1. 3. State Space Reduction

2. Plan
• Stubborn sets [Petri Nets 1999]
• Symmetry [Acta Informatica 2000]
• Invariants [TACAS 2003]
• Sweep-Line [TACAS 2004]

3. The Stubborn Set Method
3

4. Diamonds from concurrency
a b
b
a
4

5. Diamonds from concurrency
s1
a b
s s’
b
s2 a
4

6. State Explosion by Concurrency
Process A Process B Process C
internal internal 1
internal
internal internal 2
internal
sync sync sync 3
4
111
211 121 112
311 221 131 212 122 113
312 321 231 222 132 213 123
322 331 232 313 133 223
332 323 233
333
444 5

7. Stubborn Sets
111
211 121 112
311 221 131 212 122 113
312 321 231 222 132 213 123
322 331 232 313 133 223
332 323 233
333
6
444

8. Stubborn Sets
In every marking m:
stubborn(m) ⊆ T
fire only activated transitions in stubborn(m)
111
211 121 112
311 221 131 212 122 113
312 321 231 222 132 213 123
322 331 232 313 133 223
332 323 233
333
6
444

9. Stubborn Sets
In every marking m:
stubborn(m) ⊆ T
fire only activated transitions in stubborn(m)
reduced transition system:
111
211 121 112
311 221 131 212 122 113
312 321 231 222 132 213 123
322 331 232 313 133 223
332 323 233
333
6
444

10. Stubborn Sets
In every marking m:
stubborn(m) ⊆ T
fire only activated transitions in stubborn(m)
reduced transition system:
111
211 121 112
311 221 131 212 122 113
312 321 231 222 132 213 123
322 331 232 313 133 223
332 323 233
333
6
444

11. Stubborn Sets
In every marking m:
stubborn(m) ⊆ T
fire only activated transitions in stubborn(m)
reduced transition system:
111
211 121 112
311 221 131 212 122 113
312 321 231 222 132 213 123
322 331 232 313 133 223
332 323 233
333
6
444

12. Stubborn Sets
In every marking m:
stubborn(m) ⊆ T
fire only activated transitions in stubborn(m)
reduced transition system:
111
211 121 112
311 221 131 212 122 113
312 321 231 222 132 213 123
322 331 232 313 133 223
332 323 233
333
6
444

13. Stubborn Sets
In every marking m:
stubborn(m) ⊆ T
fire only activated transitions in stubborn(m)
reduced transition system:
111
211 121 112
311 221 131 212 122 113
312 321 231 222 132 213 123
322 331 232 313 133 223
332 323 233
333
6
444

14. Reduced Transition System
111
121
122
222
223
323
333
7
444

15. How to Preserve Properties
Core principle:
outside stubborn(m)
m2 implies
in stubborn(m)
plus property specific requirements
presence of right path justifies absence of left path
8

16. How to Preserve Properties
Core principle:
outside stubborn(m)
m w1 m1 t m2 implies
in stubborn(m)
plus property specific requirements
presence of right path justifies absence of left path
8

17. How to Preserve Properties
Core principle:
outside stubborn(m)
m w1 m1 t m2 implies m t m1 ’ w1 m2
in stubborn(m)
plus property specific requirements
presence of right path justifies absence of left path
8

18. Preservation of Deadlocks
Core principle +
implies
Proof:
9

19. Preservation of Deadlocks
Core principle +
implies
Proof:
Let m w d length(w) = min
9

20. Preservation of Deadlocks
Core principle +
implies
Proof:
Let m w d length(w) = min
1st case: some t of stubborn(m) occurs in w
9

21. Preservation of Deadlocks
Core principle +
implies
Proof:
Let m w d length(w) = min
1st case: some t of stubborn(m) occurs in w
m w1 s1 t m2 w2 d
9

22. Preservation of Deadlocks
Core principle +
implies
Proof:
Let m w d length(w) = min
1st case: some t of stubborn(m) occurs in w
m w1 s1 t m2 w2 d
m t m1 ’ w1 m2 w2 d
9

23. Preservation of Deadlocks
Core principle +
implies
Proof:
Let m w d length(w) = min
1st case: some t of stubborn(m) occurs in w
m w1 s1 t m2 w2 d m1’ in red. TS,
m t m1 ’ w1 m2 w2 d closer to d!
9

24. Preservation of Deadlocks
Core principle +
implies
Proof:
Let m w d length(w) = min
1st case: some t of stubborn(m) occurs in w
m w1 s1 t m2 w2 d m1’ in red. TS,
m t m1 ’ w1 m2 w2 d closer to d!
2nd case: no t of stubborn(m) occurs in w
9

25. Preservation of Deadlocks
Core principle +
implies
Proof:
Let m w d length(w) = min
1st case: some t of stubborn(m) occurs in w
m w1 s1 t m2 w2 d m1’ in red. TS,
m t m1 ’ w1 m2 w2 d closer to d!
2nd case: no t of stubborn(m) occurs in w
m w d
9

26. Preservation of Deadlocks
Core principle +
implies
Proof:
Let m w d length(w) = min
1st case: some t of stubborn(m) occurs in w
m w1 s1 t m2 w2 d m1’ in red. TS,
m t m1 ’ w1 m2 w2 d closer to d!
2nd case: no t of stubborn(m) occurs in w
m w d
t 9

27. Preservation of Deadlocks
Core principle +
implies
Proof:
Let m w d length(w) = min
1st case: some t of stubborn(m) occurs in w
m w1 s1 t m2 w2 d m1’ in red. TS,
m t m1 ’ w1 m2 w2 d closer to d!
2nd case: no t of stubborn(m) occurs in w
m w d
t t 9

28. Preservation of Deadlocks
Core principle +
implies
Proof:
Let m w d length(w) = min
1st case: some t of stubborn(m) occurs in w
m w1 s1 t m2 w2 d m1’ in red. TS,
m t m1 ’ w1 m2 w2 d closer to d!
2nd case: no t of stubborn(m) occurs in w
m w d
t d not a
t 9
deadlock!

29. Preservation of Deadlocks
Core principle + m w m’
implies
Proof:
Let m w d length(w) = min
1st case: some t of stubborn(m) occurs in w
m w1 s1 t m2 w2 d m1’ in red. TS,
m t m1 ’ w1 m2 w2 d closer to d!
2nd case: no t of stubborn(m) occurs in w
m w d
t d not a
t 9
deadlock!

30. Preservation of Deadlocks
Core principle + m w m’
implies
t
Proof:
Let m w d length(w) = min
1st case: some t of stubborn(m) occurs in w
m w1 s1 t m2 w2 d m1’ in red. TS,
m t m1 ’ w1 m2 w2 d closer to d!
2nd case: no t of stubborn(m) occurs in w
m w d
t d not a
t 9
deadlock!

31. Preservation of Deadlocks
Core principle + m w m’
implies t
t
Proof:
Let m w d length(w) = min
1st case: some t of stubborn(m) occurs in w
m w1 s1 t m2 w2 d m1’ in red. TS,
m t m1 ’ w1 m2 w2 d closer to d!
2nd case: no t of stubborn(m) occurs in w
m w d
t d not a
t 9
deadlock!

32. Preservation of LTL/CTL
LTLX:
Core principle
+Visibility: all transitions in stubborn(m) invisible to φ or
stubborn(m) = T
+Proviso: Once in every cycle: stubborn(m) = T
CTLX:
LTL
+ |stubborn(m)| = 1 or stubborn(m) = T
Consequences:
- only local properties yield reduction
- Proviso avoids infinite stuttering
- Proviso known to cause explosion
- Proviso requires cycle detection (e.g. depth first)
- CTL only performant when number of conflicts is small

33. LoLA’s Approaches
Let φ be state predicate Assume m does not satisfy φ
wrup(m, φ ) = some set of transitions such that every path
to an m’ that satisfies φ contains at least
one transition of wrup(m, φ ).
Examples:
wrup(m, “m* reached”) = •p, for some p with m(p) < m*(p)
= p•, for some p with m(p) > m*(p)
wrup(m,p>k) = wrup(m,p≥k) = •p
wrup(m,p<k) = wrup(m,p≤k) = p•
wrup(m, φ1 ∧φ2) = wrup(m, φ1) if m does not satisfy φ1
= wrup(m, φ2) if m does not satisfy φ2
wrup(m, φ1 ∨φ2) = wrup(m, φ1)∪ wrup(m, φ2)
wrup(m, t not dead) = {t} 11

34. Theorem
Reachability of φ:
core principle
+ wrup(m, φ) ⊆ stubborn(m)
orig. φ
red.
m
m0 12

35. Theorem
Reachability of φ:
core principle
+ wrup(m, φ) ⊆ stubborn(m)
orig. φ
red.
m
m0 12

36. Theorem
Reachability of φ:
core principle
+ wrup(m, φ) ⊆ stubborn(m)
orig. φ
red.
in wrup(m, φ)
m
m0 12

37. Theorem
Reachability of φ:
core principle
+ wrup(m, φ) ⊆ stubborn(m)
orig. φ
red.
in wrup(m, φ)
t 1st in ample(m)
m
m0 12

38. Theorem
Reachability of φ:
core principle
+ wrup(m, φ) ⊆ stubborn(m)
orig. φ
red.
in wrup(m, φ)
m1
t 1st in ample(m)
m
m0 12

39. Theorem
Reachability of φ:
core principle
+ wrup(m, φ) ⊆ stubborn(m)
orig. φ
red.
in wrup(m, φ)
m1
t 1st in ample(m)
m
m1 closer to m’ than m
m0 12

40. Effect
• Can be applied to global predicates
• Astonishing goal-orientation
• Has been relaxed by Kristensen/Valmari (wrup must
be contained only once in an scc)
• They perform better if predicate unreachable
• Unrelaxed method better if predicate reachable
• Can be extended to boundedness:
• Bounded net: wrup(m) = {t : |t•|>|•t|}
• Bounded place: wrup(m,p) = •p
relaxed

41. TSCC based properties
Valmari:
core principle
+ weak proviso: Every transition in stubborn(m) at
least once in every tscc of reduced system:
every tscc of original state space visited in reduced
state space

42. TSCC based properties
Idea:
- Construct Valmari’s tscc-preserving state space
- Pick one element of each tscc of reduced state space
- check mutual reachability for home state
- check reachability of m0 for reversibility
- check rechability of φ for liveness of φ
userconfig.H:
twophase TWOPHASE

43. CTL/LTL properties
• CTL: Separate search space for each subformula
• Use wrup for EF and AG
• Use traditional CTL method for other
operators
• LTL: search counterexample path: F φ ➪ G¬φ,
GF φ ➪ FG¬φ, FGφ ➪ GF¬φ
• G ¬φ LTL preserving, but drop Proviso
• FG¬φ,GF¬φ:
• drop Proviso if m satisfies ¬φ
• wrup(m,¬φ) if m satisfies φ

44. Symmetries

45. Symmetric Behavior
Goal: symmetry in transition system
σ is symmetry if: ΣTS: set of all
σ is bijection R(m0)  R(m0) symmetries in R(m0)
m [t> m’ iff ex. t’: σ(m) t’> σ(m’)
σ(m0) = m0
by induction:
m0 m1 m2 ... path 
σ(m0) σ(m1) σ(m2) ... path as well
-Id is always symmetry [ΣTS,o] is
-If σ symmetry, so is σ-1 group
-If σ1 and σ2 symmetries, so is σ1 o σ2
18

46. Equivalence of States
19

47. Equivalence of States
Have to detect symmetries prior to state space generation,
typically cannot deduce all of them
but: can always close under inversion and composition
19

48. Equivalence of States
Have to detect symmetries prior to state space generation,
typically cannot deduce all of them
but: can always close under inversion and composition
fix some subgroup Σ ⊆ ΣTS
19

49. Equivalence of States
Have to detect symmetries prior to state space generation,
typically cannot deduce all of them
but: can always close under inversion and composition
fix some subgroup Σ ⊆ ΣTS
m ~ m’ iff ex. σ ∈ Σ such that σ(m) = m’
19

50. Equivalence of States
Have to detect symmetries prior to state space generation,
typically cannot deduce all of them
but: can always close under inversion and composition
fix some subgroup Σ ⊆ ΣTS
m ~ m’ iff ex. σ ∈ Σ such that σ(m) = m’
~ is equivalence relation
19

51. Reduced Transition System
TSΣ = [R(m0)/~ , EΣ , [m0]Σ]
EΣ = { [ [s],[s’] ] | ex. s ∈ [s], ex. s’ ∈ [s’] : [s,s’] ∈ E}
Size of reduced system:
| R(m0)/~ | ≥ | R(m0) | / | Σ |
|Σ | can be exponential in size of Petri net
20

52. Σ = { Id, σ}
Example
σ([x,y,z]) =
[y,x,z]
(i,i,1)
(r,i,1) (i,r,1)
g1
(c,i,0) (r,r,1) (i,c,0)
(c,r,0) (r,c,0)
21

53. Example
Σ = { Id, σ} σ([x,y,z]) =
[y,x,z]
(i,i,1)
(r,i,1)
(c,i,0) (r,r,1)
(c,r,0)
22

54. Construction of reduced
R := E := ø; dfs(m0);
dfs(m) Approximation
R := R ∪ {m};
FOR ALL t: activated in m DO
m’ = m + Δt;
IF can find σ with σ(m’)∈ R THEN
E := E ∪{[m, t, σ(m’) ]}; The “Orbit-
ELSE Problem”
E := E ∪{[m,t, m’ ]};
dfs(m’);
END
END
23

55. “Traditional” Symmetry
Tools
• Depend on “scalar set” data type
• =, ≠, arrays, for each, no constant
• Cannot model networks other than cliques
• LoLA: can handle all kinds of symmetry in
the net structure

56. PN automorphisms
Bijection σ: P∪T → P∪T is PN automorphism,
iff, for all x,y ∈ P∪T:
- m0(x) = m0(σ(x))
- If [x,y] ∈ F then [σ(x),σ(y)] ∈ F and W([x,y]) = W([σ(x),σ(y)])
Every PN automorphism induces symmetry in state space:
σ(m)(σ(p)) = m(p)
25

57. Example
2 3
1 4
11 11 12 12 13 13 14 14
22 24 21 23 22 24 21 23
33 33 34 34 31 31 32 32
44 42 43 41 44 42 43 41
id
26

58. Schreier-Sims generating set
U1
U2
U3 subgroup induces partition of whole group
pick one element of each class (“orbit”)
Group: all automorphisms
U1: all automorphisms that map p1 to p1
U2: all automorphisms that map p1 to p1, p2 to p2
...
Un: Id
has O(n^2) elements

59. Example
2 3
1 4
11 11 12 12 13 13 14 14
22 24 21 23 22 24 21 23
33 33 34 34 31 31 32 32
44 42 43 41 44 42 43 41
id
U1
U2 28

60. 2 3
Example
1 4
E={2 id, 3 2 ,3 2 3, 2 3
;
id, }
1 g1 4 1 g2 4 1 g3 4 1 g4 4
id o id = id g2 o id =
id o g4 = g2 o g4 =
g1 o id = g3 o id =
g1 o g4 = g3 o g4 =
29

61. Another Example
8 7
5 6
4 3 g = g1 o g2 o g3
1 2
1. Layer: 1 →1 ... 8
2. Layer 1 → 1, 2 → 2,4,5
3. Layer 1 → 1, 2 → 2, 3 → 3,6
7 + 2 + 1 = 10 generators for
8 x 3 x 2 = 48 automorphisms
30

62. Orbit Problem: Approximation
id id
g11 g12 g13
g14-1 g21 g22 g23 g31 g32
g14
given: m searched: canonical representative(m)
31

63. Orbit Problem: Approximation
id id
g11 g12 g13
g14-1 g21 g22 g23 g31 g32
g14
given: m searched: canonical representative(m)
1. m1 := MIN{g1i-1(m), i = ...}
31

64. Orbit Problem: Approximation
id id
g11 g12 g13
g14-1 g21 g22 g23 g31 g32
g14
given: m searched: canonical representative(m)
1. m1 := MIN{g1i-1(m), i = ...}
2. m2 := MIN{g2i-1(m1), i = ...}
31

65. Orbit Problem: Approximation
id id
g11 g12 g13
g14-1 g21 g22 g23 g31 g32
g14
given: m searched: canonical representative(m)
1. m1 := MIN{g1i-1(m), i = ...}
2. m2 := MIN{g2i-1(m1), i = ...}
3. m3 := MIN{g3i-1(m2), i = ...}
31

66. Orbit Problem: Approximation
id id
g11 g12 g13
g14-1 g21 g22 g23 g31 g32
g14
given: m searched: canonical representative(m)
1. m1 := MIN{g1i-1(m), i = ...} ........
2. m2 := MIN{g2i-1(m1), i = ...} n. mn := MIN{gni-1(mn-1), i = ...}
3. m3 := MIN{g3i-1(m2), i = ...}
31

67. Orbit Problem: Approximation
id id
g11 g12 g13
g14-1 g21 g22 g23 g31 g32
g14
given: m searched: canonical representative(m)
1. m1 := MIN{g1i-1(m), i = ...} ........
2. m2 := MIN{g2i-1(m1), i = ...} n. mn := MIN{gni-1(mn-1), i = ...}
3. m3 := MIN{g3i-1(m2), i = ...} canrep(m) := mn
31

68. 2 3
Example
2 2 2
2 3 3 3 3
1 4
E={ , , ; }
1 4 1 4 1 4 1 4
g12 g13 g14 g22
3 2
2 3
m 32 32
id-1(m) = id(m) =
1 4 11 41
1 1 12 31
-1(m) = (m) =
31 42
32 31
-1(m) = (m) =
21 41
12 33
-1(m) = (m) =
11 4 2
32

69. 2 3
Example
2 2 2
2 3 3 3 3
1 4
E={ , , ; }
1 4 1 4 1 4 1 4
g12 g13 g14 g22
3 2
2 3
m 32 32
id-1(m) = id(m) =
1 4 11 41
1 1 12 31
-1(m) = (m) =
31 42
32 31
-1(m) = (m) =
21 41
12 33
-1(m) = (m) = ≠ m1
11 4 2
32

70. 2 3
2
Example
2 2 2
3 3 3 3
1 4
E={ , , ; }
1 4 1 4 1 4 1 4
g12 g13 g14 g22
3 2
2 3
s 12 33
id-1(m1) = id(m1) =
1 4 11 42
1 1
x = 12 3x = 3 22 33
s1 -1(m) = (m) =
x=11 4x = 2 11 41
12 32
Result ≠ canrep(m) = (m) =
1 1 43

71. 2 3
2
Example
2 2 2
3 3 3 3
1 4
E={ , , ; }
1 4 1 4 1 4 1 4
g12 g13 g14 g22
3 2
2 3
s 12 33
id-1(m1) = id(m1) = Result
1 4 11 42
1 1
x = 12 3x = 3 22 33
s1 -1(m) = (m) =
x=11 4x = 2 11 41
12 32
Result ≠ canrep(m) = (m) =
1 1 43

72. Summary Symmetries
symmetries 34

73. Summary Symmetries
calculation of symmetries, exact solution of orbit problem:
equivalent to graph isomorphism (NP)
symmetries 34

74. Summary Symmetries
calculation of symmetries, exact solution of orbit problem:
equivalent to graph isomorphism (NP)
Many other orbit algorithms available in LoLA, even more by
Tommi Junttila
best choice depends on structure of symmetry group
symmetries 34

75. Using Petri net invariants
in state space

76. Two approaches
compress states (use place invariants)
save space and time
exempt states from storage (use transition invariants)
space/time tradeoff
36

77. First approach: use place invariants
37

78. First approach: use place invariants
Let i be place invariant:.
For all reachable m:
i • m = i • m0
37

79. First approach: use place invariants
Let i be place invariant:.
For all reachable m:
i • m = i • m0
i • m0 – Σp’≠p i(p’) • m(p’)
.... and, for a place p with i(p) ≠ 0: m(p) = i(p)
37

80. Example
3 2
invariant 1: [ 1 1 0 0 0 ] invariant 2: [ 0 0 0 1 1 ]
that is, for all reachable markings m:
m(p1) = 1 – m(p2) m(p5) = 2 – m(p4)
only p2,p3,p4 need to be stored (40 % compression)
38

81. Overhead
preprocessing
- time
- space
state space
construction
- time
39

82. Overhead
appears to be:
preprocessing
- time compute invariants
- space |inv| • |places|
state space
construction
- time recover saved
components
39

83. Overhead
appears to be: actually is:
preprocessing
- time compute invariants compute upper triangular
form
- space |inv| • |places| 1bit • |places|
state space
construction
- time recover saved search, insert performed
components on smaller vectors
39

84. State space construction
state
yes/no state
pointer depository
(short
vectors)
state (recover
removed components)
1 0 1
0 0 0
= 1
0 - -2
-1 = 3
1
2 1 1
40

85. State space construction
state
yes/no state
pointer depository
(short
vectors)
state (recover
removed components)
1 0 1 Observe:
0 0 0
= 1
0 - -2
-1 = 3
1
values of i
irrelevant,
2 1 1
supp(i) sufficient!
40

86. Upper triangular form
1 -1 0 0 1 0 0 0 . .
-1 1 0 0 -1 0 0 0 0 1
0 3 -2 0 0 1 0 0 . .
0 0 -1 1 0 0 -1 0 . .
0 0 1 -1 0 0 1 0 1 0
incidence matrix triangular form invariants
m(p2),m(p5) can be calculated from m(p1), m(p3), m(p4)
41

87. Results
1. Space reduction 30% - 55%
2. Preprocessing time insignificant
3. Run time reduction proportional to space reduction
Reason: search and insert operations take
80 – 95 % of overall run time
... are now performed on shorter vectors
4. combination with most other reduction techniques
possible
preduction 42

88. Second approach:
43

89. Second approach:
what happens if some states are
removed from the depository?
43

90. Second approach:
what happens if some states are
removed from the depository?
43

91. Second approach:
what happens if some states are
removed from the depository?
construction still terminates as long as
removed states do not form cycles!
43

92. Second approach:
what happens if some states are
removed from the depository?
construction still terminates as long as
removed states do not form cycles!
use structural knowledge about cycles
43

93. Transition invariants
cycle in state space corresponds to transition
invariant
44

94. Transition invariants
cycle in state space corresponds to transition
invariant
Assume: Set U of transitions s.t. for every transition
invariant i:
U ∩ supp(i) ≠∅
Then: store states that enable transitions in U
do not store other states
U can be determined from triangular form
44

95. Example
3 2
transition invariant: [2,2,3,3]
 U = {t}
store only states where t is enabled
45

96. Problems:
1. Too many states enable transitions in U
Solution: combine with partial order reduction
2. Unacceptable run time overhead
Solution 1: heuristically store additional states
Solution 2: remove only non-branching states
46

97. Ad 1: Full vs. Partial
full state space
47

98. Ad 1: Full vs. Partial
stubborn set reduced state space
48

99. Ad 2: store additional states
k
k
49

100. Results
1. Controllable space/time trade-off
2. Combination with partial order reduction compulsory
3. Combination with a few other reduction techniques
possible
4. Only simple properties can be verified (no access to
graph structure of the state space)
50

101. The Sweep-Line Method

102. Road map
The sweep-line method (basic/extended)
Calculation of a progress measure
Discussion
- Combination with other reduction techniques

103. The sweep-line method (Basic)
Idea: state s → progress value p(s)
with
s [t> s‘ p(s) > p(s’)
Unprocessed
sweep-line

104. The sweep-line method (Basic)
Idea: state s → progress value p(s)
with
s [t> s‘ p(s) > p(s’)
Unprocessed
sweep-line p

105. The sweep-line method (Basic)
Idea: state s → progress value p(s)
with
s [t> s‘ p(s) > p(s’)
Unprocessed
Processed
sweep-line p

106. The sweep-line method (Basic)
Idea: state s → progress value p(s)
with
s [t> s‘ p(s) > p(s’)
Unprocessed
Not yet seen
Processed
sweep-line p

107. The sweep-line method (Basic)
Idea: state s → progress value p(s)
with
s [t> s‘ p(s) > p(s’)
Unprocessed
 Not yet seen
Processed
sweep-line p

108. The sweep-line method (extended)
If p is not monotonous:
t
s’
s p(s’) < p(s)

109. The sweep-line method (extended)
If p is not monotonous:
t
s’
s p(s’) < p(s)
-mark s’ “persistent”
-start new sweep from s’

110. The sweep-line method (extended)
If p is not monotonous:
t
s’
s p(s’) < p(s)
-mark s’ “persistent”
-start new sweep from s’
Consequently: not too often p(s’) < p(s)

111. Setting for LoLA’s measure
-incremental: “transition offsets”
Δ p(t) : m [t> m‘ p(m’) = p(m) + Δ p(t)
-not necessarily monotonous
(in every cycle: one negative Δ p or all Δ p = 0)

112. The measure
partition T into U and TU
in U: all transitions linear independent
in TU: all transitions linear dependent of U
i.e. |U| = rank(C)
-for t in U: Δ p (t) := 1
-for t in TU: Δ p(t) determined by (unique) lin. combination of U
(for t in TU: Δ p(t) >0, =0, <0 )
typical size: |U| 60% - 100% of |T|

113. U
Examples
TU
1 1 1
2 -2 1
1 1 0

114. Geometric interpretation
p2
s
p3
p1
sweep

115. Geometric interpretation
p2
s
p3
U
p1
sweep

116. Geometric interpretation
p2
s
p3
U
p1
sweep

117. Geometric interpretation
progress
p2
s
p3
U
p1
sweep

118. Geometric interpretation
progress
p2
s
p(s)
p3
U
p1
sweep

119. Geometric interpretation
progress
p2
s
p(s)
p3
1
U
p1
sweep