Preprocessing
Eliminate variables [X₂] that do not contribute to the problem
Found invariant 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₀ for location h
Found invariant 0 ≤ X₁ for location g
Found invariant 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₀ for location i
Probabilistic Analysis
Probabilistic Program after Preprocessing
Start: f
Program_Vars: X₀, X₁, X₂
Temp_Vars:
Locations: f, g, h, i
Transitions:
g₀:f(X₀,X₁,X₂) -{0}> t₁:g(X₀,X₁,X₂) :|: 0 ≤ X₁
g₂:g(X₀,X₁,X₂) -{0}> [1/4]:t₃:h(X₀-1,X₁,X₂) :+: [3/4]:t₄:i(X₀-1,X₁,X₂) :|: 1 ≤ X₀ ∧ 0 ≤ X₁
g₅:h(X₀,X₁,X₂) → t₆:g(X₀,1+X₁,X₂) :|: 0 ≤ X₀ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₁
g₇:i(X₀,X₁,X₂) → t₈:i(X₀,X₁-1,X₂) :|: 1 ≤ X₁ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₁
g₉:i(X₀,X₁,X₂) → t₁₀:g(X₀,X₁,X₂) :|: X₁ ≤ 0 ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ 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: inf {Infinity}
Costbounds
Overall costbound: inf {Infinity}
g₀: inf {Infinity}
g₂: inf {Infinity}
g₅: inf {Infinity}
g₇: inf {Infinity}
g₉: inf {Infinity}
Sizebounds
(g₀,g), X₀: X₀ {O(n)}
(g₀,g), X₁: X₁ {O(n)}
(g₀,g), X₂: X₂ {O(n)}
Run probabilistic analysis on SCC: [f]
Run classical analysis on SCC: [g; h; i]
MPRF for transition t₃: g(X₀,X₁,X₂) -{0}> h(X₀-1,X₁,X₂) :|: 0 ≤ X₁ ∧ 1 ≤ X₀ of depth 1:
new bound:
X₀ {O(n)}
MPRF:
• g: [X₀]
• h: [X₀]
• i: [X₀]
MPRF for transition t₄: g(X₀,X₁,X₂) -{0}> i(X₀-1,X₁,X₂) :|: 0 ≤ X₁ ∧ 1 ≤ X₀ of depth 1:
new bound:
X₀ {O(n)}
MPRF:
• g: [X₀]
• h: [1+X₀]
• i: [X₀]
MPRF for transition t₆: h(X₀,X₁,X₂) → g(X₀,1+X₁,X₂) :|: 0 ≤ X₀ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₁ of depth 1:
new bound:
X₀ {O(n)}
MPRF:
• g: [X₀]
• h: [1+X₀]
• i: [X₀]
MPRF for transition t₈: i(X₀,X₁,X₂) → i(X₀,X₁-1,X₂) :|: 0 ≤ X₀ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₁ ∧ 1 ≤ X₁ of depth 1:
new bound:
X₀+X₁+1 {O(n)}
MPRF:
• g: [1+X₀+X₁]
• h: [2+X₀+X₁]
• i: [1+X₀+X₁]
MPRF for transition t₁₀: i(X₀,X₁,X₂) → g(X₀,X₁,X₂) :|: 0 ≤ X₀ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₁ ∧ X₁ ≤ 0 of depth 1:
new bound:
X₀ {O(n)}
MPRF:
• g: [X₀]
• h: [X₀]
• i: [1+X₀]
Classical Approximation after Lifting Classical Results
All Bounds
Timebounds
Overall timebound:5⋅X₀+X₁+2 {O(n)}
g₀: 1 {O(1)}
g₂: 2⋅X₀ {O(n)}
g₅: X₀ {O(n)}
g₇: X₀+X₁+1 {O(n)}
g₉: X₀ {O(n)}
Costbounds
Overall costbound: inf {Infinity}
g₀: inf {Infinity}
g₂: inf {Infinity}
g₅: inf {Infinity}
g₇: inf {Infinity}
g₉: inf {Infinity}
Sizebounds
(g₀,g), X₀: X₀ {O(n)}
(g₀,g), X₁: X₁ {O(n)}
(g₀,g), X₂: X₂ {O(n)}
(g₂,h), X₀: 2⋅X₀ {O(n)}
(g₂,h), X₁: 2⋅X₀+3⋅X₁ {O(n)}
(g₂,h), X₂: 2⋅X₂ {O(n)}
(g₂,i), X₀: 2⋅X₀ {O(n)}
(g₂,i), X₁: 2⋅X₀+3⋅X₁ {O(n)}
(g₂,i), X₂: 2⋅X₂ {O(n)}
(g₅,g), X₀: X₀ {O(n)}
(g₅,g), X₁: X₀+X₁ {O(n)}
(g₅,g), X₂: X₂ {O(n)}
(g₇,i), X₀: X₀ {O(n)}
(g₇,i), X₁: 2⋅X₁+X₀ {O(n)}
(g₇,i), X₂: X₂ {O(n)}
(g₉,g), X₀: X₀ {O(n)}
(g₉,g), X₁: 0 {O(1)}
(g₉,g), X₂: X₂ {O(n)}
Run probabilistic analysis on SCC: [g; h; i]
Results of Probabilistic Analysis
All Bounds
Timebounds
Overall timebound:5⋅X₀+X₁+2 {O(n)}
g₀: 1 {O(1)}
g₂: 2⋅X₀ {O(n)}
g₅: X₀ {O(n)}
g₇: X₀+X₁+1 {O(n)}
g₉: X₀ {O(n)}
Costbounds
Overall costbound: 3⋅X₀+X₁+1 {O(n)}
g₀: 0 {O(1)}
g₂: 0 {O(1)}
g₅: X₀ {O(n)}
g₇: X₀+X₁+1 {O(n)}
g₉: X₀ {O(n)}
Sizebounds
(g₀,g), X₀: X₀ {O(n)}
(g₀,g), X₁: X₁ {O(n)}
(g₀,g), X₂: X₂ {O(n)}
(g₂,h), X₀: X₀ {O(n)}
(g₂,h), X₁: X₀+X₁ {O(n)}
(g₂,h), X₂: X₂ {O(n)}
(g₂,i), X₀: X₀ {O(n)}
(g₂,i), X₁: X₀+X₁ {O(n)}
(g₂,i), X₂: X₂ {O(n)}
(g₅,g), X₀: X₀ {O(n)}
(g₅,g), X₁: X₀+X₁ {O(n)}
(g₅,g), X₂: X₂ {O(n)}
(g₇,i), X₀: X₀ {O(n)}
(g₇,i), X₁: X₀+X₁ {O(n)}
(g₇,i), X₂: X₂ {O(n)}
(g₉,g), X₀: X₀ {O(n)}
(g₉,g), X₁: 0 {O(1)}
(g₉,g), X₂: X₂ {O(n)}