Preprocessing

Found invariant 1 ≤ X₃ ∧ 1 ≤ X₂+X₃ ∧ 1 ≤ X₁+X₃ ∧ 1 ≤ X₀+X₃ ∧ X₀ ≤ 2+X₃ ∧ X₂ ≤ X₁ ∧ 0 ≤ X₂ ∧ 0 ≤ X₁+X₂ ∧ 0 ≤ X₀+X₂ ∧ X₀ ≤ 3+X₂ ∧ 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ X₀ ≤ 2+X₁ ∧ X₀ ≤ 3 ∧ 0 ≤ X₀ for location h

Found invariant 0 ≤ X₂ for location g

Probabilistic Analysis

Probabilistic Program after Preprocessing

Start: f
Program_Vars: X₀, X₁, X₂, X₃
Temp_Vars:
Locations: f, g, h
Transitions:
g₁:f(X₀,X₁,X₂,X₃) -{0}> t₂:g(X₀,X₁,X₂,X₃) :|: 0 ≤ X₂
g₃:g(X₀,X₁,X₂,X₃) -{0}> t₄:h(0,X₂,X₂,X₃) :|: 1 ≤ X₃ ∧ 0 ≤ X₂
g₅:h(X₀,X₁,X₂,X₃) → [1/4]:t₆:h(X₀,1+X₁,X₂,X₃) :+: [3/4]:t₇:h(UNIFORM(1, 3),1+X₁,X₂,X₃) :|: X₀ ≤ X₁ ∧ X₀ ≤ 3 ∧ X₀ ≤ 3+X₂ ∧ X₀ ≤ 2+X₁ ∧ X₀ ≤ 2+X₃ ∧ 0 ≤ 1 ∧ 1 ≤ X₀+X₃ ∧ 1 ≤ X₁+X₃ ∧ 1 ≤ X₂+X₃ ∧ 1 ≤ X₃ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₀+X₂ ∧ 0 ≤ X₁ ∧ 0 ≤ X₁+X₂ ∧ X₂ ≤ X₁ ∧ 0 ≤ X₂
g₈:h(X₀,X₁,X₂,X₃) -{0}> t₉:g(X₀,X₁,X₂,X₃-1) :|: 1+X₁ ≤ X₀ ∧ X₀ ≤ 3 ∧ X₀ ≤ 3+X₂ ∧ X₀ ≤ 2+X₁ ∧ X₀ ≤ 2+X₃ ∧ 1 ≤ X₀+X₃ ∧ 1 ≤ X₁+X₃ ∧ 1 ≤ X₂+X₃ ∧ 1 ≤ X₃ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₀+X₂ ∧ 0 ≤ X₁ ∧ 0 ≤ X₁+X₂ ∧ X₂ ≤ X₁ ∧ 0 ≤ X₂

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}
g₈: inf {Infinity}

Costbounds

Overall costbound: inf {Infinity}
g₁: inf {Infinity}
g₃: inf {Infinity}
g₅: inf {Infinity}
g₈: inf {Infinity}

Sizebounds

(g₁,g), X₀: X₀ {O(n)}
(g₁,g), X₁: X₁ {O(n)}
(g₁,g), X₂: X₂ {O(n)}
(g₁,g), X₃: X₃ {O(n)}

Run probabilistic analysis on SCC: [f]

Run classical analysis on SCC: [g; h]

MPRF for transition t₄: g(X₀,X₁,X₂,X₃) -{0}> h(Temp_Int₉₈,X₂,X₂,X₃) :|: 0 ≤ X₂ ∧ 1 ≤ X₃ ∧ 0 ≤ X₂ ∧ Temp_Int₉₈ ≤ 0 ∧ 0 ≤ Temp_Int₉₈ ∧ 1 ≤ X₃ of depth 1:

new bound:

X₃ {O(n)}

MPRF:

• g: [X₃]
• h: [X₃-1]

MPRF for transition t₉: h(X₀,X₁,X₂,X₃) -{0}> g(X₀,X₁,X₂,X₃-1) :|: X₀ ≤ 3 ∧ X₀ ≤ 3+X₂ ∧ X₀ ≤ 2+X₁ ∧ X₀ ≤ 2+X₃ ∧ 1 ≤ X₀+X₃ ∧ 1 ≤ X₁+X₃ ∧ 1 ≤ X₂+X₃ ∧ 1 ≤ X₃ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₀+X₂ ∧ 0 ≤ X₁ ∧ 0 ≤ X₁+X₂ ∧ X₂ ≤ X₁ ∧ 0 ≤ X₂ ∧ 1+X₁ ≤ X₀ of depth 1:

new bound:

X₃ {O(n)}

MPRF:

• g: [X₃]
• h: [X₃]

Classical Approximation after Lifting Classical Results

All Bounds
Timebounds

Overall timebound:inf {Infinity}
g₁: 1 {O(1)}
g₃: X₃ {O(n)}
g₅: inf {Infinity}
g₈: X₃ {O(n)}

Costbounds

Overall costbound: inf {Infinity}
g₁: inf {Infinity}
g₃: inf {Infinity}
g₅: inf {Infinity}
g₈: inf {Infinity}

Sizebounds

(g₁,g), X₀: X₀ {O(n)}
(g₁,g), X₁: X₁ {O(n)}
(g₁,g), X₂: X₂ {O(n)}
(g₁,g), X₃: X₃ {O(n)}
(g₃,h), X₀: 0 {O(1)}
(g₃,h), X₁: X₂+2 {O(n)}
(g₃,h), X₂: X₂+2 {O(n)}
(g₃,h), X₃: X₃ {O(n)}
(g₅,h), X₀: 6 {O(1)}
(g₅,h), X₂: 2⋅X₂+4 {O(n)}
(g₅,h), X₃: 2⋅X₃ {O(n)}
(g₈,g), X₀: 3 {O(1)}
(g₈,g), X₁: 2 {O(1)}
(g₈,g), X₂: 2 {O(1)}
(g₈,g), X₃: X₃ {O(n)}

Run probabilistic analysis on SCC: [g; h]

Results of Probabilistic Analysis

All Bounds

Timebounds

Overall timebound:inf {Infinity}
g₁: 1 {O(1)}
g₃: X₃ {O(n)}
g₅: inf {Infinity}
g₈: X₃ {O(n)}

Costbounds

Overall costbound: inf {Infinity}
g₁: 0 {O(1)}
g₃: 0 {O(1)}
g₅: inf {Infinity}
g₈: 0 {O(1)}

Sizebounds

(g₁,g), X₀: X₀ {O(n)}
(g₁,g), X₁: X₁ {O(n)}
(g₁,g), X₂: X₂ {O(n)}
(g₁,g), X₃: X₃ {O(n)}
(g₃,h), X₀: 0 {O(1)}
(g₃,h), X₁: X₂+2 {O(n)}
(g₃,h), X₂: X₂ {O(n)}
(g₃,h), X₃: X₃ {O(n)}
(g₅,h), X₀: 6 {O(1)}
(g₅,h), X₂: X₂ {O(n)}
(g₅,h), X₃: X₃ {O(n)}
(g₈,g), X₀: 3 {O(1)}
(g₈,g), X₁: 2 {O(1)}
(g₈,g), X₂: 2 {O(1)}
(g₈,g), X₃: X₃ {O(n)}