Preprocessing
Eliminate variables [X₃; X₄] that do not contribute to the problem
Found invariant 0 ≤ X₅ ∧ 3 ≤ X₂+X₅ ∧ X₂ ≤ 3+X₅ ∧ 10 ≤ X₁+X₅ ∧ X₁ ≤ 10+X₅ ∧ X₂ ≤ 3 ∧ 7+X₂ ≤ X₁ ∧ X₁+X₂ ≤ 13 ∧ 3 ≤ X₂ ∧ 13 ≤ X₁+X₂ ∧ X₁ ≤ 7+X₂ ∧ X₁ ≤ 10 ∧ 10 ≤ X₁ for location h
Found invariant X₅ ≤ 10+X₀ ∧ 0 ≤ X₅ ∧ 3 ≤ X₂+X₅ ∧ X₂ ≤ 3+X₅ ∧ 10 ≤ X₁+X₅ ∧ X₁ ≤ 10+X₅ ∧ 9 ≤ X₀+X₅ ∧ X₂ ≤ 3 ∧ 7+X₂ ≤ X₁ ∧ X₁+X₂ ≤ 13 ∧ X₂ ≤ 3+X₀ ∧ 3 ≤ X₂ ∧ 13 ≤ X₁+X₂ ∧ X₁ ≤ 7+X₂ ∧ 3 ≤ X₀+X₂ ∧ X₁ ≤ 10 ∧ X₁ ≤ 10+X₀ ∧ 10 ≤ X₁ ∧ 10 ≤ X₀+X₁ ∧ 0 ≤ X₀ for location j
Found invariant X₅ ≤ 0 ∧ 3+X₅ ≤ X₂ ∧ X₂+X₅ ≤ 3 ∧ 10+X₅ ≤ X₁ ∧ X₁+X₅ ≤ 10 ∧ 0 ≤ X₅ ∧ 3 ≤ X₂+X₅ ∧ X₂ ≤ 3+X₅ ∧ 10 ≤ X₁+X₅ ∧ X₁ ≤ 10+X₅ ∧ X₂ ≤ 3 ∧ 7+X₂ ≤ X₁ ∧ X₁+X₂ ≤ 13 ∧ 3 ≤ X₂ ∧ 13 ≤ X₁+X₂ ∧ X₁ ≤ 7+X₂ ∧ X₁ ≤ 10 ∧ 10 ≤ X₁ for location g
Found invariant X₅ ≤ X₀ ∧ 0 ≤ X₅ ∧ 3 ≤ X₂+X₅ ∧ X₂ ≤ 3+X₅ ∧ 10 ≤ X₁+X₅ ∧ X₁ ≤ 10+X₅ ∧ 0 ≤ X₀+X₅ ∧ X₂ ≤ 3 ∧ 7+X₂ ≤ X₁ ∧ X₁+X₂ ≤ 13 ∧ X₂ ≤ 3+X₀ ∧ 3 ≤ X₂ ∧ 13 ≤ X₁+X₂ ∧ X₁ ≤ 7+X₂ ∧ 3 ≤ X₀+X₂ ∧ X₁ ≤ 10 ∧ X₁ ≤ 10+X₀ ∧ 10 ≤ X₁ ∧ 10 ≤ X₀+X₁ ∧ 0 ≤ X₀ for location i
Probabilistic Analysis
Probabilistic Program after Preprocessing
Start: f
Program_Vars: X₀, X₁, X₂, X₃, X₄, X₅
Temp_Vars: valToFill
Locations: f, g, h, i, j
Transitions:
g₅:f(X₀,X₁,X₂,X₃,X₄,X₅) -{0}> t₆:g(X₀,10,3,1,X₄,0) :|:
g₇:g(X₀,X₁,X₂,X₃,X₄,X₅) -{0}> t₈:h(X₀,X₁,X₂,X₃,X₄,X₅+UNIFORM(0, 1)) :|: X₁+X₂ ≤ 13 ∧ X₁ ≤ 10 ∧ X₁ ≤ 10+X₅ ∧ X₁+X₅ ≤ 10 ∧ X₁ ≤ 7+X₂ ∧ X₂ ≤ 3 ∧ X₂ ≤ 3+X₅ ∧ X₂+X₅ ≤ 3 ∧ 0 ≤ 1 ∧ 3 ≤ X₂ ∧ 3 ≤ X₂+X₅ ∧ 3+X₅ ≤ X₂ ∧ 7+X₂ ≤ X₁ ∧ 10 ≤ X₁ ∧ 10 ≤ X₁+X₅ ∧ 10+X₅ ≤ X₁ ∧ 13 ≤ X₁+X₂ ∧ 0 ≤ X₅ ∧ X₅ ≤ 0
g₉:h(X₀,X₁,X₂,X₃,X₄,X₅) -{0}> t₁₀:i(X₀,X₁,X₂,X₃,X₄,X₅) :|: X₅ ≤ X₀ ∧ X₁+X₂ ≤ 13 ∧ X₁ ≤ 10 ∧ X₁ ≤ 10+X₅ ∧ X₁ ≤ 7+X₂ ∧ X₂ ≤ 3 ∧ X₂ ≤ 3+X₅ ∧ 3 ≤ X₂ ∧ 3 ≤ X₂+X₅ ∧ 7+X₂ ≤ X₁ ∧ 10 ≤ X₁ ∧ 10 ≤ X₁+X₅ ∧ 13 ≤ X₁+X₂ ∧ 0 ≤ X₅
g₁₁:i(X₀,X₁,X₂,X₃,X₄,X₅) -{0}> t₁₂:j(X₀,X₁,X₂,X₃,X₄,X₅+UNIFORM(9, 10)) :|: 1+valToFill ≤ X₁+X₅ ∧ X₁+X₂ ≤ 13 ∧ X₁ ≤ 10+X₀ ∧ X₁ ≤ 10 ∧ X₁ ≤ 10+X₅ ∧ X₁ ≤ 7+X₂ ∧ X₂ ≤ 3+X₀ ∧ X₂ ≤ 3 ∧ X₂ ≤ 3+X₅ ∧ 0 ≤ 1 ∧ 3 ≤ X₀+X₂ ∧ 3 ≤ X₂ ∧ 3 ≤ X₂+X₅ ∧ 7+X₂ ≤ X₁ ∧ 10 ≤ X₀+X₁ ∧ 10 ≤ X₁ ∧ 10 ≤ X₁+X₅ ∧ 13 ≤ X₁+X₂ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₅ ∧ X₅ ≤ X₀ ∧ 0 ≤ X₅
g₁₃:i(X₀,X₁,X₂,X₃,X₄,X₅) -{0}> t₁₄:j(X₀,X₁,X₂,X₃,X₄,X₅+UNIFORM(2, 4)) :|: 1+valToFill ≤ X₂+X₅ ∧ X₁+X₅ ≤ X₀ ∧ X₁+X₂ ≤ 13 ∧ X₁ ≤ 10+X₀ ∧ X₁ ≤ 10 ∧ X₁ ≤ 10+X₅ ∧ X₁ ≤ 7+X₂ ∧ X₂ ≤ 3+X₀ ∧ X₂ ≤ 3 ∧ X₂ ≤ 3+X₅ ∧ 0 ≤ 1 ∧ 3 ≤ X₀+X₂ ∧ 3 ≤ X₂ ∧ 3 ≤ X₂+X₅ ∧ 7+X₂ ≤ X₁ ∧ 10 ≤ X₀+X₁ ∧ 10 ≤ X₁ ∧ 10 ≤ X₁+X₅ ∧ 13 ≤ X₁+X₂ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₅ ∧ X₅ ≤ X₀ ∧ 0 ≤ X₅
g₁₅:i(X₀,X₁,X₂,X₃,X₄,X₅) -{0}> t₁₆:j(X₀,X₁,X₂,X₃,X₄,X₅+UNIFORM(0, 4)) :|: X₂+X₅ ≤ valToFill ∧ X₁+X₅ ≤ X₀ ∧ X₁+X₂ ≤ 13 ∧ X₁ ≤ 10+X₀ ∧ X₁ ≤ 10 ∧ X₁ ≤ 10+X₅ ∧ X₁ ≤ 7+X₂ ∧ X₂ ≤ 3+X₀ ∧ X₂ ≤ 3 ∧ X₂ ≤ 3+X₅ ∧ 0 ≤ 1 ∧ 3 ≤ X₀+X₂ ∧ 3 ≤ X₂ ∧ 3 ≤ X₂+X₅ ∧ 7+X₂ ≤ X₁ ∧ 10 ≤ X₀+X₁ ∧ 10 ≤ X₁ ∧ 10 ≤ X₁+X₅ ∧ 13 ≤ X₁+X₂ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₅ ∧ X₅ ≤ X₀ ∧ 0 ≤ X₅
g₁₇:j(X₀,X₁,X₂,X₃,X₄,X₅) → t₁₈:h(X₀,X₁,X₂,X₃,X₄,X₅+UNIFORM(0, 1)) :|: X₁+X₂ ≤ 13 ∧ X₁ ≤ 10+X₀ ∧ X₅ ≤ 10+X₀ ∧ X₁ ≤ 10 ∧ X₁ ≤ 10+X₅ ∧ X₁ ≤ 7+X₂ ∧ X₂ ≤ 3+X₀ ∧ X₂ ≤ 3 ∧ X₂ ≤ 3+X₅ ∧ 0 ≤ 1 ∧ 3 ≤ X₀+X₂ ∧ 3 ≤ X₂ ∧ 3 ≤ X₂+X₅ ∧ 7+X₂ ≤ X₁ ∧ 9 ≤ X₀+X₅ ∧ 10 ≤ X₀+X₁ ∧ 10 ≤ X₁ ∧ 10 ≤ X₁+X₅ ∧ 13 ≤ X₁+X₂ ∧ 0 ≤ X₀ ∧ 0 ≤ X₅
Show Graph
G
f
f
g
g
f->g
p = 1
t₆ ∈ g₅
η (X₁) = 10
η (X₂) = 3
η (X₅) = 0
{0}
h
h
g->h
p = 1
t₈ ∈ g₇
η (X₅) = X₅+UNIFORM(0, 1)
τ = X₁+X₂ ≤ 13 ∧ X₁ ≤ 10 ∧ X₁ ≤ 10+X₅ ∧ X₁+X₅ ≤ 10 ∧ X₁ ≤ 7+X₂ ∧ X₂ ≤ 3 ∧ X₂ ≤ 3+X₅ ∧ X₂+X₅ ≤ 3 ∧ 0 ≤ 1 ∧ 3 ≤ X₂ ∧ 3 ≤ X₂+X₅ ∧ 3+X₅ ≤ X₂ ∧ 7+X₂ ≤ X₁ ∧ 10 ≤ X₁ ∧ 10 ≤ X₁+X₅ ∧ 10+X₅ ≤ X₁ ∧ 13 ≤ X₁+X₂ ∧ 0 ≤ X₅ ∧ X₅ ≤ 0
{0}
i
i
h->i
p = 1
t₁₀ ∈ g₉
τ = X₁+X₂ ≤ 13 ∧ X₁ ≤ 10 ∧ X₁ ≤ 10+X₅ ∧ X₁ ≤ 7+X₂ ∧ X₂ ≤ 3 ∧ X₂ ≤ 3+X₅ ∧ 3 ≤ X₂ ∧ 3 ≤ X₂+X₅ ∧ 7+X₂ ≤ X₁ ∧ 10 ≤ X₁ ∧ 10 ≤ X₁+X₅ ∧ 13 ≤ X₁+X₂ ∧ 0 ≤ X₅ ∧ X₅ ≤ X₀
{0}
j
j
i->j
p = 1
t₁₂ ∈ g₁₁
η (X₅) = X₅+UNIFORM(9, 10)
τ = X₁+X₂ ≤ 13 ∧ X₁ ≤ 10+X₀ ∧ X₁ ≤ 10 ∧ X₁ ≤ 10+X₅ ∧ X₁ ≤ 7+X₂ ∧ X₂ ≤ 3+X₀ ∧ X₂ ≤ 3 ∧ X₂ ≤ 3+X₅ ∧ 0 ≤ 1 ∧ 3 ≤ X₀+X₂ ∧ 3 ≤ X₂ ∧ 3 ≤ X₂+X₅ ∧ 7+X₂ ≤ X₁ ∧ 10 ≤ X₀+X₁ ∧ 10 ≤ X₁ ∧ 10 ≤ X₁+X₅ ∧ 13 ≤ X₁+X₂ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₅ ∧ X₅ ≤ X₀ ∧ 0 ≤ X₅ ∧ 1+valToFill ≤ X₁+X₅
{0}
i->j
p = 1
t₁₄ ∈ g₁₃
η (X₅) = X₅+UNIFORM(2, 4)
τ = X₁+X₂ ≤ 13 ∧ X₁ ≤ 10+X₀ ∧ X₁ ≤ 10 ∧ X₁ ≤ 10+X₅ ∧ X₁ ≤ 7+X₂ ∧ X₂ ≤ 3+X₀ ∧ X₂ ≤ 3 ∧ X₂ ≤ 3+X₅ ∧ 0 ≤ 1 ∧ 3 ≤ X₀+X₂ ∧ 3 ≤ X₂ ∧ 3 ≤ X₂+X₅ ∧ 7+X₂ ≤ X₁ ∧ 10 ≤ X₀+X₁ ∧ 10 ≤ X₁ ∧ 10 ≤ X₁+X₅ ∧ 13 ≤ X₁+X₂ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₅ ∧ X₅ ≤ X₀ ∧ 0 ≤ X₅ ∧ 1+valToFill ≤ X₂+X₅ ∧ X₁+X₅ ≤ X₀
{0}
i->j
p = 1
t₁₆ ∈ g₁₅
η (X₅) = X₅+UNIFORM(0, 4)
τ = X₁+X₂ ≤ 13 ∧ X₁ ≤ 10+X₀ ∧ X₁ ≤ 10 ∧ X₁ ≤ 10+X₅ ∧ X₁ ≤ 7+X₂ ∧ X₂ ≤ 3+X₀ ∧ X₂ ≤ 3 ∧ X₂ ≤ 3+X₅ ∧ 0 ≤ 1 ∧ 3 ≤ X₀+X₂ ∧ 3 ≤ X₂ ∧ 3 ≤ X₂+X₅ ∧ 7+X₂ ≤ X₁ ∧ 10 ≤ X₀+X₁ ∧ 10 ≤ X₁ ∧ 10 ≤ X₁+X₅ ∧ 13 ≤ X₁+X₂ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₅ ∧ X₅ ≤ X₀ ∧ 0 ≤ X₅ ∧ X₂+X₅ ≤ valToFill ∧ X₁+X₅ ≤ X₀
{0}
j->h
p = 1
t₁₈ ∈ g₁₇
η (X₅) = X₅+UNIFORM(0, 1)
τ = X₁+X₂ ≤ 13 ∧ X₁ ≤ 10+X₀ ∧ X₅ ≤ 10+X₀ ∧ X₁ ≤ 10 ∧ X₁ ≤ 10+X₅ ∧ X₁ ≤ 7+X₂ ∧ X₂ ≤ 3+X₀ ∧ X₂ ≤ 3 ∧ X₂ ≤ 3+X₅ ∧ 0 ≤ 1 ∧ 3 ≤ X₀+X₂ ∧ 3 ≤ X₂ ∧ 3 ≤ X₂+X₅ ∧ 7+X₂ ≤ X₁ ∧ 9 ≤ X₀+X₅ ∧ 10 ≤ X₀+X₁ ∧ 10 ≤ X₁ ∧ 10 ≤ X₁+X₅ ∧ 13 ≤ X₁+X₂ ∧ 0 ≤ X₀ ∧ 0 ≤ X₅
Run classical analysis on SCC: [f]
Classical Approximation after Lifting Classical Results
All Bounds
Timebounds
Overall timebound:inf {Infinity}
g₅: 1 {O(1)}
g₇: 1 {O(1)}
g₉: inf {Infinity}
g₁₁: inf {Infinity}
g₁₃: inf {Infinity}
g₁₅: inf {Infinity}
g₁₇: inf {Infinity}
Costbounds
Overall costbound: inf {Infinity}
g₅: inf {Infinity}
g₇: inf {Infinity}
g₉: inf {Infinity}
g₁₁: inf {Infinity}
g₁₃: inf {Infinity}
g₁₅: inf {Infinity}
g₁₇: inf {Infinity}
Sizebounds
(g₅,g), X₀: X₀ {O(n)}
(g₅,g), X₁: 10 {O(1)}
(g₅,g), X₂: 3 {O(1)}
(g₅,g), X₃: 1 {O(1)}
(g₅,g), X₄: X₄ {O(n)}
(g₅,g), X₅: 0 {O(1)}
Run probabilistic analysis on SCC: [f]
Run classical analysis on SCC: [g]
Classical Approximation after Lifting Classical Results
All Bounds
Timebounds
Overall timebound:inf {Infinity}
g₅: 1 {O(1)}
g₇: 1 {O(1)}
g₉: inf {Infinity}
g₁₁: inf {Infinity}
g₁₃: inf {Infinity}
g₁₅: inf {Infinity}
g₁₇: inf {Infinity}
Costbounds
Overall costbound: inf {Infinity}
g₅: inf {Infinity}
g₇: inf {Infinity}
g₉: inf {Infinity}
g₁₁: inf {Infinity}
g₁₃: inf {Infinity}
g₁₅: inf {Infinity}
g₁₇: inf {Infinity}
Sizebounds
(g₅,g), X₀: X₀ {O(n)}
(g₅,g), X₁: 10 {O(1)}
(g₅,g), X₂: 3 {O(1)}
(g₅,g), X₃: 1 {O(1)}
(g₅,g), X₄: X₄ {O(n)}
(g₅,g), X₅: 0 {O(1)}
(g₇,h), X₀: X₀ {O(n)}
(g₇,h), X₁: 10 {O(1)}
(g₇,h), X₂: 3 {O(1)}
(g₇,h), X₃: 1 {O(1)}
(g₇,h), X₄: X₄ {O(n)}
(g₇,h), X₅: 1 {O(1)}
Run probabilistic analysis on SCC: [g]
Run classical analysis on SCC: [h; i; j]
MPRF for transition t₁₂: i(X₀,X₁,X₂,X₃,X₄,X₅) -{0}> j(X₀,X₁,X₂,X₃,X₄,Temp_Int₁₅₈) :|: X₁+X₂ ≤ 13 ∧ X₁ ≤ 10+X₀ ∧ X₁ ≤ 10 ∧ X₁ ≤ 10+X₅ ∧ X₁ ≤ 7+X₂ ∧ X₂ ≤ 3+X₀ ∧ X₂ ≤ 3 ∧ X₂ ≤ 3+X₅ ∧ 0 ≤ 1 ∧ 3 ≤ X₀+X₂ ∧ 3 ≤ X₂ ∧ 3 ≤ X₂+X₅ ∧ 7+X₂ ≤ X₁ ∧ 10 ≤ X₀+X₁ ∧ 10 ≤ X₁ ∧ 10 ≤ X₁+X₅ ∧ 13 ≤ X₁+X₂ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₅ ∧ X₅ ≤ X₀ ∧ 0 ≤ X₅ ∧ 9+Temp_Int₃ ≤ Temp_Int₄+Temp_Int₁₅₈ ∧ 9 ≤ Temp_Int₄ ∧ Temp_Int₄ ≤ 10 ∧ Temp_Int₄+Temp_Int₁₅₈ ≤ 10+Temp_Int₃ ∧ valToFill+Temp_Int₄ ≤ 9+Temp_Int₃ ∧ Temp_Int₃ ≤ Temp_Int₄+X₀ ∧ Temp_Int₄ ≤ Temp_Int₃ ∧ X₁ ≤ 10 ∧ 10 ≤ X₁ ∧ Temp_Int₄+X₅ ≤ Temp_Int₃ ∧ Temp_Int₃ ≤ Temp_Int₄+X₅ ∧ X₂ ≤ 3 ∧ 3 ≤ X₂ ∧ Temp_Int₃ ≤ Temp_Int₄+X₅ ∧ Temp_Int₄+X₅ ≤ Temp_Int₃ ∧ 9 ≤ Temp_Int₄ ∧ Temp_Int₄ ≤ 10 ∧ 1+valToFill ≤ X₁+X₅ of depth 1:
new bound:
X₀+2 {O(n)}
MPRF:
• h: [1+X₀-X₅]
• i: [1+X₀-X₅]
• j: [1+X₀-X₅]
Show Graph
G
f
f
g
g
f->g
t₆
η (X₁) = 10
η (X₂) = 3
η (X₃) = 1
η (X₅) = 0
{0}
h
h
g->h
t₈
η (X₅) = Temp_Int₁
τ = X₁+X₂ ≤ 13 ∧ X₁ ≤ 10 ∧ X₁ ≤ 10+X₅ ∧ X₁+X₅ ≤ 10 ∧ X₁ ≤ 7+X₂ ∧ X₂ ≤ 3 ∧ X₂ ≤ 3+X₅ ∧ X₂+X₅ ≤ 3 ∧ 0 ≤ 1 ∧ 3 ≤ X₂ ∧ 3 ≤ X₂+X₅ ∧ 3+X₅ ≤ X₂ ∧ 7+X₂ ≤ X₁ ∧ 10 ≤ X₁ ∧ 10 ≤ X₁+X₅ ∧ 10+X₅ ≤ X₁ ∧ 13 ≤ X₁+X₂ ∧ 0 ≤ X₅ ∧ X₅ ≤ 0 ∧ Temp_Int₁ ≤ Temp_Int₂+X₅ ∧ Temp_Int₂+X₅ ≤ Temp_Int₁ ∧ 0 ≤ Temp_Int₂ ∧ Temp_Int₂ ≤ 1
{0}
i
i
h->i
t₁₀
τ = X₁+X₂ ≤ 13 ∧ X₁ ≤ 10 ∧ X₁ ≤ 10+X₅ ∧ X₁ ≤ 7+X₂ ∧ X₂ ≤ 3 ∧ X₂ ≤ 3+X₅ ∧ 3 ≤ X₂ ∧ 3 ≤ X₂+X₅ ∧ 7+X₂ ≤ X₁ ∧ 10 ≤ X₁ ∧ 10 ≤ X₁+X₅ ∧ 13 ≤ X₁+X₂ ∧ 0 ≤ X₅ ∧ X₅ ≤ X₀
{0}
j
j
i->j
t₁₂
η (X₅) = Temp_Int₃
τ = X₁+X₂ ≤ 13 ∧ X₁ ≤ 10+X₀ ∧ X₁ ≤ 10 ∧ X₁ ≤ 10+X₅ ∧ X₁ ≤ 7+X₂ ∧ X₂ ≤ 3+X₀ ∧ X₂ ≤ 3 ∧ X₂ ≤ 3+X₅ ∧ 0 ≤ 1 ∧ 3 ≤ X₀+X₂ ∧ 3 ≤ X₂ ∧ 3 ≤ X₂+X₅ ∧ 7+X₂ ≤ X₁ ∧ 10 ≤ X₀+X₁ ∧ 10 ≤ X₁ ∧ 10 ≤ X₁+X₅ ∧ 13 ≤ X₁+X₂ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₅ ∧ X₅ ≤ X₀ ∧ 0 ≤ X₅ ∧ Temp_Int₃ ≤ Temp_Int₄+X₅ ∧ Temp_Int₄+X₅ ≤ Temp_Int₃ ∧ 9 ≤ Temp_Int₄ ∧ Temp_Int₄ ≤ 10 ∧ 1+valToFill ≤ X₁+X₅
{0}
i->j
t₁₄
η (X₅) = Temp_Int₅
τ = X₁+X₂ ≤ 13 ∧ X₁ ≤ 10+X₀ ∧ X₁ ≤ 10 ∧ X₁ ≤ 10+X₅ ∧ X₁ ≤ 7+X₂ ∧ X₂ ≤ 3+X₀ ∧ X₂ ≤ 3 ∧ X₂ ≤ 3+X₅ ∧ 0 ≤ 1 ∧ 3 ≤ X₀+X₂ ∧ 3 ≤ X₂ ∧ 3 ≤ X₂+X₅ ∧ 7+X₂ ≤ X₁ ∧ 10 ≤ X₀+X₁ ∧ 10 ≤ X₁ ∧ 10 ≤ X₁+X₅ ∧ 13 ≤ X₁+X₂ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₅ ∧ X₅ ≤ X₀ ∧ 0 ≤ X₅ ∧ Temp_Int₅ ≤ Temp_Int₆+X₅ ∧ Temp_Int₆+X₅ ≤ Temp_Int₅ ∧ 2 ≤ Temp_Int₆ ∧ Temp_Int₆ ≤ 4 ∧ 1+valToFill ≤ X₂+X₅ ∧ X₁+X₅ ≤ X₀
{0}
i->j
t₁₆
η (X₅) = Temp_Int₇
τ = X₁+X₂ ≤ 13 ∧ X₁ ≤ 10+X₀ ∧ X₁ ≤ 10 ∧ X₁ ≤ 10+X₅ ∧ X₁ ≤ 7+X₂ ∧ X₂ ≤ 3+X₀ ∧ X₂ ≤ 3 ∧ X₂ ≤ 3+X₅ ∧ 0 ≤ 1 ∧ 3 ≤ X₀+X₂ ∧ 3 ≤ X₂ ∧ 3 ≤ X₂+X₅ ∧ 7+X₂ ≤ X₁ ∧ 10 ≤ X₀+X₁ ∧ 10 ≤ X₁ ∧ 10 ≤ X₁+X₅ ∧ 13 ≤ X₁+X₂ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₅ ∧ X₅ ≤ X₀ ∧ 0 ≤ X₅ ∧ Temp_Int₇ ≤ Temp_Int₈+X₅ ∧ Temp_Int₈+X₅ ≤ Temp_Int₇ ∧ 0 ≤ Temp_Int₈ ∧ Temp_Int₈ ≤ 4 ∧ X₂+X₅ ≤ valToFill ∧ X₁+X₅ ≤ X₀
{0}
j->h
t₁₈
η (X₅) = Temp_Int₉
τ = X₁+X₂ ≤ 13 ∧ X₁ ≤ 10+X₀ ∧ X₅ ≤ 10+X₀ ∧ X₁ ≤ 10 ∧ X₁ ≤ 10+X₅ ∧ X₁ ≤ 7+X₂ ∧ X₂ ≤ 3+X₀ ∧ X₂ ≤ 3 ∧ X₂ ≤ 3+X₅ ∧ 0 ≤ 1 ∧ 3 ≤ X₀+X₂ ∧ 3 ≤ X₂ ∧ 3 ≤ X₂+X₅ ∧ 7+X₂ ≤ X₁ ∧ 9 ≤ X₀+X₅ ∧ 10 ≤ X₀+X₁ ∧ 10 ≤ X₁ ∧ 10 ≤ X₁+X₅ ∧ 13 ≤ X₁+X₂ ∧ 0 ≤ X₀ ∧ 0 ≤ X₅ ∧ Temp_Int₉ ≤ Temp_Int₁₀+X₅ ∧ Temp_Int₁₀+X₅ ≤ Temp_Int₉ ∧ 0 ≤ Temp_Int₁₀ ∧ Temp_Int₁₀ ≤ 1
MPRF for transition t₁₄: i(X₀,X₁,X₂,X₃,X₄,X₅) -{0}> j(X₀,X₁,X₂,X₃,X₄,Temp_Int₁₆₀) :|: X₁+X₂ ≤ 13 ∧ X₁ ≤ 10+X₀ ∧ X₁ ≤ 10 ∧ X₁ ≤ 10+X₅ ∧ X₁ ≤ 7+X₂ ∧ X₂ ≤ 3+X₀ ∧ X₂ ≤ 3 ∧ X₂ ≤ 3+X₅ ∧ 0 ≤ 1 ∧ 3 ≤ X₀+X₂ ∧ 3 ≤ X₂ ∧ 3 ≤ X₂+X₅ ∧ 7+X₂ ≤ X₁ ∧ 10 ≤ X₀+X₁ ∧ 10 ≤ X₁ ∧ 10 ≤ X₁+X₅ ∧ 13 ≤ X₁+X₂ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₅ ∧ X₅ ≤ X₀ ∧ 0 ≤ X₅ ∧ 2+X₅ ≤ Temp_Int₁₆₀ ∧ 2 ≤ Temp_Int₆ ∧ Temp_Int₆ ≤ 4 ∧ Temp_Int₁₆₀ ≤ 4+X₅ ∧ 10+X₅ ≤ X₀ ∧ 0 ≤ X₅ ∧ valToFill ≤ 2+X₅ ∧ X₁ ≤ 10 ∧ 10 ≤ X₁ ∧ Temp_Int₆+X₅ ≤ Temp_Int₅ ∧ Temp_Int₅ ≤ Temp_Int₆+X₅ ∧ X₂ ≤ 3 ∧ 3 ≤ X₂ ∧ Temp_Int₅ ≤ Temp_Int₆+X₅ ∧ Temp_Int₆+X₅ ≤ Temp_Int₅ ∧ 2 ≤ Temp_Int₆ ∧ Temp_Int₆ ≤ 4 ∧ 1+valToFill ≤ X₂+X₅ ∧ X₁+X₅ ≤ X₀ of depth 1:
new bound:
X₀+2 {O(n)}
MPRF:
• h: [1+X₀-X₅]
• i: [1+X₀-X₅]
• j: [1+X₀-X₅]
Show Graph
G
f
f
g
g
f->g
t₆
η (X₁) = 10
η (X₂) = 3
η (X₃) = 1
η (X₅) = 0
{0}
h
h
g->h
t₈
η (X₅) = Temp_Int₁
τ = X₁+X₂ ≤ 13 ∧ X₁ ≤ 10 ∧ X₁ ≤ 10+X₅ ∧ X₁+X₅ ≤ 10 ∧ X₁ ≤ 7+X₂ ∧ X₂ ≤ 3 ∧ X₂ ≤ 3+X₅ ∧ X₂+X₅ ≤ 3 ∧ 0 ≤ 1 ∧ 3 ≤ X₂ ∧ 3 ≤ X₂+X₅ ∧ 3+X₅ ≤ X₂ ∧ 7+X₂ ≤ X₁ ∧ 10 ≤ X₁ ∧ 10 ≤ X₁+X₅ ∧ 10+X₅ ≤ X₁ ∧ 13 ≤ X₁+X₂ ∧ 0 ≤ X₅ ∧ X₅ ≤ 0 ∧ Temp_Int₁ ≤ Temp_Int₂+X₅ ∧ Temp_Int₂+X₅ ≤ Temp_Int₁ ∧ 0 ≤ Temp_Int₂ ∧ Temp_Int₂ ≤ 1
{0}
i
i
h->i
t₁₀
τ = X₁+X₂ ≤ 13 ∧ X₁ ≤ 10 ∧ X₁ ≤ 10+X₅ ∧ X₁ ≤ 7+X₂ ∧ X₂ ≤ 3 ∧ X₂ ≤ 3+X₅ ∧ 3 ≤ X₂ ∧ 3 ≤ X₂+X₅ ∧ 7+X₂ ≤ X₁ ∧ 10 ≤ X₁ ∧ 10 ≤ X₁+X₅ ∧ 13 ≤ X₁+X₂ ∧ 0 ≤ X₅ ∧ X₅ ≤ X₀
{0}
j
j
i->j
t₁₂
η (X₅) = Temp_Int₃
τ = X₁+X₂ ≤ 13 ∧ X₁ ≤ 10+X₀ ∧ X₁ ≤ 10 ∧ X₁ ≤ 10+X₅ ∧ X₁ ≤ 7+X₂ ∧ X₂ ≤ 3+X₀ ∧ X₂ ≤ 3 ∧ X₂ ≤ 3+X₅ ∧ 0 ≤ 1 ∧ 3 ≤ X₀+X₂ ∧ 3 ≤ X₂ ∧ 3 ≤ X₂+X₅ ∧ 7+X₂ ≤ X₁ ∧ 10 ≤ X₀+X₁ ∧ 10 ≤ X₁ ∧ 10 ≤ X₁+X₅ ∧ 13 ≤ X₁+X₂ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₅ ∧ X₅ ≤ X₀ ∧ 0 ≤ X₅ ∧ Temp_Int₃ ≤ Temp_Int₄+X₅ ∧ Temp_Int₄+X₅ ≤ Temp_Int₃ ∧ 9 ≤ Temp_Int₄ ∧ Temp_Int₄ ≤ 10 ∧ 1+valToFill ≤ X₁+X₅
{0}
i->j
t₁₄
η (X₅) = Temp_Int₅
τ = X₁+X₂ ≤ 13 ∧ X₁ ≤ 10+X₀ ∧ X₁ ≤ 10 ∧ X₁ ≤ 10+X₅ ∧ X₁ ≤ 7+X₂ ∧ X₂ ≤ 3+X₀ ∧ X₂ ≤ 3 ∧ X₂ ≤ 3+X₅ ∧ 0 ≤ 1 ∧ 3 ≤ X₀+X₂ ∧ 3 ≤ X₂ ∧ 3 ≤ X₂+X₅ ∧ 7+X₂ ≤ X₁ ∧ 10 ≤ X₀+X₁ ∧ 10 ≤ X₁ ∧ 10 ≤ X₁+X₅ ∧ 13 ≤ X₁+X₂ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₅ ∧ X₅ ≤ X₀ ∧ 0 ≤ X₅ ∧ Temp_Int₅ ≤ Temp_Int₆+X₅ ∧ Temp_Int₆+X₅ ≤ Temp_Int₅ ∧ 2 ≤ Temp_Int₆ ∧ Temp_Int₆ ≤ 4 ∧ 1+valToFill ≤ X₂+X₅ ∧ X₁+X₅ ≤ X₀
{0}
i->j
t₁₆
η (X₅) = Temp_Int₇
τ = X₁+X₂ ≤ 13 ∧ X₁ ≤ 10+X₀ ∧ X₁ ≤ 10 ∧ X₁ ≤ 10+X₅ ∧ X₁ ≤ 7+X₂ ∧ X₂ ≤ 3+X₀ ∧ X₂ ≤ 3 ∧ X₂ ≤ 3+X₅ ∧ 0 ≤ 1 ∧ 3 ≤ X₀+X₂ ∧ 3 ≤ X₂ ∧ 3 ≤ X₂+X₅ ∧ 7+X₂ ≤ X₁ ∧ 10 ≤ X₀+X₁ ∧ 10 ≤ X₁ ∧ 10 ≤ X₁+X₅ ∧ 13 ≤ X₁+X₂ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₅ ∧ X₅ ≤ X₀ ∧ 0 ≤ X₅ ∧ Temp_Int₇ ≤ Temp_Int₈+X₅ ∧ Temp_Int₈+X₅ ≤ Temp_Int₇ ∧ 0 ≤ Temp_Int₈ ∧ Temp_Int₈ ≤ 4 ∧ X₂+X₅ ≤ valToFill ∧ X₁+X₅ ≤ X₀
{0}
j->h
t₁₈
η (X₅) = Temp_Int₉
τ = X₁+X₂ ≤ 13 ∧ X₁ ≤ 10+X₀ ∧ X₅ ≤ 10+X₀ ∧ X₁ ≤ 10 ∧ X₁ ≤ 10+X₅ ∧ X₁ ≤ 7+X₂ ∧ X₂ ≤ 3+X₀ ∧ X₂ ≤ 3 ∧ X₂ ≤ 3+X₅ ∧ 0 ≤ 1 ∧ 3 ≤ X₀+X₂ ∧ 3 ≤ X₂ ∧ 3 ≤ X₂+X₅ ∧ 7+X₂ ≤ X₁ ∧ 9 ≤ X₀+X₅ ∧ 10 ≤ X₀+X₁ ∧ 10 ≤ X₁ ∧ 10 ≤ X₁+X₅ ∧ 13 ≤ X₁+X₂ ∧ 0 ≤ X₀ ∧ 0 ≤ X₅ ∧ Temp_Int₉ ≤ Temp_Int₁₀+X₅ ∧ Temp_Int₁₀+X₅ ≤ Temp_Int₉ ∧ 0 ≤ Temp_Int₁₀ ∧ Temp_Int₁₀ ≤ 1
Classical Approximation after Lifting Classical Results
All Bounds
Timebounds
Overall timebound:inf {Infinity}
g₅: 1 {O(1)}
g₇: 1 {O(1)}
g₉: inf {Infinity}
g₁₁: X₀+2 {O(n)}
g₁₃: X₀+2 {O(n)}
g₁₅: inf {Infinity}
g₁₇: inf {Infinity}
Costbounds
Overall costbound: inf {Infinity}
g₅: inf {Infinity}
g₇: inf {Infinity}
g₉: inf {Infinity}
g₁₁: inf {Infinity}
g₁₃: inf {Infinity}
g₁₅: inf {Infinity}
g₁₇: inf {Infinity}
Sizebounds
(g₅,g), X₀: X₀ {O(n)}
(g₅,g), X₁: 10 {O(1)}
(g₅,g), X₂: 3 {O(1)}
(g₅,g), X₃: 1 {O(1)}
(g₅,g), X₄: X₄ {O(n)}
(g₅,g), X₅: 0 {O(1)}
(g₇,h), X₀: X₀ {O(n)}
(g₇,h), X₁: 10 {O(1)}
(g₇,h), X₂: 3 {O(1)}
(g₇,h), X₃: 1 {O(1)}
(g₇,h), X₄: X₄ {O(n)}
(g₇,h), X₅: 1 {O(1)}
(g₉,i), X₀: X₀ {O(n)}
(g₉,i), X₁: 10 {O(1)}
(g₉,i), X₂: 3 {O(1)}
(g₉,i), X₃: 1 {O(1)}
(g₉,i), X₄: X₄ {O(n)}
(g₁₁,j), X₀: X₀ {O(n)}
(g₁₁,j), X₁: 10 {O(1)}
(g₁₁,j), X₂: 3 {O(1)}
(g₁₁,j), X₃: 1 {O(1)}
(g₁₁,j), X₄: X₄ {O(n)}
(g₁₃,j), X₀: X₀ {O(n)}
(g₁₃,j), X₁: 10 {O(1)}
(g₁₃,j), X₂: 3 {O(1)}
(g₁₃,j), X₃: 1 {O(1)}
(g₁₃,j), X₄: X₄ {O(n)}
(g₁₅,j), X₀: X₀ {O(n)}
(g₁₅,j), X₁: 10 {O(1)}
(g₁₅,j), X₂: 3 {O(1)}
(g₁₅,j), X₃: 1 {O(1)}
(g₁₅,j), X₄: X₄ {O(n)}
(g₁₇,h), X₀: X₀ {O(n)}
(g₁₇,h), X₁: 10 {O(1)}
(g₁₇,h), X₂: 3 {O(1)}
(g₁₇,h), X₃: 1 {O(1)}
(g₁₇,h), X₄: X₄ {O(n)}
Run probabilistic analysis on SCC: [h; i; j]
Plrf for transition g₉:h(X₀,X₁,X₂,X₃,X₄,X₅) -{0}> t₁₀:i(X₀,X₁,X₂,X₃,X₄,X₅) :|: X₅ ≤ X₀ ∧ X₁+X₂ ≤ 13 ∧ X₁ ≤ 10 ∧ X₁ ≤ 10+X₅ ∧ X₁ ≤ 7+X₂ ∧ X₂ ≤ 3 ∧ X₂ ≤ 3+X₅ ∧ 3 ≤ X₂ ∧ 3 ≤ X₂+X₅ ∧ 7+X₂ ≤ X₁ ∧ 10 ≤ X₁ ∧ 10 ≤ X₁+X₅ ∧ 13 ≤ X₁+X₂ ∧ 0 ≤ X₅:
new bound:
2/5⋅X₀+24/5 {O(n)}
PLRF:
• h: 1+2/5⋅X₀+17/50⋅X₁-2/5⋅X₅
• i: 2/5⋅X₀+17/50⋅X₁-2/5⋅X₅
• j: 2/5⋅X₀+21/50⋅X₁-2/5⋅X₅
Show Graph
G
f
f
g
g
f->g
p = 1
t₆ ∈ g₅
η (X₁) = 10
η (X₂) = 3
η (X₅) = 0
{0}
h
h
g->h
p = 1
t₈ ∈ g₇
η (X₅) = X₅+UNIFORM(0, 1)
τ = X₁+X₂ ≤ 13 ∧ X₁ ≤ 10 ∧ X₁ ≤ 10+X₅ ∧ X₁+X₅ ≤ 10 ∧ X₁ ≤ 7+X₂ ∧ X₂ ≤ 3 ∧ X₂ ≤ 3+X₅ ∧ X₂+X₅ ≤ 3 ∧ 0 ≤ 1 ∧ 3 ≤ X₂ ∧ 3 ≤ X₂+X₅ ∧ 3+X₅ ≤ X₂ ∧ 7+X₂ ≤ X₁ ∧ 10 ≤ X₁ ∧ 10 ≤ X₁+X₅ ∧ 10+X₅ ≤ X₁ ∧ 13 ≤ X₁+X₂ ∧ 0 ≤ X₅ ∧ X₅ ≤ 0
{0}
i
i
h->i
p = 1
t₁₀ ∈ g₉
τ = X₁+X₂ ≤ 13 ∧ X₁ ≤ 10 ∧ X₁ ≤ 10+X₅ ∧ X₁ ≤ 7+X₂ ∧ X₂ ≤ 3 ∧ X₂ ≤ 3+X₅ ∧ 3 ≤ X₂ ∧ 3 ≤ X₂+X₅ ∧ 7+X₂ ≤ X₁ ∧ 10 ≤ X₁ ∧ 10 ≤ X₁+X₅ ∧ 13 ≤ X₁+X₂ ∧ 0 ≤ X₅ ∧ X₅ ≤ X₀
{0}
j
j
i->j
p = 1
t₁₂ ∈ g₁₁
η (X₅) = X₅+UNIFORM(9, 10)
τ = X₁+X₂ ≤ 13 ∧ X₁ ≤ 10+X₀ ∧ X₁ ≤ 10 ∧ X₁ ≤ 10+X₅ ∧ X₁ ≤ 7+X₂ ∧ X₂ ≤ 3+X₀ ∧ X₂ ≤ 3 ∧ X₂ ≤ 3+X₅ ∧ 0 ≤ 1 ∧ 3 ≤ X₀+X₂ ∧ 3 ≤ X₂ ∧ 3 ≤ X₂+X₅ ∧ 7+X₂ ≤ X₁ ∧ 10 ≤ X₀+X₁ ∧ 10 ≤ X₁ ∧ 10 ≤ X₁+X₅ ∧ 13 ≤ X₁+X₂ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₅ ∧ X₅ ≤ X₀ ∧ 0 ≤ X₅ ∧ 1+valToFill ≤ X₁+X₅
{0}
i->j
p = 1
t₁₄ ∈ g₁₃
η (X₅) = X₅+UNIFORM(2, 4)
τ = X₁+X₂ ≤ 13 ∧ X₁ ≤ 10+X₀ ∧ X₁ ≤ 10 ∧ X₁ ≤ 10+X₅ ∧ X₁ ≤ 7+X₂ ∧ X₂ ≤ 3+X₀ ∧ X₂ ≤ 3 ∧ X₂ ≤ 3+X₅ ∧ 0 ≤ 1 ∧ 3 ≤ X₀+X₂ ∧ 3 ≤ X₂ ∧ 3 ≤ X₂+X₅ ∧ 7+X₂ ≤ X₁ ∧ 10 ≤ X₀+X₁ ∧ 10 ≤ X₁ ∧ 10 ≤ X₁+X₅ ∧ 13 ≤ X₁+X₂ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₅ ∧ X₅ ≤ X₀ ∧ 0 ≤ X₅ ∧ 1+valToFill ≤ X₂+X₅ ∧ X₁+X₅ ≤ X₀
{0}
i->j
p = 1
t₁₆ ∈ g₁₅
η (X₅) = X₅+UNIFORM(0, 4)
τ = X₁+X₂ ≤ 13 ∧ X₁ ≤ 10+X₀ ∧ X₁ ≤ 10 ∧ X₁ ≤ 10+X₅ ∧ X₁ ≤ 7+X₂ ∧ X₂ ≤ 3+X₀ ∧ X₂ ≤ 3 ∧ X₂ ≤ 3+X₅ ∧ 0 ≤ 1 ∧ 3 ≤ X₀+X₂ ∧ 3 ≤ X₂ ∧ 3 ≤ X₂+X₅ ∧ 7+X₂ ≤ X₁ ∧ 10 ≤ X₀+X₁ ∧ 10 ≤ X₁ ∧ 10 ≤ X₁+X₅ ∧ 13 ≤ X₁+X₂ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₅ ∧ X₅ ≤ X₀ ∧ 0 ≤ X₅ ∧ X₂+X₅ ≤ valToFill ∧ X₁+X₅ ≤ X₀
{0}
j->h
p = 1
t₁₈ ∈ g₁₇
η (X₅) = X₅+UNIFORM(0, 1)
τ = X₁+X₂ ≤ 13 ∧ X₁ ≤ 10+X₀ ∧ X₅ ≤ 10+X₀ ∧ X₁ ≤ 10 ∧ X₁ ≤ 10+X₅ ∧ X₁ ≤ 7+X₂ ∧ X₂ ≤ 3+X₀ ∧ X₂ ≤ 3 ∧ X₂ ≤ 3+X₅ ∧ 0 ≤ 1 ∧ 3 ≤ X₀+X₂ ∧ 3 ≤ X₂ ∧ 3 ≤ X₂+X₅ ∧ 7+X₂ ≤ X₁ ∧ 9 ≤ X₀+X₅ ∧ 10 ≤ X₀+X₁ ∧ 10 ≤ X₁ ∧ 10 ≤ X₁+X₅ ∧ 13 ≤ X₁+X₂ ∧ 0 ≤ X₀ ∧ 0 ≤ X₅
Use expected size bounds for entry point (g₇:g→[t₈:1:h],h)
Use expected size bounds for entry point (g₇:g→[t₈:1:h],h)
Use expected size bounds for entry point (g₇:g→[t₈:1:h],h)
Use classical time bound for entry point (g₇:g→[t₈:1:h],h)
Plrf for transition g₁₅:i(X₀,X₁,X₂,X₃,X₄,X₅) -{0}> t₁₆:j(X₀,X₁,X₂,X₃,X₄,X₅+UNIFORM(0, 4)) :|: X₂+X₅ ≤ valToFill ∧ X₁+X₅ ≤ X₀ ∧ X₁+X₂ ≤ 13 ∧ X₁ ≤ 10+X₀ ∧ X₁ ≤ 10 ∧ X₁ ≤ 10+X₅ ∧ X₁ ≤ 7+X₂ ∧ X₂ ≤ 3+X₀ ∧ X₂ ≤ 3 ∧ X₂ ≤ 3+X₅ ∧ 0 ≤ 1 ∧ 3 ≤ X₀+X₂ ∧ 3 ≤ X₂ ∧ 3 ≤ X₂+X₅ ∧ 7+X₂ ≤ X₁ ∧ 10 ≤ X₀+X₁ ∧ 10 ≤ X₁ ∧ 10 ≤ X₁+X₅ ∧ 13 ≤ X₁+X₂ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₅ ∧ X₅ ≤ X₀ ∧ 0 ≤ X₅:
new bound:
2/5⋅X₀+24/5 {O(n)}
PLRF:
• h: 1/5+2/5⋅X₀+7/5⋅X₂-2/5⋅X₅
• i: 1/5+2/5⋅X₀+7/5⋅X₂-2/5⋅X₅
• j: 2/5⋅X₀+7/5⋅X₂-2/5⋅X₅
Show Graph
G
f
f
g
g
f->g
p = 1
t₆ ∈ g₅
η (X₁) = 10
η (X₂) = 3
η (X₅) = 0
{0}
h
h
g->h
p = 1
t₈ ∈ g₇
η (X₅) = X₅+UNIFORM(0, 1)
τ = X₁+X₂ ≤ 13 ∧ X₁ ≤ 10 ∧ X₁ ≤ 10+X₅ ∧ X₁+X₅ ≤ 10 ∧ X₁ ≤ 7+X₂ ∧ X₂ ≤ 3 ∧ X₂ ≤ 3+X₅ ∧ X₂+X₅ ≤ 3 ∧ 0 ≤ 1 ∧ 3 ≤ X₂ ∧ 3 ≤ X₂+X₅ ∧ 3+X₅ ≤ X₂ ∧ 7+X₂ ≤ X₁ ∧ 10 ≤ X₁ ∧ 10 ≤ X₁+X₅ ∧ 10+X₅ ≤ X₁ ∧ 13 ≤ X₁+X₂ ∧ 0 ≤ X₅ ∧ X₅ ≤ 0
{0}
i
i
h->i
p = 1
t₁₀ ∈ g₉
τ = X₁+X₂ ≤ 13 ∧ X₁ ≤ 10 ∧ X₁ ≤ 10+X₅ ∧ X₁ ≤ 7+X₂ ∧ X₂ ≤ 3 ∧ X₂ ≤ 3+X₅ ∧ 3 ≤ X₂ ∧ 3 ≤ X₂+X₅ ∧ 7+X₂ ≤ X₁ ∧ 10 ≤ X₁ ∧ 10 ≤ X₁+X₅ ∧ 13 ≤ X₁+X₂ ∧ 0 ≤ X₅ ∧ X₅ ≤ X₀
{0}
j
j
i->j
p = 1
t₁₂ ∈ g₁₁
η (X₅) = X₅+UNIFORM(9, 10)
τ = X₁+X₂ ≤ 13 ∧ X₁ ≤ 10+X₀ ∧ X₁ ≤ 10 ∧ X₁ ≤ 10+X₅ ∧ X₁ ≤ 7+X₂ ∧ X₂ ≤ 3+X₀ ∧ X₂ ≤ 3 ∧ X₂ ≤ 3+X₅ ∧ 0 ≤ 1 ∧ 3 ≤ X₀+X₂ ∧ 3 ≤ X₂ ∧ 3 ≤ X₂+X₅ ∧ 7+X₂ ≤ X₁ ∧ 10 ≤ X₀+X₁ ∧ 10 ≤ X₁ ∧ 10 ≤ X₁+X₅ ∧ 13 ≤ X₁+X₂ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₅ ∧ X₅ ≤ X₀ ∧ 0 ≤ X₅ ∧ 1+valToFill ≤ X₁+X₅
{0}
i->j
p = 1
t₁₄ ∈ g₁₃
η (X₅) = X₅+UNIFORM(2, 4)
τ = X₁+X₂ ≤ 13 ∧ X₁ ≤ 10+X₀ ∧ X₁ ≤ 10 ∧ X₁ ≤ 10+X₅ ∧ X₁ ≤ 7+X₂ ∧ X₂ ≤ 3+X₀ ∧ X₂ ≤ 3 ∧ X₂ ≤ 3+X₅ ∧ 0 ≤ 1 ∧ 3 ≤ X₀+X₂ ∧ 3 ≤ X₂ ∧ 3 ≤ X₂+X₅ ∧ 7+X₂ ≤ X₁ ∧ 10 ≤ X₀+X₁ ∧ 10 ≤ X₁ ∧ 10 ≤ X₁+X₅ ∧ 13 ≤ X₁+X₂ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₅ ∧ X₅ ≤ X₀ ∧ 0 ≤ X₅ ∧ 1+valToFill ≤ X₂+X₅ ∧ X₁+X₅ ≤ X₀
{0}
i->j
p = 1
t₁₆ ∈ g₁₅
η (X₅) = X₅+UNIFORM(0, 4)
τ = X₁+X₂ ≤ 13 ∧ X₁ ≤ 10+X₀ ∧ X₁ ≤ 10 ∧ X₁ ≤ 10+X₅ ∧ X₁ ≤ 7+X₂ ∧ X₂ ≤ 3+X₀ ∧ X₂ ≤ 3 ∧ X₂ ≤ 3+X₅ ∧ 0 ≤ 1 ∧ 3 ≤ X₀+X₂ ∧ 3 ≤ X₂ ∧ 3 ≤ X₂+X₅ ∧ 7+X₂ ≤ X₁ ∧ 10 ≤ X₀+X₁ ∧ 10 ≤ X₁ ∧ 10 ≤ X₁+X₅ ∧ 13 ≤ X₁+X₂ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₅ ∧ X₅ ≤ X₀ ∧ 0 ≤ X₅ ∧ X₂+X₅ ≤ valToFill ∧ X₁+X₅ ≤ X₀
{0}
j->h
p = 1
t₁₈ ∈ g₁₇
η (X₅) = X₅+UNIFORM(0, 1)
τ = X₁+X₂ ≤ 13 ∧ X₁ ≤ 10+X₀ ∧ X₅ ≤ 10+X₀ ∧ X₁ ≤ 10 ∧ X₁ ≤ 10+X₅ ∧ X₁ ≤ 7+X₂ ∧ X₂ ≤ 3+X₀ ∧ X₂ ≤ 3 ∧ X₂ ≤ 3+X₅ ∧ 0 ≤ 1 ∧ 3 ≤ X₀+X₂ ∧ 3 ≤ X₂ ∧ 3 ≤ X₂+X₅ ∧ 7+X₂ ≤ X₁ ∧ 9 ≤ X₀+X₅ ∧ 10 ≤ X₀+X₁ ∧ 10 ≤ X₁ ∧ 10 ≤ X₁+X₅ ∧ 13 ≤ X₁+X₂ ∧ 0 ≤ X₀ ∧ 0 ≤ X₅
Use expected size bounds for entry point (g₇:g→[t₈:1:h],h)
Use expected size bounds for entry point (g₇:g→[t₈:1:h],h)
Use expected size bounds for entry point (g₇:g→[t₈:1:h],h)
Use classical time bound for entry point (g₇:g→[t₈:1:h],h)
Plrf for transition g₁₇:j(X₀,X₁,X₂,X₃,X₄,X₅) → t₁₈:h(X₀,X₁,X₂,X₃,X₄,X₅+UNIFORM(0, 1)) :|: X₁+X₂ ≤ 13 ∧ X₁ ≤ 10+X₀ ∧ X₅ ≤ 10+X₀ ∧ X₁ ≤ 10 ∧ X₁ ≤ 10+X₅ ∧ X₁ ≤ 7+X₂ ∧ X₂ ≤ 3+X₀ ∧ X₂ ≤ 3 ∧ X₂ ≤ 3+X₅ ∧ 0 ≤ 1 ∧ 3 ≤ X₀+X₂ ∧ 3 ≤ X₂ ∧ 3 ≤ X₂+X₅ ∧ 7+X₂ ≤ X₁ ∧ 9 ≤ X₀+X₅ ∧ 10 ≤ X₀+X₁ ∧ 10 ≤ X₁ ∧ 10 ≤ X₁+X₅ ∧ 13 ≤ X₁+X₂ ∧ 0 ≤ X₀ ∧ 0 ≤ X₅:
new bound:
2/5⋅X₀+24/5 {O(n)}
PLRF:
• h: 2/5⋅X₀+11/25⋅X₁-2/5⋅X₅
• i: 2/5⋅X₀+11/25⋅X₁-2/5⋅X₅
• j: 4/5+2/5⋅X₀+11/25⋅X₁-2/5⋅X₅
Show Graph
G
f
f
g
g
f->g
p = 1
t₆ ∈ g₅
η (X₁) = 10
η (X₂) = 3
η (X₅) = 0
{0}
h
h
g->h
p = 1
t₈ ∈ g₇
η (X₅) = X₅+UNIFORM(0, 1)
τ = X₁+X₂ ≤ 13 ∧ X₁ ≤ 10 ∧ X₁ ≤ 10+X₅ ∧ X₁+X₅ ≤ 10 ∧ X₁ ≤ 7+X₂ ∧ X₂ ≤ 3 ∧ X₂ ≤ 3+X₅ ∧ X₂+X₅ ≤ 3 ∧ 0 ≤ 1 ∧ 3 ≤ X₂ ∧ 3 ≤ X₂+X₅ ∧ 3+X₅ ≤ X₂ ∧ 7+X₂ ≤ X₁ ∧ 10 ≤ X₁ ∧ 10 ≤ X₁+X₅ ∧ 10+X₅ ≤ X₁ ∧ 13 ≤ X₁+X₂ ∧ 0 ≤ X₅ ∧ X₅ ≤ 0
{0}
i
i
h->i
p = 1
t₁₀ ∈ g₉
τ = X₁+X₂ ≤ 13 ∧ X₁ ≤ 10 ∧ X₁ ≤ 10+X₅ ∧ X₁ ≤ 7+X₂ ∧ X₂ ≤ 3 ∧ X₂ ≤ 3+X₅ ∧ 3 ≤ X₂ ∧ 3 ≤ X₂+X₅ ∧ 7+X₂ ≤ X₁ ∧ 10 ≤ X₁ ∧ 10 ≤ X₁+X₅ ∧ 13 ≤ X₁+X₂ ∧ 0 ≤ X₅ ∧ X₅ ≤ X₀
{0}
j
j
i->j
p = 1
t₁₂ ∈ g₁₁
η (X₅) = X₅+UNIFORM(9, 10)
τ = X₁+X₂ ≤ 13 ∧ X₁ ≤ 10+X₀ ∧ X₁ ≤ 10 ∧ X₁ ≤ 10+X₅ ∧ X₁ ≤ 7+X₂ ∧ X₂ ≤ 3+X₀ ∧ X₂ ≤ 3 ∧ X₂ ≤ 3+X₅ ∧ 0 ≤ 1 ∧ 3 ≤ X₀+X₂ ∧ 3 ≤ X₂ ∧ 3 ≤ X₂+X₅ ∧ 7+X₂ ≤ X₁ ∧ 10 ≤ X₀+X₁ ∧ 10 ≤ X₁ ∧ 10 ≤ X₁+X₅ ∧ 13 ≤ X₁+X₂ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₅ ∧ X₅ ≤ X₀ ∧ 0 ≤ X₅ ∧ 1+valToFill ≤ X₁+X₅
{0}
i->j
p = 1
t₁₄ ∈ g₁₃
η (X₅) = X₅+UNIFORM(2, 4)
τ = X₁+X₂ ≤ 13 ∧ X₁ ≤ 10+X₀ ∧ X₁ ≤ 10 ∧ X₁ ≤ 10+X₅ ∧ X₁ ≤ 7+X₂ ∧ X₂ ≤ 3+X₀ ∧ X₂ ≤ 3 ∧ X₂ ≤ 3+X₅ ∧ 0 ≤ 1 ∧ 3 ≤ X₀+X₂ ∧ 3 ≤ X₂ ∧ 3 ≤ X₂+X₅ ∧ 7+X₂ ≤ X₁ ∧ 10 ≤ X₀+X₁ ∧ 10 ≤ X₁ ∧ 10 ≤ X₁+X₅ ∧ 13 ≤ X₁+X₂ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₅ ∧ X₅ ≤ X₀ ∧ 0 ≤ X₅ ∧ 1+valToFill ≤ X₂+X₅ ∧ X₁+X₅ ≤ X₀
{0}
i->j
p = 1
t₁₆ ∈ g₁₅
η (X₅) = X₅+UNIFORM(0, 4)
τ = X₁+X₂ ≤ 13 ∧ X₁ ≤ 10+X₀ ∧ X₁ ≤ 10 ∧ X₁ ≤ 10+X₅ ∧ X₁ ≤ 7+X₂ ∧ X₂ ≤ 3+X₀ ∧ X₂ ≤ 3 ∧ X₂ ≤ 3+X₅ ∧ 0 ≤ 1 ∧ 3 ≤ X₀+X₂ ∧ 3 ≤ X₂ ∧ 3 ≤ X₂+X₅ ∧ 7+X₂ ≤ X₁ ∧ 10 ≤ X₀+X₁ ∧ 10 ≤ X₁ ∧ 10 ≤ X₁+X₅ ∧ 13 ≤ X₁+X₂ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₅ ∧ X₅ ≤ X₀ ∧ 0 ≤ X₅ ∧ X₂+X₅ ≤ valToFill ∧ X₁+X₅ ≤ X₀
{0}
j->h
p = 1
t₁₈ ∈ g₁₇
η (X₅) = X₅+UNIFORM(0, 1)
τ = X₁+X₂ ≤ 13 ∧ X₁ ≤ 10+X₀ ∧ X₅ ≤ 10+X₀ ∧ X₁ ≤ 10 ∧ X₁ ≤ 10+X₅ ∧ X₁ ≤ 7+X₂ ∧ X₂ ≤ 3+X₀ ∧ X₂ ≤ 3 ∧ X₂ ≤ 3+X₅ ∧ 0 ≤ 1 ∧ 3 ≤ X₀+X₂ ∧ 3 ≤ X₂ ∧ 3 ≤ X₂+X₅ ∧ 7+X₂ ≤ X₁ ∧ 9 ≤ X₀+X₅ ∧ 10 ≤ X₀+X₁ ∧ 10 ≤ X₁ ∧ 10 ≤ X₁+X₅ ∧ 13 ≤ X₁+X₂ ∧ 0 ≤ X₀ ∧ 0 ≤ X₅
Use expected size bounds for entry point (g₇:g→[t₈:1:h],h)
Use expected size bounds for entry point (g₇:g→[t₈:1:h],h)
Use expected size bounds for entry point (g₇:g→[t₈:1:h],h)
Use classical time bound for entry point (g₇:g→[t₈:1:h],h)
Results of Probabilistic Analysis
All Bounds
Timebounds
Overall timebound:16/5⋅X₀+102/5 {O(n)}
g₅: 1 {O(1)}
g₇: 1 {O(1)}
g₉: 2/5⋅X₀+24/5 {O(n)}
g₁₁: X₀+2 {O(n)}
g₁₃: X₀+2 {O(n)}
g₁₅: 2/5⋅X₀+24/5 {O(n)}
g₁₇: 2/5⋅X₀+24/5 {O(n)}
Costbounds
Overall costbound: 2/5⋅X₀+24/5 {O(n)}
g₅: 0 {O(1)}
g₇: 0 {O(1)}
g₉: 0 {O(1)}
g₁₁: 0 {O(1)}
g₁₃: 0 {O(1)}
g₁₅: 0 {O(1)}
g₁₇: 2/5⋅X₀+24/5 {O(n)}
Sizebounds
(g₅,g), X₀: X₀ {O(n)}
(g₅,g), X₁: 10 {O(1)}
(g₅,g), X₂: 3 {O(1)}
(g₅,g), X₃: 1 {O(1)}
(g₅,g), X₄: X₄ {O(n)}
(g₅,g), X₅: 0 {O(1)}
(g₇,h), X₀: X₀ {O(n)}
(g₇,h), X₁: 10 {O(1)}
(g₇,h), X₂: 3 {O(1)}
(g₇,h), X₃: 1 {O(1)}
(g₇,h), X₄: X₄ {O(n)}
(g₇,h), X₅: 1 {O(1)}
(g₉,i), X₀: X₀ {O(n)}
(g₉,i), X₁: 10 {O(1)}
(g₉,i), X₂: 3 {O(1)}
(g₉,i), X₃: 1 {O(1)}
(g₉,i), X₄: X₄ {O(n)}
(g₉,i), X₅: 27/2⋅X₀+38 {O(n)}
(g₁₁,j), X₀: X₀ {O(n)}
(g₁₁,j), X₁: 10 {O(1)}
(g₁₁,j), X₂: 3 {O(1)}
(g₁₁,j), X₃: 1 {O(1)}
(g₁₁,j), X₄: X₄ {O(n)}
(g₁₁,j), X₅: 27/2⋅X₀+38 {O(n)}
(g₁₃,j), X₀: X₀ {O(n)}
(g₁₃,j), X₁: 10 {O(1)}
(g₁₃,j), X₂: 3 {O(1)}
(g₁₃,j), X₃: 1 {O(1)}
(g₁₃,j), X₄: X₄ {O(n)}
(g₁₃,j), X₅: 27/2⋅X₀+38 {O(n)}
(g₁₅,j), X₀: X₀ {O(n)}
(g₁₅,j), X₁: 10 {O(1)}
(g₁₅,j), X₂: 3 {O(1)}
(g₁₅,j), X₃: 1 {O(1)}
(g₁₅,j), X₄: X₄ {O(n)}
(g₁₅,j), X₅: 27/2⋅X₀+38 {O(n)}
(g₁₇,h), X₀: X₀ {O(n)}
(g₁₇,h), X₁: 10 {O(1)}
(g₁₇,h), X₂: 3 {O(1)}
(g₁₇,h), X₃: 1 {O(1)}
(g₁₇,h), X₄: X₄ {O(n)}
(g₁₇,h), X₅: 27/2⋅X₀+38 {O(n)}