Preprocessing

Found invariant 11 ≤ X₁ ∧ 7+X₀ ≤ X₁ ∧ X₀ ≤ 4 for location stop

Found invariant 11 ≤ X₁ for location i

Probabilistic Analysis

Probabilistic Program after Preprocessing

Start: f
Program_Vars: X₀, X₁
Temp_Vars:
Locations: f, g, h, i, stop
Transitions:
g₀:f(X₀,X₁) -{0}> t₁:g(X₀,X₁) :|:
g₂:g(X₀,X₁) -{0}> [1/2]:t₃:h(X₀,2+X₁) :+: [1/2]:t₄:h(X₀,X₁-1) :|:
g₅:h(X₀,X₁) -{0}> t₆:i(X₀-1,X₁) :|: 11 ≤ X₁
g₇:h(X₀,X₁) → t₈:g(X₀,X₁) :|: X₁ ≤ 10
g₉:i(X₀,X₁) → t₁₀:stop(X₀,X₁) :|: X₀ ≤ 4 ∧ 11 ≤ X₁
g₁₁:i(X₀,X₁) → t₁₂:g(X₀,X₁) :|: 5 ≤ X₀ ∧ 11 ≤ X₁

Run classical analysis on SCC: [f]

Classical Approximation after Lifting Classical Results

All Bounds
Timebounds

Overall timebound:inf {Infinity}
g₀: 1 {O(1)}
g₂: inf {Infinity}
g₅: inf {Infinity}
g₇: inf {Infinity}
g₉: 1 {O(1)}
g₁₁: inf {Infinity}

Costbounds

Overall costbound: inf {Infinity}
g₀: inf {Infinity}
g₂: inf {Infinity}
g₅: inf {Infinity}
g₇: inf {Infinity}
g₉: inf {Infinity}
g₁₁: inf {Infinity}

Sizebounds

(g₀,g), X₀: X₀ {O(n)}
(g₀,g), X₁: X₁ {O(n)}

Run probabilistic analysis on SCC: [f]

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

MPRF for transition t₁₂: i(X₀,X₁) → g(X₀,X₁) :|: 11 ≤ X₁ ∧ 5 ≤ X₀ of depth 1:

new bound:

X₀+5 {O(n)}

MPRF:

• g: [X₀-5]
• h: [X₀-5]
• i: [X₀-4]

MPRF for transition t₆: h(X₀,X₁) -{0}> i(X₀-1,X₁) :|: 11 ≤ X₁ of depth 1:

new bound:

X₀+6 {O(n)}

MPRF:

• g: [1]
• h: [1]
• i: [0]

Classical Approximation after Lifting Classical Results

All Bounds
Timebounds

Overall timebound:inf {Infinity}
g₀: 1 {O(1)}
g₂: inf {Infinity}
g₅: X₀+6 {O(n)}
g₇: inf {Infinity}
g₉: 1 {O(1)}
g₁₁: X₀+5 {O(n)}

Costbounds

Overall costbound: inf {Infinity}
g₀: inf {Infinity}
g₂: inf {Infinity}
g₅: inf {Infinity}
g₇: inf {Infinity}
g₉: inf {Infinity}
g₁₁: inf {Infinity}

Sizebounds

(g₀,g), X₀: X₀ {O(n)}
(g₀,g), X₁: X₁ {O(n)}
(g₂,h), X₀: 4⋅X₀+12 {O(n)}
(g₅,i), X₀: 2⋅X₀+6 {O(n)}
(g₇,g), X₀: 2⋅X₀+6 {O(n)}
(g₉,stop), X₀: 2⋅X₀+6 {O(n)}
(g₁₁,g), X₀: 2⋅X₀+6 {O(n)}

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

Run classical analysis on SCC: [stop]

Classical Approximation after Lifting Classical Results

All Bounds
Timebounds

Overall timebound:inf {Infinity}
g₀: 1 {O(1)}
g₂: inf {Infinity}
g₅: X₀+6 {O(n)}
g₇: inf {Infinity}
g₉: 1 {O(1)}
g₁₁: X₀+5 {O(n)}

Costbounds

Overall costbound: inf {Infinity}
g₀: inf {Infinity}
g₂: inf {Infinity}
g₅: inf {Infinity}
g₇: inf {Infinity}
g₉: inf {Infinity}
g₁₁: inf {Infinity}

Sizebounds

(g₀,g), X₀: X₀ {O(n)}
(g₀,g), X₁: X₁ {O(n)}
(g₂,h), X₀: 2⋅X₀+6 {O(n)}
(g₅,i), X₀: 2⋅X₀+6 {O(n)}
(g₇,g), X₀: 2⋅X₀+6 {O(n)}
(g₉,stop), X₀: 2⋅X₀+6 {O(n)}
(g₁₁,g), X₀: 2⋅X₀+6 {O(n)}

Run probabilistic analysis on SCC: [stop]

Results of Probabilistic Analysis

All Bounds

Timebounds

Overall timebound:inf {Infinity}
g₀: 1 {O(1)}
g₂: inf {Infinity}
g₅: X₀+6 {O(n)}
g₇: inf {Infinity}
g₉: 1 {O(1)}
g₁₁: X₀+5 {O(n)}

Costbounds

Overall costbound: inf {Infinity}
g₀: 0 {O(1)}
g₂: 0 {O(1)}
g₅: 0 {O(1)}
g₇: inf {Infinity}
g₉: 1 {O(1)}
g₁₁: X₀+5 {O(n)}

Sizebounds

(g₀,g), X₀: X₀ {O(n)}
(g₀,g), X₁: X₁ {O(n)}
(g₂,h), X₀: 2⋅X₀+6 {O(n)}
(g₅,i), X₀: 2⋅X₀+6 {O(n)}
(g₇,g), X₀: 2⋅X₀+6 {O(n)}
(g₉,stop), X₀: 2⋅X₀+6 {O(n)}
(g₁₁,g), X₀: 2⋅X₀+6 {O(n)}