Memory-Efficient Reachability Graph Representations Using Sweep-Line Method
TACAS 2004
Obtaining Memory-Efficient
Reachability Graph Representations
Using the Sweep-Line Method
Thomas Mailund and Michael Westergaard
Department of Computer Science
University of Aarhus
Denmark
March 30, 2004
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Memory-Efficient Reachability Graph Representations Using Sweep-Line Method
TACAS 2004
Setting (1)
• We want to explicitely store the entire reachability
graph, but conserve space by storing each
state/node using as few bits as possible
• The most naїve representation of a state would
store a pair (n, m) as two independent numbers,
potentially wasting space (if n,m  {0, ... ,4}, we
would use 2 · ceil(log(5)) = 3 + 3 = 6 bits per state,
or even 32 + 32 = 64 bits or more, if we store them
as integers)
• A better approach is to enumerate the
syntactically possible states, and use only enough
bits to distinguish between them (ceil(log(5 · 5)) =
5 bits per state)
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Memory-Efficient Reachability Graph Representations Using Sweep-Line Method
TACAS 2004
Setting (2)
• The optimal solution is to enumerate the
reachable states only, and store each state using
the number of bits required to distinguish them
(in the example, if only 13 of the states were
actually reachable, we could store each state using
ceil(log(13)) = 4 bits only)
• Unfortunately we do not know the number of
reachable states until after we have constructed
the entire reachability graph. We try to address
this problem here
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Memory-Efficient Reachability Graph Representations Using Sweep-Line Method
TACAS 2004
Outline
• Condensed Representation
• The Sweep-Line Method
• On-the-fly Construction of the Condensed
Representation
• Experimental Results
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Memory-Efficient Reachability Graph Representations Using Sweep-Line Method
TACAS 2004
Example (1)
•
•
2 runners in a race with 2 laps
Either
t1) runner 1 finishes a lap before runner 2,
t2) runner 2 finishes a lap before runner 1, or
t3) both runners finish a lap at the same time
•
•
•
A runner can be at most 1 lap ahead of the other
runner
In the beginning neither of the runners have
finished any laps
The state of the system is a pair (n, m), where n is
the number of laps completed by runner 1 and m
is the number of laps completed by runner 2
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Memory-Efficient Reachability Graph Representations Using Sweep-Line Method
TACAS 2004
Example (2)
•
•
•
•
March 30, 2004
Only some (7) of the syntactically
possible states (9) are reachable
At least ceil(log(9))=4 bits are
used to store each state, although
ceil(log(7))=3 bits would suffice
In realistic examples, the number
of syntactically possible states is
much larger than the number of
reachable states, so distinguishing
only between reachable states
yields a good reduction
Alas, we first know the number of
reachable states, when we have
constructed the reachability graph
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Memory-Efficient Reachability Graph Representations Using Sweep-Line Method
TACAS 2004
Condensed Representation
• We assume that we can
Transition
enumerate the transitions
• Assign to each reachable
Destination
Numberstate
of transitions
a number, 0…R-1state
•
4
2
3
6
5
State
number
0
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Memory-Efficient Reachability Graph Representations Using Sweep-Line Method
TACAS 2004
Traversal of Condensed Representation
2
0
4
6
3
5
1
• We assume the existence of a
(partial) mapping next that for a state
and a transition gives the next state
(e.g. next((1,0), t3)=(2,1))
• We have not lost any information
with this reduction, so analysis is still
possible; for example a depth-first
traversal would look like:
DFS(0, sI)
proc DFS(i, m)
if (visited(i)) return
analyse(m)
for each (t, i’) in E[i]
DFS(i’, next(m, t))
end for
end proc
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Memory-Efficient Reachability Graph Representations Using Sweep-Line Method
TACAS 2004
The Sweep-Line Method
sweep-line
Not yet discovered state
• We assume a progress
measure, ψ, that assigns to
each state a progress
value, such
thatbuts->s’ =>
Discovered
unprocessed state
ψ(s)≤ψ(s’)
• Here, we take ψ(n,m)=n
• All states to be processed
are in front of the sweepline
• All new states are added in
front of Processed
the sweep-line
state
• We do not need the states
behind the sweep-line;
they can safely be removed
from memory
0
1
2
progress
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Memory-Efficient Reachability Graph Representations Using Sweep-Line Method
TACAS 2004
New header:
On-the-fly Construction
Number of bits used to
of the Condensod
Representation
represent
the successor states
sweep-line
4
State
number
6
Number of
transitions
2
3
0
1
Transition
Destination
state
5Now 3 bits are
required
progress
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Memory-Efficient Reachability Graph Representations Using Sweep-Line Method
TACAS 2004
Experimental Results
• A prototypical implementation of the described
method has been made in Design/CPN, a tool for
editing, simulating and analysing Coloured Petri
Nets
• The implementation does not take into account
the different sizes of the numbers, and encodes
everything in a machine word
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Memory-Efficient Reachability Graph Representations Using Sweep-Line Method
TACAS 2004
Experimental Results (Runner Example)
size/kb
r a t io
8
1.20
Full size
7
1.05
Condensed size
6
0.90
Ratio
5
0.75
4
0.60
3
0.45
2
0.30
1
0.15
0
0.00
100
0
March 30, 2004
20
40
La ps
60
80
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Memory-Efficient Reachability Graph Representations Using Sweep-Line Method
TACAS 2004
Experimental Results
(Stop and Wait Protocol)
ratio
2,00
size/mb
4,0
3,5
1,75
3,0
1,50
2,5
1,25
2,0
1,00
Full size
Condensed size
Size ratio
Time ratio
1,5
1,0
0,5
0,75
0,50
0,25
0,0
0,00
0
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20
40
60
Packets
80
100
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Memory-Efficient Reachability Graph Representations Using Sweep-Line Method
TACAS 2004
Experimental Results
(Dining Philosophers)
size/mb
4,0
3,5
ratio
4,0
3,5
Full size
Condensed size
Size ratio
Time ratio
3,0
2,5
2,0
3,0
2,5
2,0
1,5
1,5
1,0
1,0
0,5
0,5
0,0
0,0
0
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5
10
Philosophers
15
20
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Memory-Efficient Reachability Graph Representations Using Sweep-Line Method
TACAS 2004
Conclusion
• We have seen an efficient representation of
reachability graphs, and how this representation
can be traversed for analysis
• We have seen how the efficient representation can
be calculated efficiently using the sweep-line
method
• We have seen how the method performs on some
examples – basically, the method performs well
when the sweep-line method performs well, i.e.
for systems with a clear notion of progress
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Memory-Efficient Reachability Graph Representations Using Sweep-Line Method
TACAS 2004
Future Work
• Implementation of LTL/CTL checker
• Use on more realistic examples
• Use the described method with methods reducing
the reachability graph itself, e.g. symmetry
reduction
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Memory-Efficient Reachability Graph Representations Using Sweep-Line Method
TACAS 2004
Non-monotone Progress Measures
• If s->s’ => ψ(s)≤ψ(s’), we say that ψ is monotone
• If the progress measure we use is not monotone,
the algorithm still works, but the generated
reachability graph may be an unfolding of the real
reachability graph
• The unfolded reachability graph and the real
reachability graph are bisimilar (so CTL* is
preserved)
• This actually happens in the Dining Philosophers
example
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Memory-Efficient Reachability Graph Representations Using Sweep-Line Method
TACAS 2004
Non-monotone Progress Measures (2)
•
To the runner example,
add a transition:
t4) When both runners have
completed the race, start
over
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