Preprocessing

Found invariant X₂ ≤ X₀ ∧ X₀ ≤ X₂ for location h

Found invariant X₂ ≤ 0 ∧ 0 ≤ X₂ for location g

Found invariant X₂ ≤ X₁ ∧ X₂ ≤ 1+X₀ ∧ X₀ ≤ X₂ ∧ 1+X₀ ≤ X₁ for location i

Probabilistic Analysis

Probabilistic Program after Preprocessing

Start: f
Program_Vars: X₀, X₁, X₂
Temp_Vars:
Locations: f, g, h, i
Transitions:
g₁:f(X₀,X₁,X₂) -{0}> t₂:g(X₀,X₁,0) :|:
g₃:g(X₀,X₁,X₂) -{0}> t₄:h(X₀,X₁,X₀) :|: 0 ≤ X₂ ∧ X₂ ≤ 0
g₅:h(X₀,X₁,X₂) -{0}> t₆:i(X₀,X₁,X₂+UNIFORM(0, 1)) :|: 1+X₂ ≤ X₁ ∧ 0 ≤ 1 ∧ X₂ ≤ X₀ ∧ X₀ ≤ X₂
g₇:i(X₀,X₁,X₂) → t₈:g(X₂,X₁,0) :|: X₂ ≤ 1+X₀ ∧ 1+X₀ ≤ X₁ ∧ X₀ ≤ X₂ ∧ X₂ ≤ X₁

Run classical analysis on SCC: [f]

Classical Approximation after Lifting Classical Results

All Bounds
Timebounds

Overall timebound:inf {Infinity}
g₁: 1 {O(1)}
g₃: inf {Infinity}
g₅: inf {Infinity}
g₇: inf {Infinity}

Costbounds

Overall costbound: inf {Infinity}
g₁: inf {Infinity}
g₃: inf {Infinity}
g₅: inf {Infinity}
g₇: inf {Infinity}

Sizebounds

(g₁,g), X₀: X₀ {O(n)}
(g₁,g), X₁: X₁ {O(n)}
(g₁,g), X₂: 0 {O(1)}

Run probabilistic analysis on SCC: [f]

Run classical analysis on SCC: [g; h; i]

Classical Approximation after Lifting Classical Results

All Bounds
Timebounds

Overall timebound:inf {Infinity}
g₁: 1 {O(1)}
g₃: inf {Infinity}
g₅: inf {Infinity}
g₇: inf {Infinity}

Costbounds

Overall costbound: inf {Infinity}
g₁: inf {Infinity}
g₃: inf {Infinity}
g₅: inf {Infinity}
g₇: inf {Infinity}

Sizebounds

(g₁,g), X₀: X₀ {O(n)}
(g₁,g), X₁: X₁ {O(n)}
(g₁,g), X₂: 0 {O(1)}
(g₃,h), X₁: X₁ {O(n)}
(g₅,i), X₁: X₁ {O(n)}
(g₇,g), X₁: X₁ {O(n)}
(g₇,g), X₂: 0 {O(1)}

Run probabilistic analysis on SCC: [g; h; i]

Results of Probabilistic Analysis

All Bounds

Timebounds

Overall timebound:inf {Infinity}
g₁: 1 {O(1)}
g₃: inf {Infinity}
g₅: inf {Infinity}
g₇: inf {Infinity}

Costbounds

Overall costbound: inf {Infinity}
g₁: 0 {O(1)}
g₃: 0 {O(1)}
g₅: 0 {O(1)}
g₇: inf {Infinity}

Sizebounds

(g₁,g), X₀: X₀ {O(n)}
(g₁,g), X₁: X₁ {O(n)}
(g₁,g), X₂: 0 {O(1)}
(g₃,h), X₁: X₁ {O(n)}
(g₅,i), X₁: X₁ {O(n)}
(g₇,g), X₁: X₁ {O(n)}
(g₇,g), X₂: 0 {O(1)}