Preprocessing

Probabilistic Analysis

Probabilistic Program after Preprocessing

Start: f
Program_Vars: X₀, X₁
Temp_Vars:
Locations: f, g
Transitions:
g₀:f(X₀,X₁) -{0}> t₁:g(X₀,X₁) :|:
g₂:g(X₀,X₁) → [1/2]:t₃:g(X₀-1,X₁) :+: [1/2]:t₄:g(X₀,X₁-1) :|: 1 ≤ X₀ ∧ 1 ≤ 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}

Costbounds

Overall costbound: inf {Infinity}
g₀: inf {Infinity}
g₂: 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]

MPRF for transition t₃: g(X₀,X₁) → g(X₀-1,X₁) :|: 1 ≤ X₀ ∧ 1 ≤ X₁ of depth 1:

new bound:

X₀ {O(n)}

MPRF:

• g: [X₀]

MPRF for transition t₄: g(X₀,X₁) → g(X₀,X₁-1) :|: 1 ≤ X₀ ∧ 1 ≤ X₁ of depth 1:

new bound:

X₁ {O(n)}

MPRF:

• g: [X₁]

Classical Approximation after Lifting Classical Results

All Bounds
Timebounds

Overall timebound:X₀+X₁+1 {O(n)}
g₀: 1 {O(1)}
g₂: X₀+X₁ {O(n)}

Costbounds

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

Sizebounds

(g₀,g), X₀: X₀ {O(n)}
(g₀,g), X₁: X₁ {O(n)}
(g₂,g), X₀: 2⋅X₀ {O(n)}
(g₂,g), X₁: 2⋅X₁ {O(n)}

Run probabilistic analysis on SCC: [g]

Results of Probabilistic Analysis

All Bounds

Timebounds

Overall timebound:X₀+X₁+1 {O(n)}
g₀: 1 {O(1)}
g₂: X₀+X₁ {O(n)}

Costbounds

Overall costbound: 2⋅X₀+2⋅X₁ {O(n)}
g₀: 0 {O(1)}
g₂: 2⋅X₀+2⋅X₁ {O(n)}

Sizebounds

(g₀,g), X₀: X₀ {O(n)}
(g₀,g), X₁: X₁ {O(n)}
(g₂,g), X₀: X₀ {O(n)}
(g₂,g), X₁: X₁ {O(n)}