On the optimal reachability problem in weighted timed automata and

Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
On the optimal reachability problem
in weighted timed automata and games
Patricia Bouyer-Decitre
LSV, CNRS & ENS Cachan, France
Based on former works with Thomas Brihaye, Kim G. Larsen, Nicolas Markey, etc...
And on recent work with Samy Jaziri and Nicolas Markey
1/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Time-dependent systems
We are interested in timed systems
... and in their analysis and control
2/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Time-dependent systems
We are interested in timed systems
... and in their analysis and control
2/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Time-dependent systems
We are interested in timed systems
... and in their analysis and control
2/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
An example: The task graph scheduling problem
Compute
P1
using two processors:
D×(C ×(A+B))+(A+B)+(C ×D)
(fast):
P2
+
2 picoseconds
+
5 picoseconds
3 picoseconds
×
7 picoseconds
idle
20 Watts
in use
90 Watts
in use
30 Watts
Sch1
Sch2
Sch3
P2
P1
P1
P2
10
T3
T1
T1
P2
×
×
20
T5
T4
T3
T2
T4
T5
25
12 p
ic
1.39 osecon
ds
nano
joule
s
T6
T2
T1
T6
13 p
ic
1.37 osecon
ds
nano
joule
s
T6
T4
T3
+
T5
15
T5
+
T4
D
10 Watt
5
T2
T3
energy
idle
T2
D
×
C
×
0
C
T1
time
energy
B
+
(slow):
time
P1
A
T6
19 p
ic
1.32 osecon
ds
nano
joule
s
[BFLM10] Bouyer, Fahrenberg, Larsen, Markey. Quantitative Analysis of Real-Time Systems using Priced Timed Automata.
3/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
An example: The task graph scheduling problem
Compute
P1
using two processors:
D×(C ×(A+B))+(A+B)+(C ×D)
(fast):
P2
+
2 picoseconds
+
5 picoseconds
3 picoseconds
×
7 picoseconds
idle
20 Watts
in use
90 Watts
in use
30 Watts
Sch1
Sch2
Sch3
P2
P1
P1
P2
10
T3
T1
T1
P2
×
×
20
T5
T4
T3
T2
T4
T5
25
12 p
ic
1.39 osecon
ds
nano
joule
s
T6
T2
T1
T6
13 p
ic
1.37 osecon
ds
nano
joule
s
T6
T4
T3
+
T5
15
T5
+
T4
D
10 Watt
5
T2
T3
energy
idle
T2
D
×
C
×
0
C
T1
time
energy
B
+
(slow):
time
P1
A
T6
19 p
ic
1.32 osecon
ds
nano
joule
s
[BFLM10] Bouyer, Fahrenberg, Larsen, Markey. Quantitative Analysis of Real-Time Systems using Priced Timed Automata.
3/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
An example: The task graph scheduling problem
Compute
P1
using two processors:
D×(C ×(A+B))+(A+B)+(C ×D)
(fast):
P2
+
2 picoseconds
+
5 picoseconds
3 picoseconds
×
7 picoseconds
idle
20 Watts
in use
90 Watts
in use
30 Watts
Sch1
Sch2
Sch3
P2
P1
P1
P2
10
T3
T1
T1
P2
×
×
20
T5
T4
T3
T2
T4
T5
25
12 p
ic
1.39 osecon
ds
nano
joule
s
T6
T2
T1
T6
13 p
ic
1.37 osecon
ds
nano
joule
s
T6
T4
T3
+
T5
15
T5
+
T4
D
10 Watt
5
T2
T3
energy
idle
T2
D
×
C
×
0
C
T1
time
energy
B
+
(slow):
time
P1
A
T6
19 p
ic
1.32 osecon
ds
nano
joule
s
[BFLM10] Bouyer, Fahrenberg, Larsen, Markey. Quantitative Analysis of Real-Time Systems using Priced Timed Automata.
3/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
An example: The task graph scheduling problem
Compute
P1
using two processors:
D×(C ×(A+B))+(A+B)+(C ×D)
(fast):
P2
+
2 picoseconds
+
5 picoseconds
3 picoseconds
×
7 picoseconds
idle
20 Watts
in use
90 Watts
in use
30 Watts
Sch1
Sch2
Sch3
P2
P1
P1
P2
10
T3
T1
T1
P2
×
×
20
T5
T4
T3
T2
T4
T5
25
12 p
ic
1.39 osecon
ds
nano
joule
s
T6
T2
T1
T6
13 p
ic
1.37 osecon
ds
nano
joule
s
T6
T4
T3
+
T5
15
T5
+
T4
D
10 Watt
5
T2
T3
energy
idle
T2
D
×
C
×
0
C
T1
time
energy
B
+
(slow):
time
P1
A
T6
19 p
ic
1.32 osecon
ds
nano
joule
s
[BFLM10] Bouyer, Fahrenberg, Larsen, Markey. Quantitative Analysis of Real-Time Systems using Priced Timed Automata.
3/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Outline
1
Timed automata
2
Weighted timed automata
3
Timed games
4
Weighted timed games
5
Conclusion
4/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
The model of timed automata
y
, 22≤
done
≤25
≤
,x
air
rep
problem,
safe
x:=0
15
0
y :=
alarm
repairing
repair
2≤y ∧x≤56
del
15≤
aye
x≤
16
d,
y :=
0
y :=0
failsafe
5/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
The model of timed automata
y
, 22≤
done
≤25
≤
,x
air
rep
problem,
x:=0
safe
15
0
y :=
alarm
aye
x≤
16
d,
y :=
0
23
−→
safe
problem
−−−−−→
repair
2≤y ∧x≤56
15≤
del
safe
repairing
alarm
15.6
−−→
y :=0
failsafe
alarm
delayed
−−−−−→
failsafe
x
0
23
0
15.6
15.6
y
0
23
23
38.6
0
failsafe
···
2.3
−−→
failsafe
repair
−−−−→
repairing
22.1
−−→
repairing
done
−−−→
···
safe
15.6
17.9
17.9
40
40
0
2.3
0
22.1
22.1
5/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Modelling the task graph scheduling problem
6/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Modelling the task graph scheduling problem
Processors
x=2
P1 :
+
(x≤2)
P2 :
+
(y ≤5)
done1
add1
x=3
idle
x:=0
done1
mult1
×
x:=0
(x≤3)
y =5
y =7
done2
add2
done2
mult2
x:=0
idle
x:=0
×
(y ≤7)
6/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Modelling the task graph scheduling problem
Processors
x=2
P1 :
done1
mult1
×
x:=0
x:=0
(x≤3)
y =5
y =7
+
done1
add1
(x≤2)
P2 :
+
(y ≤5)
Tasks
T4 :
x=3
done2
add2
x:=0
idle
idle
done2
mult2
x:=0
T5 :
t1 ∧t2
t4 :=1
addi
donei
addi
donei
t3
t5 :=1
×
(y ≤7)
A schedule is a path in the product automaton
6/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Analyzing timed automata
x≤2, x:=0
x=1
x=0 ∧
y ≥2
y :=0
y ≥2, y :=0
7/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Analyzing timed automata
y
2
x≤2, x:=0
x=1
x=0 ∧
y ≥2
1
y :=0
y ≥2, y :=0
x
0
1
2
7/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Analyzing timed automata
y
2
x≤2, x:=0
x=1
x=0 ∧
y ≥2
1
y :=0
y ≥2, y :=0
x
0
1
2
7/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Analyzing timed automata
y
2
x≤2, x:=0
x=1
x=0 ∧
y ≥2
1
y :=0
y ≥2, y :=0
x
0
1
2
7/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Analyzing timed automata
y
2
x≤2, x:=0
x=1
x=0 ∧
y ≥2
1
y :=0
y ≥2, y :=0
x
0
1
2
7/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Analyzing timed automata
y
2
x≤2, x:=0
x=1
x=0 ∧
y ≥2
1
y :=0
y ≥2, y :=0
x
0
1
2
7/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Analyzing timed automata
y
2
x≤2, x:=0
x=1
x=0 ∧
y ≥2
1
y :=0
y ≥2, y :=0
x
0
1
2
7/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Analyzing timed automata
y
2
x≤2, x:=0
x=1
x=0 ∧
y ≥2
1
y :=0
y ≥2, y :=0
x
0
1
2
7/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Analyzing timed automata
y
2
x≤2, x:=0
x=1
x=0 ∧
y ≥2
1
y :=0
y ≥2, y :=0
x
0
1
2
7/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Analyzing timed automata
y
2
x≤2, x:=0
x=1
x=0 ∧
y ≥2
1
y :=0
y ≥2, y :=0
x
0
1
2
7/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Analyzing timed automata
y
2
x≤2, x:=0
x=1
x=0 ∧
y ≥2
1
y :=0
y ≥2, y :=0
x
0
1
2
7/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Analyzing timed automata
y
2
x≤2, x:=0
x=1
x=0 ∧
y ≥2
1
y :=0
y ≥2, y :=0
x
0
1
2
7/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Analyzing timed automata
y
2
x≤2, x:=0
x=1
x=0 ∧
y ≥2
1
y :=0
y ≥2, y :=0
x
0
1
2
7/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Analyzing timed automata
y
2
x≤2, x:=0
x=1
x=0 ∧
y ≥2
1
y :=0
y ≥2, y :=0
x
0
1
2
Theorem [AD94]
Reachability in timed automata is decidable (as well as many other
important properties). It is PSPACE-complete.
Technical tool: region abstraction
Efficient symbolic technics based on zones, implemented in tools
7/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Analyzing timed automata
y
2
x≤2, x:=0
x=1
x=0 ∧
y ≥2
1
y :=0
y ≥2, y :=0
x
0
1
2
Theorem [AD94]
Reachability in timed automata is decidable (as well as many other
important properties). It is PSPACE-complete.
Technical tool: region abstraction
Efficient symbolic technics based on zones, implemented in tools
7/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Analyzing timed automata
y
2
x≤2, x:=0
x=1
x=0 ∧
y ≥2
1
y :=0
y ≥2, y :=0
x
0
1
2
Theorem [AD94]
Reachability in timed automata is decidable (as well as many other
important properties). It is PSPACE-complete.
Technical tool: region abstraction
Efficient symbolic technics based on zones, implemented in tools
Skip regions
7/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Technical tool: Region abstraction
clock y
clock x
“compatibility” between regions and constraints
“compatibility” between regions and time elapsing
8/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Technical tool: Region abstraction
clock y
only constraints: x ∼ c with c ∈ {0, 1, 2}
y ∼ c with c ∈ {0, 1, 2}
2
1
0
clock x
0
1
2
“compatibility” between regions and constraints
“compatibility” between regions and time elapsing
8/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Technical tool: Region abstraction
clock y
x=1
y =1
The path
- can be fired from
- cannot be fired from
2
1
0
clock x
0
1
2
“compatibility” between regions and constraints
“compatibility” between regions and time elapsing
8/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Technical tool: Region abstraction
clock y
2
1
0
clock x
0
1
2
“compatibility” between regions and constraints
“compatibility” between regions and time elapsing
8/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Technical tool: Region abstraction
clock y
2
1
0
clock x
0
1
2
“compatibility” between regions and constraints
“compatibility” between regions and time elapsing
; This is a finite time-abstract bisimulation!
8/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Time-abstract bisimulation
This is a relation between • and • such that:
9/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Time-abstract bisimulation
This is a relation between • and • such that:
∀
a
9/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Time-abstract bisimulation
This is a relation between • and • such that:
∀
∃
a
a
9/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Time-abstract bisimulation
This is a relation between • and • such that:
∀
∃
a
∀d ≥ 0
δ(d)
a
9/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Time-abstract bisimulation
This is a relation between • and • such that:
∀
∃
a
a
∀d ≥ 0
∃d 0 ≥ 0
δ(d)
δ(d 0 )
9/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Time-abstract bisimulation
This is a relation between • and • such that:
∀
∃
a
a
∀d ≥ 0
∃d 0 ≥ 0
δ(d)
δ(d 0 )
... and vice-versa (swap • and •).
9/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Time-abstract bisimulation
This is a relation between • and • such that:
a
∀
a
∃
δ(d)
∀d ≥ 0
δ(d 0 )
∃d 0 ≥ 0
... and vice-versa (swap • and •).
Consequence
∀
(`1 , v1 )
d1 ,a1
(`2 , v2 )
d2 ,a2
(`3 , v3 )
d3 ,a3
...
9/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Time-abstract bisimulation
This is a relation between • and • such that:
a
∀
a
∃
δ(d)
∀d ≥ 0
δ(d 0 )
∃d 0 ≥ 0
... and vice-versa (swap • and •).
Consequence
∀
(`1 , v1 )
(`1 , R1 )
d1 ,a1
a1
(`2 , v2 )
(`2 , R2 )
d2 ,a2
a2
(`3 , v3 )
(`3 , R3 )
d3 ,a3
...
a3
...
with vi ∈ Ri
9/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Time-abstract bisimulation
This is a relation between • and • such that:
a
∀
a
∃
δ(d)
∀d ≥ 0
δ(d 0 )
∃d 0 ≥ 0
... and vice-versa (swap • and •).
Consequence
∀
(`1 , v1 )
(`1 , R1 )
d1 ,a1
a1
(`2 , v2 )
(`2 , R2 )
d2 ,a2
a2
(`3 , v3 )
(`3 , R3 )
d3 ,a3
...
a3
...
with vi ∈ Ri
∀v10 ∈ R1
9/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Time-abstract bisimulation
This is a relation between • and • such that:
a
∀
a
∃
δ(d)
∀d ≥ 0
δ(d 0 )
∃d 0 ≥ 0
... and vice-versa (swap • and •).
Consequence
∀
(`1 , v1 )
(`1 , R1 )
∀v10 ∈ R1 ∃ (`1 , v10 )
d1 ,a1
a1
d10 ,a1
(`2 , v2 )
(`2 , R2 )
(`2 , v20 )
d2 ,a2
a2
d20 ,a2
(`3 , v3 )
(`3 , R3 )
(`3 , v30 )
d3 ,a3
...
a3
...
with vi ∈ Ri
...
with vi0 ∈ Ri
d30 ,a3
9/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Technical tool: Region abstraction – An example [AD94]
s2
y =1,b x<1,c x<1,c
x>0,a
s0
s1
s3
x>1,d
y :=0
y <1,a,y :=0
10/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Technical tool: Region abstraction – An example [AD94]
s2
y =1,b x<1,c x<1,c
x>0,a
s0
s1
s3
x>1,d
y :=0
y <1,a,y :=0
y
x
10/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Technical tool: Region abstraction – An example [AD94]
s2
y =1,b x<1,c x<1,c
x>0,a
s0
s1
s3
x>1,d
y :=0
y
y <1,a,y :=0
s0
x=y =0
b
a
a
a
x
s1
s1
0=y <x<1
c
a
s3
0<y <x<1
a
b
y =0,x=1
a
a
s1
b
y =0,x>1
s2
1=y <x
d
d
s3
d
d
0<y <1<x
d
s3
1=y <x
d
d
s3
x>1,y >1
d
10/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Outline
1
Timed automata
2
Weighted timed automata
3
Timed games
4
Weighted timed games
5
Conclusion
11/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Modelling resources in timed systems
System resources might be relevant and even crucial information
12/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Modelling resources in timed systems
System resources might be relevant and even crucial information
energy consumption,
memory usage,
price to pay,
bandwidth,
...
12/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Modelling resources in timed systems
System resources might be relevant and even crucial information
energy consumption,
memory usage,
price to pay,
bandwidth,
...
; timed automata are not powerful enough!
12/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Modelling resources in timed systems
System resources might be relevant and even crucial information
energy consumption,
memory usage,
price to pay,
bandwidth,
...
; timed automata are not powerful enough!
A possible solution: use hybrid automata
a discrete control (the mode of the system)
+ continuous evolution of the variables within a mode
12/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Modelling resources in timed systems
System resources might be relevant and even crucial information
energy consumption,
memory usage,
price to pay,
bandwidth,
...
; timed automata are not powerful enough!
A possible solution: use hybrid automata
The thermostat example
T ≤19
Off
On
Ṫ =−0.5T
Ṫ =2.25−0.5T
(T ≥18)
(T ≤22)
T ≥21
12/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Modelling resources in timed systems
System resources might be relevant and even crucial information
energy consumption,
price to pay,
memory usage,
bandwidth,
...
; timed automata are not powerful enough!
A possible solution: use hybrid automata
The thermostat example
T ≤19
Off
On
Ṫ =−0.5T
Ṫ =2.25−0.5T
(T ≥18)
(T ≤22)
T ≥21
22
21
19
18
2
4
6
8
10
time
12/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Ok...
13/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Ok...
Easy...
13/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Ok...
Easy...
13/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Ok...
Easy...
Easy...
13/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Ok... but?
Easy...
Easy...
constraint
constraint
13/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Ok... but?
Easy...
Easy...
constraint
constraint
Hard!
13/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Modelling resources in timed systems
System resources might be relevant and even crucial information
energy consumption,
memory usage,
price to pay,
bandwidth,
...
; timed automata are not powerful enough!
A possible solution: use hybrid automata
Theorem [HKPV95]
The reachability problem is undecidable in hybrid automata. Even for the
simplest, the so-called stopwatch automata (clocks can be stopped).
[HKPV95] Henzinger, Kopke, Puri, Varaiya. What’s decidable wbout hybrid automata? (SToC’95).
14/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Modelling resources in timed systems
System resources might be relevant and even crucial information
energy consumption,
memory usage,
price to pay,
bandwidth,
...
; timed automata are not powerful enough!
A possible solution: use hybrid automata
Theorem [HKPV95]
The reachability problem is undecidable in hybrid automata. Even for the
simplest, the so-called stopwatch automata (clocks can be stopped).
An alternative: weighted/priced timed automata [ALP01,BFH+01]
; hybrid variables do not constrain the system
hybrid variables are observer variables
[HKPV95] Henzinger, Kopke, Puri, Varaiya. What’s decidable wbout hybrid automata? (SToC’95).
[ALP01] Alur, La Torre, Pappas. Optimal paths in weighted timed automata (HSCC’01).
[BFH+01] Behrmann, Fehnker, Hune, Larsen, Pettersson, Romijn, Vaandrager. Minimum-cost reachability in priced timed automata (HSCC’01).
14/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Modelling the task graph scheduling problem
Processors
x=2
P1 :
done1
mult1
×
x:=0
x:=0
(x≤3)
y =5
y =7
+
done1
add1
(x≤2)
P2 :
+
(y ≤5)
Tasks
T4 :
x=3
done2
add2
x:=0
idle
idle
done2
mult2
x:=0
T5 :
t1 ∧t2
t4 :=1
addi
donei
addi
donei
t3
t5 :=1
×
(y ≤7)
15/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Modelling the task graph scheduling problem
Processors
x=2
P1 :
done1
mult1
×
x:=0
x:=0
(x≤3)
y =5
y =7
+
done1
add1
(x≤2)
P2 :
+
(y ≤5)
done2
add2
idle
idle
x:=0
+90
P2 :
+30
(y ≤5)
done1
add1
+10
x:=0
y =5
done2
add2
x:=0
done2
mult2
x:=0
Modelling energy
x=2
P1 :
(x≤2)
Tasks
T4 :
x=3
t4 :=1
addi
donei
addi
donei
t3
t5 :=1
×
(y ≤7)
x=3
done1
mult1
x:=0
y =7
+20
T5 :
t1 ∧t2
done2
mult2
x:=0
+90
(x≤3)
A good schedule is a path in the
product automaton with a low cost
+30
(y ≤7)
15/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Weighted/priced timed automata
[ALP01,BFH+01]
+10
`2
u
x =2
,c
u
,c
x =2
x≤2,c,y :=0
`0
`1
+5
(y =0)
`3
+1
+1
+7
,
[ALP01] Alur, La Torre, Pappas. Optimal paths in weighted timed automata (HSCC’01).
[BFH+01] Behrmann, Fehnker, Hune, Larsen, Pettersson, Romijn, Vaandrager. Minimum-cost reachability in priced timed automata (HSCC’01).
16/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Weighted/priced timed automata
[ALP01,BFH+01]
+10
`2
u
x =2
,c
u
,c
x =2
+1
x≤2,c,y :=0
`0
`1
+5
(y =0)
`3
+7
+1
x
y
`0
0
0
1.3
−−→
`0
1.3
1.3
c
−−→
`1
1.3
0
u
−−→
`3
1.3
0
0.7
−−−→
`3
2
0.7
c
−−→
,
,
[ALP01] Alur, La Torre, Pappas. Optimal paths in weighted timed automata (HSCC’01).
[BFH+01] Behrmann, Fehnker, Hune, Larsen, Pettersson, Romijn, Vaandrager. Minimum-cost reachability in priced timed automata (HSCC’01).
16/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Weighted/priced timed automata
[ALP01,BFH+01]
+10
`2
u
x =2
,c
u
,c
x =2
+1
x≤2,c,y :=0
`0
`1
+5
(y =0)
`3
+7
+1
x
y
`0
0
0
1.3
−−→
`0
1.3
1.3
c
−−→
`1
1.3
0
u
−−→
`3
1.3
0
0.7
−−−→
`3
2
0.7
c
−−→
,
,
cost :
[ALP01] Alur, La Torre, Pappas. Optimal paths in weighted timed automata (HSCC’01).
[BFH+01] Behrmann, Fehnker, Hune, Larsen, Pettersson, Romijn, Vaandrager. Minimum-cost reachability in priced timed automata (HSCC’01).
16/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Weighted/priced timed automata
[ALP01,BFH+01]
+10
`2
u
x =2
,c
u
,c
x =2
+1
x≤2,c,y :=0
`0
`1
+5
(y =0)
`3
+7
+1
x
y
cost :
`0
0
0
1.3
−−→
`0
1.3
1.3
c
−−→
`1
1.3
0
u
−−→
`3
1.3
0
0.7
−−−→
`3
2
0.7
c
−−→
,
,
6.5
[ALP01] Alur, La Torre, Pappas. Optimal paths in weighted timed automata (HSCC’01).
[BFH+01] Behrmann, Fehnker, Hune, Larsen, Pettersson, Romijn, Vaandrager. Minimum-cost reachability in priced timed automata (HSCC’01).
16/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Weighted/priced timed automata
[ALP01,BFH+01]
+10
`2
u
x =2
,c
u
,c
x =2
+1
x≤2,c,y :=0
`0
`1
+5
(y =0)
`3
+7
+1
x
y
cost :
`0
0
0
1.3
c
−−→
`0
1.3
1.3
−−→
6.5
+
0
`1
1.3
0
u
−−→
`3
1.3
0
0.7
−−−→
`3
2
0.7
c
−−→
,
,
[ALP01] Alur, La Torre, Pappas. Optimal paths in weighted timed automata (HSCC’01).
[BFH+01] Behrmann, Fehnker, Hune, Larsen, Pettersson, Romijn, Vaandrager. Minimum-cost reachability in priced timed automata (HSCC’01).
16/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Weighted/priced timed automata
[ALP01,BFH+01]
+10
`2
u
x =2
,c
u
,c
x =2
+1
x≤2,c,y :=0
`0
`1
+5
(y =0)
`3
+7
+1
x
y
cost :
`0
0
0
1.3
c
u
−−→
`0
1.3
1.3
−−→
`1
1.3
0
−−→
6.5
+
0
+
0
`3
1.3
0
0.7
−−−→
`3
2
0.7
c
−−→
,
,
[ALP01] Alur, La Torre, Pappas. Optimal paths in weighted timed automata (HSCC’01).
[BFH+01] Behrmann, Fehnker, Hune, Larsen, Pettersson, Romijn, Vaandrager. Minimum-cost reachability in priced timed automata (HSCC’01).
16/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Weighted/priced timed automata
[ALP01,BFH+01]
+10
`2
u
x =2
,c
u
,c
x =2
+1
x≤2,c,y :=0
`0
`1
+5
(y =0)
`3
+7
+1
x
y
cost :
`0
0
0
1.3
c
u
0.7
−−→
`0
1.3
1.3
−−→
`1
1.3
0
−−→
`3
1.3
0
−−−→
6.5
+
0
+
0
+
0.7
`3
2
0.7
c
−−→
,
,
[ALP01] Alur, La Torre, Pappas. Optimal paths in weighted timed automata (HSCC’01).
[BFH+01] Behrmann, Fehnker, Hune, Larsen, Pettersson, Romijn, Vaandrager. Minimum-cost reachability in priced timed automata (HSCC’01).
16/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Weighted/priced timed automata
[ALP01,BFH+01]
+10
`2
u
x =2
,c
u
,c
x =2
+1
x≤2,c,y :=0
`0
`1
+5
(y =0)
`3
+7
+1
x
y
cost :
`0
0
0
1.3
c
u
0.7
c
−−→
`0
1.3
1.3
−−→
`1
1.3
0
−−→
`3
1.3
0
−−−→
`3
2
0.7
−−→
6.5
+
0
+
0
+
0.7
+
7
,
,
[ALP01] Alur, La Torre, Pappas. Optimal paths in weighted timed automata (HSCC’01).
[BFH+01] Behrmann, Fehnker, Hune, Larsen, Pettersson, Romijn, Vaandrager. Minimum-cost reachability in priced timed automata (HSCC’01).
16/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Weighted/priced timed automata
[ALP01,BFH+01]
+10
`2
u
x =2
,c
u
,c
x =2
+1
x≤2,c,y :=0
`0
`1
+5
(y =0)
`3
+7
+1
x
y
cost :
`0
0
0
1.3
c
u
0.7
c
−−→
`0
1.3
1.3
−−→
`1
1.3
0
−−→
`3
1.3
0
−−−→
`3
2
0.7
−−→
6.5
+
0
+
0
+
0.7
+
7
,
,
= 14.2
[ALP01] Alur, La Torre, Pappas. Optimal paths in weighted timed automata (HSCC’01).
[BFH+01] Behrmann, Fehnker, Hune, Larsen, Pettersson, Romijn, Vaandrager. Minimum-cost reachability in priced timed automata (HSCC’01).
16/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Weighted/priced timed automata
[ALP01,BFH+01]
+10
`2
u
x =2
,c
u
,c
x =2
x≤2,c,y :=0
`0
`1
+5
(y =0)
`3
+1
Question: what is the optimal cost for reaching
+1
+7
,
,?
[ALP01] Alur, La Torre, Pappas. Optimal paths in weighted timed automata (HSCC’01).
[BFH+01] Behrmann, Fehnker, Hune, Larsen, Pettersson, Romijn, Vaandrager. Minimum-cost reachability in priced timed automata (HSCC’01).
16/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Weighted/priced timed automata
[ALP01,BFH+01]
+10
`2
u
x =2
,c
u
,c
x =2
x≤2,c,y :=0
`0
`1
+5
(y =0)
`3
+1
Question: what is the optimal cost for reaching
inf
0≤t≤2
+1
+7
,
,?
min ( 5t + 10(2 − t) + 1 , 5t + (2 − t) + 7 ) = 9
; strategy: leave immediately `0 , go to `3 , and wait there 2 t.u.
[ALP01] Alur, La Torre, Pappas. Optimal paths in weighted timed automata (HSCC’01).
[BFH+01] Behrmann, Fehnker, Hune, Larsen, Pettersson, Romijn, Vaandrager. Minimum-cost reachability in priced timed automata (HSCC’01).
16/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Weighted/priced timed automata
[ALP01,BFH+01]
+10
`2
u
x =2
,c
u
,c
x =2
x≤2,c,y :=0
`0
`1
+5
(y =0)
`3
+1
Question: what is the optimal cost for reaching
inf
0≤t≤2
+1
+7
,
,?
min ( 5t + 10(2 − t) + 1 , 5t + (2 − t) + 7 ) = 9
; strategy: leave immediately `0 , go to `3 , and wait there 2 t.u.
[ALP01] Alur, La Torre, Pappas. Optimal paths in weighted timed automata (HSCC’01).
[BFH+01] Behrmann, Fehnker, Hune, Larsen, Pettersson, Romijn, Vaandrager. Minimum-cost reachability in priced timed automata (HSCC’01).
16/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Weighted/priced timed automata
[ALP01,BFH+01]
+10
`2
u
x =2
,c
u
,c
x =2
x≤2,c,y :=0
`0
`1
+5
(y =0)
`3
+1
Question: what is the optimal cost for reaching
inf
0≤t≤2
+1
+7
,
,?
min ( 5t + 10(2 − t) + 1 , 5t + (2 − t) + 7 ) = 9
; strategy: leave immediately `0 , go to `3 , and wait there 2 t.u.
[ALP01] Alur, La Torre, Pappas. Optimal paths in weighted timed automata (HSCC’01).
[BFH+01] Behrmann, Fehnker, Hune, Larsen, Pettersson, Romijn, Vaandrager. Minimum-cost reachability in priced timed automata (HSCC’01).
16/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Weighted/priced timed automata
[ALP01,BFH+01]
+10
`2
u
x =2
,c
u
,c
x =2
x≤2,c,y :=0
`0
`1
+5
(y =0)
`3
+1
Question: what is the optimal cost for reaching
inf
0≤t≤2
+1
+7
,
,?
min ( 5t + 10(2 − t) + 1 , 5t + (2 − t) + 7 ) = 9
; strategy: leave immediately `0 , go to `3 , and wait there 2 t.u.
[ALP01] Alur, La Torre, Pappas. Optimal paths in weighted timed automata (HSCC’01).
[BFH+01] Behrmann, Fehnker, Hune, Larsen, Pettersson, Romijn, Vaandrager. Minimum-cost reachability in priced timed automata (HSCC’01).
16/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Weighted/priced timed automata
[ALP01,BFH+01]
+10
`2
u
x =2
,c
u
,c
x =2
x≤2,c,y :=0
`0
`1
+5
(y =0)
`3
+1
Question: what is the optimal cost for reaching
inf
0≤t≤2
+1
+7
,
,?
min ( 5t + 10(2 − t) + 1 , 5t + (2 − t) + 7 ) = 9
; strategy: leave immediately `0 , go to `3 , and wait there 2 t.u.
[ALP01] Alur, La Torre, Pappas. Optimal paths in weighted timed automata (HSCC’01).
[BFH+01] Behrmann, Fehnker, Hune, Larsen, Pettersson, Romijn, Vaandrager. Minimum-cost reachability in priced timed automata (HSCC’01).
16/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Optimal-cost reachability
Theorem [ALP01,BFH+01,BBBR07]
In weighted timed automata, the optimal cost is an integer and can be
computed in PSPACE.
Technical tool: a refinement of the regions, the corner-point
abstraction
0
3
0 0
7
0
0
7
3
[ALP01] Alur, La Torre, Pappas. Optimal paths in weighted timed automata (HSCC’01).
[BFH+01] Behrmann, Fehnker, Hune, Larsen, Pettersson, Romijn, Vaandrager. Minimum-cost reachability in priced timed automata (HSCC’01).
[BBBR07] Bouyer, Brihaye, Bruyère, Raskin. On the optimal reachability problem (Formal Methods in System Design).
17/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Technical tool: the corner-point abstraction
y
x
18/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Technical tool: the corner-point abstraction
y
x
18/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Technical tool: the corner-point abstraction
y
Abstract time successors:
x
18/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Technical tool: the corner-point abstraction
y
Abstract time successors:
x
Concrete time successors:
18/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Technical tool: the corner-point abstraction
y
Abstract time successors:
x
Concrete time successors:
18/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Technical tool: the corner-point abstraction
y
Abstract time successors:
x
Concrete time successors:
18/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Technical tool: the corner-point abstraction
y
Abstract time successors:
x
Concrete time successors:
18/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Technical tool: the corner-point abstraction
y
Abstract time successors:
x
Concrete time successors:
18/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Technical tool: the corner-point abstraction
Idea
only record information about the extremal points
Time elapsing
Discrete transition
18/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Technical tool: the corner-point abstraction
Idea
only record information about the extremal points
0
3
0
0
7
0
0
3
7
Cost rate 3
Discrete cost 7
18/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
From timed to discrete behaviours
Optimal reachability as a linear programming problem
t1
t2
t3
t4
t5
···
Lemma
Let Z be a bounded zone and f be a function
f : (T1 , ..., Tn ) 7→
n
X
c i Ti + c
i=1
well-defined on Z . Then infZ f is obtained on the border of Z with integer coordinates.
; for every finite path π in A, there exists a path Π in Acp such that
cost(Π) ≤ cost(π)
[Π is a “corner-point projection” of π]
19/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
From timed to discrete behaviours
Optimal reachability as a linear programming problem
t1
t2
t3
t4
t5
···
Lemma
Let Z be a bounded zone and f be a function
f : (T1 , ..., Tn ) 7→
n
X
c i Ti + c
i=1
well-defined on Z . Then infZ f is obtained on the border of Z with integer coordinates.
; for every finite path π in A, there exists a path Π in Acp such that
cost(Π) ≤ cost(π)
[Π is a “corner-point projection” of π]
19/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
From timed to discrete behaviours
Optimal reachability as a linear programming problem
t1
t2
t3
t4
t5
ß
···
t1 +t2 ≤2
x≤2
Lemma
Let Z be a bounded zone and f be a function
f : (T1 , ..., Tn ) 7→
n
X
c i Ti + c
i=1
well-defined on Z . Then infZ f is obtained on the border of Z with integer coordinates.
; for every finite path π in A, there exists a path Π in Acp such that
cost(Π) ≤ cost(π)
[Π is a “corner-point projection” of π]
19/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
From timed to discrete behaviours
Optimal reachability as a linear programming problem
t1
t2
y :=0
x≤2
t3
t4
t5
ß
···
y ≥5
t1 +t2 ≤2
t2 +t3 +t4 ≥5
Lemma
Let Z be a bounded zone and f be a function
f : (T1 , ..., Tn ) 7→
n
X
c i Ti + c
i=1
well-defined on Z . Then infZ f is obtained on the border of Z with integer coordinates.
; for every finite path π in A, there exists a path Π in Acp such that
cost(Π) ≤ cost(π)
[Π is a “corner-point projection” of π]
19/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
From timed to discrete behaviours
Optimal reachability as a linear programming problem
T1
T2
T3
T4
T5
t1
t2
t3
t4
t5
y :=0
x≤2
ß
···
y ≥5
t1 +t2 ≤2
T2 ≤2
t2 +t3 +t4 ≥5
T4 −T1 ≥5
Lemma
Let Z be a bounded zone and f be a function
f : (T1 , ..., Tn ) 7→
n
X
c i Ti + c
i=1
well-defined on Z . Then infZ f is obtained on the border of Z with integer coordinates.
; for every finite path π in A, there exists a path Π in Acp such that
cost(Π) ≤ cost(π)
[Π is a “corner-point projection” of π]
19/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
From timed to discrete behaviours
Optimal reachability as a linear programming problem
T1
T2
T3
T4
T5
t1
t2
t3
t4
t5
y :=0
x≤2
ß
···
y ≥5
t1 +t2 ≤2
T2 ≤2
t2 +t3 +t4 ≥5
T4 −T1 ≥5
Lemma
Let Z be a bounded zone and f be a function
f : (T1 , ..., Tn ) 7→
n
X
c i Ti + c
i=1
well-defined on Z . Then infZ f is obtained on the border of Z with integer coordinates.
; for every finite path π in A, there exists a path Π in Acp such that
cost(Π) ≤ cost(π)
[Π is a “corner-point projection” of π]
19/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
From timed to discrete behaviours
Optimal reachability as a linear programming problem
T1
T2
T3
T4
T5
t1
t2
t3
t4
t5
y :=0
x≤2
ß
···
y ≥5
t1 +t2 ≤2
T2 ≤2
t2 +t3 +t4 ≥5
T4 −T1 ≥5
Lemma
Let Z be a bounded zone and f be a function
f : (T1 , ..., Tn ) 7→
n
X
c i Ti + c
i=1
well-defined on Z . Then infZ f is obtained on the border of Z with integer coordinates.
; for every finite path π in A, there exists a path Π in Acp such that
cost(Π) ≤ cost(π)
[Π is a “corner-point projection” of π]
19/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
From discrete to timed behaviours
Approximation of abstract paths:
For any path Π of Acp , for any ε > 0, there exists a path πε of A s.t.
kΠ − πε k∞ < ε
For every η > 0, there exists ε > 0 s.t.
kΠ − πε k∞ < ε ⇒ |cost(Π) − cost(πε )| < η
20/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
From discrete to timed behaviours
Approximation of abstract paths:
For any path Π of Acp , for any ε > 0, there exists a path πε of A s.t.
kΠ − πε k∞ < ε
For every η > 0, there exists ε > 0 s.t.
kΠ − πε k∞ < ε ⇒ |cost(Π) − cost(πε )| < η
20/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
From discrete to timed behaviours
Approximation of abstract paths:
For any path Π of Acp , for any ε > 0, there exists a path πε of A s.t.
kΠ − πε k∞ < ε
For every η > 0, there exists ε > 0 s.t.
kΠ − πε k∞ < ε ⇒ |cost(Π) − cost(πε )| < η
20/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
From discrete to timed behaviours
Approximation of abstract paths:
For any path Π of Acp , for any ε > 0, there exists a path πε of A s.t.
kΠ − πε k∞ < ε
For every η > 0, there exists ε > 0 s.t.
kΠ − πε k∞ < ε ⇒ |cost(Π) − cost(πε )| < η
20/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Use of the corner-point abstraction
It is a very interesting abstraction, that can be used in several other
contexts:
for mean-cost optimization
for discounted-cost optimization
for all concavely-priced timed automata
for deciding frequency objectives
[BBL04,BBL08]
[FL08]
[JT08]
[BBBS11,Sta12]
...
[BBL04] Bouyer, Brinksma, Larsen. Staying Alive As Cheaply As Possible (HSCC’04).
[BBL08] Bouyer, Brinksma, Larsen. Optimal infinite scheduling for multi-priced timed automata (Formal Methods in System Designs).
[FL08] Fahrenberg, Larsen. Discount-optimal infinite runs in priced timed automata (INFINITY’08).
[JT08] Judziński, Trivedi. Concavely-priced timed automata (FORMATS’08).
[BBBS11] Bertrand, Bouyer, Brihaye, Stainer. Emptiness and universality problems in timed automata with positive frequency (ICALP’11).
[Sta12] Stainer. Frequencies in forgetful timed automata (FORMATS’12).
21/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Going further 1: mean-cost optimization
att?,x:=0
att!
x=D
z≥S
(x≤D)
Ċ =P
Ṙ=G
High
Low
x :=0
Ċ =p
Ṙ=g
z:=0
Op
att?
; compute optimal infinite schedules that minimize
mean-cost(π) = lim sup
n→+∞
cost(πn )
reward(πn )
[BBL08] Bouyer, Brinksma, Larsen. Optimal infinite scheduling for multi-priced timed automata (Formal Methods in System Designs).
22/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Going further 1: mean-cost optimization
att?,x:=0
att!
x=D
z≥S
(x≤D)
Ċ =P
Ṙ=G
High
Low
x :=0
Ċ =p
Ṙ=g
z:=0
Op
att?
; compute optimal infinite schedules that minimize
mean-cost(π) = lim sup
n→+∞
cost(πn )
reward(πn )
[BBL08] Bouyer, Brinksma, Larsen. Optimal infinite scheduling for multi-priced timed automata (Formal Methods in System Designs).
22/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Going further 1: mean-cost optimization
att?,x:=0
att!
x=D
z≥S
(x≤D)
Ċ =P
Ṙ=G
High
Low
x :=0
Ċ =p
Ṙ=g
z:=0
Op
att?
; compute optimal infinite schedules that minimize
mean-cost(π) = lim sup
n→+∞
M1 H
L
M2 H
L
O
Time
1
4
1
8
2
1
12 16
Schedule with ratio
≈1.455
cost(πn )
reward(πn )
M1 H
L
M2 H
L
O
Time
1
1
1
4
8
12 16
Schedule with ratio
1
≈1.478
[BBL08] Bouyer, Brinksma, Larsen. Optimal infinite scheduling for multi-priced timed automata (Formal Methods in System Designs).
22/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Going further 1: mean-cost optimization
att?,x:=0
att!
x=D
z≥S
(x≤D)
Ċ =P
Ṙ=G
High
Low
x :=0
Ċ =p
Ṙ=g
z:=0
Op
att?
; compute optimal infinite schedules that minimize
mean-cost(π) = lim sup
n→+∞
cost(πn )
reward(πn )
Theorem [BBL08]
In weighted timed automata, the optimal mean-cost can be compute in
PSPACE.
; the corner-point abstraction can be used
[BBL08] Bouyer, Brinksma, Larsen. Optimal infinite scheduling for multi-priced timed automata (Formal Methods in System Designs).
22/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
From timed to discrete behaviours
Finite behaviours: based on the following property
Lemma
Let Z be a bounded zone and f be a function
f : (t1 , ..., tn ) 7→
Pn
ci ti + c
Pi=1
n
rt
i=1 i i
+r
well-defined on Z . Then infZ f is obtained on the border of Z with integer coordinates.
; for every finite path π in A, there exists a path Π in Acp s.t.
mean-cost(Π) ≤ mean-cost(π)
Infinite behaviours: decompose each sufficiently long projection
into cycles:
The (acyclic) linear part will be negligible!
; the optimal cycle of Acp is better than any infinite path of A!
23/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
From timed to discrete behaviours
Finite behaviours: based on the following property
Lemma
Let Z be a bounded zone and f be a function
f : (t1 , ..., tn ) 7→
Pn
ci ti + c
Pi=1
n
rt
i=1 i i
+r
well-defined on Z . Then infZ f is obtained on the border of Z with integer coordinates.
; for every finite path π in A, there exists a path Π in Acp s.t.
mean-cost(Π) ≤ mean-cost(π)
Infinite behaviours: decompose each sufficiently long projection
into cycles:
The (acyclic) linear part will be negligible!
; the optimal cycle of Acp is better than any infinite path of A!
23/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
From timed to discrete behaviours
Finite behaviours: based on the following property
Lemma
Let Z be a bounded zone and f be a function
f : (t1 , ..., tn ) 7→
Pn
ci ti + c
Pi=1
n
rt
i=1 i i
+r
well-defined on Z . Then infZ f is obtained on the border of Z with integer coordinates.
; for every finite path π in A, there exists a path Π in Acp s.t.
mean-cost(Π) ≤ mean-cost(π)
Infinite behaviours: decompose each sufficiently long projection
into cycles:
The (acyclic) linear part will be negligible!
; the optimal cycle of Acp is better than any infinite path of A!
23/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
From timed to discrete behaviours
Finite behaviours: based on the following property
Lemma
Let Z be a bounded zone and f be a function
f : (t1 , ..., tn ) 7→
Pn
ci ti + c
Pi=1
n
rt
i=1 i i
+r
well-defined on Z . Then infZ f is obtained on the border of Z with integer coordinates.
; for every finite path π in A, there exists a path Π in Acp s.t.
mean-cost(Π) ≤ mean-cost(π)
Infinite behaviours: decompose each sufficiently long projection
into cycles:
The (acyclic) linear part will be negligible!
; the optimal cycle of Acp is better than any infinite path of A!
23/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
From discrete to timed behaviours
Approximation of abstract paths:
For any path Π of Acp , for any ε > 0, there exists a path πε of A s.t.
kΠ − πε k∞ < ε
For every η > 0, there exists ε > 0 s.t.
kΠ − πε k∞ < ε ⇒ |mean-cost(Π) − mean-cost(πε )| < η
24/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
From discrete to timed behaviours
Approximation of abstract paths:
For any path Π of Acp , for any ε > 0, there exists a path πε of A s.t.
kΠ − πε k∞ < ε
For every η > 0, there exists ε > 0 s.t.
kΠ − πε k∞ < ε ⇒ |mean-cost(Π) − mean-cost(πε )| < η
24/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
From discrete to timed behaviours
Approximation of abstract paths:
For any path Π of Acp , for any ε > 0, there exists a path πε of A s.t.
kΠ − πε k∞ < ε
For every η > 0, there exists ε > 0 s.t.
kΠ − πε k∞ < ε ⇒ |mean-cost(Π) − mean-cost(πε )| < η
24/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
From discrete to timed behaviours
Approximation of abstract paths:
For any path Π of Acp , for any ε > 0, there exists a path πε of A s.t.
kΠ − πε k∞ < ε
For every η > 0, there exists ε > 0 s.t.
kΠ − πε k∞ < ε ⇒ |mean-cost(Π) − mean-cost(πε )| < η
24/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Going further 2: concavely-priced cost functions
; A general abstract framework for quantitative timed systems
Theorem [JT08]
In concavely-priced timed automata, optimal cost is computable, if we
restrict to quasi-concave cost functions. For the following cost functions,
the (decision) problem is even PSPACE-complete:
optimal-time and optimal-cost reachability;
optimal discrete discounted cost;
optimal mean-cost.
; the corner-point abstraction can be used
[JT08] Judziński, Trivedi. Concavely-priced timed automata (FORMATS’08).
25/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Going further 3: discounted-time cost optimization
Globally,
(z≤8)
x=3,x:=0
(x≤3)
x=3
(x≤3)
deg
High
+2
deg
Med
+2
att
z≥2,x,z:=0
+1
Low
+5
+9
att
z≥2,z:=0
; compute optimal infinite schedules that minimize
[FL08] Fahrenberg, Larsen. Discount-optimal infinite runs in priced timed automata (INFINITY’08).
26/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Going further 3: discounted-time cost optimization
Globally,
(z≤8)
x=3,x:=0
(x≤3)
x=3
(x≤3)
deg
High
+2
deg
Med
+2
att
z≥2,x,z:=0
+1
Low
+5
+9
att
z≥2,z:=0
; compute optimal infinite schedules that minimize
discounted cost over time
[FL08] Fahrenberg, Larsen. Discount-optimal infinite runs in priced timed automata (INFINITY’08).
26/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Going further 3: discounted-time cost optimization
Globally,
(z≤8)
x=3,x:=0
(x≤3)
x=3
(x≤3)
deg
High
deg
Med
+2
+2
att
+1
z≥2,x,z:=0
Low
+5
+9
att
z≥2,z:=0
; compute optimal infinite schedules that minimize
Z τn+1
X
an+1
λt cost(`n ) dt+λTn+1 cost(`n −−→ `n+1 )
discounted-costλ (π) =
λ Tn
n≥0
τ1 ,a1
t=0
τ2 ,a2
if π = (`0 , v0 ) −−−→ (`1 , v1 ) −−−→ · · · and Tn =
P
i≤n τi
[FL08] Fahrenberg, Larsen. Discount-optimal infinite runs in priced timed automata (INFINITY’08).
26/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Going further 3: discounted-time cost optimization
Globally,
(z≤8)
x=3,x:=0
(x≤3)
x=3
(x≤3)
deg
High
+2
deg
Med
+2
att
z≥2,x,z:=0
+1
Low
+5
+9
att
z≥2,z:=0
; compute optimal infinite schedules that minimize
discounted cost over time
[FL08] Fahrenberg, Larsen. Discount-optimal infinite runs in priced timed automata (INFINITY’08).
26/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Going further 3: discounted-time cost optimization
Globally,
(z≤8)
x=3,x:=0
(x≤3)
x=3
(x≤3)
deg
High
+2
deg
Med
+2
att
z≥2,x,z:=0
+1
Low
+5
+9
att
z≥2,z:=0
; compute optimal infinite schedules that minimize
discounted cost over time
if λ = e −1 , the discounted cost of
that infinite schedule is ≈ 2.16
0
3
6 7
9
[FL08] Fahrenberg, Larsen. Discount-optimal infinite runs in priced timed automata (INFINITY’08).
26/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Going further 3: discounted-time cost optimization
Globally,
(z≤8)
x=3,x:=0
(x≤3)
x=3
(x≤3)
deg
High
deg
Med
+2
+2
att
z≥2,x,z:=0
+1
Low
+5
+9
att
z≥2,z:=0
; compute optimal infinite schedules that minimize
discounted cost over time
Theorem [FL08]
In weighted timed automata, the optimal discounted cost is computable
in EXPTIME.
; the corner-point abstraction can be used
[FL08] Fahrenberg, Larsen. Discount-optimal infinite runs in priced timed automata (INFINITY’08).
26/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Outline
1
Timed automata
2
Weighted timed automata
3
Timed games
4
Weighted timed games
5
Conclusion
27/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Modelling the task graph scheduling problem
Processors
x=2
P1 :
done1
mult1
×
x:=0
x:=0
(x≤3)
y =5
y =7
+
done1
add1
(x≤2)
P2 :
+
(y ≤5)
done2
add2
idle
idle
x:=0
+90
P2 :
+30
(y ≤5)
done1
add1
+10
x:=0
y =5
done2
add2
x:=0
done2
mult2
x:=0
Modelling energy
x=2
P1 :
(x≤2)
Tasks
T4 :
x=3
t4 :=1
addi
donei
addi
donei
t3
t5 :=1
×
(y ≤7)
x=3
done1
mult1
x:=0
y =7
+20
T5 :
t1 ∧t2
done2
mult2
x:=0
+90
(x≤3)
+30
(y ≤7)
28/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Modelling the task graph scheduling problem
Processors
x=2
P1 :
done1
mult1
×
x:=0
x:=0
(x≤3)
y =5
y =7
+
done1
add1
(x≤2)
P2 :
+
(y ≤5)
done2
add2
idle
idle
x:=0
+90
P2 :
+30
(y ≤5)
done1
add1
+10
x:=0
y =5
done2
add2
x:=0
done2
mult2
x:=0
Modelling energy
x=2
P1 :
(x≤2)
Tasks
T4 :
x=3
T5 :
t4 :=1
addi
donei
addi
donei
t3
t5 :=1
×
(y ≤7)
Modelling uncertainty
x=3
done1
mult1
x:=0
y =7
+20
t1 ∧t2
done2
mult2
x:=0
P1 :
+90
+
(x≤3)
(x≤2)
+30
P2 :
+
(y ≤7)
(x≤2)
x≥1
x≥1
done1
done1
add1
idle
x:=0
mult1
x:=0
y ≥3
y ≥2
done2
done2
add2
x:=0
idle
mult2
x:=0
×
(x≤3)
×
(x≤3)
28/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Modelling the task graph scheduling problem
Processors
x=2
P1 :
done1
mult1
×
x:=0
x:=0
(x≤3)
y =5
y =7
+
done1
add1
(x≤2)
P2 :
+
(y ≤5)
done2
add2
idle
idle
x:=0
+90
P2 :
+30
(y ≤5)
done1
add1
+10
x:=0
y =5
done2
add2
x:=0
done2
mult2
x:=0
Modelling energy
x=2
P1 :
(x≤2)
Tasks
T4 :
x=3
(y ≤7)
T5 :
t4 :=1
addi
donei
addi
donei
t3
t5 :=1
A (good) schedule is a strategy in
the product game (with a low cost)
Modelling uncertainty
x=3
done1
mult1
x:=0
y =7
+20
×
t1 ∧t2
done2
mult2
x:=0
P1 :
+90
+
(x≤3)
(x≤2)
+30
P2 :
+
(y ≤7)
(x≤2)
x≥1
x≥1
done1
done1
add1
idle
x:=0
mult1
x:=0
y ≥3
y ≥2
done2
done2
add2
x:=0
idle
mult2
x:=0
×
(x≤3)
×
(x≤3)
28/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
An example of a timed game
Rule of the game
(x≤2)
x≥1,u3
`0
x≤1,c1
x≥2,c4
x<1,u2 ,x:=0
`1
x<1,u1
`2
/
Aim: avoid
/ and reach ,
How do we play? According to a
strategy:
f : history 7→ (delay, cont. transition)
,
x≤1,c3
c2
`3
29/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
An example of a timed game
Rule of the game
(x≤2)
x≥1,u3
`0
x≤1,c1
x≥2,c4
x<1,u2 ,x:=0
`1
x<1,u1
`2
/
Aim: avoid
/ and reach ,
How do we play? According to a
strategy:
f : history 7→ (delay, cont. transition)
,
x≤1,c3
c2
`3
29/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
An example of a timed game
Rule of the game
(x≤2)
x≥1,u3
`0
x≤1,c1
x≥2,c4
x<1,u2 ,x:=0
`1
x<1,u1
`2
/
Aim: avoid
/ and reach ,
How do we play? According to a
strategy:
f : history 7→ (delay, cont. transition)
,
x≤1,c3
c2
`3
29/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
An example of a timed game
Rule of the game
(x≤2)
x≥1,u3
`0
x≤1,c1
x≥2,c4
x<1,u2 ,x:=0
`1
x<1,u1
`2
/
,
x≤1,c3
Aim: avoid
/ and reach ,
How do we play? According to a
strategy:
f : history 7→ (delay, cont. transition)
A (memoryless) winning strategy
from (`0 , 0), play (0.5, c1 )
c2
`3
29/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
An example of a timed game
Rule of the game
(x≤2)
x≥1,u3
`0
x≤1,c1
x≥2,c4
x<1,u2 ,x:=0
`1
x<1,u1
`2
c2
/
,
x≤1,c3
Aim: avoid
/ and reach ,
How do we play? According to a
strategy:
f : history 7→ (delay, cont. transition)
A (memoryless) winning strategy
from (`0 , 0), play (0.5, c1 )
; can be preempted by u2
`3
29/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
An example of a timed game
Rule of the game
(x≤2)
x≥1,u3
`0
x≤1,c1
x≥2,c4
x<1,u2 ,x:=0
`1
x<1,u1
`2
/
,
x≤1,c3
Aim: avoid
/ and reach ,
How do we play? According to a
strategy:
f : history 7→ (delay, cont. transition)
A (memoryless) winning strategy
from (`0 , 0), play (0.5, c1 )
; can be preempted by u2
from (`2 , ?), play (1 − ?, c2 )
c2
`3
29/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
An example of a timed game
Rule of the game
(x≤2)
x≥1,u3
`0
x≤1,c1
x≥2,c4
x<1,u2 ,x:=0
`1
x<1,u1
`2
/
,
x≤1,c3
Aim: avoid
/ and reach ,
How do we play? According to a
strategy:
f : history 7→ (delay, cont. transition)
A (memoryless) winning strategy
from (`0 , 0), play (0.5, c1 )
; can be preempted by u2
from (`2 , ?), play (1 − ?, c2 )
c2
from (`3 , 1), play (0, c3 )
`3
29/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
An example of a timed game
Rule of the game
(x≤2)
x≥1,u3
`0
x≤1,c1
x≥2,c4
x<1,u2 ,x:=0
`1
x<1,u1
`2
/
,
x≤1,c3
Aim: avoid
/ and reach ,
How do we play? According to a
strategy:
f : history 7→ (delay, cont. transition)
A (memoryless) winning strategy
from (`0 , 0), play (0.5, c1 )
; can be preempted by u2
from (`2 , ?), play (1 − ?, c2 )
c2
from (`3 , 1), play (0, c3 )
`3
from (`1 , 1), play (1, c4 )
29/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
An example of a timed game
Rule of the game
(x≤2)
x≥1,u3
`0
x≤1,c1
x≥2,c4
x<1,u2 ,x:=0
`1
x<1,u1
`2
/
Aim: avoid
/ and reach ,
How do we play? According to a
strategy:
f : history 7→ (delay, cont. transition)
,
x≤1,c3
Problems to be considered
c2
`3
29/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
An example of a timed game
Rule of the game
(x≤2)
x≥1,u3
`0
x≤1,c1
x≥2,c4
x<1,u2 ,x:=0
`1
x<1,u1
`2
/
Aim: avoid
/ and reach ,
How do we play? According to a
strategy:
f : history 7→ (delay, cont. transition)
,
x≤1,c3
Problems to be considered
Does there exist a winning strategy?
c2
`3
29/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
An example of a timed game
Rule of the game
(x≤2)
x≥1,u3
`0
x≤1,c1
x≥2,c4
x<1,u2 ,x:=0
`1
x<1,u1
`2
/
Aim: avoid
/ and reach ,
How do we play? According to a
strategy:
f : history 7→ (delay, cont. transition)
,
x≤1,c3
Problems to be considered
Does there exist a winning strategy?
c2
If yes, compute one (as simple as possible).
`3
29/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Decidability of timed games
Theorem [AMPS98,HK99]
Reachability and safety timed games are decidable and
EXPTIME-complete. Furthermore memoryless and “region-based”
strategies are sufficient.
; classical regions are sufficient for solving such problems
Theorem [AM99,BHPR07,JT07]
Optimal-time reachability timed games are decidable and
EXPTIME-complete.
[AMPS98] Asarin, Maler, Pnueli, Sifakis. Controller synthesis for timed automata (SSC’98).
[HK99] Henzinger, Kopke. Discrete-time control for rectangular hybrid automata (Theoretical Computer Science).
30/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Decidability of timed games
Theorem [AMPS98,HK99]
Reachability and safety timed games are decidable and
EXPTIME-complete. Furthermore memoryless and “region-based”
strategies are sufficient.
; classical regions are sufficient for solving such problems
Theorem [AM99,BHPR07,JT07]
Optimal-time reachability timed games are decidable and
EXPTIME-complete.
[AMPS98] Asarin, Maler, Pnueli, Sifakis. Controller synthesis for timed automata (SSC’98).
[HK99] Henzinger, Kopke. Discrete-time control for rectangular hybrid automata (Theoretical Computer Science).
30/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Decidability of timed games
Theorem [AMPS98,HK99]
Reachability and safety timed games are decidable and
EXPTIME-complete. Furthermore memoryless and “region-based”
strategies are sufficient.
; classical regions are sufficient for solving such problems
Theorem [AM99,BHPR07,JT07]
Optimal-time reachability timed games are decidable and
EXPTIME-complete.
[AM99] Asarin, Maler. As soon as possible: time optimal control for timed automata (HSCC’99).
[BHPR07] Brihaye, Henzinger, Prabhu, Raskin. Minimum-time reachability in timed games (ICALP’07).
[JT07] Jurdziński, Trivedi. Reachability-time games on timed automata (ICALP’07).
30/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Back to the example: computing winning states
(x≤2)
x≥1,u3
`0
x≤1,c1
x≥2,c4
x<1,u2 ,x:=0
`1
x<1,u1
`2
/
,
x≤1,c3
c2
`3
31/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Back to the example: computing winning states
(x≤2)
x≥1,u3
`0
x≤1,c1
x≥2,c4
x<1,u2 ,x:=0
`1
x<1,u1
`2
/
,
x≤1,c3
`0
0
1
2
3
0
1
2
3
0
1
2
3
0
1
2
3
`1
`2
c2
`3
`3
31/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Back to the example: computing winning states
Attrac1
(x≤2)
x≥1,u3
`0
x≤1,c1
x≥2,c4
x<1,u2 ,x:=0
`1
x<1,u1
`2
/
,
x≤1,c3
`0
0
1
2
3
0
1
2
3
0
1
2
3
0
1
2
3
`1
`2
c2
`3
`3
31/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Back to the example: computing winning states
(x≤2)
x≥1,u3
`0
x≤1,c1
x≥2,c4
x<1,u2 ,x:=0
`1
x<1,u1
`2
/
,
x≤1,c3
`0
0
1
2
3
0
1
2
3
0
1
2
3
0
1
2
3
`1
`2
c2
`3
`3
31/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Back to the example: computing winning states
(x≤2)
x≥1,u3
`0
x≤1,c1
x≥2,c4
x<1,u2 ,x:=0
`1
x<1,u1
`2
/
,
x≤1,c3
`0
0
1
2
3
0
1
2
3
0
1
2
3
0
1
2
3
`1
`2
c2
`3
`3
31/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Back to the example: computing winning states
(x≤2)
x≥1,u3
`0
x≤1,c1
x≥2,c4
x<1,u2 ,x:=0
`1
x<1,u1
`2
/
,
x≤1,c3
`0
0
1
2
3
0
1
2
3
0
1
2
3
0
1
2
3
`1
`2
c2
`3
`3
31/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Back to the example: computing winning states
(x≤2)
x≥1,u3
`0
x≤1,c1
x≥2,c4
x<1,u2 ,x:=0
`1
x<1,u1
`2
/
,
x≤1,c3
`0
0
1
2
3
0
1
2
3
0
1
2
3
0
1
2
3
`1
`2
c2
`3
`3
31/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Back to the example: computing winning states
Winning states
(x≤2)
x≥1,u3
`0
x≤1,c1
x≥2,c4
x<1,u2 ,x:=0
`1
x<1,u1
`2
/
,
x≤1,c3
Losing states
`0
0
1
2
3
0
1
2
3
0
1
2
3
0
1
2
3
`1
`2
c2
`3
`3
31/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Decidability via attractors
Skip attractors
a
a
Pred (X ) = {• | • −
→ • ∈ X}
controllable and uncontrollable discrete predecessors:
cPred(X ) =
S
Preda (X )
uPred(X ) =
a cont.
S
Preda (X )
a uncont.
time controllable predecessors:
•
delay t t.u.
•
32/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Decidability via attractors
a
Preda (X ) = {• | • −
→ • ∈ X}
controllable and uncontrollable discrete predecessors:
cPred(X ) =
S
Preda (X )
uPred(X ) =
a cont.
S
Preda (X )
a uncont.
time controllable predecessors:
•
delay t t.u.
•
32/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Decidability via attractors
a
Preda (X ) = {• | • −
→ • ∈ X}
controllable and uncontrollable discrete predecessors:
cPred(X ) =
S
Preda (X )
uPred(X ) =
a cont.
S
Preda (X )
a uncont.
time controllable predecessors:
•
delay t t.u.
•
32/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Decidability via attractors
a
Preda (X ) = {• | • −
→ • ∈ X}
controllable and uncontrollable discrete predecessors:
cPred(X ) =
S
Preda (X )
uPred(X ) =
a cont.
S
Preda (X )
a uncont.
time controllable predecessors:
•
delay t t.u.
•
•
should be safe
32/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Decidability via attractors
a
Preda (X ) = {• | • −
→ • ∈ X}
controllable and uncontrollable discrete predecessors:
cPred(X ) =
S
Preda (X )
uPred(X ) =
a cont.
S
Preda (X )
a uncont.
time controllable predecessors:
•
delay t t.u.
•
•
should be safe
δ(t)
Predδ (X , Safe) = {• | ∃t ≥ 0, • −−→ •
δ(t 0 )
and ∀0 ≤ t 0 ≤ t, • −−−→ • ∈ Safe}
32/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Timed games with a reachability objective
We write:
π(X ) = X ∪ Predδ (cPred(X ), ¬uPred(¬X ))
, in no more than 1 step is:
Attr1 (,) = π(,)
The states from which one can ensure
, in no more than 2 steps is:
Attr2 (,) = π(Attr1 (,))
The states from which one can ensure
...
The states from which one can ensure
Attrn (,)
=
, in no more than n steps is:
π(Attrn−1 (,))
33/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Timed games with a reachability objective
We write:
π(X ) = X ∪ Predδ (cPred(X ), ¬uPred(¬X ))
, in no more than 1 step is:
Attr1 (,) = π(,)
The states from which one can ensure
, in no more than 2 steps is:
Attr2 (,) = π(Attr1 (,))
The states from which one can ensure
...
The states from which one can ensure
Attrn (,)
=
, in no more than n steps is:
π(Attrn−1 (,))
33/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Timed games with a reachability objective
We write:
π(X ) = X ∪ Predδ (cPred(X ), ¬uPred(¬X ))
, in no more than 1 step is:
Attr1 (,) = π(,)
The states from which one can ensure
, in no more than 2 steps is:
Attr2 (,) = π(Attr1 (,))
The states from which one can ensure
...
The states from which one can ensure
Attrn (,)
=
, in no more than n steps is:
π(Attrn−1 (,))
33/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Timed games with a reachability objective
We write:
π(X ) = X ∪ Predδ (cPred(X ), ¬uPred(¬X ))
, in no more than 1 step is:
Attr1 (,) = π(,)
The states from which one can ensure
, in no more than 2 steps is:
Attr2 (,) = π(Attr1 (,))
The states from which one can ensure
...
The states from which one can ensure
Attrn (,)
=
, in no more than n steps is:
π(Attrn−1 (,))
33/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Timed games with a reachability objective
We write:
π(X ) = X ∪ Predδ (cPred(X ), ¬uPred(¬X ))
, in no more than 1 step is:
Attr1 (,) = π(,)
The states from which one can ensure
, in no more than 2 steps is:
Attr2 (,) = π(Attr1 (,))
The states from which one can ensure
...
The states from which one can ensure
Attrn (,)
=
, in no more than n steps is:
π(Attrn−1 (,))
33/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Timed games with a reachability objective
We write:
π(X ) = X ∪ Predδ (cPred(X ), ¬uPred(¬X ))
, in no more than 1 step is:
Attr1 (,) = π(,)
The states from which one can ensure
, in no more than 2 steps is:
Attr2 (,) = π(Attr1 (,))
The states from which one can ensure
...
The states from which one can ensure
Attrn (,)
=
=
, in no more than n steps is:
π(Attrn−1 (,))
π n (,)
33/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Stability w.r.t. regions
if X is a union of regions, then:
Preda (X ) is a union of regions,
and so are cPred(X ) and uPred(X ).
34/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Stability w.r.t. regions
if X is a union of regions, then:
Preda (X ) is a union of regions,
and so are cPred(X ) and uPred(X ).
Does π also preserve unions of regions?
34/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Stability w.r.t. regions
if X is a union of regions, then:
Preda (X ) is a union of regions,
and so are cPred(X ) and uPred(X ).
Does π also preserve unions of regions?
cPred(X )
34/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Stability w.r.t. regions
if X is a union of regions, then:
Preda (X ) is a union of regions,
and so are cPred(X ) and uPred(X ).
Does π also preserve unions of regions?
cPred(X )
uPred(¬X )
34/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Stability w.r.t. regions
if X is a union of regions, then:
Preda (X ) is a union of regions,
and so are cPred(X ) and uPred(X ).
Does π also preserve unions of regions?
cPred(X )
uPred(¬X )
Predδ (cPred(X ), ¬uPred(¬X ))
34/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Stability w.r.t. regions
if X is a union of regions, then:
Preda (X ) is a union of regions,
and so are cPred(X ) and uPred(X ).
Does π also preserve unions of regions? Yes!
cPred(X )
uPred(¬X )
Predδ (cPred(X ), ¬uPred(¬X ))
34/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Stability w.r.t. regions
if X is a union of regions, then:
Preda (X ) is a union of regions,
and so are cPred(X ) and uPred(X ).
Does π also preserve unions of regions? Yes!
cPred(X )
uPred(¬X )
Predδ (cPred(X ), ¬uPred(¬X ))
(but it generates non-convex unions of regions...)
34/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Stability w.r.t. regions
if X is a union of regions, then:
Preda (X ) is a union of regions,
and so are cPred(X ) and uPred(X ).
Does π also preserve unions of regions? Yes!
cPred(X )
uPred(¬X )
Predδ (cPred(X ), ¬uPred(¬X ))
(but it generates non-convex unions of regions...)
; the computation of π ∗ (,) terminates!
34/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Stability w.r.t. regions
if X is a union of regions, then:
Preda (X ) is a union of regions,
and so are cPred(X ) and uPred(X ).
Does π also preserve unions of regions? Yes!
cPred(X )
uPred(¬X )
Predδ (cPred(X ), ¬uPred(¬X ))
(but it generates non-convex unions of regions...)
; the computation of π ∗ (,) terminates!
... and is correct
34/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Timed games with a safety objective
We can use operator π
e defined by
π
e(X ) = Predδ (X ∩ cPred(X ), ¬uPred(¬X ))
instead of π, and compute π
e∗ (¬/)
It is also stable w.r.t. regions.
35/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Timed games with a safety objective
We can use operator π
e defined by
π
e(X ) = Predδ (X ∩ cPred(X ), ¬uPred(¬X ))
instead of π, and compute π
e∗ (¬/)
It is also stable w.r.t. regions.
35/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Outline
1
Timed automata
2
Weighted timed automata
3
Timed games
4
Weighted timed games
5
Conclusion
36/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
timed game
A simple
`2
u
x =2
,c
u
,c
x =2
x≤2,c,y :=0
`0
`1
(y =0)
`3
,
; strategy: wait in `0 , and when t = 43 , go to `1
37/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
A simple weighted timed game
+10
`2
u
x =2
,c
u
c
=2,
x≤2,c,y :=0
`0
`1
+5
(y =0)
`3
+1
x
+1
+7
,
; strategy: wait in `0 , and when t = 43 , go to `1
37/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
A simple weighted timed game
+10
`2
u
x =2
,c
u
c
=2,
x≤2,c,y :=0
`0
`1
+5
(y =0)
`3
+1
x
+1
+7
,
Question: what is the optimal cost we can ensure while reaching
,?
; strategy: wait in `0 , and when t = 43 , go to `1
37/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
A simple weighted timed game
+10
`2
u
x =2
,c
u
c
=2,
x≤2,c,y :=0
`0
`1
+5
(y =0)
`3
+1
x
+1
+7
,
Question: what is the optimal cost we can ensure while reaching
inf
0≤t≤2
,?
max ( 5t + 10(2 − t) + 1 , 5t + (2 − t) + 7 ) = 14 +
1
3
; strategy: wait in `0 , and when t = 43 , go to `1
37/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
A simple weighted timed game
+10
`2
u
x =2
,c
u
c
=2,
x≤2,c,y :=0
`0
`1
+5
(y =0)
`3
+1
x
+1
+7
,
Question: what is the optimal cost we can ensure while reaching
inf
0≤t≤2
,?
max ( 5t + 10(2 − t) + 1 , 5t + (2 − t) + 7 ) = 14 +
1
3
; strategy: wait in `0 , and when t = 43 , go to `1
37/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
A simple weighted timed game
+10
`2
u
x =2
,c
u
c
=2,
x≤2,c,y :=0
`0
`1
+5
(y =0)
`3
+1
x
+1
+7
,
Question: what is the optimal cost we can ensure while reaching
inf
0≤t≤2
,?
max ( 5t + 10(2 − t) + 1 , 5t + (2 − t) + 7 ) = 14 +
1
3
; strategy: wait in `0 , and when t = 43 , go to `1
37/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
A simple weighted timed game
+10
`2
u
x =2
,c
u
c
=2,
x≤2,c,y :=0
`0
`1
+5
(y =0)
`3
+1
x
+1
+7
,
Question: what is the optimal cost we can ensure while reaching
inf
0≤t≤2
,?
max ( 5t + 10(2 − t) + 1 , 5t + (2 − t) + 7 ) = 14 +
1
3
; strategy: wait in `0 , and when t = 43 , go to `1
37/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
A simple weighted timed game
+10
`2
u
x =2
,c
u
c
=2,
x≤2,c,y :=0
`0
`1
+5
(y =0)
`3
+1
x
+1
+7
,
Question: what is the optimal cost we can ensure while reaching
inf
0≤t≤2
,?
max ( 5t + 10(2 − t) + 1 , 5t + (2 − t) + 7 ) = 14 +
1
3
; strategy: wait in `0 , and when t = 43 , go to `1
37/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Optimal reachability in weighted timed games (1)
This topic has been fairly hot these last fifteen years...
[LMM02,ABM04,BCFL04,BBR05,BBM06,BLMR06,Rut11,HIM13,BGK+14]
[LMM02] La Torre, Mukhopadhyay, Murano. Optimal-reachability and control for acyclic weighted timed automata (TCS@02).
[ABM04] Alur, Bernardsky, Madhusudan. Optimal reachability in weighted timed games (ICALP’04).
[BCFL04] Bouyer, Cassez, Fleury, Larsen. Optimal strategies in priced timed game automata (FSTTCS’04).
[BBR05] Brihaye, Bruyère, Raskin. On optimal timed strategies (FORMATS’05).
[BBM06] Bouyer, Brihaye, Markey. Improved undecidability results on weighted timed automata (Information Processing Letters).
[BLMR06] Bouyer, Larsen, Markey, Rasmussen. Almost-optimal strategies in one-clock priced timed automata (FSTTCS’06).
[Rut11] Rutkowski. Two-player reachability-price games on single-clock timed automata (QAPL’11).
[HIM13] Hansen, Ibsen-Jensen, Miltersen. A faster algorithm for solving one-clock priced timed games (CONCUR’13).
[BGK+14] Brihaye, Geeraerts, Krishna, Manasa, Monmege, Trivedi. Adding Negative Prices to Priced Timed Games (CONCUR’14).
38/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Optimal reachability in weighted timed games (1)
This topic has been fairly hot these last fifteen years...
[LMM02,ABM04,BCFL04,BBR05,BBM06,BLMR06,Rut11,HIM13,BGK+14]
[LMM02]
Tree-like weighted timed games can be solved in 2EXPTIME.
38/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Optimal reachability in weighted timed games (1)
This topic has been fairly hot these last fifteen years...
[LMM02,ABM04,BCFL04,BBR05,BBM06,BLMR06,Rut11,HIM13,BGK+14]
[LMM02]
Tree-like weighted timed games can be solved in 2EXPTIME.
[ABM04,BCFL04]
Depth-k weighted timed games can be solved in EXPTIME. There is a
symbolic algorithm to solve weighted timed games with a strongly
non-Zeno cost.
38/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Optimal reachability in weighted timed games (2)
[BBR05,BBM06]
In weighted timed games, the optimal cost cannot be computed, as soon
as games have three clocks or more.
39/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Optimal reachability in weighted timed games (2)
[BBR05,BBM06]
In weighted timed games, the optimal cost cannot be computed, as soon
as games have three clocks or more.
[BLMR06,Rut11,HIM13,BGK+14]
Turn-based optimal timed games are decidable in EXPTIME (resp.
PTIME) when automata have a single clock (resp. with two rates). They
are PTIME-hard.
39/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
What is easier with a single clock?
Memoryless strategies can be non-optimal...
+2
(x≤1)
`0
x=1
x<1
x:=0
,
x>0
`1
+1
... but memoryless almost-optimal strategies will be sufficient.
Key: resetting the clock somehow resets the history...
By unfolding and removing one by one the locations,we can
synthesize memoryless almost-optimal winning strategies.
Rather involved proofs of correctness
40/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
What is easier with a single clock?
Memoryless strategies can be non-optimal...
+2
(x≤1)
`0
x=1
x<1
x:=0
,
x>0
`1
+1
... but memoryless almost-optimal strategies will be sufficient.
Key: resetting the clock somehow resets the history...
By unfolding and removing one by one the locations,we can
synthesize memoryless almost-optimal winning strategies.
Rather involved proofs of correctness
40/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
What is easier with a single clock?
Memoryless strategies can be non-optimal...
+2
(x≤1)
`0
x=1
x<1
x:=0
,
x>0
`1
+1
... but memoryless almost-optimal strategies will be sufficient.
Key: resetting the clock somehow resets the history...
By unfolding and removing one by one the locations,we can
synthesize memoryless almost-optimal winning strategies.
Rather involved proofs of correctness
40/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
What is easier with a single clock?
Memoryless strategies can be non-optimal...
+2
(x≤1)
`0
x=1
x<1
x:=0
,
x>0
`1
+1
... but memoryless almost-optimal strategies will be sufficient.
Key: resetting the clock somehow resets the history...
By unfolding and removing one by one the locations,we can
synthesize memoryless almost-optimal winning strategies.
Rather involved proofs of correctness
40/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
What is easier with a single clock?
Memoryless strategies can be non-optimal...
+2
(x≤1)
`0
x=1
x<1
x:=0
,
x>0
`1
+1
... but memoryless almost-optimal strategies will be sufficient.
Key: resetting the clock somehow resets the history...
By unfolding and removing one by one the locations,we can
synthesize memoryless almost-optimal winning strategies.
Rather involved proofs of correctness
40/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
41/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Computing the optimal cost: why is that hard?
Given two clocks x and y , we can check whether y = 2x.
42/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Computing the optimal cost: why is that hard?
Given two clocks x and y , we can check whether y = 2x.
Add− (x)
Add+ (x)
y =1,y :=0
y =1,y :=0
x=1,x:=0
z:=0
0
z=1,z:=0
1
The cost is increased by
y =1,y :=0
x=1,x:=0
z:=0
1
x0
y =1,y :=0
z=1,z:=0
0
The cost is increased by
1−x0
42/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Computing the optimal cost: why is that hard?
Given two clocks x and y , we can check whether y = 2x.
x=x0
y =y0
z :=
z :=
0
0
z=0
Add+ (x)
Add+ (x)
Add− (y )
+2
z=0
Add− (x)
Add− (x)
Add+ (y )
+1
,
,
42/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Computing the optimal cost: why is that hard?
Given two clocks x and y , we can check whether y = 2x.
x=x0
y =y0
z :=
z :=
0
0
z=0
Add+ (x)
Add+ (x)
Add− (y )
+2
z=0
Add− (x)
Add− (x)
Add+ (y )
+1
In
,, cost = 2x
0
+ (1 − y0 ) + 2
,
,
42/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Computing the optimal cost: why is that hard?
Given two clocks x and y , we can check whether y = 2x.
x=x0
y =y0
z :=
z :=
0
0
z=0
Add+ (x)
Add+ (x)
Add− (y )
+2
z=0
Add− (x)
Add− (x)
Add+ (y )
+1
,, cost = 2x + (1 − y ) + 2
In ,, cost = 2(1 − x ) + y + 1
In
0
0
0
,
,
0
42/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Computing the optimal cost: why is that hard?
Given two clocks x and y , we can check whether y = 2x.
x=x0
y =y0
z :=
z :=
0
0
z=0
Add+ (x)
Add+ (x)
Add− (y )
+2
z=0
Add− (x)
Add− (x)
Add+ (y )
+1
,, cost = 2x + (1 − y ) + 2
In ,, cost = 2(1 − x ) + y + 1
In
0
0
0
,
,
0
if y0 < 2x0 , player 2 chooses the first branch: cost > 3
42/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Computing the optimal cost: why is that hard?
Given two clocks x and y , we can check whether y = 2x.
x=x0
y =y0
z :=
z :=
0
0
z=0
Add+ (x)
Add+ (x)
Add− (y )
+2
z=0
Add− (x)
Add− (x)
Add+ (y )
+1
,, cost = 2x + (1 − y ) + 2
In ,, cost = 2(1 − x ) + y + 1
In
0
0
0
,
,
0
if y0 < 2x0 , player 2 chooses the first branch: cost > 3
if y0 > 2x0 , player 2 chooses the second branch: cost > 3
42/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Computing the optimal cost: why is that hard?
Given two clocks x and y , we can check whether y = 2x.
x=x0
y =y0
z :=
z :=
0
0
z=0
Add+ (x)
Add+ (x)
Add− (y )
+2
z=0
Add− (x)
Add− (x)
Add+ (y )
+1
,, cost = 2x + (1 − y ) + 2
In ,, cost = 2(1 − x ) + y + 1
In
0
0
0
,
,
0
if y0 < 2x0 , player 2 chooses the first branch: cost > 3
if y0 > 2x0 , player 2 chooses the second branch: cost > 3
if y0 = 2x0 , in both branches, cost = 3
42/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Computing the optimal cost: why is that hard?
Given two clocks x and y , we can check whether y = 2x.
x=x0
y =y0
z :=
z :=
0
0
z=0
Add+ (x)
Add+ (x)
Add− (y )
+2
z=0
Add− (x)
Add− (x)
Add+ (y )
+1
,, cost = 2x + (1 − y ) + 2
In ,, cost = 2(1 − x ) + y + 1
In
0
0
0
,
,
0
if y0 < 2x0 , player 2 chooses the first branch: cost > 3
if y0 > 2x0 , player 2 chooses the second branch: cost > 3
if y0 = 2x0 , in both branches, cost = 3
; player 2 can enforce cost 3 + |y0 − 2x0 |
42/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Computing the optimal cost: why is that hard?
Given two clocks x and y , we can check whether y = 2x.
x=x0
y =y0
z :=
z :=
0
0
z=0
Add+ (x)
Add+ (x)
Add− (y )
+2
z=0
Add− (x)
Add− (x)
Add+ (y )
+1
,, cost = 2x + (1 − y ) + 2
In ,, cost = 2(1 − x ) + y + 1
In
0
0
0
,
,
0
if y0 < 2x0 , player 2 chooses the first branch: cost > 3
if y0 > 2x0 , player 2 chooses the second branch: cost > 3
if y0 = 2x0 , in both branches, cost = 3
; player 2 can enforce cost 3 + |y0 − 2x0 |
Player 1 has a winning strategy with cost ≤ 3 iff y0 = 2x0
42/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Computing the optimal cost: why is that hard?
Player 1 will simulate a two-counter machine:
each instruction is encoded as a module;
the counter values c1 and c2 are encoded by two clocks:
1
1
x = c1
and
y = c2
2
2
43/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Computing the optimal cost: why is that hard?
Player 1 will simulate a two-counter machine:
each instruction is encoded as a module;
the counter values c1 and c2 are encoded by two clocks:
1
1
x = c1
and
y = c2
2
2
The two-counter machine has a halting computation iff player 1 has a
winning strategy to ensure a cost no more than 3.
43/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Computing the optimal cost: why is that hard?
Player 1 will simulate a two-counter machine:
each instruction is encoded as a module;
the counter values c1 and c2 are encoded by two clocks:
1
1
x = c1
and
y = c2
2
2
The two-counter machine has a halting computation iff player 1 has a
winning strategy to ensure a cost no more than 3.
Globally,
(x≤1,y ≤1,u≤1)
x=1,x:=0
∨ y =1,y :=0
u:=0
x=1,x:=0
∨ y =1,y :=0
z:=0
Testy (x=2z)
u=1,u:=0
(u=0)
43/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Computing the optimal cost: why is that hard?
Player 1 will simulate a two-counter machine:
each instruction is encoded as a module;
the counter values c1 and c2 are encoded by two clocks:
1
1
x = c1
and
y = c2
2
2
The two-counter machine has a halting computation iff player 1 has a
winning strategy to ensure a cost no more than 3.
x=1,x:=0
∨ y =1,y :=0
u:=0
x=1,x:=0
∨ y =1,y :=0
z:=0
Testy (x=2z)
u=1,u:=0
(u=0)
43/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Computing the optimal cost: why is that hard?
Player 1 will simulate a two-counter machine:
each instruction is encoded as a module;
the counter values c1 and c2 are encoded by two clocks:
1
1
x = c1
and
y = c2
2
2
The two-counter machine has a halting computation iff player 1 has a
winning strategy to ensure a cost no more than 3.
x=1,x:=0
∨ y =1,y :=0
u:=0
x=1,x:=0
∨ y =1,y :=0
z:=0
Ñ
x= 2c11
Testy (x=2z)
u=1,u:=0
(u=0)
é
y = 2c12
z=?
43/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Computing the optimal cost: why is that hard?
Player 1 will simulate a two-counter machine:
each instruction is encoded as a module;
the counter values c1 and c2 are encoded by two clocks:
1
1
x = c1
and
y = c2
2
2
The two-counter machine has a halting computation iff player 1 has a
winning strategy to ensure a cost no more than 3.
x=1,x:=0
∨ y =1,y :=0
u:=0
x=1,x:=0
∨ y =1,y :=0
u=1,u:=0
z:=0
Ñ
x= 2c11
é
Testy (x=2z)
Ñ
x= 2c11 +α
(u=0)
é
y = 2c12
y = 2c12 +α
z=?
z=0
43/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Computing the optimal cost: why is that hard?
Player 1 will simulate a two-counter machine:
each instruction is encoded as a module;
the counter values c1 and c2 are encoded by two clocks:
1
1
x = c1
and
y = c2
2
2
The two-counter machine has a halting computation iff player 1 has a
winning strategy to ensure a cost no more than 3.
x=1,x:=0
∨ y =1,y :=0
u:=0
x=1,x:=0
∨ y =1,y :=0
x= 2c11
é
(u=0)
u=1,u:=0
z:=0
Ñ
Testy (x=2z)
Ñ
x= 2c11 +α
é
Ñ
x= 2c11
y = 2c12
y = 2c12 +α
y = 2c12
z=?
z=0
z=α
é
43/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Computing the optimal cost: why is that hard?
Player 1 will simulate a two-counter machine:
each instruction is encoded as a module;
the counter values c1 and c2 are encoded by two clocks:
1
1
x = c1
and
y = c2
2
2
The two-counter machine has a halting computation iff player 1 has a
winning strategy to ensure a cost no more than 3.
x=1,x:=0
∨ y =1,y :=0
u:=0
x=1,x:=0
∨ y =1,y :=0
x= 2c11
é
(u=0)
u=1,u:=0
z:=0
Ñ
Testy (x=2z)
Ñ
x= 2c11 +α
é
y = 2c12
y = 2c12 +α
z=?
z=0
Ñ
x= 2c11
é
y = 2c12
z=
1
2c1 +1
43/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Shape of the reduction
44/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Shape of the reduction
;
Instruction
Test module
44/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Are we done?
a strategy σ is winning whenever all its outcomes are winning;
Cost of a winning strategy σ:
cost(σ) = sup{cost(ρ) | ρ outcome of σ up to the target}
Optimal cost:
optcostG =
inf
σ winning strat.
cost(σ)
(set it to +∞ if there is no winning strategy)
45/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Are we done? No! Let’s be a bit more precise!
a strategy σ is winning whenever all its outcomes are winning;
Cost of a winning strategy σ:
cost(σ) = sup{cost(ρ) | ρ outcome of σ up to the target}
Optimal cost:
optcostG =
inf
σ winning strat.
cost(σ)
(set it to +∞ if there is no winning strategy)
45/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Are we done? No! Let’s be a bit more precise!
Given G a weighted timed game,
a strategy σ is winning whenever all its outcomes are winning;
Cost of a winning strategy σ:
cost(σ) = sup{cost(ρ) | ρ outcome of σ up to the target}
Optimal cost:
optcostG =
inf
σ winning strat.
cost(σ)
(set it to +∞ if there is no winning strategy)
45/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Are we done? No! Let’s be a bit more precise!
Given G a weighted timed game,
a strategy σ is winning whenever all its outcomes are winning;
Cost of a winning strategy σ:
cost(σ) = sup{cost(ρ) | ρ outcome of σ up to the target}
Optimal cost:
optcostG =
inf
σ winning strat.
cost(σ)
(set it to +∞ if there is no winning strategy)
45/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Are we done? No! Let’s be a bit more precise!
Given G a weighted timed game,
a strategy σ is winning whenever all its outcomes are winning;
Cost of a winning strategy σ:
cost(σ) = sup{cost(ρ) | ρ outcome of σ up to the target}
Optimal cost:
optcostG =
inf
σ winning strat.
cost(σ)
(set it to +∞ if there is no winning strategy)
45/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Are we done? No! Let’s be a bit more precise!
Given G a weighted timed game,
a strategy σ is winning whenever all its outcomes are winning;
Cost of a winning strategy σ:
cost(σ) = sup{cost(ρ) | ρ outcome of σ up to the target}
Optimal cost:
optcostG =
inf
σ winning strat.
cost(σ)
(set it to +∞ if there is no winning strategy)
Two problems of interest
The value problem asks, given G and a threshold ./ c, whether
optcostG ./ c?
The existence problem asks, given G and a threshold ./ c, whether
there exists a winning strategy in G such that cost(σ) ./ c?
45/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Are we done? No! Let’s be a bit more precise!
Given G a weighted timed game,
a strategy σ is winning whenever all its outcomes are winning;
Cost of a winning strategy σ:
cost(σ) = sup{cost(ρ) | ρ outcome of σ up to the target}
Optimal cost:
optcostG =
inf
σ winning strat.
cost(σ)
(set it to +∞ if there is no winning strategy)
Two problems of interest
The value problem asks, given G and a threshold ./ c, whether
optcostG ./ c?
The existence problem asks, given G and a threshold ./ c, whether
there exists a winning strategy in G such that cost(σ) ./ c?
45/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Are we done? No! Let’s be a bit more precise!
Given G a weighted timed game,
a strategy σ is winning whenever all its outcomes are winning;
Cost of a winning strategy σ:
cost(σ) = sup{cost(ρ) | ρ outcome of σ up to the target}
Optimal cost:
optcostG =
inf
σ winning strat.
cost(σ)
(set it to +∞ if there is no winning strategy)
Two problems of interest
The value problem asks, given G and a threshold ./ c, whether
optcostG ./ c?
The existence problem asks, given G and a threshold ./ c, whether
there exists a winning strategy in G such that cost(σ) ./ c?
Note: These problems are distinct...
45/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
The value of the game is 3, but no strategy has cost 3.
q5
c1 +=2
q6
c2 >0
q8
c2 −−
q4
c2 ==0
c2 −−
q0
c2 +=2
c2 ==0
c1 −−
,
q3
q2
c1 ==0
q7
c1 −−
c1 ==0
c2 +=2
q9
c1 >0
q1
c1 +=2
46/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
The value of the game is 3, but no strategy has cost 3.
q5
c1 +=2
q6
c2 >0
q8
c2 −−
q4
c2 ==0
c2 −−
q0
c2 +=2
c1 −−
,
q3
c1 ==0
q7
c1 −−
c1 ==0
c2
•
q7
c2 +=2
•
q7
q9
q4 •
c1 >0
•
q7
q1
•
q7
q4 •
c2 ==0
q2
c1 +=2
q0
c1
46/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
The value of the game is 3, but no strategy has cost 3.
q5
c1 +=2
q6
c2 >0
q8
c2 −−
q4
c2 ==0
c2 −−
q0
c2 +=2
c1 −−
,
q3
c1 ==0
q7
c1 −−
c1 ==0
c2
•
q7
c2 +=2
•
q7
q9
q4 •
c1 >0
•
q7
q1
•
q7
q4 •
c2 ==0
q2
c1 +=2
q0
c1
46/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Weighted timed automata
In weighted timed automata, the optimal cost is an integer, and can be
computed in PSPACE.
Weighted timed games
47/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Weighted timed automata
In weighted timed automata, the optimal cost is an integer, and can be
computed in PSPACE.
The value problem is PSPACE-complete in weighted timed automata.
Almost-optimal winning schedules can be computed.
Weighted timed games
47/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Weighted timed automata
In weighted timed automata, the optimal cost is an integer, and can be
computed in PSPACE.
The value problem is PSPACE-complete in weighted timed automata.
Almost-optimal winning schedules can be computed.
Weighted timed games
Turn-based optimal timed games are decidable in EXPTIME when automata
have a single clock.
47/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Weighted timed automata
In weighted timed automata, the optimal cost is an integer, and can be
computed in PSPACE.
The value problem is PSPACE-complete in weighted timed automata.
Almost-optimal winning schedules can be computed.
Weighted timed games
Turn-based optimal timed games are decidable in EXPTIME when automata
have a single clock.
The value problem is decidable in EXPTIME in single-clock weighted timed
games. Almost-optimal memoryless winning strategies can be computed.
47/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Weighted timed automata
In weighted timed automata, the optimal cost is an integer, and can be
computed in PSPACE.
The value problem is PSPACE-complete in weighted timed automata.
Almost-optimal winning schedules can be computed.
Weighted timed games
Turn-based optimal timed games are decidable in EXPTIME when automata
have a single clock.
The value problem is decidable in EXPTIME in single-clock weighted timed
games. Almost-optimal memoryless winning strategies can be computed.
There is a symbolic algorithm to solve weighted timed games with a strongly
non-Zeno cost.
47/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Weighted timed automata
In weighted timed automata, the optimal cost is an integer, and can be
computed in PSPACE.
The value problem is PSPACE-complete in weighted timed automata.
Almost-optimal winning schedules can be computed.
Weighted timed games
Turn-based optimal timed games are decidable in EXPTIME when automata
have a single clock.
The value problem is decidable in EXPTIME in single-clock weighted timed
games. Almost-optimal memoryless winning strategies can be computed.
There is a symbolic algorithm to solve weighted timed games with a strongly
non-Zeno cost.
The value problem can be decided in EXPTIME in weighted timed games with
a strongly non-Zeno cost. Almost-optimal winning strategies can be computed.
47/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Weighted timed automata
In weighted timed automata, the optimal cost is an integer, and can be
computed in PSPACE.
The value problem is PSPACE-complete in weighted timed automata.
Almost-optimal winning schedules can be computed.
Weighted timed games
Turn-based optimal timed games are decidable in EXPTIME when automata
have a single clock.
The value problem is decidable in EXPTIME in single-clock weighted timed
games. Almost-optimal memoryless winning strategies can be computed.
There is a symbolic algorithm to solve weighted timed games with a strongly
non-Zeno cost.
The value problem can be decided in EXPTIME in weighted timed games with
a strongly non-Zeno cost. Almost-optimal winning strategies can be computed.
In weighted timed games, the optimal cost cannot be computed, as soon as
games have three clocks or more.
47/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Weighted timed automata
In weighted timed automata, the optimal cost is an integer, and can be
computed in PSPACE.
The value problem is PSPACE-complete in weighted timed automata.
Almost-optimal winning schedules can be computed.
Weighted timed games
Turn-based optimal timed games are decidable in EXPTIME when automata
have a single clock.
The value problem is decidable in EXPTIME in single-clock weighted timed
games. Almost-optimal memoryless winning strategies can be computed.
There is a symbolic algorithm to solve weighted timed games with a strongly
non-Zeno cost.
The value problem can be decided in EXPTIME in weighted timed games with
a strongly non-Zeno cost. Almost-optimal winning strategies can be computed.
In weighted timed games, the optimal cost cannot be computed, as soon as
games have three clocks or more.
The existence problem is undecidable in weighted timed games.
47/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Outline of the rest of the talk
1
Show that the value problem is undecidable in weighted timed
games
; This is intellectually satisfactory to not have this discrepancy in the
set of results
; A first proof based on a diagonal construction (originally proposed in
the context of quantitative temporal logics [BMM14])
; A second direct proof
2
Propose an approximation algorithm for a large class of weighted
timed games (that comprises the class of games used for proving the
above undecidability)
Almost-optimality in practice should be sufficient
Even when we know how to compute the value, we are only able to
synthesize almost-optimal strategies...
48/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Outline of the rest of the talk
1
Show that the value problem is undecidable in weighted timed
games
; This is intellectually satisfactory to not have this discrepancy in the
set of results
; A first proof based on a diagonal construction (originally proposed in
the context of quantitative temporal logics [BMM14])
; A second direct proof
2
Propose an approximation algorithm for a large class of weighted
timed games (that comprises the class of games used for proving the
above undecidability)
Almost-optimality in practice should be sufficient
Even when we know how to compute the value, we are only able to
synthesize almost-optimal strategies...
48/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Outline of the rest of the talk
1
Show that the value problem is undecidable in weighted timed
games
; This is intellectually satisfactory to not have this discrepancy in the
set of results
; A first proof based on a diagonal construction (originally proposed in
the context of quantitative temporal logics [BMM14])
; A second direct proof
2
Propose an approximation algorithm for a large class of weighted
timed games (that comprises the class of games used for proving the
above undecidability)
Almost-optimality in practice should be sufficient
Even when we know how to compute the value, we are only able to
synthesize almost-optimal strategies...
[BMM14] Bouyer, Markey, Matteplackel. Averaging in LTL (CONCUR’14).
48/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Outline of the rest of the talk
1
Show that the value problem is undecidable in weighted timed
games
; This is intellectually satisfactory to not have this discrepancy in the
set of results
; A first proof based on a diagonal construction (originally proposed in
the context of quantitative temporal logics [BMM14])
; A second direct proof
2
Propose an approximation algorithm for a large class of weighted
timed games (that comprises the class of games used for proving the
above undecidability)
Almost-optimality in practice should be sufficient
Even when we know how to compute the value, we are only able to
synthesize almost-optimal strategies...
48/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Outline of the rest of the talk
1
Show that the value problem is undecidable in weighted timed
games
; This is intellectually satisfactory to not have this discrepancy in the
set of results
; A first proof based on a diagonal construction (originally proposed in
the context of quantitative temporal logics [BMM14])
; A second direct proof
2
Propose an approximation algorithm for a large class of weighted
timed games (that comprises the class of games used for proving the
above undecidability)
Almost-optimality in practice should be sufficient
Even when we know how to compute the value, we are only able to
synthesize almost-optimal strategies...
48/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
A snapshot on the undecidability proof
;
Instruction
Test module
49/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
A snapshot on the undecidability proof
;
Instruction
Test module
49/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
A snapshot on the undecidability proof
Leave
Leave
;
Leave
Leave
Leave
Leave
Leave with cost 3 + 1/2n (n: length of the path)
49/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
A snapshot on the undecidability proof
Leave
M does not halt iff the
value of GM is 3
Leave
;
Leave
Leave
Leave
Leave
Leave with cost 3 + 1/2n (n: length of the path)
49/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Theorem [BJM15]
The value problem is undecidable in weighted timed games (with four
clocks or more).
Remark on the reduction:
Cost 0 within the core of the game
The rest of the game is acyclic
[BJM15] Bouyer, Jaziri, Markey. On the value problem in weighted timed games.
50/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Optimal cost is computable...
... when cost is strongly non-zeno.
[AM04,BCFL04]
That is, there exists κ > 0 such that for every region cycle C , for every real run
% read on C ,
cost(%) ≥ κ
Optimal cost is not computable...
... when cost is almost-strongly non-zeno.
[BJM15]
That is, there exists κ > 0 such that for every region cycle C , for every real run
% read on C ,
cost(%) ≥ κ or cost(%) = 0
Note: In both cases, we can assume κ = 1.
[BJM15] Bouyer, Jaziri, Markey. On the value problem in weighted timed games.
51/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Optimal cost is computable...
... when cost is strongly non-zeno.
[AM04,BCFL04]
That is, there exists κ > 0 such that for every region cycle C , for every real run
% read on C ,
cost(%) ≥ κ
Optimal cost is not computable... but is approximable!
... when cost is almost-strongly non-zeno.
[BJM15]
That is, there exists κ > 0 such that for every region cycle C , for every real run
% read on C ,
cost(%) ≥ κ or cost(%) = 0
Note: In both cases, we can assume κ = 1.
[BJM15] Bouyer, Jaziri, Markey. On the value problem in weighted timed games.
51/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Approximation of the optimal cost
Theorem
Let G be a weighted timed game, in which the cost is almost-strongly
non-zeno. For every > 0, one can compute:
two values v− and v+ such that
|v+ − v− | < and v− ≤ optcostG ≤ v+
one strategy σ such that
optcostG ≤ cost(σ ) ≤ optcostG + [it is an -optimal winning strategy]
Standard technics: unfold the game to get more precision, and
compute two adjacency sequences
52/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Approximation of the optimal cost
Theorem
Let G be a weighted timed game, in which the cost is almost-strongly
non-zeno. For every > 0, one can compute:
two values v− and v+ such that
|v+ − v− | < and v− ≤ optcostG ≤ v+
one strategy σ such that
optcostG ≤ cost(σ ) ≤ optcostG + [it is an -optimal winning strategy]
Standard technics: unfold the game to get more precision, and
compute two adjacency sequences
52/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Approximation of the optimal cost
Theorem
Let G be a weighted timed game, in which the cost is almost-strongly
non-zeno. For every > 0, one can compute:
two values v− and v+ such that
|v+ − v− | < and v− ≤ optcostG ≤ v+
one strategy σ such that
optcostG ≤ cost(σ ) ≤ optcostG + [it is an -optimal winning strategy]
Skip approximation scheme
Standard technics: unfold the game to get more precision, and
compute two adjacency sequences
52/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Approximation of the optimal cost
Theorem
Let G be a weighted timed game, in which the cost is almost-strongly
non-zeno. For every > 0, one can compute:
two values v− and v+ such that
|v+ − v− | < and v− ≤ optcostG ≤ v+
one strategy σ such that
optcostG ≤ cost(σ ) ≤ optcostG + [it is an -optimal winning strategy]
Standard technics: unfold the game to get more precision, and
compute two adjacency sequences
52/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Approximation of the optimal cost
Theorem
Let G be a weighted timed game, in which the cost is almost-strongly
non-zeno. For every > 0, one can compute:
two values v− and v+ such that
|v+ − v− | < and v− ≤ optcostG ≤ v+
one strategy σ such that
optcostG ≤ cost(σ ) ≤ optcostG + [it is an -optimal winning strategy]
Standard technics: unfold the game to get more precision, and
compute two adjacency sequences
; This is not possible here
There might be runs with prefixes of arbitrary length and cost 0 (e.g. the
game of the undecidability proof)
52/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Idea for approximation
Idea
Only partially unfold the game:
Keep components with cost 0 untouched – we call it the kernel
Unfold the rest of the game
53/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Idea for approximation
Idea
Only partially unfold the game:
Keep components with cost 0 untouched – we call it the kernel
Unfold the rest of the game
First: split the game along regions!
r1 , Y := 0
g , Y := 0
;
r2 , Y := 0
r3 , Y = 0
r4 , Y := 0
r5 , Y := 0
53/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Semi-unfolding
Only cost 0
Kernel K
Only cost 0
Kernel K
54/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Semi-unfolding
(`,r )
Only cost 0
Kernel K
Only cost 0
Kernel K
(`,r )
54/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Semi-unfolding
(`,r )
Only cost 0
Kernel K
Only cost 0
Kernel K
(`,r )
54/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Semi-unfolding
(`,r )
Only cost 0
Kernel K
Hypothesis:
cost > 0 implies cost ≥ κ
Only cost 0
Kernel K
(`,r )
54/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Semi-unfolding
(`,r )
Only cost 0
Kernel K
Hypothesis:
cost > 0 implies cost ≥ κ
Only cost 0
Kernel K
(`,r )
Conclusion: we can stop unfolding the game after N steps
(e.g. N = (M + 2) · |R(A)|, where M is a pre-computed bound on optcostG )
54/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Approximation scheme
55/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Approximation scheme
55/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Approximation scheme
Exact computation
55/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Approximation scheme
Exact computation
55/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Approximation scheme
Approximation
Exact computation
55/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
First step: Tree-like parts
; Goes back to [LMM02]
[LMM02] La Torre, Mukhopadhyay, Murano. Optimal-reachability and control for acyclic weighted timed automata (TCS@02).
56/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
First step: Tree-like parts
; Goes back to [LMM02]
c `
g 0, Y 0
c0
g 00 , Y 00
c00
`0
`00
[LMM02] La Torre, Mukhopadhyay, Murano. Optimal-reachability and control for acyclic weighted timed automata (TCS@02).
56/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
First step: Tree-like parts
; Goes back to [LMM02]
c `
g 0, Y 0
c0
O(`, v ) =
g 00 , Y 00
c00
`0
`00
O(`0 , v 0 )
O(`00 , v 00 )
[LMM02] La Torre, Mukhopadhyay, Murano. Optimal-reachability and control for acyclic weighted timed automata (TCS@02).
56/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
First step: Tree-like parts
; Goes back to [LMM02]
c `
g 0, Y 0
c0
O(`, v ) =
inf
t 0 |v +t 0 |=g 0
g 00 , Y 00
c00
`0
`00
O(`0 , v 0 )
O(`00 , v 00 )
[LMM02] La Torre, Mukhopadhyay, Murano. Optimal-reachability and control for acyclic weighted timed automata (TCS@02).
56/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
First step: Tree-like parts
; Goes back to [LMM02]
c `
g 0, Y 0
c0
O(`, v ) =
inf
t 0 |v +t 0 |=g 0
max
,
g 00 , Y 00
c00
`0
`00
O(`0 , v 0 )
O(`00 , v 00 )
[LMM02] La Torre, Mukhopadhyay, Murano. Optimal-reachability and control for acyclic weighted timed automata (TCS@02).
56/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
First step: Tree-like parts
; Goes back to [LMM02]
c `
g 0, Y 0
c0
O(`, v ) =
inf
t 0 |v +t 0 |=g 0
max (α),
g 00 , Y 00
c00
`0
`00
O(`0 , v 0 )
O(`00 , v 00 )
(α) = t 0 c + c0 + O(`0 , v 0 )
v 0 =[Y 0 ←0](v +t 0 )
[LMM02] La Torre, Mukhopadhyay, Murano. Optimal-reachability and control for acyclic weighted timed automata (TCS@02).
56/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
First step: Tree-like parts
; Goes back to [LMM02]
c `
g 0, Y 0
c0
O(`, v ) =
inf
t 0 |v +t 0 |=g 0
max (α), (β)
g 00 , Y 00
c00
`0
`00
O(`0 , v 0 )
O(`00 , v 00 )
(α) = t 0 c + c0 + O(`0 , v 0 )
(β) =
sup
t 00 c + c00 + O(`00 , v 00 )
t 00 ≤t 0 |v +t 00 |=g 00
v 0 =[Y 0 ←0](v +t 0 )
v 00 =[Y 00 ←0](v +t 00 )
[LMM02] La Torre, Mukhopadhyay, Murano. Optimal-reachability and control for acyclic weighted timed automata (TCS@02).
56/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Second step: Kernels
1
Refine the regions such that f differs
of at most within a small region
Output cost functions f
57/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Second step: Kernels
1
Refine the regions such that f differs
of at most within a small region
Output cost functions f
57/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Second step: Kernels
1
Refine the regions such that f differs
of at most within a small region
Output cost functions f
57/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Second step: Kernels
1
Refine the regions such that f differs
of at most within a small region
Output cost functions f
57/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Second step: Kernels
1
Refine the regions such that f differs
of at most within a small region
2
Under- and over-approximate by
piecewise constant functions f− and
f+
Output cost functions f
57/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Second step: Kernels
3
Refine/split the kernel along the new
small regions and fix f− or f+ , write f
4
Since cost is 0 everywhere, the
resulting game is nothing more than a
reachability timed game with an order
on target (output) edges (given by f )
Those can be solved using standard
technics based on attractors: small
regions are sufficient, and the local
optimal cost (for output f ) is constant
within a small region
5
f : constant
f : constant
; We have computed -approximations of
the optimal cost, which are constant
within small regions. Corresponding
strategies can be inferred
58/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Second step: Kernels
3
Refine/split the kernel along the new
small regions and fix f− or f+ , write f
4
Since cost is 0 everywhere, the
resulting game is nothing more than a
reachability timed game with an order
on target (output) edges (given by f )
Those can be solved using standard
technics based on attractors: small
regions are sufficient, and the local
optimal cost (for output f ) is constant
within a small region
5
f : constant
f : constant
; We have computed -approximations of
the optimal cost, which are constant
within small regions. Corresponding
strategies can be inferred
58/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Second step: Kernels
3
Refine/split the kernel along the new
small regions and fix f− or f+ , write f
4
Since cost is 0 everywhere, the
resulting game is nothing more than a
reachability timed game with an order
on target (output) edges (given by f )
Those can be solved using standard
technics based on attractors: small
regions are sufficient, and the local
optimal cost (for output f ) is constant
within a small region
5
f : constant
f : constant
; We have computed -approximations of
the optimal cost, which are constant
within small regions. Corresponding
strategies can be inferred
58/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Second step: Kernels
3
Refine/split the kernel along the new
small regions and fix f− or f+ , write f
4
Since cost is 0 everywhere, the
resulting game is nothing more than a
reachability timed game with an order
on target (output) edges (given by f )
Those can be solved using standard
technics based on attractors: small
regions are sufficient, and the local
optimal cost (for output f ) is constant
within a small region
constant
5
f : constant
f : constant
; We have computed -approximations of
the optimal cost, which are constant
within small regions. Corresponding
strategies can be inferred
58/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Second step: Kernels
3
Refine/split the kernel along the new
small regions and fix f− or f+ , write f
4
Since cost is 0 everywhere, the
resulting game is nothing more than a
reachability timed game with an order
on target (output) edges (given by f )
5
Those can be solved using standard
technics based on attractors: small
regions are sufficient, and the local
optimal cost (for output f ) is constant
within a small region
; We have computed -approximations of
the optimal cost, which are constant
within small regions. Corresponding
strategies can be inferred
58/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Outline
1
Timed automata
2
Weighted timed automata
3
Timed games
4
Weighted timed games
5
Conclusion
59/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Conclusion
Summary of the talk
Overview of results concerning the optimal reachability problem in
weighted timed automata and games
New insight into the value problem for this model
Future work
|X |
) and
Improve the approximation scheme (2EXP(|G|) · 1/
extend it to the whole class of weighted timed games
Assume stochastic uncertainty
Is the value of any game a rational number?
Understand the multiplayer setting
60/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Conclusion
Summary of the talk
Overview of results concerning the optimal reachability problem in
weighted timed automata and games
New insight into the value problem for this model
Future work
|X |
) and
Improve the approximation scheme (2EXP(|G|) · 1/
extend it to the whole class of weighted timed games
Assume stochastic uncertainty
Is the value of any game a rational number?
Understand the multiplayer setting
60/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Conclusion
Summary of the talk
Overview of results concerning the optimal reachability problem in
weighted timed automata and games
New insight into the value problem for this model
Future work
|X |
) and
Improve the approximation scheme (2EXP(|G|) · 1/
extend it to the whole class of weighted timed games
Assume stochastic uncertainty
Is the value of any game a rational number?
Understand the multiplayer setting
60/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Conclusion
Summary of the talk
Overview of results concerning the optimal reachability problem in
weighted timed automata and games
New insight into the value problem for this model
Future work
|X |
) and
Improve the approximation scheme (2EXP(|G|) · 1/
extend it to the whole class of weighted timed games
Assume stochastic uncertainty
Is the value of any game a rational number?
Understand the multiplayer setting
60/60
Timed automata Weighted timed automata Timed games Weighted timed games Conclusion
Conclusion
Summary of the talk
Overview of results concerning the optimal reachability problem in
weighted timed automata and games
New insight into the value problem for this model
Future work
|X |
) and
Improve the approximation scheme (2EXP(|G|) · 1/
extend it to the whole class of weighted timed games
Assume stochastic uncertainty
Is the value of any game a rational number?
Understand the multiplayer setting
60/60

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