Preprocessing
Probabilistic Analysis
Probabilistic Program after Preprocessing
Start: f
Program_Vars: X₀, X₁
Temp_Vars:
Locations: f, g
Transitions:
g₇:f(X₀,X₁) -{0}> t₈:g(10,X₁) :|:
g₉:g(X₀,X₁) → [6/37]:t₁₀:g(X₀+UNIFORM(3, 6),X₁) :+: [6/37]:t₁₁:g(X₀+UNIFORM(1, 2),X₁) :+: [4/37]:t₁₂:g(X₀+UNIFORM(3, 6),X₁) :+: [8/37]:t₁₃:g(X₀+UNIFORM(1, 2),X₁) :+: [8/37]:t₁₄:g(X₀+UNIFORM(-6, -3),X₁) :+: [4/37]:t₁₅:g(X₀+UNIFORM(-10, -5),X₁) :+: [1/37]:t₁₆:g(X₀+UNIFORM(-10, -5),X₁) :|: X₁ ≤ X₀ ∧ 0 ≤ 1 ∧ 0 ≤ 1 ∧ 0 ≤ 1 ∧ 0 ≤ 1 ∧ 0 ≤ 1 ∧ 0 ≤ 1 ∧ 0 ≤ 1
Show Graph
G
f
f
g
g
f->g
p = 1
t₈ ∈ g₇
η (X₀) = 10
{0}
g->g
p = 6/37
t₁₀ ∈ g₉
η (X₀) = X₀+UNIFORM(3, 6)
τ = 0 ≤ 1 ∧ 0 ≤ 1 ∧ 0 ≤ 1 ∧ 0 ≤ 1 ∧ 0 ≤ 1 ∧ 0 ≤ 1 ∧ 0 ≤ 1 ∧ X₁ ≤ X₀
g->g
p = 6/37
t₁₁ ∈ g₉
η (X₀) = X₀+UNIFORM(1, 2)
τ = 0 ≤ 1 ∧ 0 ≤ 1 ∧ 0 ≤ 1 ∧ 0 ≤ 1 ∧ 0 ≤ 1 ∧ 0 ≤ 1 ∧ 0 ≤ 1 ∧ X₁ ≤ X₀
g->g
p = 4/37
t₁₂ ∈ g₉
η (X₀) = X₀+UNIFORM(3, 6)
τ = 0 ≤ 1 ∧ 0 ≤ 1 ∧ 0 ≤ 1 ∧ 0 ≤ 1 ∧ 0 ≤ 1 ∧ 0 ≤ 1 ∧ 0 ≤ 1 ∧ X₁ ≤ X₀
g->g
p = 8/37
t₁₃ ∈ g₉
η (X₀) = X₀+UNIFORM(1, 2)
τ = 0 ≤ 1 ∧ 0 ≤ 1 ∧ 0 ≤ 1 ∧ 0 ≤ 1 ∧ 0 ≤ 1 ∧ 0 ≤ 1 ∧ 0 ≤ 1 ∧ X₁ ≤ X₀
g->g
p = 8/37
t₁₄ ∈ g₉
η (X₀) = X₀+UNIFORM(-6, -3)
τ = 0 ≤ 1 ∧ 0 ≤ 1 ∧ 0 ≤ 1 ∧ 0 ≤ 1 ∧ 0 ≤ 1 ∧ 0 ≤ 1 ∧ 0 ≤ 1 ∧ X₁ ≤ X₀
g->g
p = 4/37
t₁₅ ∈ g₉
η (X₀) = X₀+UNIFORM(-10, -5)
τ = 0 ≤ 1 ∧ 0 ≤ 1 ∧ 0 ≤ 1 ∧ 0 ≤ 1 ∧ 0 ≤ 1 ∧ 0 ≤ 1 ∧ 0 ≤ 1 ∧ X₁ ≤ X₀
g->g
p = 1/37
t₁₆ ∈ g₉
η (X₀) = X₀+UNIFORM(-10, -5)
τ = 0 ≤ 1 ∧ 0 ≤ 1 ∧ 0 ≤ 1 ∧ 0 ≤ 1 ∧ 0 ≤ 1 ∧ 0 ≤ 1 ∧ 0 ≤ 1 ∧ X₁ ≤ X₀
Run classical analysis on SCC: [f]
Classical Approximation after Lifting Classical Results
All Bounds
Timebounds
Overall timebound:inf {Infinity}
g₇: 1 {O(1)}
g₉: inf {Infinity}
Costbounds
Overall costbound: inf {Infinity}
g₇: inf {Infinity}
g₉: inf {Infinity}
Sizebounds
(g₇,g), X₀: 10 {O(1)}
(g₇,g), X₁: X₁ {O(n)}
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}
g₇: 1 {O(1)}
g₉: inf {Infinity}
Costbounds
Overall costbound: inf {Infinity}
g₇: inf {Infinity}
g₉: inf {Infinity}
Sizebounds
(g₇,g), X₀: 10 {O(1)}
(g₇,g), X₁: X₁ {O(n)}
(g₉,g), X₁: 7⋅X₁ {O(n)}
Run probabilistic analysis on SCC: [g]
Plrf for transition g₉:g(X₀,X₁) → [6/37]:t₁₀:g(X₀+UNIFORM(3, 6),X₁) :+: [6/37]:t₁₁:g(X₀+UNIFORM(1, 2),X₁) :+: [4/37]:t₁₂:g(X₀+UNIFORM(3, 6),X₁) :+: [8/37]:t₁₃:g(X₀+UNIFORM(1, 2),X₁) :+: [8/37]:t₁₄:g(X₀+UNIFORM(-6, -3),X₁) :+: [4/37]:t₁₅:g(X₀+UNIFORM(-10, -5),X₁) :+: [1/37]:t₁₆:g(X₀+UNIFORM(-10, -5),X₁) :|: X₁ ≤ X₀ ∧ 0 ≤ 1 ∧ 0 ≤ 1 ∧ 0 ≤ 1 ∧ 0 ≤ 1 ∧ 0 ≤ 1 ∧ 0 ≤ 1 ∧ 0 ≤ 1:
new bound:
74/15⋅X₁+296/3 {O(n)}
PLRF:
• g: 148/3+74/15⋅X₀-74/15⋅X₁
Show Graph
G
f
f
g
g
f->g
p = 1
t₈ ∈ g₇
η (X₀) = 10
{0}
g->g
p = 6/37
t₁₀ ∈ g₉
η (X₀) = X₀+UNIFORM(3, 6)
τ = 0 ≤ 1 ∧ 0 ≤ 1 ∧ 0 ≤ 1 ∧ 0 ≤ 1 ∧ 0 ≤ 1 ∧ 0 ≤ 1 ∧ 0 ≤ 1 ∧ X₁ ≤ X₀
g->g
p = 6/37
t₁₁ ∈ g₉
η (X₀) = X₀+UNIFORM(1, 2)
τ = 0 ≤ 1 ∧ 0 ≤ 1 ∧ 0 ≤ 1 ∧ 0 ≤ 1 ∧ 0 ≤ 1 ∧ 0 ≤ 1 ∧ 0 ≤ 1 ∧ X₁ ≤ X₀
g->g
p = 4/37
t₁₂ ∈ g₉
η (X₀) = X₀+UNIFORM(3, 6)
τ = 0 ≤ 1 ∧ 0 ≤ 1 ∧ 0 ≤ 1 ∧ 0 ≤ 1 ∧ 0 ≤ 1 ∧ 0 ≤ 1 ∧ 0 ≤ 1 ∧ X₁ ≤ X₀
g->g
p = 8/37
t₁₃ ∈ g₉
η (X₀) = X₀+UNIFORM(1, 2)
τ = 0 ≤ 1 ∧ 0 ≤ 1 ∧ 0 ≤ 1 ∧ 0 ≤ 1 ∧ 0 ≤ 1 ∧ 0 ≤ 1 ∧ 0 ≤ 1 ∧ X₁ ≤ X₀
g->g
p = 8/37
t₁₄ ∈ g₉
η (X₀) = X₀+UNIFORM(-6, -3)
τ = 0 ≤ 1 ∧ 0 ≤ 1 ∧ 0 ≤ 1 ∧ 0 ≤ 1 ∧ 0 ≤ 1 ∧ 0 ≤ 1 ∧ 0 ≤ 1 ∧ X₁ ≤ X₀
g->g
p = 4/37
t₁₅ ∈ g₉
η (X₀) = X₀+UNIFORM(-10, -5)
τ = 0 ≤ 1 ∧ 0 ≤ 1 ∧ 0 ≤ 1 ∧ 0 ≤ 1 ∧ 0 ≤ 1 ∧ 0 ≤ 1 ∧ 0 ≤ 1 ∧ X₁ ≤ X₀
g->g
p = 1/37
t₁₆ ∈ g₉
η (X₀) = X₀+UNIFORM(-10, -5)
τ = 0 ≤ 1 ∧ 0 ≤ 1 ∧ 0 ≤ 1 ∧ 0 ≤ 1 ∧ 0 ≤ 1 ∧ 0 ≤ 1 ∧ 0 ≤ 1 ∧ X₁ ≤ X₀
Use expected size bounds for entry point (g₇:f→[t₈:1:g],g)
Use expected size bounds for entry point (g₇:f→[t₈:1:g],g)
Use classical time bound for entry point (g₇:f→[t₈:1:g],g)
Results of Probabilistic Analysis
All Bounds
Timebounds
Overall timebound:74/15⋅X₁+299/3 {O(n)}
g₇: 1 {O(1)}
g₉: 74/15⋅X₁+296/3 {O(n)}
Costbounds
Overall costbound: 518/15⋅X₁+2072/3 {O(n)}
g₇: 0 {O(1)}
g₉: 518/15⋅X₁+2072/3 {O(n)}
Sizebounds
(g₇,g), X₀: 10 {O(1)}
(g₇,g), X₁: X₁ {O(n)}
(g₉,g), X₀: 93/5⋅X₁+382 {O(n)}
(g₉,g), X₁: X₁ {O(n)}