Initial Problem
Start: l0
Program_Vars: X₀, X₁, X₂
Temp_Vars:
Locations: l0, l1, l2, l3, l4, l5, l6, l7
Transitions:
t₀: l0(X₀, X₁, X₂) → l6(X₀, X₁, X₂)
t₄: l1(X₀, X₁, X₂) → l2(X₀, X₁, X₂) :|: 1 ≤ X₂
t₅: l1(X₀, X₁, X₂) → l3(X₀, X₁, X₂) :|: X₂ ≤ 0
t₆: l2(X₀, X₁, X₂) → l1(X₀, X₁, X₂-1)
t₇: l3(X₀, X₁, X₂) → l4(X₀, X₁-1, X₂)
t₂: l4(X₀, X₁, X₂) → l1(X₀, X₁, X₀) :|: 1 ≤ X₁
t₃: l4(X₀, X₁, X₂) → l5(X₀, X₁, X₂) :|: X₁ ≤ 0
t₈: l5(X₀, X₁, X₂) → l7(X₀, X₁, X₂)
t₁: l6(X₀, X₁, X₂) → l4(X₁, X₀, X₂)
Preprocessing
Found invariant X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 2 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location l2
Found invariant X₁ ≤ 0 for location l7
Found invariant X₁ ≤ 0 for location l5
Found invariant X₂ ≤ X₀ ∧ 1 ≤ X₁ for location l1
Found invariant X₂ ≤ 0 ∧ 1+X₂ ≤ X₁ ∧ X₂ ≤ X₀ ∧ 1 ≤ X₁ for location l3
Problem after Preprocessing
Start: l0
Program_Vars: X₀, X₁, X₂
Temp_Vars:
Locations: l0, l1, l2, l3, l4, l5, l6, l7
Transitions:
t₀: l0(X₀, X₁, X₂) → l6(X₀, X₁, X₂)
t₄: l1(X₀, X₁, X₂) → l2(X₀, X₁, X₂) :|: 1 ≤ X₂ ∧ X₂ ≤ X₀ ∧ 1 ≤ X₁
t₅: l1(X₀, X₁, X₂) → l3(X₀, X₁, X₂) :|: X₂ ≤ 0 ∧ X₂ ≤ X₀ ∧ 1 ≤ X₁
t₆: l2(X₀, X₁, X₂) → l1(X₀, X₁, X₂-1) :|: X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 2 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀
t₇: l3(X₀, X₁, X₂) → l4(X₀, X₁-1, X₂) :|: X₂ ≤ 0 ∧ 1+X₂ ≤ X₁ ∧ X₂ ≤ X₀ ∧ 1 ≤ X₁
t₂: l4(X₀, X₁, X₂) → l1(X₀, X₁, X₀) :|: 1 ≤ X₁
t₃: l4(X₀, X₁, X₂) → l5(X₀, X₁, X₂) :|: X₁ ≤ 0
t₈: l5(X₀, X₁, X₂) → l7(X₀, X₁, X₂) :|: X₁ ≤ 0
t₁: l6(X₀, X₁, X₂) → l4(X₁, X₀, X₂)
MPRF for transition t₅: l1(X₀, X₁, X₂) → l3(X₀, X₁, X₂) :|: X₂ ≤ 0 ∧ X₂ ≤ X₀ ∧ 1 ≤ X₁ of depth 1:
new bound:
X₀ {O(n)}
MPRF:
l2 [X₁ ]
l3 [X₁-1 ]
l4 [X₁ ]
l1 [X₁ ]
MPRF for transition t₇: l3(X₀, X₁, X₂) → l4(X₀, X₁-1, X₂) :|: X₂ ≤ 0 ∧ 1+X₂ ≤ X₁ ∧ X₂ ≤ X₀ ∧ 1 ≤ X₁ of depth 1:
new bound:
X₀+1 {O(n)}
MPRF:
l2 [X₁+1 ]
l3 [X₁+1 ]
l4 [X₁+1 ]
l1 [X₁+1 ]
MPRF for transition t₂: l4(X₀, X₁, X₂) → l1(X₀, X₁, X₀) :|: 1 ≤ X₁ of depth 1:
new bound:
X₀ {O(n)}
MPRF:
l2 [X₁-1 ]
l3 [X₁-1 ]
l4 [X₁ ]
l1 [X₁-1 ]
Found invariant X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 2 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location l2
Found invariant X₁ ≤ 0 for location l7
Found invariant X₁ ≤ 0 for location l5
Found invariant X₂ ≤ X₀ ∧ 1 ≤ X₁ for location l1
Found invariant X₂ ≤ 0 ∧ 1+X₂ ≤ X₁ ∧ X₂ ≤ X₀ ∧ 1 ≤ X₁ for location l3
Time-Bound by TWN-Loops:
TWN-Loops: t₄ 4⋅X₀⋅X₁+4⋅X₀ {O(n^2)}
TWN-Loops:
entry: t₂: l4(X₀, X₁, X₂) → l1(X₀, X₁, X₀) :|: 1 ≤ X₁
results in twn-loop: twn:Inv: [X₂ ≤ X₀ ∧ 1 ≤ X₁ ∧ X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 2 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀] , (X₀,X₁,X₂) -> (X₀,X₁,X₂-1) :|: 1 ≤ X₂
order: [X₀; X₁; X₂]
closed-form:
X₀: X₀
X₁: X₁
X₂: X₂ + [[n != 0]] * -1 * n^1
Termination: true
Formula:
1 < 0
∨ 1 < X₂ ∧ 1 ≤ 0 ∧ 0 ≤ 1
∨ 1 ≤ 0 ∧ 0 ≤ 1 ∧ 1 ≤ X₂ ∧ X₂ ≤ 1
Stabilization-Threshold for: 1 ≤ X₂
alphas_abs: X₂
M: 0
N: 1
Bound: 2⋅X₂+2 {O(n)}
relevant size-bounds w.r.t. t₂:
X₂: 2⋅X₁ {O(n)}
Runtime-bound of t₂: X₀ {O(n)}
Results in: 4⋅X₀⋅X₁+4⋅X₀ {O(n^2)}
4⋅X₀⋅X₁+4⋅X₀ {O(n^2)}
Time-Bound by TWN-Loops:
TWN-Loops: t₆ 4⋅X₀⋅X₁+4⋅X₀ {O(n^2)}
relevant size-bounds w.r.t. t₂:
X₂: 2⋅X₁ {O(n)}
Runtime-bound of t₂: X₀ {O(n)}
Results in: 4⋅X₀⋅X₁+4⋅X₀ {O(n^2)}
4⋅X₀⋅X₁+4⋅X₀ {O(n^2)}
Analysing control-flow refined program
Found invariant 1+X₂ ≤ X₀ ∧ 0 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 1 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location n_l1___2
Found invariant 1+X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 3 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 2 ≤ X₀ for location n_l2___1
Found invariant X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location n_l2___3
Found invariant X₁ ≤ 0 for location l7
Found invariant X₁ ≤ 0 for location l5
Found invariant X₂ ≤ X₀ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₁ for location l1
Found invariant X₂ ≤ 0 ∧ 1+X₂ ≤ X₁ ∧ X₂ ≤ X₀ ∧ 1 ≤ X₁ for location l3
knowledge_propagation leads to new time bound X₀ {O(n)} for transition t₇₃: l1(X₀, X₁, X₂) → n_l2___3(X₀, X₁, X₂) :|: 1 ≤ X₁ ∧ X₂ ≤ X₀ ∧ X₀ ≤ X₂ ∧ X₂ ≤ X₀ ∧ 1 ≤ X₁ ∧ X₂ ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₂ ∧ X₂ ≤ X₀ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₁
knowledge_propagation leads to new time bound X₀ {O(n)} for transition t₇₅: n_l2___3(X₀, X₁, X₂) → n_l1___2(X₀, X₁, X₂-1) :|: 1 ≤ X₁ ∧ 1 ≤ X₀ ∧ X₀ ≤ X₂ ∧ X₂ ≤ X₀ ∧ X₂ ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₂ ∧ X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀
MPRF for transition t₇₂: n_l1___2(X₀, X₁, X₂) → n_l2___1(X₀, X₁, X₂) :|: 1 ≤ X₁ ∧ X₂ ≤ X₀ ∧ 0 ≤ X₂ ∧ 1+X₂ ≤ X₀ ∧ 1 ≤ X₁ ∧ X₂ ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₂ ∧ 1+X₂ ≤ X₀ ∧ 0 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 1 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ of depth 1:
new bound:
2⋅X₀⋅X₁+X₀ {O(n^2)}
MPRF:
n_l2___3 [0 ]
l4 [0 ]
l1 [0 ]
l3 [0 ]
n_l2___1 [X₂ ]
n_l1___2 [X₂+1 ]
MPRF for transition t₇₉: n_l1___2(X₀, X₁, X₂) → l3(X₀, X₁, X₂) :|: X₂ ≤ 0 ∧ X₂ ≤ X₀ ∧ 1 ≤ X₁ ∧ 1+X₂ ≤ X₀ ∧ 0 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 1 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ of depth 1:
new bound:
X₀ {O(n)}
MPRF:
l4 [X₁ ]
l1 [X₁ ]
l3 [X₁-1 ]
n_l2___1 [X₁ ]
n_l2___3 [X₁ ]
n_l1___2 [X₁ ]
MPRF for transition t₇₄: n_l2___1(X₀, X₁, X₂) → n_l1___2(X₀, X₁, X₂-1) :|: 1 ≤ X₁ ∧ 1+X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ X₂ ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₂ ∧ 1+X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 3 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 2 ≤ X₀ of depth 1:
new bound:
2⋅X₀⋅X₁ {O(n^2)}
MPRF:
n_l2___3 [0 ]
l4 [0 ]
l1 [0 ]
l3 [0 ]
n_l2___1 [X₂ ]
n_l1___2 [X₂ ]
CFR did not improve the program. Rolling back
All Bounds
Timebounds
Overall timebound:8⋅X₀⋅X₁+11⋅X₀+5 {O(n^2)}
t₀: 1 {O(1)}
t₄: 4⋅X₀⋅X₁+4⋅X₀ {O(n^2)}
t₅: X₀ {O(n)}
t₆: 4⋅X₀⋅X₁+4⋅X₀ {O(n^2)}
t₇: X₀+1 {O(n)}
t₂: X₀ {O(n)}
t₃: 1 {O(1)}
t₈: 1 {O(1)}
t₁: 1 {O(1)}
Costbounds
Overall costbound: 8⋅X₀⋅X₁+11⋅X₀+5 {O(n^2)}
t₀: 1 {O(1)}
t₄: 4⋅X₀⋅X₁+4⋅X₀ {O(n^2)}
t₅: X₀ {O(n)}
t₆: 4⋅X₀⋅X₁+4⋅X₀ {O(n^2)}
t₇: X₀+1 {O(n)}
t₂: X₀ {O(n)}
t₃: 1 {O(1)}
t₈: 1 {O(1)}
t₁: 1 {O(1)}
Sizebounds
t₀, X₀: X₀ {O(n)}
t₀, X₁: X₁ {O(n)}
t₀, X₂: X₂ {O(n)}
t₄, X₀: X₁ {O(n)}
t₄, X₁: X₀ {O(n)}
t₄, X₂: 2⋅X₁ {O(n)}
t₅, X₀: X₁ {O(n)}
t₅, X₁: X₀ {O(n)}
t₅, X₂: 4⋅X₁ {O(n)}
t₆, X₀: X₁ {O(n)}
t₆, X₁: X₀ {O(n)}
t₆, X₂: 2⋅X₁ {O(n)}
t₇, X₀: X₁ {O(n)}
t₇, X₁: X₀ {O(n)}
t₇, X₂: 4⋅X₁ {O(n)}
t₂, X₀: X₁ {O(n)}
t₂, X₁: X₀ {O(n)}
t₂, X₂: 2⋅X₁ {O(n)}
t₃, X₀: 2⋅X₁ {O(n)}
t₃, X₁: 2⋅X₀ {O(n)}
t₃, X₂: 4⋅X₁+X₂ {O(n)}
t₈, X₀: 2⋅X₁ {O(n)}
t₈, X₁: 2⋅X₀ {O(n)}
t₈, X₂: 4⋅X₁+X₂ {O(n)}
t₁, X₀: X₁ {O(n)}
t₁, X₁: X₀ {O(n)}
t₁, X₂: X₂ {O(n)}