Preprocessing
Found invariant 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₀ for location l2
Probabilistic Analysis
Probabilistic Program after Preprocessing
Start: l0
Program_Vars: X₀, X₁, X₂
Temp_Vars:
Locations: l0, l1, l2
Transitions:
g₀:l0(X₀,X₁,X₂) → t₁:l1(X₀,X₁,X₂) :|:
g₂:l1(X₀,X₁,X₂) → t₃:l2(X₀,X₁,X₂) :|: 1 ≤ X₀ ∧ X₀ ≤ X₂
g₄:l2(X₀,X₁,X₂) → t₅:l1(3⋅X₀,X₁,2⋅X₂) :|: X₁ ≤ 0 ∧ 1 ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂
g₆:l2(X₀,X₁,X₂) → [1/2]:t₇:l1(X₀-1,X₁,X₂) :+: [1/2]:t₈:l1(X₀,X₁,X₂) :|: 1 ≤ X₁ ∧ 1 ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂
Show Graph
G
l0
l0
l1
l1
l0->l1
p = 1
t₁ ∈ g₀
l2
l2
l1->l2
p = 1
t₃ ∈ g₂
τ = 1 ≤ X₀ ∧ X₀ ≤ X₂
l2->l1
p = 1
t₅ ∈ g₄
η (X₀) = 3⋅X₀
η (X₂) = 2⋅X₂
τ = 1 ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ X₁ ≤ 0
l2->l1
p = 1/2
t₇ ∈ g₆
η (X₀) = X₀-1
τ = 1 ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₁
l2->l1
p = 1/2
t₈ ∈ g₆
τ = 1 ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₁
Run classical analysis on SCC: [l0]
Classical Approximation after Lifting Classical Results
All Bounds
Timebounds
Overall timebound:inf {Infinity}
g₀: 1 {O(1)}
g₂: inf {Infinity}
g₄: inf {Infinity}
g₆: inf {Infinity}
Costbounds
Overall costbound: inf {Infinity}
g₀: inf {Infinity}
g₂: inf {Infinity}
g₄: inf {Infinity}
g₆: inf {Infinity}
Sizebounds
(g₀,l1), X₀: X₀ {O(n)}
(g₀,l1), X₁: X₁ {O(n)}
(g₀,l1), X₂: X₂ {O(n)}
Run probabilistic analysis on SCC: [l0]
Run classical analysis on SCC: [l1; l2]
Classical Approximation after Lifting Classical Results
All Bounds
Timebounds
Overall timebound:inf {Infinity}
g₀: 1 {O(1)}
g₂: inf {Infinity}
g₄: inf {Infinity}
g₆: inf {Infinity}
Costbounds
Overall costbound: inf {Infinity}
g₀: inf {Infinity}
g₂: inf {Infinity}
g₄: inf {Infinity}
g₆: inf {Infinity}
Sizebounds
(g₀,l1), X₀: X₀ {O(n)}
(g₀,l1), X₁: X₁ {O(n)}
(g₀,l1), X₂: X₂ {O(n)}
(g₂,l2), X₁: X₁ {O(n)}
(g₄,l1), X₁: X₁ {O(n)}
(g₆,l1), X₁: 2⋅X₁ {O(n)}
Run probabilistic analysis on SCC: [l1; l2]
Analysing control-flow refined program
Run classical analysis on SCC: [l1]
Classical Approximation after Lifting Classical Results
All Bounds
Timebounds
Overall timebound:inf {Infinity}
g₀: 1 {O(1)}
g₁₀: 1 {O(1)}
g₁₂: 1 {O(1)}
g₁₅: 1 {O(1)}
g₁₇: inf {Infinity}
g₂₀: inf {Infinity}
g₂₂: inf {Infinity}
g₂₄: inf {Infinity}
g₂₇: inf {Infinity}
g₂₉: inf {Infinity}
g₃₁: inf {Infinity}
Costbounds
Overall costbound: inf {Infinity}
g₀: inf {Infinity}
g₁₀: inf {Infinity}
g₁₂: inf {Infinity}
g₁₅: inf {Infinity}
g₁₇: inf {Infinity}
g₂₀: inf {Infinity}
g₂₂: inf {Infinity}
g₂₄: inf {Infinity}
g₂₇: inf {Infinity}
g₂₉: inf {Infinity}
g₃₁: inf {Infinity}
Sizebounds
(g₀,l1), X₀: X₀ {O(n)}
(g₀,l1), X₁: X₁ {O(n)}
(g₀,l1), X₂: X₂ {O(n)}
(g₁₀,l2_v1), X₀: X₀ {O(n)}
(g₁₀,l2_v1), X₁: X₁ {O(n)}
(g₁₀,l2_v1), X₂: X₂ {O(n)}
Run probabilistic analysis on SCC: [l1]
Run classical analysis on SCC: [l2_v1]
Classical Approximation after Lifting Classical Results
All Bounds
Timebounds
Overall timebound:inf {Infinity}
g₀: 1 {O(1)}
g₁₀: 1 {O(1)}
g₁₂: 1 {O(1)}
g₁₅: 2 {O(1)}
g₁₇: inf {Infinity}
g₂₀: inf {Infinity}
g₂₂: inf {Infinity}
g₂₄: inf {Infinity}
g₂₇: inf {Infinity}
g₂₉: inf {Infinity}
g₃₁: inf {Infinity}
Costbounds
Overall costbound: inf {Infinity}
g₀: inf {Infinity}
g₁₀: inf {Infinity}
g₁₂: inf {Infinity}
g₁₅: inf {Infinity}
g₁₇: inf {Infinity}
g₂₀: inf {Infinity}
g₂₂: inf {Infinity}
g₂₄: inf {Infinity}
g₂₇: inf {Infinity}
g₂₉: inf {Infinity}
g₃₁: inf {Infinity}
Sizebounds
(g₀,l1), X₀: X₀ {O(n)}
(g₀,l1), X₁: X₁ {O(n)}
(g₀,l1), X₂: X₂ {O(n)}
(g₁₀,l2_v1), X₀: X₀ {O(n)}
(g₁₀,l2_v1), X₁: X₁ {O(n)}
(g₁₀,l2_v1), X₂: X₂ {O(n)}
(g₁₂,l1_v1), X₀: 3⋅X₀ {O(n)}
(g₁₂,l1_v1), X₁: X₁ {O(n)}
(g₁₂,l1_v1), X₂: 2⋅X₂ {O(n)}
(g₁₅,l1_v2), X₀: 2⋅X₀ {O(n)}
(g₁₅,l1_v2), X₁: 2⋅X₁ {O(n)}
(g₁₅,l1_v2), X₂: 2⋅X₂ {O(n)}
(g₁₅,l1_v3), X₀: 2⋅X₀ {O(n)}
(g₁₅,l1_v3), X₁: 2⋅X₁ {O(n)}
(g₁₅,l1_v3), X₂: 2⋅X₂ {O(n)}
Run probabilistic analysis on SCC: [l2_v1]
Run classical analysis on SCC: [l1_v1; l2_v4]
TWN: t₂₈: l1_v1→l2_v4
Show Graph
G
l0
l0
l1
l1
l0->l1
t₁
l2_v1
l2_v1
l1->l2_v1
t₉
τ = 1 ≤ X₀ ∧ X₀ ≤ X₂
l1_v1
l1_v1
l2_v4
l2_v4
l1_v1->l2_v4
t₂₈
τ = 1 ≤ X₂ ∧ 3 ≤ X₀ ∧ 0 ≤ X₀ ∧ 2⋅X₀ ≤ 3⋅X₂ ∧ X₁ ≤ 0 ∧ 1 ≤ X₀ ∧ X₀ ≤ X₂
l1_v2
l1_v2
l2_v2
l2_v2
l1_v2->l2_v2
t₁₆
τ = 1+X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₂ ∧ 0 ≤ X₀ ∧ 2⋅X₀ ≤ 3⋅X₂ ∧ 1 ≤ X₀ ∧ X₀ ≤ X₂
l1_v3
l1_v3
l2_v3
l2_v3
l1_v3->l2_v3
t₂₃
τ = 1 ≤ X₁ ∧ 1 ≤ X₂ ∧ 0 ≤ X₀ ∧ 2⋅X₀ ≤ 3⋅X₂ ∧ 1 ≤ X₀ ∧ X₀ ≤ X₂
l1_v4
l1_v4
l1_v4->l2_v2
t₂₁
τ = 1+X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₂ ∧ 0 ≤ X₀ ∧ 2⋅X₀ ≤ 3⋅X₂ ∧ 1 ≤ X₀ ∧ X₀ ≤ X₂
l2_v1->l1_v1
t₁₁
η (X₀) = 3⋅X₀
η (X₂) = 2⋅X₂
τ = 1 ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ X₁ ≤ 0
l2_v1->l1_v2
t₁₃
η (X₀) = X₀-1
τ = 1 ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₁
l2_v1->l1_v3
t₁₄
τ = 1 ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₁
l2_v2->l1_v2
t₁₈
η (X₀) = X₀-1
τ = 1 ≤ X₀ ∧ 1+X₀ ≤ X₂ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₁
l2_v2->l1_v4
t₁₉
τ = 1 ≤ X₀ ∧ 1+X₀ ≤ X₂ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₁
l2_v3->l1_v2
t₂₅
η (X₀) = X₀-1
τ = 1 ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₁
l2_v3->l1_v3
t₂₆
τ = 1 ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₁
l2_v4->l1_v1
t₃₀
η (X₀) = 3⋅X₀
η (X₂) = 2⋅X₂
τ = 1 ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ 3 ≤ X₀ ∧ X₀ ≤ X₂ ∧ X₁ ≤ 0
cycle: [t₂₈: l1_v1→l2_v4; t₃₀: l2_v4→l1_v1]
loop: (1 ≤ X₀ ∧ X₀ ≤ X₂ ∧ X₁ ≤ 0,(X₀,X₁,X₂) -> (3⋅X₀,X₁,2⋅X₂)
order: [X₀; X₁; X₂]
closed-form:X₀: X₀ * 3^n
X₁: X₁
X₂: X₂ * 2^n
Termination: true
Formula:
0 ≤ 1 ∧ 1 ≤ 0 ∧ 1+X₀ ≤ 0 ∧ 1+X₁ ≤ 0 ∧ 0 ≤ X₀ ∧ X₀ ≤ 0
∨ 0 ≤ 1 ∧ 1 ≤ 0 ∧ 1+X₀ ≤ 0 ∧ 0 ≤ X₀ ∧ X₀ ≤ 0 ∧ 0 ≤ X₁ ∧ X₁ ≤ 0
∨ 0 ≤ 1 ∧ 1 ≤ 0 ∧ 1+X₁ ≤ 0 ∧ 1 ≤ X₂ ∧ 0 ≤ X₀ ∧ X₀ ≤ 0
∨ 0 ≤ 1 ∧ 1 ≤ 0 ∧ 1+X₁ ≤ 0 ∧ 0 ≤ X₀ ∧ X₀ ≤ 0 ∧ 0 ≤ X₂ ∧ X₂ ≤ 0
∨ 0 ≤ 1 ∧ 1 ≤ 0 ∧ 1 ≤ X₂ ∧ 0 ≤ X₀ ∧ X₀ ≤ 0 ∧ 0 ≤ X₁ ∧ X₁ ≤ 0
∨ 0 ≤ 1 ∧ 1 ≤ 0 ∧ 0 ≤ X₀ ∧ X₀ ≤ 0 ∧ 0 ≤ X₁ ∧ X₁ ≤ 0 ∧ 0 ≤ X₂ ∧ X₂ ≤ 0
∨ 1 ≤ 0 ∧ 1+X₀ ≤ 0 ∧ 1+X₁ ≤ 0 ∧ 0 ≤ X₀ ∧ X₀ ≤ 0
∨ 1 ≤ 0 ∧ 1+X₀ ≤ 0 ∧ 0 ≤ X₀ ∧ X₀ ≤ 0 ∧ 0 ≤ X₁ ∧ X₁ ≤ 0
∨ 1 ≤ 0 ∧ 1+X₁ ≤ 0 ∧ 1 ≤ X₂ ∧ 0 ≤ X₀ ∧ X₀ ≤ 0
∨ 1 ≤ 0 ∧ 1+X₁ ≤ 0 ∧ 0 ≤ X₀ ∧ X₀ ≤ 0 ∧ 0 ≤ X₂ ∧ X₂ ≤ 0
∨ 1 ≤ 0 ∧ 1 ≤ X₂ ∧ 0 ≤ X₀ ∧ X₀ ≤ 0 ∧ 0 ≤ X₁ ∧ X₁ ≤ 0
∨ 1 ≤ 0 ∧ 0 ≤ X₀ ∧ X₀ ≤ 0 ∧ 0 ≤ X₁ ∧ X₁ ≤ 0 ∧ 0 ≤ X₂ ∧ X₂ ≤ 0
∨ 1 ≤ X₀ ∧ 1+X₀ ≤ 0 ∧ 1+X₁ ≤ 0
∨ 1 ≤ X₀ ∧ 1+X₀ ≤ 0 ∧ 0 ≤ X₁ ∧ X₁ ≤ 0
∨ 1 ≤ X₀ ∧ 1+X₁ ≤ 0 ∧ 1 ≤ X₂ ∧ 0 ≤ X₀ ∧ X₀ ≤ 0
∨ 1 ≤ X₀ ∧ 1+X₁ ≤ 0 ∧ 0 ≤ X₀ ∧ X₀ ≤ 0 ∧ 0 ≤ X₂ ∧ X₂ ≤ 0
∨ 1 ≤ X₀ ∧ 1 ≤ X₂ ∧ 0 ≤ X₀ ∧ X₀ ≤ 0 ∧ 0 ≤ X₁ ∧ X₁ ≤ 0
∨ 1 ≤ X₀ ∧ 0 ≤ X₀ ∧ X₀ ≤ 0 ∧ 0 ≤ X₁ ∧ X₁ ≤ 0 ∧ 0 ≤ X₂ ∧ X₂ ≤ 0
Stabilization-Threshold for: X₀ ≤ X₂
alphas_abs: 1+X₂
M: 0
N: 1
Bound: 2⋅X₂+4 {O(n)}
TWN - Lifting for [28: l1_v1->l2_v4; 30: l2_v4->l1_v1] of 2⋅X₂+8 {O(n)}
relevant size-bounds w.r.t. t₁₁: l2_v1→l1_v1:
X₂: 2⋅X₂ {O(n)}
Runtime-bound of t₁₁: 1 {O(1)}
Results in: 4⋅X₂+8 {O(n)}
Classical Approximation after Lifting Classical Results
All Bounds
Timebounds
Overall timebound:inf {Infinity}
g₀: 1 {O(1)}
g₁₀: 1 {O(1)}
g₁₂: 1 {O(1)}
g₁₅: 2 {O(1)}
g₁₇: inf {Infinity}
g₂₀: inf {Infinity}
g₂₂: inf {Infinity}
g₂₄: inf {Infinity}
g₂₇: inf {Infinity}
g₂₉: 4⋅X₂+8 {O(n)}
g₃₁: 4⋅X₂+8 {O(n)}
Costbounds
Overall costbound: inf {Infinity}
g₀: inf {Infinity}
g₁₀: inf {Infinity}
g₁₂: inf {Infinity}
g₁₅: inf {Infinity}
g₁₇: inf {Infinity}
g₂₀: inf {Infinity}
g₂₂: inf {Infinity}
g₂₄: inf {Infinity}
g₂₇: inf {Infinity}
g₂₉: inf {Infinity}
g₃₁: inf {Infinity}
Sizebounds
(g₀,l1), X₀: X₀ {O(n)}
(g₀,l1), X₁: X₁ {O(n)}
(g₀,l1), X₂: X₂ {O(n)}
(g₁₀,l2_v1), X₀: X₀ {O(n)}
(g₁₀,l2_v1), X₁: X₁ {O(n)}
(g₁₀,l2_v1), X₂: X₂ {O(n)}
(g₁₂,l1_v1), X₀: 3⋅X₀ {O(n)}
(g₁₂,l1_v1), X₁: X₁ {O(n)}
(g₁₂,l1_v1), X₂: 2⋅X₂ {O(n)}
(g₁₅,l1_v2), X₀: X₀ {O(n)}
(g₁₅,l1_v2), X₁: X₁ {O(n)}
(g₁₅,l1_v2), X₂: X₂ {O(n)}
(g₁₅,l1_v3), X₀: X₀ {O(n)}
(g₁₅,l1_v3), X₁: X₁ {O(n)}
(g₁₅,l1_v3), X₂: X₂ {O(n)}
(g₂₉,l2_v4), X₀: 3⋅3^(4⋅X₂+8)⋅X₀ {O(EXP)}
(g₂₉,l2_v4), X₁: X₁ {O(n)}
(g₂₉,l2_v4), X₂: 2⋅2^(4⋅X₂+8)⋅X₂ {O(EXP)}
(g₃₁,l1_v1), X₀: 3⋅3^(4⋅X₂+8)⋅X₀ {O(EXP)}
(g₃₁,l1_v1), X₁: X₁ {O(n)}
(g₃₁,l1_v1), X₂: 2⋅2^(4⋅X₂+8)⋅X₂ {O(EXP)}
Run probabilistic analysis on SCC: [l1_v1; l2_v4]
Run classical analysis on SCC: [l1_v3; l2_v3]
Classical Approximation after Lifting Classical Results
All Bounds
Timebounds
Overall timebound:inf {Infinity}
g₀: 1 {O(1)}
g₁₀: 1 {O(1)}
g₁₂: 1 {O(1)}
g₁₅: 2 {O(1)}
g₁₇: inf {Infinity}
g₂₀: inf {Infinity}
g₂₂: inf {Infinity}
g₂₄: inf {Infinity}
g₂₇: inf {Infinity}
g₂₉: 4⋅X₂+8 {O(n)}
g₃₁: 4⋅X₂+8 {O(n)}
Costbounds
Overall costbound: inf {Infinity}
g₀: inf {Infinity}
g₁₀: inf {Infinity}
g₁₂: inf {Infinity}
g₁₅: inf {Infinity}
g₁₇: inf {Infinity}
g₂₀: inf {Infinity}
g₂₂: inf {Infinity}
g₂₄: inf {Infinity}
g₂₇: inf {Infinity}
g₂₉: inf {Infinity}
g₃₁: inf {Infinity}
Sizebounds
(g₀,l1), X₀: X₀ {O(n)}
(g₀,l1), X₁: X₁ {O(n)}
(g₀,l1), X₂: X₂ {O(n)}
(g₁₀,l2_v1), X₀: X₀ {O(n)}
(g₁₀,l2_v1), X₁: X₁ {O(n)}
(g₁₀,l2_v1), X₂: X₂ {O(n)}
(g₁₂,l1_v1), X₀: 3⋅X₀ {O(n)}
(g₁₂,l1_v1), X₁: X₁ {O(n)}
(g₁₂,l1_v1), X₂: 2⋅X₂ {O(n)}
(g₁₅,l1_v2), X₀: X₀ {O(n)}
(g₁₅,l1_v2), X₁: X₁ {O(n)}
(g₁₅,l1_v2), X₂: X₂ {O(n)}
(g₁₅,l1_v3), X₀: X₀ {O(n)}
(g₁₅,l1_v3), X₁: X₁ {O(n)}
(g₁₅,l1_v3), X₂: X₂ {O(n)}
(g₂₄,l2_v3), X₀: X₀ {O(n)}
(g₂₄,l2_v3), X₁: X₁ {O(n)}
(g₂₄,l2_v3), X₂: X₂ {O(n)}
(g₂₇,l1_v2), X₀: 2⋅X₀ {O(n)}
(g₂₇,l1_v2), X₁: 2⋅X₁ {O(n)}
(g₂₇,l1_v2), X₂: 2⋅X₂ {O(n)}
(g₂₇,l1_v3), X₀: 2⋅X₀ {O(n)}
(g₂₇,l1_v3), X₁: 2⋅X₁ {O(n)}
(g₂₇,l1_v3), X₂: 2⋅X₂ {O(n)}
(g₂₉,l2_v4), X₀: 3⋅3^(4⋅X₂+8)⋅X₀ {O(EXP)}
(g₂₉,l2_v4), X₁: X₁ {O(n)}
(g₂₉,l2_v4), X₂: 2⋅2^(4⋅X₂+8)⋅X₂ {O(EXP)}
(g₃₁,l1_v1), X₀: 3⋅3^(4⋅X₂+8)⋅X₀ {O(EXP)}
(g₃₁,l1_v1), X₁: X₁ {O(n)}
(g₃₁,l1_v1), X₂: 2⋅2^(4⋅X₂+8)⋅X₂ {O(EXP)}
Run probabilistic analysis on SCC: [l1_v3; l2_v3]
Plrf for transition g₂₄:l1_v3(X₀,X₁,X₂) → t₂₃:l2_v3(X₀,X₁,X₂) :|: 1 ≤ X₀ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₂ ∧ 0 ≤ X₀ ∧ 2⋅X₀ ≤ 3⋅X₂:
new bound:
X₂+1 {O(n)}
PLRF:
• l1_v2: X₂-1
• l1_v3: 1+X₂
• l2_v3: X₂
Show Graph
G
l0
l0
l1
l1
l0->l1
p = 1
t₁ ∈ g₀
l2_v1
l2_v1
l1->l2_v1
p = 1
t₉ ∈ g₁₀
τ = 1 ≤ X₀ ∧ X₀ ≤ X₂
l1_v1
l1_v1
l2_v4
l2_v4
l1_v1->l2_v4
p = 1
t₂₈ ∈ g₂₉
τ = 1 ≤ X₂ ∧ 3 ≤ X₀ ∧ 0 ≤ X₀ ∧ 2⋅X₀ ≤ 3⋅X₂ ∧ X₁ ≤ 0 ∧ 1 ≤ X₀ ∧ X₀ ≤ X₂
l1_v2
l1_v2
l2_v2
l2_v2
l1_v2->l2_v2
p = 1
t₁₆ ∈ g₁₇
τ = 1+X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₂ ∧ 0 ≤ X₀ ∧ 2⋅X₀ ≤ 3⋅X₂ ∧ 1 ≤ X₀ ∧ X₀ ≤ X₂
l1_v3
l1_v3
l2_v3
l2_v3
l1_v3->l2_v3
p = 1
t₂₃ ∈ g₂₄
τ = 1 ≤ X₁ ∧ 1 ≤ X₂ ∧ 0 ≤ X₀ ∧ 2⋅X₀ ≤ 3⋅X₂ ∧ 1 ≤ X₀ ∧ X₀ ≤ X₂
l1_v4
l1_v4
l1_v4->l2_v2
p = 1
t₂₁ ∈ g₂₂
τ = 1+X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₂ ∧ 0 ≤ X₀ ∧ 2⋅X₀ ≤ 3⋅X₂ ∧ 1 ≤ X₀ ∧ X₀ ≤ X₂
l2_v1->l1_v1
p = 1
t₁₁ ∈ g₁₂
η (X₀) = 3⋅X₀
η (X₂) = 2⋅X₂
τ = 1 ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ X₁ ≤ 0
l2_v1->l1_v2
p = 1/2
t₁₃ ∈ g₁₅
η (X₀) = X₀-1
τ = 1 ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₁
l2_v1->l1_v3
p = 1/2
t₁₄ ∈ g₁₅
τ = 1 ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₁
l2_v2->l1_v2
p = 1/2
t₁₈ ∈ g₂₀
η (X₀) = X₀-1
τ = 1 ≤ X₀ ∧ 1+X₀ ≤ X₂ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₁
l2_v2->l1_v4
p = 1/2
t₁₉ ∈ g₂₀
τ = 1 ≤ X₀ ∧ 1+X₀ ≤ X₂ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₁
l2_v3->l1_v2
p = 1/2
t₂₅ ∈ g₂₇
η (X₀) = X₀-1
τ = 1 ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₁
l2_v3->l1_v3
p = 1/2
t₂₆ ∈ g₂₇
τ = 1 ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₁
l2_v4->l1_v1
p = 1
t₃₀ ∈ g₃₁
η (X₀) = 3⋅X₀
η (X₂) = 2⋅X₂
τ = 1 ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ 3 ≤ X₀ ∧ X₀ ≤ X₂ ∧ X₁ ≤ 0
Use expected size bounds for entry point (g₁₅:l2_v1→[t₁₃:1/2:l1_v2; t₁₄:1/2:l1_v3],l1_v3)
Use classical time bound for entry point (g₁₅:l2_v1→[t₁₃:1/2:l1_v2; t₁₄:1/2:l1_v3],l1_v3)
Plrf for transition g₂₇:l2_v3(X₀,X₁,X₂) → [1/2]:t₂₅:l1_v2(X₀-1,X₁,X₂) :+: [1/2]:t₂₆:l1_v3(X₀,X₁,X₂) :|: 1 ≤ X₁ ∧ 1 ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂:
new bound:
2⋅X₂ {O(n)}
PLRF:
• l1_v2: 0
• l1_v3: 2⋅X₂
• l2_v3: 1+X₂
Show Graph
G
l0
l0
l1
l1
l0->l1
p = 1
t₁ ∈ g₀
l2_v1
l2_v1
l1->l2_v1
p = 1
t₉ ∈ g₁₀
τ = 1 ≤ X₀ ∧ X₀ ≤ X₂
l1_v1
l1_v1
l2_v4
l2_v4
l1_v1->l2_v4
p = 1
t₂₈ ∈ g₂₉
τ = 1 ≤ X₂ ∧ 3 ≤ X₀ ∧ 0 ≤ X₀ ∧ 2⋅X₀ ≤ 3⋅X₂ ∧ X₁ ≤ 0 ∧ 1 ≤ X₀ ∧ X₀ ≤ X₂
l1_v2
l1_v2
l2_v2
l2_v2
l1_v2->l2_v2
p = 1
t₁₆ ∈ g₁₇
τ = 1+X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₂ ∧ 0 ≤ X₀ ∧ 2⋅X₀ ≤ 3⋅X₂ ∧ 1 ≤ X₀ ∧ X₀ ≤ X₂
l1_v3
l1_v3
l2_v3
l2_v3
l1_v3->l2_v3
p = 1
t₂₃ ∈ g₂₄
τ = 1 ≤ X₁ ∧ 1 ≤ X₂ ∧ 0 ≤ X₀ ∧ 2⋅X₀ ≤ 3⋅X₂ ∧ 1 ≤ X₀ ∧ X₀ ≤ X₂
l1_v4
l1_v4
l1_v4->l2_v2
p = 1
t₂₁ ∈ g₂₂
τ = 1+X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₂ ∧ 0 ≤ X₀ ∧ 2⋅X₀ ≤ 3⋅X₂ ∧ 1 ≤ X₀ ∧ X₀ ≤ X₂
l2_v1->l1_v1
p = 1
t₁₁ ∈ g₁₂
η (X₀) = 3⋅X₀
η (X₂) = 2⋅X₂
τ = 1 ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ X₁ ≤ 0
l2_v1->l1_v2
p = 1/2
t₁₃ ∈ g₁₅
η (X₀) = X₀-1
τ = 1 ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₁
l2_v1->l1_v3
p = 1/2
t₁₄ ∈ g₁₅
τ = 1 ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₁
l2_v2->l1_v2
p = 1/2
t₁₈ ∈ g₂₀
η (X₀) = X₀-1
τ = 1 ≤ X₀ ∧ 1+X₀ ≤ X₂ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₁
l2_v2->l1_v4
p = 1/2
t₁₉ ∈ g₂₀
τ = 1 ≤ X₀ ∧ 1+X₀ ≤ X₂ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₁
l2_v3->l1_v2
p = 1/2
t₂₅ ∈ g₂₇
η (X₀) = X₀-1
τ = 1 ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₁
l2_v3->l1_v3
p = 1/2
t₂₆ ∈ g₂₇
τ = 1 ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₁
l2_v4->l1_v1
p = 1
t₃₀ ∈ g₃₁
η (X₀) = 3⋅X₀
η (X₂) = 2⋅X₂
τ = 1 ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ 3 ≤ X₀ ∧ X₀ ≤ X₂ ∧ X₁ ≤ 0
Use expected size bounds for entry point (g₁₅:l2_v1→[t₁₃:1/2:l1_v2; t₁₄:1/2:l1_v3],l1_v3)
Use classical time bound for entry point (g₁₅:l2_v1→[t₁₃:1/2:l1_v2; t₁₄:1/2:l1_v3],l1_v3)
Run classical analysis on SCC: [l1_v2; l1_v4; l2_v2]
MPRF for transition t₁₆: l1_v2(X₀,X₁,X₂) → l2_v2(X₀,X₁,X₂) :|: 1+X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₂ ∧ 0 ≤ X₀ ∧ 2⋅X₀ ≤ 3⋅X₂ ∧ 1 ≤ X₀ ∧ X₀ ≤ X₂ of depth 1:
new bound:
2⋅X₀+2 {O(n)}
MPRF:
• l1_v2: [1+X₀]
• l1_v4: [X₀]
• l2_v2: [X₀]
Show Graph
G
l0
l0
l1
l1
l0->l1
t₁
l2_v1
l2_v1
l1->l2_v1
t₉
τ = 1 ≤ X₀ ∧ X₀ ≤ X₂
l1_v1
l1_v1
l2_v4
l2_v4
l1_v1->l2_v4
t₂₈
τ = 1 ≤ X₂ ∧ 3 ≤ X₀ ∧ 0 ≤ X₀ ∧ 2⋅X₀ ≤ 3⋅X₂ ∧ X₁ ≤ 0 ∧ 1 ≤ X₀ ∧ X₀ ≤ X₂
l1_v2
l1_v2
l2_v2
l2_v2
l1_v2->l2_v2
t₁₆
τ = 1+X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₂ ∧ 0 ≤ X₀ ∧ 2⋅X₀ ≤ 3⋅X₂ ∧ 1 ≤ X₀ ∧ X₀ ≤ X₂
l1_v3
l1_v3
l2_v3
l2_v3
l1_v3->l2_v3
t₂₃
τ = 1 ≤ X₁ ∧ 1 ≤ X₂ ∧ 0 ≤ X₀ ∧ 2⋅X₀ ≤ 3⋅X₂ ∧ 1 ≤ X₀ ∧ X₀ ≤ X₂
l1_v4
l1_v4
l1_v4->l2_v2
t₂₁
τ = 1+X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₂ ∧ 0 ≤ X₀ ∧ 2⋅X₀ ≤ 3⋅X₂ ∧ 1 ≤ X₀ ∧ X₀ ≤ X₂
l2_v1->l1_v1
t₁₁
η (X₀) = 3⋅X₀
η (X₂) = 2⋅X₂
τ = 1 ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ X₁ ≤ 0
l2_v1->l1_v2
t₁₃
η (X₀) = X₀-1
τ = 1 ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₁
l2_v1->l1_v3
t₁₄
τ = 1 ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₁
l2_v2->l1_v2
t₁₈
η (X₀) = X₀-1
τ = 1 ≤ X₀ ∧ 1+X₀ ≤ X₂ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₁
l2_v2->l1_v4
t₁₉
τ = 1 ≤ X₀ ∧ 1+X₀ ≤ X₂ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₁
l2_v3->l1_v2
t₂₅
η (X₀) = X₀-1
τ = 1 ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₁
l2_v3->l1_v3
t₂₆
τ = 1 ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₁
l2_v4->l1_v1
t₃₀
η (X₀) = 3⋅X₀
η (X₂) = 2⋅X₂
τ = 1 ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ 3 ≤ X₀ ∧ X₀ ≤ X₂ ∧ X₁ ≤ 0
MPRF for transition t₁₈: l2_v2(X₀,X₁,X₂) → l1_v2(X₀-1,X₁,X₂) :|: 1 ≤ X₀ ∧ 1+X₀ ≤ X₂ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₁ of depth 1:
new bound:
2⋅X₀ {O(n)}
MPRF:
• l1_v2: [X₀]
• l1_v4: [X₀]
• l2_v2: [X₀]
Show Graph
G
l0
l0
l1
l1
l0->l1
t₁
l2_v1
l2_v1
l1->l2_v1
t₉
τ = 1 ≤ X₀ ∧ X₀ ≤ X₂
l1_v1
l1_v1
l2_v4
l2_v4
l1_v1->l2_v4
t₂₈
τ = 1 ≤ X₂ ∧ 3 ≤ X₀ ∧ 0 ≤ X₀ ∧ 2⋅X₀ ≤ 3⋅X₂ ∧ X₁ ≤ 0 ∧ 1 ≤ X₀ ∧ X₀ ≤ X₂
l1_v2
l1_v2
l2_v2
l2_v2
l1_v2->l2_v2
t₁₆
τ = 1+X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₂ ∧ 0 ≤ X₀ ∧ 2⋅X₀ ≤ 3⋅X₂ ∧ 1 ≤ X₀ ∧ X₀ ≤ X₂
l1_v3
l1_v3
l2_v3
l2_v3
l1_v3->l2_v3
t₂₃
τ = 1 ≤ X₁ ∧ 1 ≤ X₂ ∧ 0 ≤ X₀ ∧ 2⋅X₀ ≤ 3⋅X₂ ∧ 1 ≤ X₀ ∧ X₀ ≤ X₂
l1_v4
l1_v4
l1_v4->l2_v2
t₂₁
τ = 1+X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₂ ∧ 0 ≤ X₀ ∧ 2⋅X₀ ≤ 3⋅X₂ ∧ 1 ≤ X₀ ∧ X₀ ≤ X₂
l2_v1->l1_v1
t₁₁
η (X₀) = 3⋅X₀
η (X₂) = 2⋅X₂
τ = 1 ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ X₁ ≤ 0
l2_v1->l1_v2
t₁₃
η (X₀) = X₀-1
τ = 1 ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₁
l2_v1->l1_v3
t₁₄
τ = 1 ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₁
l2_v2->l1_v2
t₁₈
η (X₀) = X₀-1
τ = 1 ≤ X₀ ∧ 1+X₀ ≤ X₂ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₁
l2_v2->l1_v4
t₁₉
τ = 1 ≤ X₀ ∧ 1+X₀ ≤ X₂ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₁
l2_v3->l1_v2
t₂₅
η (X₀) = X₀-1
τ = 1 ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₁
l2_v3->l1_v3
t₂₆
τ = 1 ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₁
l2_v4->l1_v1
t₃₀
η (X₀) = 3⋅X₀
η (X₂) = 2⋅X₂
τ = 1 ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ 3 ≤ X₀ ∧ X₀ ≤ X₂ ∧ X₁ ≤ 0
Classical Approximation after Lifting Classical Results
All Bounds
Timebounds
Overall timebound:inf {Infinity}
g₀: 1 {O(1)}
g₁₀: 1 {O(1)}
g₁₂: 1 {O(1)}
g₁₅: 2 {O(1)}
g₁₇: 2⋅X₀+2 {O(n)}
g₂₀: inf {Infinity}
g₂₂: inf {Infinity}
g₂₄: X₂+1 {O(n)}
g₂₇: 2⋅X₂ {O(n)}
g₂₉: 4⋅X₂+8 {O(n)}
g₃₁: 4⋅X₂+8 {O(n)}
Costbounds
Overall costbound: inf {Infinity}
g₀: inf {Infinity}
g₁₀: inf {Infinity}
g₁₂: inf {Infinity}
g₁₅: inf {Infinity}
g₁₇: inf {Infinity}
g₂₀: inf {Infinity}
g₂₂: inf {Infinity}
g₂₄: inf {Infinity}
g₂₇: inf {Infinity}
g₂₉: inf {Infinity}
g₃₁: inf {Infinity}
Sizebounds
(g₀,l1), X₀: X₀ {O(n)}
(g₀,l1), X₁: X₁ {O(n)}
(g₀,l1), X₂: X₂ {O(n)}
(g₁₀,l2_v1), X₀: X₀ {O(n)}
(g₁₀,l2_v1), X₁: X₁ {O(n)}
(g₁₀,l2_v1), X₂: X₂ {O(n)}
(g₁₂,l1_v1), X₀: 3⋅X₀ {O(n)}
(g₁₂,l1_v1), X₁: X₁ {O(n)}
(g₁₂,l1_v1), X₂: 2⋅X₂ {O(n)}
(g₁₅,l1_v2), X₀: X₀ {O(n)}
(g₁₅,l1_v2), X₁: X₁ {O(n)}
(g₁₅,l1_v2), X₂: X₂ {O(n)}
(g₁₅,l1_v3), X₀: X₀ {O(n)}
(g₁₅,l1_v3), X₁: X₁ {O(n)}
(g₁₅,l1_v3), X₂: X₂ {O(n)}
(g₁₇,l2_v2), X₀: 2⋅X₀ {O(n)}
(g₁₇,l2_v2), X₁: 2⋅X₁ {O(n)}
(g₁₇,l2_v2), X₂: 2⋅X₂ {O(n)}
(g₂₀,l1_v2), X₀: 4⋅X₀ {O(n)}
(g₂₀,l1_v2), X₁: 4⋅X₁ {O(n)}
(g₂₀,l1_v2), X₂: 4⋅X₂ {O(n)}
(g₂₀,l1_v4), X₀: 4⋅X₀ {O(n)}
(g₂₀,l1_v4), X₁: 4⋅X₁ {O(n)}
(g₂₀,l1_v4), X₂: 4⋅X₂ {O(n)}
(g₂₂,l2_v2), X₀: 2⋅X₀ {O(n)}
(g₂₂,l2_v2), X₁: 2⋅X₁ {O(n)}
(g₂₂,l2_v2), X₂: 2⋅X₂ {O(n)}
(g₂₄,l2_v3), X₀: X₀ {O(n)}
(g₂₄,l2_v3), X₁: X₁ {O(n)}
(g₂₄,l2_v3), X₂: X₂ {O(n)}
(g₂₇,l1_v2), X₀: X₀ {O(n)}
(g₂₇,l1_v2), X₁: X₁ {O(n)}
(g₂₇,l1_v2), X₂: X₂ {O(n)}
(g₂₇,l1_v3), X₀: X₀ {O(n)}
(g₂₇,l1_v3), X₁: X₁ {O(n)}
(g₂₇,l1_v3), X₂: X₂ {O(n)}
(g₂₉,l2_v4), X₀: 3⋅3^(4⋅X₂+8)⋅X₀ {O(EXP)}
(g₂₉,l2_v4), X₁: X₁ {O(n)}
(g₂₉,l2_v4), X₂: 2⋅2^(4⋅X₂+8)⋅X₂ {O(EXP)}
(g₃₁,l1_v1), X₀: 3⋅3^(4⋅X₂+8)⋅X₀ {O(EXP)}
(g₃₁,l1_v1), X₁: X₁ {O(n)}
(g₃₁,l1_v1), X₂: 2⋅2^(4⋅X₂+8)⋅X₂ {O(EXP)}
Run probabilistic analysis on SCC: [l1_v2; l1_v4; l2_v2]
Plrf for transition g₂₀:l2_v2(X₀,X₁,X₂) → [1/2]:t₁₈:l1_v2(X₀-1,X₁,X₂) :+: [1/2]:t₁₉:l1_v4(X₀,X₁,X₂) :|: 1 ≤ X₁ ∧ 1 ≤ X₀ ∧ 1+X₀ ≤ X₂ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂:
new bound:
4⋅X₀ {O(n)}
PLRF:
• l1_v2: 2⋅X₀
• l1_v4: 2⋅X₀
• l2_v2: 2⋅X₀
Show Graph
G
l0
l0
l1
l1
l0->l1
p = 1
t₁ ∈ g₀
l2_v1
l2_v1
l1->l2_v1
p = 1
t₉ ∈ g₁₀
τ = 1 ≤ X₀ ∧ X₀ ≤ X₂
l1_v1
l1_v1
l2_v4
l2_v4
l1_v1->l2_v4
p = 1
t₂₈ ∈ g₂₉
τ = 1 ≤ X₂ ∧ 3 ≤ X₀ ∧ 0 ≤ X₀ ∧ 2⋅X₀ ≤ 3⋅X₂ ∧ X₁ ≤ 0 ∧ 1 ≤ X₀ ∧ X₀ ≤ X₂
l1_v2
l1_v2
l2_v2
l2_v2
l1_v2->l2_v2
p = 1
t₁₆ ∈ g₁₇
τ = 1+X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₂ ∧ 0 ≤ X₀ ∧ 2⋅X₀ ≤ 3⋅X₂ ∧ 1 ≤ X₀ ∧ X₀ ≤ X₂
l1_v3
l1_v3
l2_v3
l2_v3
l1_v3->l2_v3
p = 1
t₂₃ ∈ g₂₄
τ = 1 ≤ X₁ ∧ 1 ≤ X₂ ∧ 0 ≤ X₀ ∧ 2⋅X₀ ≤ 3⋅X₂ ∧ 1 ≤ X₀ ∧ X₀ ≤ X₂
l1_v4
l1_v4
l1_v4->l2_v2
p = 1
t₂₁ ∈ g₂₂
τ = 1+X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₂ ∧ 0 ≤ X₀ ∧ 2⋅X₀ ≤ 3⋅X₂ ∧ 1 ≤ X₀ ∧ X₀ ≤ X₂
l2_v1->l1_v1
p = 1
t₁₁ ∈ g₁₂
η (X₀) = 3⋅X₀
η (X₂) = 2⋅X₂
τ = 1 ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ X₁ ≤ 0
l2_v1->l1_v2
p = 1/2
t₁₃ ∈ g₁₅
η (X₀) = X₀-1
τ = 1 ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₁
l2_v1->l1_v3
p = 1/2
t₁₄ ∈ g₁₅
τ = 1 ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₁
l2_v2->l1_v2
p = 1/2
t₁₈ ∈ g₂₀
η (X₀) = X₀-1
τ = 1 ≤ X₀ ∧ 1+X₀ ≤ X₂ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₁
l2_v2->l1_v4
p = 1/2
t₁₉ ∈ g₂₀
τ = 1 ≤ X₀ ∧ 1+X₀ ≤ X₂ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₁
l2_v3->l1_v2
p = 1/2
t₂₅ ∈ g₂₇
η (X₀) = X₀-1
τ = 1 ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₁
l2_v3->l1_v3
p = 1/2
t₂₆ ∈ g₂₇
τ = 1 ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₁
l2_v4->l1_v1
p = 1
t₃₀ ∈ g₃₁
η (X₀) = 3⋅X₀
η (X₂) = 2⋅X₂
τ = 1 ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ 3 ≤ X₀ ∧ X₀ ≤ X₂ ∧ X₁ ≤ 0
Use expected size bounds for entry point (g₁₅:l2_v1→[t₁₃:1/2:l1_v2; t₁₄:1/2:l1_v3],l1_v2)
Use classical time bound for entry point (g₁₅:l2_v1→[t₁₃:1/2:l1_v2; t₁₄:1/2:l1_v3],l1_v2)
Use expected size bounds for entry point (g₂₇:l2_v3→[t₂₅:1/2:l1_v2; t₂₆:1/2:l1_v3],l1_v2)
Use classical time bound for entry point (g₂₇:l2_v3→[t₂₅:1/2:l1_v2; t₂₆:1/2:l1_v3],l1_v2)
Plrf for transition g₂₂:l1_v4(X₀,X₁,X₂) → t₂₁:l2_v2(X₀,X₁,X₂) :|: 1 ≤ X₀ ∧ X₀ ≤ X₂ ∧ 1+X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₂ ∧ 0 ≤ X₀ ∧ 2⋅X₀ ≤ 3⋅X₂:
new bound:
2⋅X₀ {O(n)}
PLRF:
• l1_v2: X₀
• l1_v4: 1+X₀
• l2_v2: X₀
Show Graph
G
l0
l0
l1
l1
l0->l1
p = 1
t₁ ∈ g₀
l2_v1
l2_v1
l1->l2_v1
p = 1
t₉ ∈ g₁₀
τ = 1 ≤ X₀ ∧ X₀ ≤ X₂
l1_v1
l1_v1
l2_v4
l2_v4
l1_v1->l2_v4
p = 1
t₂₈ ∈ g₂₉
τ = 1 ≤ X₂ ∧ 3 ≤ X₀ ∧ 0 ≤ X₀ ∧ 2⋅X₀ ≤ 3⋅X₂ ∧ X₁ ≤ 0 ∧ 1 ≤ X₀ ∧ X₀ ≤ X₂
l1_v2
l1_v2
l2_v2
l2_v2
l1_v2->l2_v2
p = 1
t₁₆ ∈ g₁₇
τ = 1+X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₂ ∧ 0 ≤ X₀ ∧ 2⋅X₀ ≤ 3⋅X₂ ∧ 1 ≤ X₀ ∧ X₀ ≤ X₂
l1_v3
l1_v3
l2_v3
l2_v3
l1_v3->l2_v3
p = 1
t₂₃ ∈ g₂₄
τ = 1 ≤ X₁ ∧ 1 ≤ X₂ ∧ 0 ≤ X₀ ∧ 2⋅X₀ ≤ 3⋅X₂ ∧ 1 ≤ X₀ ∧ X₀ ≤ X₂
l1_v4
l1_v4
l1_v4->l2_v2
p = 1
t₂₁ ∈ g₂₂
τ = 1+X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₂ ∧ 0 ≤ X₀ ∧ 2⋅X₀ ≤ 3⋅X₂ ∧ 1 ≤ X₀ ∧ X₀ ≤ X₂
l2_v1->l1_v1
p = 1
t₁₁ ∈ g₁₂
η (X₀) = 3⋅X₀
η (X₂) = 2⋅X₂
τ = 1 ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ X₁ ≤ 0
l2_v1->l1_v2
p = 1/2
t₁₃ ∈ g₁₅
η (X₀) = X₀-1
τ = 1 ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₁
l2_v1->l1_v3
p = 1/2
t₁₄ ∈ g₁₅
τ = 1 ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₁
l2_v2->l1_v2
p = 1/2
t₁₈ ∈ g₂₀
η (X₀) = X₀-1
τ = 1 ≤ X₀ ∧ 1+X₀ ≤ X₂ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₁
l2_v2->l1_v4
p = 1/2
t₁₉ ∈ g₂₀
τ = 1 ≤ X₀ ∧ 1+X₀ ≤ X₂ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₁
l2_v3->l1_v2
p = 1/2
t₂₅ ∈ g₂₇
η (X₀) = X₀-1
τ = 1 ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₁
l2_v3->l1_v3
p = 1/2
t₂₆ ∈ g₂₇
τ = 1 ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₁
l2_v4->l1_v1
p = 1
t₃₀ ∈ g₃₁
η (X₀) = 3⋅X₀
η (X₂) = 2⋅X₂
τ = 1 ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ 3 ≤ X₀ ∧ X₀ ≤ X₂ ∧ X₁ ≤ 0
Use expected size bounds for entry point (g₁₅:l2_v1→[t₁₃:1/2:l1_v2; t₁₄:1/2:l1_v3],l1_v2)
Use classical time bound for entry point (g₁₅:l2_v1→[t₁₃:1/2:l1_v2; t₁₄:1/2:l1_v3],l1_v2)
Use expected size bounds for entry point (g₂₇:l2_v3→[t₂₅:1/2:l1_v2; t₂₆:1/2:l1_v3],l1_v2)
Use classical time bound for entry point (g₂₇:l2_v3→[t₂₅:1/2:l1_v2; t₂₆:1/2:l1_v3],l1_v2)
CFR: Improvement to new bound with the following program:
method: PartialEvaluationProbabilistic new bound:
O(n)
cfr-program:
Start: l0
Program_Vars: X₀, X₁, X₂
Temp_Vars:
Locations: l0, l1, l1_v1, l1_v2, l1_v3, l1_v4, l2_v1, l2_v2, l2_v3, l2_v4
Transitions:
g₀:l0(X₀,X₁,X₂) → t₁:l1(X₀,X₁,X₂) :|:
g₁₀:l1(X₀,X₁,X₂) → t₉:l2_v1(X₀,X₁,X₂) :|: 1 ≤ X₀ ∧ X₀ ≤ X₂
g₁₂:l2_v1(X₀,X₁,X₂) → t₁₁:l1_v1(3⋅X₀,X₁,2⋅X₂) :|: X₁ ≤ 0 ∧ 1 ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂
g₁₅:l2_v1(X₀,X₁,X₂) → [1/2]:t₁₃:l1_v2(X₀-1,X₁,X₂) :+: [1/2]:t₁₄:l1_v3(X₀,X₁,X₂) :|: 1 ≤ X₁ ∧ 1 ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂
g₁₇:l1_v2(X₀,X₁,X₂) → t₁₆:l2_v2(X₀,X₁,X₂) :|: 1 ≤ X₀ ∧ X₀ ≤ X₂ ∧ 1+X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₂ ∧ 0 ≤ X₀ ∧ 2⋅X₀ ≤ 3⋅X₂
g₂₀:l2_v2(X₀,X₁,X₂) → [1/2]:t₁₈:l1_v2(X₀-1,X₁,X₂) :+: [1/2]:t₁₉:l1_v4(X₀,X₁,X₂) :|: 1 ≤ X₁ ∧ 1 ≤ X₀ ∧ 1+X₀ ≤ X₂ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂
g₂₂:l1_v4(X₀,X₁,X₂) → t₂₁:l2_v2(X₀,X₁,X₂) :|: 1 ≤ X₀ ∧ X₀ ≤ X₂ ∧ 1+X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₂ ∧ 0 ≤ X₀ ∧ 2⋅X₀ ≤ 3⋅X₂
g₂₄:l1_v3(X₀,X₁,X₂) → t₂₃:l2_v3(X₀,X₁,X₂) :|: 1 ≤ X₀ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₂ ∧ 0 ≤ X₀ ∧ 2⋅X₀ ≤ 3⋅X₂
g₂₇:l2_v3(X₀,X₁,X₂) → [1/2]:t₂₅:l1_v2(X₀-1,X₁,X₂) :+: [1/2]:t₂₆:l1_v3(X₀,X₁,X₂) :|: 1 ≤ X₁ ∧ 1 ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂
g₂₉:l1_v1(X₀,X₁,X₂) → t₂₈:l2_v4(X₀,X₁,X₂) :|: 1 ≤ X₀ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₂ ∧ 3 ≤ X₀ ∧ 0 ≤ X₀ ∧ 2⋅X₀ ≤ 3⋅X₂ ∧ X₁ ≤ 0
g₃₁:l2_v4(X₀,X₁,X₂) → t₃₀:l1_v1(3⋅X₀,X₁,2⋅X₂) :|: X₁ ≤ 0 ∧ 1 ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ 3 ≤ X₀ ∧ X₀ ≤ X₂
Show Graph
G
l0
l0
l1
l1
l0->l1
t₁ ∈ g₀
l2_v1
l2_v1
l1->l2_v1
t₉ ∈ g₁₀
τ = 1 ≤ X₀ ∧ X₀ ≤ X₂
l1_v1
l1_v1
l2_v4
l2_v4
l1_v1->l2_v4
t₂₈ ∈ g₂₉
τ = 1 ≤ X₂ ∧ 3 ≤ X₀ ∧ 0 ≤ X₀ ∧ 2⋅X₀ ≤ 3⋅X₂ ∧ X₁ ≤ 0 ∧ 1 ≤ X₀ ∧ X₀ ≤ X₂
l1_v2
l1_v2
l2_v2
l2_v2
l1_v2->l2_v2
t₁₆ ∈ g₁₇
τ = 1+X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₂ ∧ 0 ≤ X₀ ∧ 2⋅X₀ ≤ 3⋅X₂ ∧ 1 ≤ X₀ ∧ X₀ ≤ X₂
l1_v3
l1_v3
l2_v3
l2_v3
l1_v3->l2_v3
t₂₃ ∈ g₂₄
τ = 1 ≤ X₁ ∧ 1 ≤ X₂ ∧ 0 ≤ X₀ ∧ 2⋅X₀ ≤ 3⋅X₂ ∧ 1 ≤ X₀ ∧ X₀ ≤ X₂
l1_v4
l1_v4
l1_v4->l2_v2
t₂₁ ∈ g₂₂
τ = 1+X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₂ ∧ 0 ≤ X₀ ∧ 2⋅X₀ ≤ 3⋅X₂ ∧ 1 ≤ X₀ ∧ X₀ ≤ X₂
l2_v1->l1_v1
t₁₁ ∈ g₁₂
η (X₀) = 3⋅X₀
η (X₂) = 2⋅X₂
τ = 1 ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ X₁ ≤ 0
l2_v1->l1_v2
t₁₃ ∈ g₁₅
η (X₀) = X₀-1
τ = 1 ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₁
l2_v1->l1_v3
t₁₄ ∈ g₁₅
τ = 1 ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₁
l2_v2->l1_v2
t₁₈ ∈ g₂₀
η (X₀) = X₀-1
τ = 1 ≤ X₀ ∧ 1+X₀ ≤ X₂ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₁
l2_v2->l1_v4
t₁₉ ∈ g₂₀
τ = 1 ≤ X₀ ∧ 1+X₀ ≤ X₂ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₁
l2_v3->l1_v2
t₂₅ ∈ g₂₇
η (X₀) = X₀-1
τ = 1 ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₁
l2_v3->l1_v3
t₂₆ ∈ g₂₇
τ = 1 ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₁
l2_v4->l1_v1
t₃₀ ∈ g₃₁
η (X₀) = 3⋅X₀
η (X₂) = 2⋅X₂
τ = 1 ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ 3 ≤ X₀ ∧ X₀ ≤ X₂ ∧ X₁ ≤ 0
Results of Probabilistic Analysis
All Bounds
Timebounds
Overall timebound:11⋅X₂+8⋅X₀+24 {O(n)}
g₀: 1 {O(1)}
g₁₀: 1 {O(1)}
g₁₂: 1 {O(1)}
g₁₅: 2 {O(1)}
g₁₇: 2⋅X₀+2 {O(n)}
g₂₀: 4⋅X₀ {O(n)}
g₂₂: 2⋅X₀ {O(n)}
g₂₄: X₂+1 {O(n)}
g₂₇: 2⋅X₂ {O(n)}
g₂₉: 4⋅X₂+8 {O(n)}
g₃₁: 4⋅X₂+8 {O(n)}
Costbounds
Overall costbound: 12⋅X₀+13⋅X₂+26 {O(n)}
g₀: 1 {O(1)}
g₁₀: 1 {O(1)}
g₁₂: 1 {O(1)}
g₁₅: 4 {O(1)}
g₁₇: 2⋅X₀+2 {O(n)}
g₂₀: 8⋅X₀ {O(n)}
g₂₂: 2⋅X₀ {O(n)}
g₂₄: X₂+1 {O(n)}
g₂₇: 4⋅X₂ {O(n)}
g₂₉: 4⋅X₂+8 {O(n)}
g₃₁: 4⋅X₂+8 {O(n)}
Sizebounds
(g₀,l1), X₀: X₀ {O(n)}
(g₀,l1), X₁: X₁ {O(n)}
(g₀,l1), X₂: X₂ {O(n)}
(g₁₀,l2_v1), X₀: X₀ {O(n)}
(g₁₀,l2_v1), X₁: X₁ {O(n)}
(g₁₀,l2_v1), X₂: X₂ {O(n)}
(g₁₂,l1_v1), X₀: 3⋅X₀ {O(n)}
(g₁₂,l1_v1), X₁: X₁ {O(n)}
(g₁₂,l1_v1), X₂: 2⋅X₂ {O(n)}
(g₁₅,l1_v2), X₀: X₀ {O(n)}
(g₁₅,l1_v2), X₁: X₁ {O(n)}
(g₁₅,l1_v2), X₂: X₂ {O(n)}
(g₁₅,l1_v3), X₀: X₀ {O(n)}
(g₁₅,l1_v3), X₁: X₁ {O(n)}
(g₁₅,l1_v3), X₂: X₂ {O(n)}
(g₁₇,l2_v2), X₀: 2⋅X₀ {O(n)}
(g₁₇,l2_v2), X₁: 2⋅X₁ {O(n)}
(g₁₇,l2_v2), X₂: 2⋅X₂ {O(n)}
(g₂₀,l1_v2), X₀: 4⋅X₀ {O(n)}
(g₂₀,l1_v2), X₁: 4⋅X₁ {O(n)}
(g₂₀,l1_v2), X₂: 4⋅X₂ {O(n)}
(g₂₀,l1_v4), X₀: 4⋅X₀ {O(n)}
(g₂₀,l1_v4), X₁: 4⋅X₁ {O(n)}
(g₂₀,l1_v4), X₂: 4⋅X₂ {O(n)}
(g₂₂,l2_v2), X₀: 2⋅X₀ {O(n)}
(g₂₂,l2_v2), X₁: 2⋅X₁ {O(n)}
(g₂₂,l2_v2), X₂: 2⋅X₂ {O(n)}
(g₂₄,l2_v3), X₀: X₀ {O(n)}
(g₂₄,l2_v3), X₁: X₁ {O(n)}
(g₂₄,l2_v3), X₂: X₂ {O(n)}
(g₂₇,l1_v2), X₀: X₀ {O(n)}
(g₂₇,l1_v2), X₁: X₁ {O(n)}
(g₂₇,l1_v2), X₂: X₂ {O(n)}
(g₂₇,l1_v3), X₀: X₀ {O(n)}
(g₂₇,l1_v3), X₁: X₁ {O(n)}
(g₂₇,l1_v3), X₂: X₂ {O(n)}
(g₂₉,l2_v4), X₀: 3⋅3^(4⋅X₂+8)⋅X₀ {O(EXP)}
(g₂₉,l2_v4), X₁: X₁ {O(n)}
(g₂₉,l2_v4), X₂: 2⋅2^(4⋅X₂+8)⋅X₂ {O(EXP)}
(g₃₁,l1_v1), X₀: 3⋅3^(4⋅X₂+8)⋅X₀ {O(EXP)}
(g₃₁,l1_v1), X₁: X₁ {O(n)}
(g₃₁,l1_v1), X₂: 2⋅2^(4⋅X₂+8)⋅X₂ {O(EXP)}