Initial Problem
Start: l0
Program_Vars: X₀, X₁, X₂
Temp_Vars:
Locations: l0, l1
Transitions:
t₀: l0(X₀, X₁, X₂) → l1(X₀, X₁, X₂) :|: 1 ≤ X₀
t₁: l1(X₀, X₁, X₂) → l1(X₀, X₁+X₀, X₂) :|: X₁+1 ≤ X₀ ∧ X₀+1 ≤ X₂ ∧ 1 ≤ X₀
t₂: l1(X₀, X₁, X₂) → l1(X₀, X₁, X₁-X₀) :|: X₁+1 ≤ X₀ ∧ X₀+1 ≤ X₂ ∧ 1 ≤ X₀
Preprocessing
Found invariant 1 ≤ X₀ for location l1
Problem after Preprocessing
Start: l0
Program_Vars: X₀, X₁, X₂
Temp_Vars:
Locations: l0, l1
Transitions:
t₀: l0(X₀, X₁, X₂) → l1(X₀, X₁, X₂) :|: 1 ≤ X₀
t₁: l1(X₀, X₁, X₂) → l1(X₀, X₁+X₀, X₂) :|: X₁+1 ≤ X₀ ∧ X₀+1 ≤ X₂ ∧ 1 ≤ X₀ ∧ 1 ≤ X₀
t₂: l1(X₀, X₁, X₂) → l1(X₀, X₁, X₁-X₀) :|: X₁+1 ≤ X₀ ∧ X₀+1 ≤ X₂ ∧ 1 ≤ X₀ ∧ 1 ≤ X₀
Solv. Size Bound: t₁: l1→l1 for X₁
cycle: [t₁: l1→l1]
loop: (X₁+1 ≤ X₀ ∧ X₀+1 ≤ X₂ ∧ 1 ≤ X₀,(X₀,X₁) -> (X₀,X₁+X₀)
overappr. closed-form: X₀⋅n+X₁ {O(n^2)}
runtime bound: X₀+X₁+1 {O(n)}
Solv. Size Bound - Lifting for t₁: l1→l1 and X₁: 2⋅X₀⋅X₀+2⋅X₀⋅X₁+2⋅X₀+2⋅X₁ {O(n^2)}
Solv. Size Bound: t₂: l1→l1 for X₁
Solv. Size Bound: t₂: l1→l1 for X₂
cycle: [t₂: l1→l1]
loop: (X₁+1 ≤ X₀ ∧ X₀+1 ≤ X₂ ∧ 1 ≤ X₀,(X₀,X₁,X₂) -> (X₀,X₁,X₁-X₀)
overappr. closed-form: 2⋅X₀+2⋅X₁ {O(n)}
runtime bound: X₀+X₂+1 {O(n)}
Solv. Size Bound - Lifting for t₂: l1→l1 and X₂: 4⋅X₀⋅X₀+4⋅X₀⋅X₁+6⋅X₁+8⋅X₀ {O(n^2)}
MPRF for transition t₁: l1(X₀, X₁, X₂) → l1(X₀, X₁+X₀, X₂) :|: X₁+1 ≤ X₀ ∧ X₀+1 ≤ X₂ ∧ 1 ≤ X₀ ∧ 1 ≤ X₀ of depth 1:
new bound:
X₀+X₁ {O(n)}
MPRF for transition t₂: l1(X₀, X₁, X₂) → l1(X₀, X₁, X₁-X₀) :|: X₁+1 ≤ X₀ ∧ X₀+1 ≤ X₂ ∧ 1 ≤ X₀ ∧ 1 ≤ X₀ of depth 1:
new bound:
X₀+X₂ {O(n)}
All Bounds
Timebounds
Overall timebound:2⋅X₀+X₁+X₂+1 {O(n)}
t₀: 1 {O(1)}
t₁: X₀+X₁ {O(n)}
t₂: X₀+X₂ {O(n)}
Costbounds
Overall costbound: 2⋅X₀+X₁+X₂+1 {O(n)}
t₀: 1 {O(1)}
t₁: X₀+X₁ {O(n)}
t₂: X₀+X₂ {O(n)}
Sizebounds
t₀, X₀: X₀ {O(n)}
t₀, X₁: X₁ {O(n)}
t₀, X₂: X₂ {O(n)}
t₁, X₀: X₀ {O(n)}
t₁, X₁: 2⋅X₀⋅X₀+2⋅X₀⋅X₁+2⋅X₀+X₁ {O(n^2)}
t₁, X₂: X₂ {O(n)}
t₂, X₀: 2⋅X₀ {O(n)}
t₂, X₁: 2⋅X₀⋅X₀+2⋅X₀⋅X₁+2⋅X₀+2⋅X₁ {O(n^2)}
t₂, X₂: 2⋅X₀⋅X₀+2⋅X₀⋅X₁+2⋅X₁+4⋅X₀ {O(n^2)}