Checking Two Structural Properties of Vector Addition Systems with States Florent Avellaneda & Rémi Morin Université d’Aix Marseille
6 décembre 2012
Florent Avellaneda
MOVEP 2012
1 / 19
VASS Definition A vector addition system with state (VASS) is a directed graph G = (Q, A, i) with : Q a finite set of nodes. A ⊆ Q × Zd × Q a finite set of edges labeled by vectors. An initial state ∈ Q × Nd .
(-1, 1)
l1 q2
(1, -1)
q1 l4 (-1, 0)
Florent Avellaneda
l2
(0, -1) l3
(5, 5)
MOVEP 2012
2 / 19
VASS Definition A vector addition system with state (VASS) is a directed graph G = (Q, A, i) with : Q a finite set of nodes. A ⊆ Q × Zd × Q a finite set of edges labeled by vectors. An initial state ∈ Q × Nd .
(-1, 1)
l1 q2
(1, -1)
q1 l4 (-1, 0)
Florent Avellaneda
l2
(0, -1) l3
(5, 5)
MOVEP 2012
2 / 19
VASS Definition A vector addition system with state (VASS) is a directed graph G = (Q, A, i) with : Q a finite set of nodes. A ⊆ Q × Zd × Q a finite set of edges labeled by vectors. An initial state ∈ Q × Nd .
(-1, 1)
l1 q2
(1, -1)
q1 l4 (-1, 0)
Florent Avellaneda
l2
(0, -1) l3
(5, 5)
MOVEP 2012
2 / 19
VASS Definition A vector addition system with state (VASS) is a directed graph G = (Q, A, i) with : Q a finite set of nodes. A ⊆ Q × Zd × Q a finite set of edges labeled by vectors. An initial state ∈ Q × Nd .
(-1, 1)
l1 q2
(1, -1)
q1 l4 (-1, 0)
Florent Avellaneda
l2
(0, -1) l3
(5, 5)
MOVEP 2012
2 / 19
VASS Definition A vector addition system with state (VASS) is a directed graph G = (Q, A, i) with : Q a finite set of nodes. A ⊆ Q × Zd × Q a finite set of edges labeled by vectors. An initial state ∈ Q × Nd .
(-1, 1)
l1 q2
(1, -1)
q1 l4 (-1, 0)
Florent Avellaneda
l2
(0, -1) l3
(5, 5)
MOVEP 2012
2 / 19
VASS Definition A vector addition system with state (VASS) is a directed graph G = (Q, A, i) with : Q a finite set of nodes. A ⊆ Q × Zd × Q a finite set of edges labeled by vectors. An initial state ∈ Q × Nd .
(-1, 1)
l1 q2
(1, -1)
q1 l4 (-1, 0)
Florent Avellaneda
l2
(0, -1) l3
(5, 5)
MOVEP 2012
2 / 19
VASS Definition A vector addition system with state (VASS) is a directed graph G = (Q, A, i) with : Q a finite set of nodes. A ⊆ Q × Zd × Q a finite set of edges labeled by vectors. An initial state ∈ Q × Nd .
(-1, 1)
l1 q2
(1, -1)
q1 l4 (-1, 0)
Florent Avellaneda
l2
(0, -1) l3
(5, 5)
MOVEP 2012
2 / 19
VASS Definition A vector addition system with state (VASS) is a directed graph G = (Q, A, i) with : Q a finite set of nodes. A ⊆ Q × Zd × Q a finite set of edges labeled by vectors. An initial state ∈ Q × Nd .
(-1, 1)
l1 q2
(1, -1)
q1 l4 (-1, 0)
Florent Avellaneda
l2
(0, -1) l3
(5, 5)
MOVEP 2012
2 / 19
VASS Definition A vector addition system with state (VASS) is a directed graph G = (Q, A, i) with : Q a finite set of nodes. A ⊆ Q × Zd × Q a finite set of edges labeled by vectors. An initial state ∈ Q × Nd .
(-1, 1)
l1 q2
(1, -1)
q1 l4 (-1, 0)
Florent Avellaneda
l2
(0, -1) l3
(5, 5)
MOVEP 2012
2 / 19
VASS Definition A vector addition system with state (VASS) is a directed graph G = (Q, A, i) with : Q a finite set of nodes. A ⊆ Q × Zd × Q a finite set of edges labeled by vectors. An initial state ∈ Q × Nd .
(-1, 1)
l1 q2
(1, -1)
q1 l4 (-1, 0)
Florent Avellaneda
l2
(0, -1) l3
(5, 5)
MOVEP 2012
2 / 19
Petri net
VASS are very close to Petri nets : A VASS can be simulated by a Petri net.
q2 a→b
b
a
l1 b→∅
l2
l3
q2
b→a
q1 l4
a→∅
∅→4a 3b
q1
Florent Avellaneda
MOVEP 2012
3 / 19
Petri net VASS are very close to Petri nets : A VASS can be simulated by a Petri net. A Petri net is a VASS with one state. 3a→4b 5c 3c→a 2b
a b→2c
b 3c→2a
Florent Avellaneda
MOVEP 2012
3 / 19
MSG
MSG (Message Sequence Graph) Automaton where each node is labeled by an MSC.
MSGs are a special case of VASSs.
a b c a b c a b c
a b c
Florent Avellaneda
VASSs are exponentially more concise than MSGs.
b c
MOVEP 2012
4 / 19
Example : simplified sliding window protocol
i j n.w j m→b j
i b→i w
i
j
i
j
i
j
i
j
...
i
j
i
j
i w→i b
j b→j m
...
n
Florent Avellaneda
MOVEP 2012
5 / 19
Termination
Carstensen 87 Termination of a given VASS is EXPSPACE-hard.
Florent Avellaneda
MOVEP 2012
6 / 19
Termination
Carstensen 87 Termination of a given VASS is EXPSPACE-hard. But, structural termination is polynomial problem.
Florent Avellaneda
MOVEP 2012
6 / 19
Property A VASS is structurally terminating if and only if there exists no closed path whose cost is ≥ ~0.
Florent Avellaneda
MOVEP 2012
7 / 19
Property A VASS is structurally terminating if and only if there exists no closed path whose cost is ≥ ~0. Problem solvable in polynomial time by linear programming.
Florent Avellaneda
MOVEP 2012
7 / 19
Minimum length of a closed path solution is potentially exponential.
(n, -n)
(-1, 1) l2
l1
l1 . . . l2 l2 l2 l2 . . . ⇒ l1 + n · l2 | {z } n times
The counter example is a multiset of edges.
Florent Avellaneda
MOVEP 2012
8 / 19
Challenge The counter example as multiset of edge is too complex.
Goal Find a simple representation of counter examples.
Solution Use a set of lassos.
Florent Avellaneda
MOVEP 2012
9 / 19
Definition Let Y0 be a circuit starting from node n. Let Y1 be a simple path from qin to n and Y2 be a simple path from n to qin . The closed path Y1 · (Y0 )k · Y2 is called a lasso with valuation k starting from qin .
k times
n
qin
Florent Avellaneda
MOVEP 2012
10 / 19
Result
Theorem Let H be a closed path given as a multiset of edges. We can compute in polynomial time a finite multiset of lassos S starting from b ∈ VH such that cost(S) = m × cost(H) with m ∈ N ∗ .
(-1, -1) (3, -1) l2
Florent Avellaneda
q2
q0 a1
a3 a2 (-1, -1)
(-1, -1) q1
l1 (-1, 2)
MOVEP 2012
11 / 19
Example
Theorem Let H be a closed path given as a multiset of edges. We can compute in polynomial time a finite multiset of lassos S starting from b ∈ VH such that cost(S) = m × cost(H) with m ∈ N ∗ .
x1 (-1, -1) (3, -1) l2 x3
Florent Avellaneda
q2
q0 a3
a1
(-1, -1) x1
a2 (-1, -1) x1
q1
l1 (-1, 2) x5
H = a1 + 5l1 + a2 + 3l2 + a3 S1 = a1 + 10l1 + a2 + a3 S2 = a1 + a2 + 6l2 + a3 S = S1 + S2 = 2 × cost(H)
MOVEP 2012
12 / 19
Example
Theorem Let H be a closed path given as a multiset of edges. We can compute in polynomial time a finite multiset of lassos S starting from b ∈ VH such that cost(S) = m × cost(H) with m ∈ N ∗ .
x1 (-1, -1) (3, -1) l2
Florent Avellaneda
q2
q0 a3
a1
(-1, -1) x1
a2 (-1, -1) x1
q1
l1 (-1, 2) x10
H = a1 + 5l1 + a2 + 3l2 + a3 S1 = a1 + 10l1 + a2 + a3 S2 = a1 + a2 + 6l2 + a3 S = S1 + S2 = 2 × cost(H)
MOVEP 2012
13 / 19
Example
Theorem Let H be a closed path given as a multiset of edges. We can compute in polynomial time a finite multiset of lassos S starting from b ∈ VH such that cost(S) = m × cost(H) with m ∈ N ∗ .
x1 (-1, -1) (3, -1) l2 x6
Florent Avellaneda
q2
q0 a3
a1
(-1, -1) x1
a2 (-1, -1) x1
q1
l1 (-1, 2)
H = a1 + 5l1 + a2 + 3l2 + a3 S1 = a1 + 10l1 + a2 + a3 S2 = a1 + a2 + 6l2 + a3 S = S1 + S2 = 2 × cost(H)
MOVEP 2012
14 / 19
Example
Theorem Let H be a closed path given as a multiset of edges. We can compute in polynomial time a finite multiset of lassos S starting from b ∈ VH such that cost(S) = m × cost(H) with m ∈ N ∗ .
(-1, -1) (3, -1) l2
Florent Avellaneda
q2
q0 a1
a3 a2 (-1, -1)
(-1, -1) q1
l1 (-1, 2)
H = a1 + 5l1 + a2 + 3l2 + a3 S1 = a1 + 10l1 + a2 + a3 S2 = a1 + a2 + 6l2 + a3 S = S1 + S2 cost(S) = 2 × cost(H)
MOVEP 2012
15 / 19
Theorem Let H be a closed path given as a multiset of edges. We can compute in polynomial time a finite multiset of lassos S starting from b ∈ VH such that cost(S) = m × cost(H) with m ∈ N ∗ . Idea :
b H
Florent Avellaneda
MOVEP 2012
16 / 19
Theorem Let H be a closed path given as a multiset of edges. We can compute in polynomial time a finite multiset of lassos S starting from b ∈ VH such that cost(S) = m × cost(H) with m ∈ N ∗ . Idea :
b L H
Florent Avellaneda
MOVEP 2012
16 / 19
Theorem Let H be a closed path given as a multiset of edges. We can compute in polynomial time a finite multiset of lassos S starting from b ∈ VH such that cost(S) = m × cost(H) with m ∈ N ∗ . Idea :
b L H
Florent Avellaneda
b'
MOVEP 2012
16 / 19
Theorem Let H be a closed path given as a multiset of edges. We can compute in polynomial time a finite multiset of lassos S starting from b ∈ VH such that cost(S) = m × cost(H) with m ∈ N ∗ . Idea :
b
L'
L H
Florent Avellaneda
b'
MOVEP 2012
16 / 19
Theorem Let H be a closed path given as a multiset of edges. We can compute in polynomial time a finite multiset of lassos S starting from b ∈ VH such that cost(S) = m × cost(H) with m ∈ N ∗ . Idea :
b
L'
H
Florent Avellaneda
MOVEP 2012
16 / 19
Theorem Let H be a closed path given as a multiset of edges. We can compute in polynomial time a finite multiset of lassos S starting from b ∈ VH such that cost(S) = m × cost(H) with m ∈ N ∗ . Idea :
b
L'
H
Florent Avellaneda
MOVEP 2012
16 / 19
Theorem Let H be a closed path given as a multiset of edges. We can compute in polynomial time a finite multiset of lassos S starting from b ∈ VH such that cost(S) = m × cost(H) with m ∈ N ∗ . Idea :
b
L'
3H
Florent Avellaneda
MOVEP 2012
16 / 19
Corollary Let H be a closed path with b ∈ VH . Then there exist a closed path H 0 with almost d lassos starting from b such that cost(H) = m × cost(H 0 ). Idea : x1 · cost(S1 ) + x2 · cost(S2 ) + ... + x|A| · cost(S|A| ) ≥ ~0 By linear programming, we can find d lassos such that xi1 · cost(Si1 ) + xi2 · cost(Si2 ) + ... + xid · cost(Sid ) ≥ ~0
Florent Avellaneda
MOVEP 2012
17 / 19
Conclusion
Structural termination of VASS is equivalent to search a positive closed path. Linear programming solve this problem in polynomial time, but with complex counter example. Polynomial algorithm to represent a counter example by d lassos.
Florent Avellaneda
MOVEP 2012
18 / 19
Thanks.
Florent Avellaneda
MOVEP 2012
19 / 19
