Preprocessing

Found invariant X₀ ≤ 5 ∧ 0 ≤ X₀ for location g

Probabilistic Analysis

Probabilistic Program after Preprocessing

Start: f
Program_Vars: X₀
Temp_Vars:
Locations: f, g
Transitions:
g₀:f(X₀) -{0}> t₁:g(0) :|:
g₂:g(X₀) → t₃:g(1+X₀) :|: X₀ ≤ 4 ∧ 0 ≤ X₀ ∧ X₀ ≤ 0 ∧ X₀ ≤ 5
g₄:g(X₀) → [1/5]:t₅:g(1) :+: [4/5]:t₆:g(1+X₀) :|: X₀ ≤ 4 ∧ X₀ ≤ 1 ∧ 1 ≤ X₀ ∧ X₀ ≤ 5 ∧ 0 ≤ X₀
g₇:g(X₀) → [2/5]:t₈:g(2) :+: [3/5]:t₉:g(1+X₀) :|: X₀ ≤ 4 ∧ X₀ ≤ 2 ∧ 2 ≤ X₀ ∧ X₀ ≤ 5 ∧ 0 ≤ X₀
g₁₀:g(X₀) → [3/5]:t₁₁:g(3) :+: [2/5]:t₁₂:g(1+X₀) :|: X₀ ≤ 4 ∧ X₀ ≤ 3 ∧ 3 ≤ X₀ ∧ X₀ ≤ 5 ∧ 0 ≤ X₀
g₁₃:g(X₀) → [4/5]:t₁₄:g(4) :+: [1/5]:t₁₅:g(1+X₀) :|: X₀ ≤ 4 ∧ 4 ≤ X₀ ∧ X₀ ≤ 5 ∧ 0 ≤ X₀

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}
g₁₀: inf {Infinity}
g₁₃: inf {Infinity}

Costbounds

Overall costbound: inf {Infinity}
g₀: inf {Infinity}
g₂: inf {Infinity}
g₄: inf {Infinity}
g₇: inf {Infinity}
g₁₀: inf {Infinity}
g₁₃: inf {Infinity}

Sizebounds

(g₀,g), X₀: 0 {O(1)}

Run probabilistic analysis on SCC: [f]

Run classical analysis on SCC: [g]

knowledge_propagation leads to new time bound 1 {O(1)} for transition t₃: g(X₀) → g(1+X₀) :|: X₀ ≤ 5 ∧ X₀ ≤ 4 ∧ X₀ ≤ 0 ∧ 0 ≤ X₀

MPRF for transition t₆: g(X₀) → g(1+X₀) :|: X₀ ≤ 5 ∧ 0 ≤ X₀ ∧ X₀ ≤ 4 ∧ X₀ ≤ 1 ∧ 1 ≤ X₀ of depth 1:

new bound:

5 {O(1)}

MPRF:

• g: [5-X₀]

MPRF for transition t₉: g(X₀) → g(1+X₀) :|: X₀ ≤ 5 ∧ 0 ≤ X₀ ∧ X₀ ≤ 4 ∧ X₀ ≤ 2 ∧ 2 ≤ X₀ of depth 1:

new bound:

5 {O(1)}

MPRF:

• g: [5-X₀]

MPRF for transition t₁₂: g(X₀) → g(1+X₀) :|: X₀ ≤ 5 ∧ 0 ≤ X₀ ∧ X₀ ≤ 4 ∧ X₀ ≤ 3 ∧ 3 ≤ X₀ of depth 1:

new bound:

5 {O(1)}

MPRF:

• g: [5-X₀]

MPRF for transition t₁₅: g(X₀) → g(1+X₀) :|: X₀ ≤ 5 ∧ 0 ≤ X₀ ∧ X₀ ≤ 4 ∧ X₀ ≤ 4 ∧ 4 ≤ X₀ of depth 1:

new bound:

5 {O(1)}

MPRF:

• g: [5-X₀]

Classical Approximation after Lifting Classical Results

All Bounds
Timebounds

Overall timebound:inf {Infinity}
g₀: 1 {O(1)}
g₂: 1 {O(1)}
g₄: inf {Infinity}
g₇: inf {Infinity}
g₁₀: inf {Infinity}
g₁₃: inf {Infinity}

Costbounds

Overall costbound: inf {Infinity}
g₀: inf {Infinity}
g₂: inf {Infinity}
g₄: inf {Infinity}
g₇: inf {Infinity}
g₁₀: inf {Infinity}
g₁₃: inf {Infinity}

Sizebounds

(g₀,g), X₀: 0 {O(1)}
(g₂,g), X₀: 1 {O(1)}
(g₄,g), X₀: 3 {O(1)}
(g₇,g), X₀: 5 {O(1)}
(g₁₀,g), X₀: 7 {O(1)}
(g₁₃,g), X₀: 9 {O(1)}

Run probabilistic analysis on SCC: [g]

Plrf for transition g₄:g(X₀) → [1/5]:t₅:g(1) :+: [4/5]:t₆:g(1+X₀) :|: X₀ ≤ 4 ∧ X₀ ≤ 1 ∧ 1 ≤ X₀ ∧ X₀ ≤ 5 ∧ 0 ≤ X₀:

new bound:

25/4 {O(1)}

PLRF:

• g: 25/4-5/4⋅X₀

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)

Plrf for transition g₇:g(X₀) → [2/5]:t₈:g(2) :+: [3/5]:t₉:g(1+X₀) :|: X₀ ≤ 4 ∧ X₀ ≤ 2 ∧ 2 ≤ X₀ ∧ X₀ ≤ 5 ∧ 0 ≤ X₀:

new bound:

25/3 {O(1)}

PLRF:

• g: 25/3-5/3⋅X₀

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)

Plrf for transition g₁₀:g(X₀) → [3/5]:t₁₁:g(3) :+: [2/5]:t₁₂:g(1+X₀) :|: X₀ ≤ 4 ∧ X₀ ≤ 3 ∧ 3 ≤ X₀ ∧ X₀ ≤ 5 ∧ 0 ≤ X₀:

new bound:

25/2 {O(1)}

PLRF:

• g: 25/2-5/2⋅X₀

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)

Plrf for transition g₁₃:g(X₀) → [4/5]:t₁₄:g(4) :+: [1/5]:t₁₅:g(1+X₀) :|: X₀ ≤ 4 ∧ 4 ≤ X₀ ∧ X₀ ≤ 5 ∧ 0 ≤ X₀:

new bound:

25 {O(1)}

PLRF:

• g: 25-5⋅X₀

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:649/12 {O(1)}
g₀: 1 {O(1)}
g₂: 1 {O(1)}
g₄: 25/4 {O(1)}
g₇: 25/3 {O(1)}
g₁₀: 25/2 {O(1)}
g₁₃: 25 {O(1)}

Costbounds

Overall costbound: 631/6 {O(1)}
g₀: 0 {O(1)}
g₂: 1 {O(1)}
g₄: 25/2 {O(1)}
g₇: 50/3 {O(1)}
g₁₀: 25 {O(1)}
g₁₃: 50 {O(1)}

Sizebounds

(g₀,g), X₀: 0 {O(1)}
(g₂,g), X₀: 1 {O(1)}
(g₄,g), X₀: 3 {O(1)}
(g₇,g), X₀: 5 {O(1)}
(g₁₀,g), X₀: 7 {O(1)}
(g₁₃,g), X₀: 9 {O(1)}