔
✔
❌

69. Error Example
13
1. let
2. function f(x) =
3. x + 1
4. in
5. (f "Hello!") * 2
ty1 == FUN(ty2, ty3)
ty2 == INT()
ty3 == INT()
ty1 == FUN(ty4, ty5)
ty4 == STRING()
ty5 == INT()
[function f(x)]
[x]
[_ + _]
[f _]
[“Hello!”]
[_ * 2]
ty1 == FUN(ty4, ty5)
ty4 == STRING()
ty5 == INT()
ty2 == INT()
ty1 == FUN(ty2, ty3)
ty3 == INT()
ty1 == FUN(ty2, ty3)
ty1 == FUN(ty4, ty5)
ty2 == INT()
ty3 == INT()
ty5 == INT()
ty4 == STRING()
✔
✔
✔
✔
❌
✔
✔
✔
✔
✔
❌
ty1 == FUN(ty2, ty3)
ty3 == INT()
ty4 == STRING()
ty5 == INT()
ty2 == INT()
ty1 == FUN(ty4, ty5)
✔
✔
✔
✔
✔
❌

70. Error Example
13
1. let
2. function f(x) =
3. x + 1
4. in
5. (f "Hello!") * 2
ty1 == FUN(ty2, ty3)
ty2 == INT()
ty3 == INT()
ty1 == FUN(ty4, ty5)
ty4 == STRING()
ty5 == INT()
[function f(x)]
[x]
[_ + _]
[f _]
[“Hello!”]
[_ * 2]
ty1 == FUN(ty4, ty5)
ty4 == STRING()
ty5 == INT()
ty2 == INT()
ty1 == FUN(ty2, ty3)
ty3 == INT()
ty1 == FUN(ty2, ty3)
ty1 == FUN(ty4, ty5)
ty2 == INT()
ty3 == INT()
ty5 == INT()
ty4 == STRING()
✔
✔
✔
✔
❌
✔
✔
✔
✔
✔
❌
ty1 == FUN(ty2, ty3)
ty3 == INT()
ty4 == STRING()
ty5 == INT()
ty2 == INT()
ty1 == FUN(ty4, ty5)
✔
✔
✔
✔
✔
❌
Order of constraint solving is relevant!

71. Conflict Sets
14
1. let
2. function f(x) =
3. x + 1
4. in
5. (f "Hello!") * 2
ty1 == FUN(ty2, ty3)
ty2 == INT()
ty3 == INT()
ty1 == FUN(ty4, ty5)
ty4 == STRING()
ty5 == INT()
[function f(x)]
[x + _]
[_ + _]
[f _]
[“Hello!”]
[_ * 2]
ty1 == FUN(ty2, ty3)
ty2 == INT()
ty3 == INT()
ty1 == FUN(ty4, ty5)
ty4 == STRING()
ty5 == INT()

72. Conflict Sets
14
1. let
2. function f(x) =
3. x + 1
4. in
5. (f "Hello!") * 2
ty1 == FUN(ty2, ty3)
ty2 == INT()
ty3 == INT()
ty1 == FUN(ty4, ty5)
ty4 == STRING()
ty5 == INT()
[function f(x)]
[x + _]
[_ + _]
[f _]
[“Hello!”]
[_ * 2]
ty1 == FUN(ty2, ty3)
ty2 == INT()
ty3 == INT()
ty1 == FUN(ty4, ty5)
ty4 == STRING()
ty5 == INT()
ty1 == FUN(ty2, ty3)

73. Conflict Sets
14
1. let
2. function f(x) =
3. x + 1
4. in
5. (f "Hello!") * 2
ty1 == FUN(ty2, ty3)
ty2 == INT()
ty3 == INT()
ty1 == FUN(ty4, ty5)
ty4 == STRING()
ty5 == INT()
[function f(x)]
[x + _]
[_ + _]
[f _]
[“Hello!”]
[_ * 2]
ty1 == FUN(ty2, ty3)
ty2 == INT()
ty3 == INT()
ty1 == FUN(ty4, ty5)
ty4 == STRING()
ty5 == INT()
ty1 == FUN(ty2, ty3)
ty2 == INT()

74. Conflict Sets
14
1. let
2. function f(x) =
3. x + 1
4. in
5. (f "Hello!") * 2
ty1 == FUN(ty2, ty3)
ty2 == INT()
ty3 == INT()
ty1 == FUN(ty4, ty5)
ty4 == STRING()
ty5 == INT()
[function f(x)]
[x + _]
[_ + _]
[f _]
[“Hello!”]
[_ * 2]
ty1 == FUN(ty2, ty3)
ty2 == INT()
ty3 == INT()
ty1 == FUN(ty4, ty5)
ty4 == STRING()
ty5 == INT()
ty1 == FUN(ty2, ty3)
ty2 == INT()

75. Conflict Sets
14
1. let
2. function f(x) =
3. x + 1
4. in
5. (f "Hello!") * 2
ty1 == FUN(ty2, ty3)
ty2 == INT()
ty3 == INT()
ty1 == FUN(ty4, ty5)
ty4 == STRING()
ty5 == INT()
[function f(x)]
[x + _]
[_ + _]
[f _]
[“Hello!”]
[_ * 2]
ty1 == FUN(ty2, ty3)
ty2 == INT()
ty3 == INT()
ty1 == FUN(ty4, ty5)
ty4 == STRING()
ty5 == INT()
ty1 == FUN(ty2, ty3)
ty2 == INT()
ty1 == FUN(ty4, ty5)

76. Conflict Sets
14
1. let
2. function f(x) =
3. x + 1
4. in
5. (f "Hello!") * 2
ty1 == FUN(ty2, ty3)
ty2 == INT()
ty3 == INT()
ty1 == FUN(ty4, ty5)
ty4 == STRING()
ty5 == INT()
[function f(x)]
[x + _]
[_ + _]
[f _]
[“Hello!”]
[_ * 2]
ty1 == FUN(ty2, ty3)
ty2 == INT()
ty3 == INT()
ty1 == FUN(ty4, ty5)
ty4 == STRING()
ty5 == INT()
ty1 == FUN(ty2, ty3)
ty2 == INT()
ty1 == FUN(ty4, ty5)
ty4 == STRING()

77. Conflict Sets
14
1. let
2. function f(x) =
3. x + 1
4. in
5. (f "Hello!") * 2
ty1 == FUN(ty2, ty3)
ty2 == INT()
ty3 == INT()
ty1 == FUN(ty4, ty5)
ty4 == STRING()
ty5 == INT()
[function f(x)]
[x + _]
[_ + _]
[f _]
[“Hello!”]
[_ * 2]
ty1 == FUN(ty2, ty3)
ty2 == INT()
ty3 == INT()
ty1 == FUN(ty4, ty5)
ty4 == STRING()
ty5 == INT()
ty1 == FUN(ty2, ty3)
ty2 == INT()
ty1 == FUN(ty4, ty5)
ty4 == STRING()

78. Conflict Sets
14
1. let
2. function f(x) =
3. x + 1
4. in
5. (f "Hello!") * 2
ty1 == FUN(ty2, ty3)
ty2 == INT()
ty3 == INT()
ty1 == FUN(ty4, ty5)
ty4 == STRING()
ty5 == INT()
[function f(x)]
[x + _]
[_ + _]
[f _]
[“Hello!”]
[_ * 2]
ty1 == FUN(ty2, ty3)
ty2 == INT()
ty3 == INT()
ty1 == FUN(ty4, ty5)
ty4 == STRING()
ty5 == INT()
ty1 == FUN(ty2, ty3)
ty2 == INT()
ty1 == FUN(ty4, ty5)
ty4 == STRING()
A conflict set is an inconsistent subset of the constraint set

79. Conflict Sets
- Sets of constraint that are inconsistent together
- Multiple conflict sets might appear in one constraint set
- Conflict sets can intersect
How to find conflict sets?
- Find minimal sets that are in conflict
‣ Combinatorial explosion (Min-SAT)
- Approximate conflict sets during constraint solving
‣ Conflict set may be too large
Where to report errors?
- Report the conflicting program slice
(using all conflicting constraints)
- Report on a few constraints in the conflict set
‣ Remove constraints until inconsistency disappears
‣ Use heuristics to efficiently select constraints
There is not always a best place to report the error!
Conflict Sets and Errors
15
1. let
2. function f(x) =
3. x + 1
4. in
5. (f "Hello!") * 2

80. Constraint Graphs
16
Zhang, Meyers. Toward General Diagnosis of Static Errors. POPL’14
ty1 == FUN(ty2, ty3)
ty2 == INT()
ty3 == INT()
ty1 == FUN(ty4, ty5)
ty4 == STRING()
ty5 == INT()
ty1
ty2
ty3
ty4
ty5
INT()
STRING()
FUN(ty2, ty3)
FUN(ty4, ty5)
1
2
1
2
INT()
INT()

81. Constraint Graphs
16
Zhang, Meyers. Toward General Diagnosis of Static Errors. POPL’14
ty1 == FUN(ty2, ty3)
ty2 == INT()
ty3 == INT()
ty1 == FUN(ty4, ty5)
ty4 == STRING()
ty5 == INT()
ty1
ty2
ty3
ty4
ty5
INT()
STRING()
FUN(ty2, ty3)
FUN(ty4, ty5)
1
2
1
2
- Consider paths between
non-variable terms
INT()
INT()

82. Constraint Graphs
16
Zhang, Meyers. Toward General Diagnosis of Static Errors. POPL’14
ty1 == FUN(ty2, ty3)
ty2 == INT()
ty3 == INT()
ty1 == FUN(ty4, ty5)
ty4 == STRING()
ty5 == INT()
ty1
ty2
ty3
ty4
ty5
INT()
STRING()
FUN(ty2, ty3)
FUN(ty4, ty5)
1
2
1
2
- Consider paths between
non-variable terms
INT()
INT()

83. Constraint Graphs
16
Zhang, Meyers. Toward General Diagnosis of Static Errors. POPL’14
ty1 == FUN(ty2, ty3)
ty2 == INT()
ty3 == INT()
ty1 == FUN(ty4, ty5)
ty4 == STRING()
ty5 == INT()
ty1
ty2
ty3
ty4
ty5
INT()
STRING()
FUN(ty2, ty3)
FUN(ty4, ty5)
1
2
1
2
- Consider paths between
non-variable terms
- Trust edges involved in
many correct paths
INT()
INT()

84. Constraint Graphs
16
Zhang, Meyers. Toward General Diagnosis of Static Errors. POPL’14
ty1 == FUN(ty2, ty3)
ty2 == INT()
ty3 == INT()
ty1 == FUN(ty4, ty5)
ty4 == STRING()
ty5 == INT()
ty1
ty2
ty3
ty4
ty5
INT()
STRING()
FUN(ty2, ty3)
FUN(ty4, ty5)
1
2
1
2
- Consider paths between
non-variable terms
- Trust edges involved in
many correct paths
INT()
INT()

85. Constraint Graphs
16
Zhang, Meyers. Toward General Diagnosis of Static Errors. POPL’14
ty1 == FUN(ty2, ty3)
ty2 == INT()
ty3 == INT()
ty1 == FUN(ty4, ty5)
ty4 == STRING()
ty5 == INT()
ty1
ty2
ty3
ty4
ty5
INT()
STRING()
FUN(ty2, ty3)
FUN(ty4, ty5)
1
2
1
2
- Consider paths between
non-variable terms
- Trust edges involved in
many correct paths
- Suspect edges involved in
many wrong paths
INT()
INT()

86. Constraint Graphs
16
Zhang, Meyers. Toward General Diagnosis of Static Errors. POPL’14
ty1 == FUN(ty2, ty3)
ty2 == INT()
ty3 == INT()
ty1 == FUN(ty4, ty5)
ty4 == STRING()
ty5 == INT()
ty1
ty2
ty3
ty4
ty5
INT()
STRING()
FUN(ty2, ty3)
FUN(ty4, ty5)
1
2
1
2
- Consider paths between
non-variable terms
- Trust edges involved in
many correct paths
- Suspect edges involved in
many wrong paths
- Restricts constraint langauge
and solver implementation
INT()
INT()

87. Use approximated conflict sets
17
1. ty1 == FUN(ty2, ty3)
2. ty2 == INT()
3. ty3 == INT()
4. ty1 == FUN(ty4, ty5)
5. ty4 == STRING()
6. ty5 == INT()
Loncaric e.a. A practical framework for type inference error explanation. OOPSLA’16

88. Use approximated conflict sets
17
1. ty1 == FUN(ty2, ty3)
2. ty2 == INT()
3. ty3 == INT()
4. ty1 == FUN(ty4, ty5)
5. ty4 == STRING()
6. ty5 == INT()
Loncaric e.a. A practical framework for type inference error explanation. OOPSLA’16
ty1 FUN(ty2, ty3)

89. Use approximated conflict sets
17
1. ty1 == FUN(ty2, ty3)
2. ty2 == INT()
3. ty3 == INT()
4. ty1 == FUN(ty4, ty5)
5. ty4 == STRING()
6. ty5 == INT()
Loncaric e.a. A practical framework for type inference error explanation. OOPSLA’16
ty1 FUN(ty2, ty3)
- Track constraints during
unification

90. Use approximated conflict sets
17
1. ty1 == FUN(ty2, ty3)
2. ty2 == INT()
3. ty3 == INT()
4. ty1 == FUN(ty4, ty5)
5. ty4 == STRING()
6. ty5 == INT()
Loncaric e.a. A practical framework for type inference error explanation. OOPSLA’16
ty1 FUN(ty2, ty3)
- Track constraints during
unification
1

91. Use approximated conflict sets
17
1. ty1 == FUN(ty2, ty3)
2. ty2 == INT()
3. ty3 == INT()
4. ty1 == FUN(ty4, ty5)
5. ty4 == STRING()
6. ty5 == INT()
Loncaric e.a. A practical framework for type inference error explanation. OOPSLA’16
ty1
ty2
FUN(ty2, ty3)
- Track constraints during
unification
INT()
1
2

92. Use approximated conflict sets
17
1. ty1 == FUN(ty2, ty3)
2. ty2 == INT()
3. ty3 == INT()
4. ty1 == FUN(ty4, ty5)
5. ty4 == STRING()
6. ty5 == INT()
Loncaric e.a. A practical framework for type inference error explanation. OOPSLA’16
ty1
ty2
ty3 INT()
FUN(ty2, ty3)
- Track constraints during
unification
INT()
1
2
3

93. Use approximated conflict sets
17
1. ty1 == FUN(ty2, ty3)
2. ty2 == INT()
3. ty3 == INT()
4. ty1 == FUN(ty4, ty5)
5. ty4 == STRING()
6. ty5 == INT()
Loncaric e.a. A practical framework for type inference error explanation. OOPSLA’16
ty1
ty2
ty3
ty4
INT()
FUN(ty2, ty3)
- Track constraints during
unification
INT()
1
2
3

94. Use approximated conflict sets
17
1. ty1 == FUN(ty2, ty3)
2. ty2 == INT()
3. ty3 == INT()
4. ty1 == FUN(ty4, ty5)
5. ty4 == STRING()
6. ty5 == INT()
Loncaric e.a. A practical framework for type inference error explanation. OOPSLA’16
ty1
ty2
ty3
ty4
INT()
FUN(ty2, ty3)
- Track constraints during
unification
INT()
1
2
3
1,4

95. Use approximated conflict sets
17
1. ty1 == FUN(ty2, ty3)
2. ty2 == INT()
3. ty3 == INT()
4. ty1 == FUN(ty4, ty5)
5. ty4 == STRING()
6. ty5 == INT()
Loncaric e.a. A practical framework for type inference error explanation. OOPSLA’16
ty1
ty2
ty3
ty4
ty5 INT()
FUN(ty2, ty3)
- Track constraints during
unification
INT()
1
2
3
1,4
1,4

96. Use approximated conflict sets
17
1. ty1 == FUN(ty2, ty3)
2. ty2 == INT()
3. ty3 == INT()
4. ty1 == FUN(ty4, ty5)
5. ty4 == STRING()
6. ty5 == INT()
Loncaric e.a. A practical framework for type inference error explanation. OOPSLA’16
ty1
ty2
ty3
ty4
ty5 INT()
FUN(ty2, ty3)
- Track constraints during
unification
INT()
1
2
3
1,4
1,4
FIND(ty4) = INT() because of {1,2,4}

97. Use approximated conflict sets
17
1. ty1 == FUN(ty2, ty3)
2. ty2 == INT()
3. ty3 == INT()
4. ty1 == FUN(ty4, ty5)
5. ty4 == STRING()
6. ty5 == INT()
Loncaric e.a. A practical framework for type inference error explanation. OOPSLA’16
ty1
ty2
ty3
ty4
ty5 INT()
FUN(ty2, ty3)
- Track constraints during
unification
INT()
1
2
3
1,4
1,4
FIND(ty4) = INT() because of {1,2,4}
constraint 5 gives conflict set {1,2,4,5}

98. Use approximated conflict sets
17
1. ty1 == FUN(ty2, ty3)
2. ty2 == INT()
3. ty3 == INT()
4. ty1 == FUN(ty4, ty5)
5. ty4 == STRING()
6. ty5 == INT()
Loncaric e.a. A practical framework for type inference error explanation. OOPSLA’16
ty1
ty2
ty3
ty4
ty5 INT()
FUN(ty2, ty3)
- Track constraints during
unification
- Conflict set may be an
over-approximation
INT()
1
2
3
1,4
1,4
FIND(ty4) = INT() because of {1,2,4}
constraint 5 gives conflict set {1,2,4,5}

99. Use approximated conflict sets
17
1. ty1 == FUN(ty2, ty3)
2. ty2 == INT()
3. ty3 == INT()
4. ty1 == FUN(ty4, ty5)
5. ty4 == STRING()
6. ty5 == INT()
Loncaric e.a. A practical framework for type inference error explanation. OOPSLA’16
ty1
ty2
ty3
ty4
ty5 INT()
FUN(ty2, ty3)
- Track constraints during
unification
- Conflict set may be an
over-approximation
- Iteratively drop constraints
until inconsistency is gone
INT()
1
2
3
1,4
1,4
FIND(ty4) = INT() because of {1,2,4}
constraint 5 gives conflict set {1,2,4,5}

100. Use approximated conflict sets
17
1. ty1 == FUN(ty2, ty3)
2. ty2 == INT()
3. ty3 == INT()
4. ty1 == FUN(ty4, ty5)
5. ty4 == STRING()
6. ty5 == INT()
Loncaric e.a. A practical framework for type inference error explanation. OOPSLA’16
ty1
ty2
ty3
ty4
ty5 INT()
FUN(ty2, ty3)
- Track constraints during
unification
- Conflict set may be an
over-approximation
- Iteratively drop constraints
until inconsistency is gone
- Solver is used as an oracle,
no requirements on its
implementation
INT()
1
2
3
1,4
1,4
FIND(ty4) = INT() because of {1,2,4}
constraint 5 gives conflict set {1,2,4,5}

101. Error reporting from conflict sets
- Select constraints that are likely causes
- Few error messages is usually preferred over many
- Avoid combinatorial explosion by using heuristics
Language independent heuristics
- Select constraints shared between multiple conflict sets
- Select constraints coming from smaller program fragments
‣ Function definition type comes from the (larger) body expression
‣ Function use type comes from the (smaller) use expression
Language specific heuristics
- Select constraint based on weights assigned in specification
‣ Assign more weight to constraints from definitions, than constraints from use
- Select constraints with explicit error messages
‣ Avoids reporting on implicit equalities that arise during constraint generation
Which constraint is to blame?
18

102. Requirements on the solver
- Report (approximated) conflict sets
‣ Conflict sets should be over-approximations
‣ The closer to the minimal conflict set, the better
- Fast failure is important
‣ Solver is called repeatedly with smaller constraint sets
How to report errors?
- Reporting ‘<long function type> expected, but got <long but
slightly different function type>’ can be improved
- Can we report that first argument is different, or that number of
arguments is different?
More Aspects
19

103. Exercises
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104. Conclusion
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105. Unification with Union-Find
- Represent unifier as a graph
- Testing term equality becomes testing node identity
- Really fast with path compression and tree balancing
Error Reporting
- Selecting constraints to report errors on
- Use conflict sets to find cause of inconsistencies
- Report on
‣ all conflicting constraints (program slice)
‣ subset of constraints that removes inconsistency
- Use heuristics to select constraints to blame
Summary
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