Preprocessing

Found invariant 1 ≤ X₀ for location h

Probabilistic Analysis

Probabilistic Program after Preprocessing

Start: f
Program_Vars: X₀
Temp_Vars: T
Locations: f, g, h
Transitions:
g₁:f(X₀) → t₂:h(T) :|: 1 ≤ T
g₃:f(X₀) → [3/4]:t₄:g(X₀) :+: [1/4]:t₅:g(X₀-1) :|:
g₆:g(X₀) → t₇:g(X₀+Geometric (1/2)) :|: X₀ ≤ 9

Run classical analysis on SCC: [f]

Classical Approximation after Lifting Classical Results

All Bounds
Timebounds

Overall timebound:inf {Infinity}
g₁: 1 {O(1)}
g₃: 2 {O(1)}
g₆: inf {Infinity}

Costbounds

Overall costbound: inf {Infinity}
g₁: inf {Infinity}
g₃: inf {Infinity}
g₆: inf {Infinity}

Sizebounds

(g₃,g), X₀: 2⋅X₀+1 {O(n)}

Run probabilistic analysis on SCC: [f]

Run classical analysis on SCC: [g]

MPRF for transition t₇: g(X₀) → g(Temp_Int₂₁) :|: X₀ ≤ 9 ∧ 1+X₀ ≤ Temp_Int₂₁ ∧ 1+X₀ ≤ Temp_Int₁ ∧ Temp_Int₂+X₀ ≤ Temp_Int₁ ∧ Temp_Int₁ ≤ Temp_Int₂+X₀ ∧ Temp_Int₁ ≤ Temp_Int₂+X₀ ∧ Temp_Int₂+X₀ ≤ Temp_Int₁ ∧ 1 ≤ Temp_Int₂ ∧ X₀ ≤ 9 of depth 1:

new bound:

2⋅X₀+21 {O(n)}

MPRF:

• g: [10-X₀]

Classical Approximation after Lifting Classical Results

All Bounds
Timebounds

Overall timebound:2⋅X₀+24 {O(n)}
g₁: 1 {O(1)}
g₃: 2 {O(1)}
g₆: 2⋅X₀+21 {O(n)}

Costbounds

Overall costbound: inf {Infinity}
g₁: inf {Infinity}
g₃: inf {Infinity}
g₆: inf {Infinity}

Sizebounds

(g₃,g), X₀: 2⋅X₀+1 {O(n)}

Run probabilistic analysis on SCC: [g]

Run classical analysis on SCC: [h]

Classical Approximation after Lifting Classical Results

All Bounds
Timebounds

Overall timebound:2⋅X₀+24 {O(n)}
g₁: 1 {O(1)}
g₃: 2 {O(1)}
g₆: 2⋅X₀+21 {O(n)}

Costbounds

Overall costbound: inf {Infinity}
g₁: inf {Infinity}
g₃: inf {Infinity}
g₆: inf {Infinity}

Sizebounds

(g₃,g), X₀: 2⋅X₀+1 {O(n)}
(g₆,g), X₀: 6⋅X₀+43 {O(n)}

Run probabilistic analysis on SCC: [h]

Results of Probabilistic Analysis

All Bounds

Timebounds

Overall timebound:2⋅X₀+24 {O(n)}
g₁: 1 {O(1)}
g₃: 2 {O(1)}
g₆: 2⋅X₀+21 {O(n)}

Costbounds

Overall costbound: 2⋅X₀+26 {O(n)}
g₁: 1 {O(1)}
g₃: 4 {O(1)}
g₆: 2⋅X₀+21 {O(n)}

Sizebounds

(g₃,g), X₀: 2⋅X₀+1 {O(n)}
(g₆,g), X₀: 6⋅X₀+43 {O(n)}