The Time-abstracting Bisimulation

1
The Time-abstracting Bisimulation
Equivalence ? on TA states
?
s1
s2
?1
s3
Preserve discrete state changes.
Abstract exact time delays.
2
The Time-abstracting Quotient Graph

• The quotient induced by the greatest
  time-abstracting
• bisimulation defined on the TA.

• Finite symbolic graph

- Nodes symbolic states (equivalence classes).
- Edges symbolic transitions (discrete and
time).

• Basic property pre-stability

?
a
?
a
s1
s2
s1
s2
Q1
Q2
Q1
Q2
3
Verification on the Quotient graphLinear-time
Every cycle in the quotient graph contains an
infinite run and vice versa.
Q1
Q4
Q3
Q2
s1
4
Verification on the Quotient graphBranching-time
If s1 ? s2, then for any TCTL formula ?, s1
satisfies ? iff s2 satisfies ?.
Due to determinism of time.
5
Regions Alternativ Definition
6
Problem with regions
Number of regions over n clocks
?
Explosion in number of clocks Explosion in
maximal constant
?
Reachability is PSPACE complete for a single TA
7
ZonesFrom infinite to finite
Symbolic state (set) (n, )
State (n, x3.2, y2.5 )
Zone conjunction of x-yltn, xltgtn
8
Symbolic Transitions
1ltxlt4 1ltylt3
y
delays to
n
x
xgt3
conjuncts to
a
y0
projects to
m
Thus (n,1ltxlt4,1ltylt3) a gt (m,3ltx, y0)
9
Fischers Protocolanalysis using zones
2

V
Criticial Section
X0
Xgt10
Xlt10
Init V1
V1
V1
A1
CS1
B1
Ylt10
Y0
Ygt10
V2
V2
CS2
B2
A2
10
Fischers cont.
X0
Xgt10
Xlt10
V1
V1
A1
CS1
B1
Ygt10
Ylt10
Y0
V2
V2
A2
CS2
B2
Untimed case
A1,A2,v1
A1,B2,v2
A1,CS2,v2
B1,CS2,v1
CS1,CS2,v1
11
Fischers cont.
X0
Xgt10
Xlt10
V1
V1
A1
CS1
B1
Ygt10
Ylt10
Y0
V2
V2
A2
CS2
B2
Untimed case
A1,A2,v1
A1,B2,v2
A1,CS2,v2
B1,CS2,v1
CS1,CS2,v1
Taking time into account
12
Fischers cont.
X0
Xgt10
Xlt10
V1
V1
A1
CS1
B1
Ygt10
Ylt10
Y0
V2
V2
A2
CS2
B2
Untimed case
A1,A2,v1
A1,B2,v2
A1,CS2,v2
B1,CS2,v1
CS1,CS2,v1
Taking time into account
Y
10
10
X
10
13
Fischers cont.
X0
Xgt10
Xlt10
V1
V1
A1
CS1
B1
Ygt10
Ylt10
Y0
V2
V2
A2
CS2
B2
Untimed case
A1,A2,v1
A1,B2,v2
A1,CS2,v2
B1,CS2,v1
CS1,CS2,v1
Taking time into account
Y
10
10
X
10
14
Fischers cont.
X0
Xgt10
Xlt10
V1
V1
A1
CS1
B1
Ygt10
Ylt10
Y0
V2
V2
A2
CS2
B2
Untimed case
A1,A2,v1
A1,B2,v2
A1,CS2,v2
B1,CS2,v1
CS1,CS2,v1
Taking time into account
Y
10
10
X
10
10
15
Fischers cont.
X0
Xgt10
Xlt10
V1
V1
A1
CS1
B1
Ygt10
Ylt10
Y0
V2
V2
A2
CS2
B2
Untimed case
A1,A2,v1
A1,B2,v2
A1,CS2,v2
B1,CS2,v1
CS1,CS2,v1
Taking time into account
Y
10
10
X
10
10
16
Forward Rechability
Init -gt Final ?
INITIAL Passed Ø Waiting
(n0,Z0) REPEAT - pick (n,Z) in Waiting
- if for some Z Z (n,Z) in Passed
then STOP - else (explore) add (m,U)
(n,Z) gt (m,U) to Waiting
Add (n,Z) to Passed UNTIL Waiting Ø
or Final is in Waiting
Final
Waiting
Init
Passed
17
Forward Rechability
Init -gt Final ?
INITIAL Passed Ø Waiting
(n0,Z0) REPEAT - pick (n,Z) in Waiting
- if for some Z Z (n,Z) in Passed
then STOP - else (explore) add (m,U)
(n,Z) gt (m,U) to Waiting
Add (n,Z) to Passed UNTIL Waiting Ø
or Final is in Waiting
Final
Waiting
n,Z
n,Z
Init
Passed
18
Forward Rechability
Init -gt Final ?
INITIAL Passed Ø Waiting
(n0,Z0) REPEAT - pick (n,Z) in Waiting
- if for some Z Z (n,Z) in Passed
then STOP - else /explore/ add (m,U)
(n,Z) gt (m,U) to Waiting
Add (n,Z) to Passed UNTIL Waiting Ø
or Final is in Waiting
Waiting
Final
m,U
n,Z
n,Z
Init
Passed
19
Forward Rechability
Init -gt Final ?
INITIAL Passed Ø Waiting
(n0,Z0) REPEAT - pick (n,Z) in Waiting
- if for some Z Z (n,Z) in Passed
then STOP - else /explore/ add (m,U)
(n,Z) gt (m,U) to Waiting
Add (n,Z) to Passed UNTIL Waiting Ø
or Final is in Waiting
Waiting
Final
m,U
n,Z
n,Z
Init
Passed
20
Canonical Dastructures for Zones Difference
Bounded Matrices
Bellman 1958, Dill 1989
Inclusion
x
1
2
xlt1 y-xlt2 z-ylt2 zlt9
D1
Graph
y
0
9
2
z
? ?
D2
xlt2 y-xlt3 ylt3 z-ylt3 zlt7
x
2
3
3
Graph
y
0
7
3
z
21
Canonical Dastructures for Zones Difference
Bounded Matrices
Bellman 1958, Dill 1989
Inclusion
x
x
1
2
xlt1 y-xlt2 z-ylt2 zlt9
1
2
Shortest Path Closure
D1
3
Graph
y
0
y
0
9
5
2
z
2
z
? ?
D2
x
xlt2 y-xlt3 ylt3 z-ylt3 zlt7
x
2
3
Shortest Path Closure
2
3
3
3
y
Graph
0
y
0
6
3
7
3
z
z
22
Canonical Dastructures for Zones Difference
Bounded Matrices
Bellman 1958, Dill 1989
Emptiness
x
1
xlt1 ygt5 y-xlt3
D
3
Graph
0
y
-5
Negative Cycle iff empty solution set
Compact
23
Canonical Dastructures for Zones Difference
Bounded Matrices
Future
y
y
Future D
D
x
x
1lt x lt4 1lt y lt3
1ltx, 1lty -2ltx-ylt3
x
4
4
x
x
Remove upper bounds on clocks
-1
Shortest Path Closure
-1
-1
3
3
0
0
0
3
3
2
2
y
-1
y
-1
y
-1
24
Canonical Dastructures for Zones Difference
Bounded Matrices
Reset
y
y
yD
D
x
x
1ltx, 1lty -2ltx-ylt3
y0, 1ltx
x
x
Remove all bounds involving y and set y to 0
-1
-1
3
0
0
0
2
y
-1
y
0
25
Improved DatastructuresCompact Datastructure for
Zones
RTSS 1997
-4
-4
x1-x2lt4 x2-x1lt10 x3-x1lt2 x2-x3lt2 x0-x1lt3 x3-x
0lt5
x1
x2
Shortest Path Closure O(n3)
x1
x2
4
10
2
3
3
2
3
-2
-2
2
2
x3
x0
x3
x0
1
5
5
26
Improved DatastructuresCompact Datastructure for
Zones
RTSS 1997
-4
-4
x1-x2lt4 x2-x1lt10 x3-x1lt2 x2-x3lt2 x0-x1lt3 x3-x
0lt5
x1
x2
Shortest Path Closure O(n3)
x1
x2
4
10
2
3
3
2
3
-2
-2
2
2
x3
x0
x3
x0
1
5
5
-4
Shortest Path Reduction O(n3)
x1
x2
Canonical wrt Space worst O(n2)
practice O(n)
3
2
3
2
x3
x0
27
Shortest Path Reduction1st attempt
Idea
An edge is REDUNDANT if there exists an
alternative path of no greater weight THUS
Remove all redundant edges!
ltw
w
Problem
v and w are both redundant Removal of one
depends on presence of other.
v
w
Observation If no zero- or negative cycles
then SAFE to remove all redundancies.
28
Over-approximation Convex Hull
y
5
3
1
x
1
3
5
Convex Hull
29
Hybrid Systems
30
Vending Machine 1
Timed Automata
31
Vending Machine 1
Behaviour
x
30
20
10
ord-cof
cup
del-cof
time
Timed Automata
32
Vending Machine 2
Clocks -gt Continuous Variables
Hybrid Automata
Maler, Manna, Pnueli91
33
Vending Machine 2
Clocks -gt Continuous Variables
Behaviour
T,H
100
50
cup
del-cof
ord-cof
t
Hybrid Automata
Maler, Manna, Pnueli91
34
Vending Machine 3
Linear Hybrid Automata
Alur, Courcouretis, Henzinger, Ho93
35
Vending Machine 3
Behaviour
T,H
100
50
cup
del-cof
ord-cof
t
HYTECH
Linear Hybrid Automata
Alur, Courcouretis, Henzinger, Ho93
36
Symbolic Analysis Polyhedra
T
H
37
Symbolic Analysis Polyhedra
T
H
38
Symbolic Analysis Polyhedra
T
H
39
Symbolic Analysis Polyhedra
T
The exploration may lead to generation of
infinitely many polyhedra gt No guarantee of
termination
?
H
?
Manipulation of polyhedra inefficient!
40
TAs versus LHAs

• TOOLS
• UPPAAL, KRONOS,CMC,...
• Decidable
• Efficient Datastructure
• DBMs, NDDs, CDDs, ..
• Expressiveness

• TOOLS
• HYTECH, POLLUX,..
• Undecidability
• Datastructures
• Plyhedra
• Expressiveness

?
?
?
?
?
?
?
?
STOPWATCH AUTOMATA
x0 or x1
41
STOPWATCH AUTOMATA
Cassez, Larsen, CONCUR00

• Extension of UPPAAL to SWA
• Reuse of efficient datastructures
• Overapproximation
• Every LHA may be translated into a SWA
• APPLICATIONS
• Scheduler
• Gasburner
• Water Level Control