Preprocessing

Found invariant 3+X₀ ≤ X₃ for location h

Probabilistic Analysis

Probabilistic Program after Preprocessing

Start: f
Program_Vars: X₀, X₁, X₂, X₃
Temp_Vars:
Locations: f, g, h
Transitions:
g₀:f(X₀,X₁,X₂,X₃) -{0}> t₁:g(X₀,X₁,X₂,X₃) :|:
g₂:g(X₀,X₁,X₂,X₃) -{0}> t₃:h(X₀,X₁,X₂,X₃) :|: 3+X₀ ≤ X₃
g₄:h(X₀,X₁,X₂,X₃) → [1/2]:t₅:g(X₀,1+X₁,X₂,X₃) :+: [1/2]:t₆:g(X₀,X₁,X₂,X₃) :|: 1+X₁ ≤ X₂ ∧ 3+X₀ ≤ X₃
g₇:h(X₀,X₁,X₂,X₃) → [1/4]:t₈:g(X₀,X₁,X₂,X₃) :+: [1/4]:t₉:g(1+X₀,X₁,X₂,X₃) :+: [1/4]:t₁₀:g(2+X₀,X₁,X₂,X₃) :+: [1/4]:t₁₁:g(3+X₀,X₁,X₂,X₃) :|: X₂ ≤ X₁ ∧ 3+X₀ ≤ X₃

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}
g₄: inf {Infinity}
g₇: inf {Infinity}

Costbounds

Overall costbound: inf {Infinity}
g₀: inf {Infinity}
g₂: inf {Infinity}
g₄: inf {Infinity}
g₇: inf {Infinity}

Sizebounds

(g₀,g), X₀: X₀ {O(n)}
(g₀,g), X₁: X₁ {O(n)}
(g₀,g), X₂: X₂ {O(n)}
(g₀,g), X₃: X₃ {O(n)}

Run probabilistic analysis on SCC: [f]

Run classical analysis on SCC: [g; h]

MPRF for transition t₅: h(X₀,X₁,X₂,X₃) → g(X₀,1+X₁,X₂,X₃) :|: 3+X₀ ≤ X₃ ∧ 1+X₁ ≤ X₂ of depth 1:

new bound:

X₁+X₂ {O(n)}

MPRF:

• g: [X₂-X₁]
• h: [X₂-X₁]

MPRF for transition t₉: h(X₀,X₁,X₂,X₃) → g(1+X₀,X₁,X₂,X₃) :|: 3+X₀ ≤ X₃ ∧ X₂ ≤ X₁ of depth 1:

new bound:

X₀+X₃+2 {O(n)}

MPRF:

• g: [X₃-2-X₀]
• h: [X₃-2-X₀]

MPRF for transition t₁₀: h(X₀,X₁,X₂,X₃) → g(2+X₀,X₁,X₂,X₃) :|: 3+X₀ ≤ X₃ ∧ X₂ ≤ X₁ of depth 1:

new bound:

X₀+X₃+2 {O(n)}

MPRF:

• g: [X₃-2-X₀]
• h: [X₃-2-X₀]

MPRF for transition t₁₁: h(X₀,X₁,X₂,X₃) → g(3+X₀,X₁,X₂,X₃) :|: 3+X₀ ≤ X₃ ∧ X₂ ≤ X₁ of depth 1:

new bound:

X₀+X₃+2 {O(n)}

MPRF:

• g: [X₃-2-X₀]
• h: [X₃-2-X₀]

Classical Approximation after Lifting Classical Results

All Bounds
Timebounds

Overall timebound:inf {Infinity}
g₀: 1 {O(1)}
g₂: inf {Infinity}
g₄: inf {Infinity}
g₇: inf {Infinity}

Costbounds

Overall costbound: inf {Infinity}
g₀: inf {Infinity}
g₂: inf {Infinity}
g₄: inf {Infinity}
g₇: inf {Infinity}

Sizebounds

(g₀,g), X₀: X₀ {O(n)}
(g₀,g), X₁: X₁ {O(n)}
(g₀,g), X₂: X₂ {O(n)}
(g₀,g), X₃: X₃ {O(n)}
(g₂,h), X₀: 6⋅X₃+7⋅X₀+12 {O(n)}
(g₂,h), X₁: 2⋅X₁+X₂ {O(n)}
(g₂,h), X₂: X₂ {O(n)}
(g₂,h), X₃: X₃ {O(n)}
(g₄,g), X₀: 12⋅X₃+14⋅X₀+24 {O(n)}
(g₄,g), X₁: 2⋅X₂+4⋅X₁ {O(n)}
(g₄,g), X₂: 2⋅X₂ {O(n)}
(g₄,g), X₃: 2⋅X₃ {O(n)}
(g₇,g), X₀: 24⋅X₃+28⋅X₀+48 {O(n)}
(g₇,g), X₁: 4⋅X₂+8⋅X₁ {O(n)}
(g₇,g), X₂: 4⋅X₂ {O(n)}
(g₇,g), X₃: 4⋅X₃ {O(n)}

Run probabilistic analysis on SCC: [g; h]

Plrf for transition g₇:h(X₀,X₁,X₂,X₃) → [1/4]:t₈:g(X₀,X₁,X₂,X₃) :+: [1/4]:t₉:g(1+X₀,X₁,X₂,X₃) :+: [1/4]:t₁₀:g(2+X₀,X₁,X₂,X₃) :+: [1/4]:t₁₁:g(3+X₀,X₁,X₂,X₃) :|: X₂ ≤ X₁ ∧ 3+X₀ ≤ X₃:

new bound:

2/3⋅X₀+2/3⋅X₃ {O(n)}

PLRF:

• g: 2/3⋅X₃-2/3⋅X₀
• h: 2/3⋅X₃-2/3⋅X₀

Use expected size bounds for entry point (g₀:f→[t₁:1:g],g)
Use expected size bounds for entry point (g₀:f→[t₁:1:g],g)
Use classical time bound for entry point (g₀:f→[t₁:1:g],g)

Results of Probabilistic Analysis

All Bounds

Timebounds

Overall timebound:inf {Infinity}
g₀: 1 {O(1)}
g₂: inf {Infinity}
g₄: inf {Infinity}
g₇: 2/3⋅X₀+2/3⋅X₃ {O(n)}

Costbounds

Overall costbound: inf {Infinity}
g₀: 0 {O(1)}
g₂: 0 {O(1)}
g₄: inf {Infinity}
g₇: 8/3⋅X₀+8/3⋅X₃ {O(n)}

Sizebounds

(g₀,g), X₀: X₀ {O(n)}
(g₀,g), X₁: X₁ {O(n)}
(g₀,g), X₂: X₂ {O(n)}
(g₀,g), X₃: X₃ {O(n)}
(g₂,h), X₀: 2⋅X₀+X₃ {O(n)}
(g₂,h), X₁: 2⋅X₁+X₂ {O(n)}
(g₂,h), X₂: X₂ {O(n)}
(g₂,h), X₃: X₃ {O(n)}
(g₄,g), X₀: 2⋅X₀+X₃ {O(n)}
(g₄,g), X₁: 2⋅X₂+4⋅X₁ {O(n)}
(g₄,g), X₂: X₂ {O(n)}
(g₄,g), X₃: X₃ {O(n)}
(g₇,g), X₀: 2⋅X₀+X₃ {O(n)}
(g₇,g), X₁: 4⋅X₂+8⋅X₁ {O(n)}
(g₇,g), X₂: X₂ {O(n)}
(g₇,g), X₃: X₃ {O(n)}