Reachability Analysis via Net Structure

Reachability Analysis via Net
Structure
HARRO WIMMEL, KARSTEN WOLF
Universität Rostock, Institut für Informatik
8. Oktober 2010 c 2010 UNIVERSITÄT ROSTOCK | FAKULTÄT FÜR INFORMATIK & ELEKTROTECHNIK, INSTITUT FÜR INFORMATIK 1 / 13

Overview
Basic Deﬁnitions
Reachability Problem
State Equation & Constraints
Solving the Reachability Problem using CEGAR
The Search Space
Example
Looking for Constraints
Finding Partial Solutions
The Algorithm
Experimental Results

Basic Deﬁnitions
Reachability Problem
Petri nets should be well-known.
• (N, m, m ) is a reachability problem; answer “yes” if m[σ N m for some
ﬁring sequence σ ∈ T∗
• N = (S, T, F) Petri net, m, m ∈ NS
markings
• m = m + Cx is the state equation
• C incidence matrix, x ∈ NT
transition vector (solution)
• from m[σ m follows m = m + C℘(σ), i.e. the Parikh image ℘(σ) solves
the state equation
• necessary condition for reachability
• ℘(σ) = x is T-invariant if Cx = 0, i.e. m[σ m

Basic Deﬁnitions
State Equation & Constraints
• The solution space of the state equation m = m + Cx is semilinear
• ∃ ﬁnite B, P ⊆ NT
: m = m + Cx ⇐⇒ x = b + i ni pi for some
b ∈ B, pi ∈ P, ni ∈ N
• IP solver, e.g. lp_solve, yields “minimal” solution
• Discrimination of solutions by adding constraints (CEGAR)
• “jump”: t < n with t ∈ T, n ∈ N
• ”increment”: k
i=1 ni ti ≥ n with ti ∈ T, ni , n ∈ N
• jumps for other minimal solutions, increments for addition of T-invariants

The Search Space
b

An Example
s
a1
b1
x1
y1
c1
b2
z
y2
c2
b3
x2a2
f
Final marking: s + 3f
State Equation’s Solutions:
3a1 +3a2 +3

An Example
s
a1
b1
x1
y1
c1
b2
z
y2
c2
b3
x2a2
f
3× 3×
3×
3a1 +3a2 +3

An Example
s
a1
b1
x1
y1
c1
b2
z
y2
c2
b3
x2a2
f
3a1 +3a2 +3
Constraints:
b2 ≥ 1 (oder a1 < 3)

An Example
s
a1
b1
x1
y1
c1
b2
z
y2
c2
b3
x2a2
f
2× 2×
3×
1× 1× 1×
3a1 +3a2 +3
2a1 +2a2 +b1 +b2 +b3 +3
Constraints:
b2 ≥ 1 (oder a1 < 3)

An Example
s
a1
b1
x1
y1
c1
b2
z
y2
c2
b3
x2a2
f
3a1 +3a2 +3
2a1 +2a2 +b1 +b2 +b3 +3
Constraints:
b2 ≥ 1 (oder a1 < 3), c1 ≥ 1

An Example
s
a1
b1
x1
y1
c1
b2
z
y2
c2
b3
x2a2
f
2× 2×
3×
1× 1× 1×
1× 1×
3a1 +3a2 +3
2a1 +2a2 +b1 +b2 +b3 +3
2a1+2a2+b1+b2+b3+c1+c2+3
Constraints:
b2 ≥ 1 (oder a1 < 3), c1 ≥ 1

An Example
s
a1
b1
x1
y1
c1
b2
z
y2
c2
b3
x2a2
f
3a1 +3a2 +3
2a1 +2a2 +b1 +b2 +b3 +3
2a1+2a2+b1+b2+b3+c1+c2+3
Constraints:
b2 ≥ 1 (oder a1 < 3), c1 ≥ 1,
b2 ≥ 2 (oder a1 < 2)

An Example
s
a1
b1
x1
y1
c1
b2
z
y2
c2
b3
x2a2
f
1× 1×
3×
2× 2× 2×
1× 1×
3a1 +3a2 +3
2a1 +2a2 +b1 +b2 +b3 +3
2a1+2a2+b1+b2+b3+c1+c2+3
a1+a2+2b1+2b2+2b3+c1+c2+3
Constraints:
b2 ≥ 1 (oder a1 < 3), c1 ≥ 1,
b2 ≥ 2 (oder a1 < 2)

An Example
s
a1
b1
x1
y1
c1
b2
z
y2
c2
b3
x2a2
f
3a1 +3a2 +3
2a1 +2a2 +b1 +b2 +b3 +3
2a1+2a2+b1+b2+b3+c1+c2+3
a1+a2+2b1+2b2+2b3+c1+c2+3
Constraints:
b2 ≥ 1 (oder a1 < 3), c1 ≥ 1,
b2 ≥ 2 (oder a1 < 2),
b2 ≥ 3 (oder a1 < 1)

An Example
s
a1
b1
x1
y1
c1
b2
z
y2
c2
b3
x2a2
f3×
3× 3× 3×
1× 1×
3a1 +3a2 +3
2a1 +2a2 +b1 +b2 +b3 +3
2a1+2a2+b1+b2+b3+c1+c2+3
a1+a2+2b1+2b2+2b3+c1+c2+3
3b1 +3b2 +3b3 +c1 +c2 +3
Constraints:
b2 ≥ 1 (oder a1 < 3), c1 ≥ 1,
b2 ≥ 2 (oder a1 < 2),
b2 ≥ 3 (oder a1 < 1)

Building a graph
Take a ﬁring sequence σ and a solution x of the state equation m = m +Cx with
• ℘(σ) ≤ x,
• ∀t ∈ T: x(t) > ℘(σ)(t) =⇒ ¬m[σt
We call σ a partial solution. Now build a graph G of:
• transitions t with x(t) > ℘(σ)(t)
• places s inhibiting the ﬁring of such a t (after σ)
• an edge from s to t if s inhibits t
• an edge from t to s if t increases token count on s

Finding Components
Get all strongly connected components (SCC) of G which have no incoming edges
(source SCCs).
Places in such SCCs cannot be marked from “inside” the graph, so tokens must
come from the outside.
=⇒ Constraint use transitions that can put tokens onto a source SCC (left side of
the constraint).
How many tokens to produce? (right side of the constraint)
• a complex problem (esp. if x(t) − ℘(σ)(t) > 1 and nets have multiarcs)
• approximation necessary
• repeated increase of the constraints by 1 token is possible

Finding Partial Solutions
• Tree of all potential firing sequences for x from m = m + Cx
• tree is finite, brute-force search possible
• depth-first-search
• enumerate partial solutions and build constraints
• Optimisations
• stubborn-set method (partial order reduction)
• additional confluence tests for x(t) − ℘(σ)(t) > n
• backtracking at repeated markings on a path
• ineffective constraints (σ is partial solution for x + y with σ = σ or
℘(σ ) = ℘(σ) + y with y a T-invariant)

The Algorithm / Conclusion
• Get solution of the state equation using an IP solver
• Get partial solutions (maximal firing sequences), stop if full solution
• Find constraints for partial solutions
• (Multiple) calls to algorithm with state equation + constraints
Conclusion:
• Positive answer is found (use “jumps” for a complete search), except in case
of insufficient memory; witness path is found
• Negative answer can be found if state equation is infeasible or if backtracking
for ineffective constraints makes search space finite; diagnosis possible
• Extensions possible, e.g. state inequations

Experimental Results
Implementation in a tool named “Sara”.
• Garavel’s challenge (LOTOS speciﬁcation): 485 places, 776 transitions, test
for dead transitions
• (Cygwin/Linux) 26/41 sec. (LoLA: 71/29 sec. + separation by hand)
• path length (medium/max) 15/28 (LoLA: 53/6232)
• SAP reference nets (business processes): 590 nets, test for relaxed
soundness
• (Cygwin/Linux) 198/110 sec. (LoLA: 24 min. + 17 unsolved)
• Boolean programs: a few nets, coverability test
• <1 second (LoLA: 1 problem with memory overﬂow (>32GB))
• Spezialized nets with increasing edge weights (self-constructed)
• Sara loses time exponentially compared to LoLA (always <3 sec.)

M. Berkelaar, K. Eikland, P. Notebaert: Lp solve Reference Guide,
http://lpsolve.sourceforge.net/5.5/, 2010.
H. Garavel: Efﬁcient Petri Net tool for computing quasi-liveness,
http://www.informatik.uni-hamburg.de/cgi-bin/TGI/pnml/getpost
?id=2003/07/2709, 2003.
L.M. Kristensen, K. Schmidt, A. Valmari: Question-guided Stubborn Set Methods for
State Properties, Formal Methods in System Design 29:3, pp.215–251, Springer,
2006.
E. Mayr: An algorithm for the general Petri net reachability problem, SIAM Journal of
Computing 13:3, pp.441–460, 1984.
H. Wimmel: Sara – Structures for Automated Reachability Analysis,
http://www.informatik.uni-rostock.de/∼nl/wiki/tools/download,
2010.
K. Wolf: LoLA – A low level analyzer, http://www.informatik.uni-
rostock.de/∼nl/wiki/tools/lola, 2010.

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Reachability Analysis via Net Structure

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