Initial Problem
Start: l0
Program_Vars: X₀, X₁, X₂, X₃, X₄, X₅
Temp_Vars:
Locations: l0, l1, l2
Transitions:
t₀: l0(X₀, X₁, X₂, X₃, X₄, X₅) → l1(X₃, X₄, X₅, X₃, X₄, X₅)
t₁: l1(X₀, X₁, X₂, X₃, X₄, X₅) → l1(X₀+X₂, X₁, X₂-1, X₃, X₄, X₅) :|: 0 < X₂
t₂: l1(X₀, X₁, X₂, X₃, X₄, X₅) → l2(X₀, X₁, X₅, X₃, X₄, X₅) :|: X₂ ≤ 0
t₄: l2(X₀, X₁, X₂, X₃, X₄, X₅) → l1(X₃, X₄, X₅, X₃, X₄, X₅-1) :|: 0 < X₅
t₃: l2(X₀, X₁, X₂, X₃, X₄, X₅) → l2(X₀, 3⋅X₁-4⋅X₂, 4⋅X₁-3⋅X₂, 5⋅X₃, 5⋅X₄-(X₀)², X₅) :|: 1 < (X₁)² ∧ 0 < X₀*X₂+2⋅X₀
Preprocessing
Found invariant X₂ ≤ 1+X₅ ∧ X₄ ≤ X₁ ∧ X₁ ≤ X₄ ∧ X₃ ≤ X₀ for location l1
Problem after Preprocessing
Start: l0
Program_Vars: X₀, X₁, X₂, X₃, X₄, X₅
Temp_Vars:
Locations: l0, l1, l2
Transitions:
t₀: l0(X₀, X₁, X₂, X₃, X₄, X₅) → l1(X₃, X₄, X₅, X₃, X₄, X₅)
t₁: l1(X₀, X₁, X₂, X₃, X₄, X₅) → l1(X₀+X₂, X₁, X₂-1, X₃, X₄, X₅) :|: 0 < X₂ ∧ X₂ ≤ 1+X₅ ∧ X₄ ≤ X₁ ∧ X₁ ≤ X₄ ∧ X₃ ≤ X₀
t₂: l1(X₀, X₁, X₂, X₃, X₄, X₅) → l2(X₀, X₁, X₅, X₃, X₄, X₅) :|: X₂ ≤ 0 ∧ X₂ ≤ 1+X₅ ∧ X₄ ≤ X₁ ∧ X₁ ≤ X₄ ∧ X₃ ≤ X₀
t₄: l2(X₀, X₁, X₂, X₃, X₄, X₅) → l1(X₃, X₄, X₅, X₃, X₄, X₅-1) :|: 0 < X₅
t₃: l2(X₀, X₁, X₂, X₃, X₄, X₅) → l2(X₀, 3⋅X₁-4⋅X₂, 4⋅X₁-3⋅X₂, 5⋅X₃, 5⋅X₄-(X₀)², X₅) :|: 1 < (X₁)² ∧ 0 < X₀*X₂+2⋅X₀
MPRF for transition t₄: l2(X₀, X₁, X₂, X₃, X₄, X₅) → l1(X₃, X₄, X₅, X₃, X₄, X₅-1) :|: 0 < X₅ of depth 1:
new bound:
X₅ {O(n)}
TWN: t₁: l1→l1
cycle: [t₁: l1→l1]
loop: (0 < X₂,(X₂) -> (X₂-1)
order: [X₂]
closed-form:
X₂: X₂ + [[n != 0]] * -1 * n^1
Termination: true
Formula:
1 < 0
∨ 0 < X₂ ∧ 1 ≤ 0 ∧ 0 ≤ 1
Stabilization-Threshold for: 0 < X₂
alphas_abs: X₂
M: 0
N: 1
Bound: 2⋅X₂+2 {O(n)}
loop: (0 < X₂,(X₂) -> (X₂-1)
order: [X₂]
closed-form:
X₂: X₂ + [[n != 0]] * -1 * n^1
Termination: true
Formula:
1 < 0
∨ 0 < X₂ ∧ 1 ≤ 0 ∧ 0 ≤ 1
Stabilization-Threshold for: 0 < X₂
alphas_abs: X₂
M: 0
N: 1
Bound: 2⋅X₂+2 {O(n)}
TWN - Lifting for t₁: l1→l1 of 2⋅X₂+4 {O(n)}
relevant size-bounds w.r.t. t₄:
X₂: 2⋅X₅ {O(n)}
Runtime-bound of t₄: X₅ {O(n)}
Results in: 4⋅X₅⋅X₅+4⋅X₅ {O(n^2)}
TWN - Lifting for t₁: l1→l1 of 2⋅X₂+4 {O(n)}
relevant size-bounds w.r.t. t₀:
X₂: X₅ {O(n)}
Runtime-bound of t₀: 1 {O(1)}
Results in: 2⋅X₅+4 {O(n)}
knowledge_propagation leads to new time bound 4⋅X₅⋅X₅+6⋅X₅+5 {O(n^2)} for transition t₂: l1(X₀, X₁, X₂, X₃, X₄, X₅) → l2(X₀, X₁, X₅, X₃, X₄, X₅) :|: X₂ ≤ 0 ∧ X₂ ≤ 1+X₅ ∧ X₄ ≤ X₁ ∧ X₁ ≤ X₄ ∧ X₃ ≤ X₀
Analysing control-flow refined program
Found invariant X₅ ≤ X₂ ∧ X₂ ≤ X₅ ∧ X₄ ≤ X₁ ∧ X₁ ≤ X₄ ∧ X₃ ≤ X₀ for location n_l2___6
Found invariant 1+X₅ ≤ X₂ ∧ 0 ≤ X₅ ∧ 1 ≤ X₂+X₅ ∧ X₂ ≤ 1+X₅ ∧ X₄ ≤ X₁ ∧ X₁ ≤ X₄ ∧ X₃ ≤ X₀ ∧ X₀ ≤ X₃ ∧ 1 ≤ X₂ for location n_l1___4
Found invariant X₅ ≤ X₂ ∧ 1 ≤ X₅ ∧ 2 ≤ X₂+X₅ ∧ X₂ ≤ X₅ ∧ X₄ ≤ X₁ ∧ X₁ ≤ X₄ ∧ 1+X₃ ≤ X₀ ∧ 1 ≤ X₂ for location n_l2___5
Found invariant 1 ≤ X₅ ∧ 1 ≤ X₂+X₅ ∧ 1+X₂ ≤ X₅ ∧ X₄ ≤ X₁ ∧ X₁ ≤ X₄ ∧ 1+X₃ ≤ X₀ ∧ 0 ≤ X₂ for location n_l1___7
Found invariant 0 ≤ X₅ ∧ 0 ≤ X₂+X₅ ∧ X₂ ≤ X₅ ∧ X₄ ≤ X₁ ∧ X₁ ≤ X₄ ∧ 1+X₃ ≤ X₀ ∧ 0 ≤ X₂ for location n_l1___2
Found invariant 1 ≤ X₅ for location n_l2___3
Found invariant X₅ ≤ X₂ ∧ X₂ ≤ X₅ ∧ X₄ ≤ X₁ ∧ X₁ ≤ X₄ ∧ X₃ ≤ X₀ ∧ X₀ ≤ X₃ for location l1
MPRF for transition t₇₄: n_l1___7(X₀, X₁, X₂, X₃, X₄, X₅) → n_l1___7(X₀+X₂, X₁, X₂-1, X₃, X₁, X₅) :|: 0 < X₅ ∧ 0 < X₂ ∧ X₃ ≤ X₀ ∧ X₂ ≤ 1+X₅ ∧ X₁ ≤ X₄ ∧ X₄ ≤ X₁ ∧ X₂ ≤ 1+X₅ ∧ X₃ ≤ X₀ ∧ X₁ ≤ X₄ ∧ X₄ ≤ X₁ ∧ 1 ≤ X₅ ∧ 1 ≤ X₂+X₅ ∧ 1+X₂ ≤ X₅ ∧ X₄ ≤ X₁ ∧ X₁ ≤ X₄ ∧ 1+X₃ ≤ X₀ ∧ 0 ≤ X₂ of depth 1:
new bound:
X₅+1 {O(n)}
MPRF for transition t₇₂: n_l1___2(X₀, X₁, X₂, X₃, X₄, X₅) → n_l2___6(X₀, X₁, X₅, X₃, X₁, X₅) :|: X₂ ≤ 0 ∧ X₃ ≤ X₀ ∧ X₂ ≤ 1+X₅ ∧ X₁ ≤ X₄ ∧ X₄ ≤ X₁ ∧ X₂ ≤ 1+X₅ ∧ X₃ ≤ X₀ ∧ X₁ ≤ X₄ ∧ X₄ ≤ X₁ ∧ 0 ≤ X₅ ∧ 0 ≤ X₂+X₅ ∧ X₂ ≤ X₅ ∧ X₄ ≤ X₁ ∧ X₁ ≤ X₄ ∧ 1+X₃ ≤ X₀ ∧ 0 ≤ X₂ of depth 1:
new bound:
7⋅X₅+2 {O(n)}
MPRF for transition t₇₃: n_l1___4(X₀, X₁, X₂, X₃, X₄, X₅) → n_l1___2(X₀+X₂, X₁, X₂-1, X₃, X₁, X₅) :|: 0 < X₂ ∧ 0 < X₂ ∧ X₃ ≤ X₀ ∧ X₂ ≤ 1+X₅ ∧ X₁ ≤ X₄ ∧ X₄ ≤ X₁ ∧ X₂ ≤ 1+X₅ ∧ X₃ ≤ X₀ ∧ X₁ ≤ X₄ ∧ X₄ ≤ X₁ ∧ 1+X₅ ≤ X₂ ∧ 0 ≤ X₅ ∧ 1 ≤ X₂+X₅ ∧ X₂ ≤ 1+X₅ ∧ X₄ ≤ X₁ ∧ X₁ ≤ X₄ ∧ X₃ ≤ X₀ ∧ X₀ ≤ X₃ ∧ 1 ≤ X₂ of depth 1:
new bound:
7⋅X₅+2 {O(n)}
MPRF for transition t₇₈: n_l2___1(X₀, X₁, X₂, X₃, X₄, X₅) → n_l1___4(X₃, X₄, X₅, X₃, X₄, X₅-1) :|: 0 < X₅ of depth 1:
new bound:
7⋅X₅+2 {O(n)}
MPRF for transition t₈₄: n_l2___6(X₀, X₁, X₂, X₃, X₄, X₅) → n_l1___4(X₃, X₄, X₅, X₃, X₄, X₅-1) :|: X₂ ≤ X₅ ∧ X₅ ≤ X₂ ∧ X₁ ≤ X₄ ∧ X₄ ≤ X₁ ∧ X₃ ≤ X₀ ∧ 0 < X₅ ∧ X₅ ≤ X₂ ∧ X₂ ≤ X₅ ∧ X₄ ≤ X₁ ∧ X₁ ≤ X₄ ∧ X₃ ≤ X₀ of depth 1:
new bound:
7⋅X₅+2 {O(n)}
TWN: t₇₁: n_l1___2→n_l1___2
cycle: [t₇₁: n_l1___2→n_l1___2]
loop: (0 < X₂,(X₂) -> (X₂-1)
order: [X₂]
closed-form:
X₂: X₂ + [[n != 0]] * -1 * n^1
Termination: true
Formula:
1 < 0
∨ 0 < X₂ ∧ 1 ≤ 0 ∧ 0 ≤ 1
Stabilization-Threshold for: 0 < X₂
alphas_abs: X₂
M: 0
N: 1
Bound: 2⋅X₂+2 {O(n)}
TWN - Lifting for t₇₁: n_l1___2→n_l1___2 of 2⋅X₂+4 {O(n)}
relevant size-bounds w.r.t. t₇₃:
X₂: 34⋅X₅ {O(n)}
Runtime-bound of t₇₃: 7⋅X₅+2 {O(n)}
Results in: 476⋅X₅⋅X₅+164⋅X₅+8 {O(n^2)}
knowledge_propagation leads to new time bound 7⋅X₅+3 {O(n)} for transition t₈₅: n_l2___6(X₀, X₁, X₂, X₃, X₄, X₅) → n_l2___1(X₀, Arg1_P, Arg2_P, 5⋅X₃, NoDet0, X₅) :|: X₂ ≤ X₅ ∧ X₅ ≤ X₂ ∧ X₁ ≤ X₄ ∧ X₄ ≤ X₁ ∧ X₃ ≤ X₀ ∧ 4⋅X₁ ≤ 3⋅X₂+Arg2_P ∧ 3⋅X₂+Arg2_P ≤ 4⋅X₁ ∧ 7⋅X₁+3⋅Arg1_P ≤ 4⋅Arg2_P ∧ 4⋅Arg2_P ≤ 7⋅X₁+3⋅Arg1_P ∧ X₅ ≤ X₂ ∧ X₂ ≤ X₅ ∧ X₄ ≤ X₁ ∧ X₁ ≤ X₄ ∧ X₃ ≤ X₀
CFR did not improve the program. Rolling back
CFR did not improve the program. Rolling back
All Bounds
Timebounds
Overall timebound:inf {Infinity}
t₀: 1 {O(1)}
t₁: 4⋅X₅⋅X₅+6⋅X₅+4 {O(n^2)}
t₂: 4⋅X₅⋅X₅+6⋅X₅+5 {O(n^2)}
t₃: inf {Infinity}
t₄: X₅ {O(n)}
Costbounds
Overall costbound: inf {Infinity}
t₀: 1 {O(1)}
t₁: 4⋅X₅⋅X₅+6⋅X₅+4 {O(n^2)}
t₂: 4⋅X₅⋅X₅+6⋅X₅+5 {O(n^2)}
t₃: inf {Infinity}
t₄: X₅ {O(n)}
Sizebounds
t₀, X₀: X₃ {O(n)}
t₀, X₁: X₄ {O(n)}
t₀, X₂: X₅ {O(n)}
t₀, X₃: X₃ {O(n)}
t₀, X₄: X₄ {O(n)}
t₀, X₅: X₅ {O(n)}
t₁, X₂: 3⋅X₅ {O(n)}
t₁, X₅: X₅ {O(n)}
t₂, X₂: 2⋅X₅ {O(n)}
t₂, X₅: X₅ {O(n)}
t₃, X₅: X₅ {O(n)}
t₄, X₂: 2⋅X₅ {O(n)}
t₄, X₅: X₅ {O(n)}