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

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

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)}