Downward Closed Language Generators

1
Downward Closed Language Generators
Parosh Aziz Abdulla Pritha Mahata Aletta Nylén
Uppsala University
2
Outline

• Reachability Approaches
• Downward-closed languages
• Recognizability of Reachable sets
• Simple Regular Expressions
• Downward closed language generators
• Hierarchical dlgs
• Timed Petri Net
• Ongoing Work

3
Systems and properties

• Transition Systems

(Set of states, set of initial states,
alphabet, transition rules)

• Safety Properties ( Nothing bad will ever
  happen)

Reachability of a bad state in the system
Verification of Safety property
4
Reachability Approaches
Forward Reachability
Reachability Analysis
Backward Reachability
5
Reachability Approaches(contd.)

• Forward Reachability set is usually not
• computable , e.g LCSCFI96.

• Backward reachability set is sometimes
  computable,
• e.g LCSAJ96b.

6
Forward Reachability
Set of reachable states of a system R

• (finite state) abstraction

Computability of R

• Symbolic graph G (V, E)

l
V partitions of R wrt some criterion
E v1 v2 iff
(e.g control states)
l
v2
v1
7
Forward Reachability
Set of reachable states of a system R

• (finite state) abstraction

Computability of R

• Symbolic graph G (V, E)

l
V partitions of R wrt some criterion
E v1 v2 iff
(e.g control states)
l
h
f
v1
v2
8
Forward Reachability (contd.)
G simulates the transition system.
If G satisfies a safetyproperty
Same result holds for the concrete system.
Verification is easier in G.
Problem R is often not computable.
But, is R recognizable !
9
Downward Closed Languages
? - finite alphabet - substring
relation on ?
L - a language over ?
If x ?L and y x gt y ? L, then L is
downward closed.
y
x - downward closed set x -
upward closed set

x

L
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Why downward closed languages ?
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Why downward closed languages ?
Timed Petri Net, N
Lossy TPN, N
Set of Bad States, Bad (upward closed)
Initial states, I
Initial states, I
Bad
M
loss
M
B
Ml
B
Bad
M
Ml
Note Considering safety properties only,
markings can be made downward-closed in TPN.
and Ml
B
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Is R recognizable ?
for each a1,a2,. ?A, there is i,j such that
(A, ) is wqo if
i lt j and ai aj
If (A, ) is wqo, (A, ) is a wqo.
(Higman)
Question Can we find some generator ? such that
R L(?) ?
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Is R recognizable ? (contd.)
14
Simple Regular Expressions
Generators simple regular expressions.

• M - a finite alphabet.
• Atomic expression e over M - a regular
  expression of the form
• (a ?) where a ? M
• (a1 a2 . am ), where a1,a2,.,am ? M
  

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R is recognizable !
w1 (bc)(a?)(ca)
atomic expressions
w2 (ca)(b?)(ab)
e c a c (b ?) b (a ? ) a
c (a ?) (a c) a
Products of atomic expressions
e sum of products an SRE
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Lossy Channel System
c?m
Control ( LTS)
c!n
Channel

• M Finite alphabet of messages
• State (s, w) s - control
  state, w ? M - channel content
• Set of reachable states of LCS is downward
  closed and can be
• expressed by SREs.

17
Well Quasi Ordering
Natural numbers
(N , )

• is wqo x1,x2natural numbers, there is i,j such
  that
• i lt j and xi xj

18
SRE
Downward Closed Language Generators
(M, ) , M finite alphabet A
wqo (A , ) (M, ) , substring


(A, ) is wqo
Atomic expressions
Let B ? A. (a ? ) s.t a ? M
B L(B) a a ? A and a
is not larger

or equal to any element of B

e.g Let A N, B 3 and L(B) 0,1,2 U
?
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Downward Closed Language Generators
Assume a wqo (A, )
Let B ? A

• Atomic expressions are of the form B or
  B
• L( B) Set of elements in A which are not
  larger or equal to
• any element in B.
• L( B) (L( B) )

• A product p over A
• L(e1 en ) w1 .. wn w1 ? L
  (e1), .. , wn ? L (en)
• 
  where e1,e2,.,en are atomic
  expressions over A.
• DLG over A L(p1 p2 . pn) L(p1) U
  .. U L(pn) ,
• 
  where p1,p2,.,pn are products over A.

20
DLG
21
DLG (contd.)
?
2. R w1 w2
0 (N U
?) 0,1 0 0,1, ? 0,1 0,1, ?
0,1
0 0, ? 0 0,1,2, ? 0,1
L( 1) L( ø) L( 2)
L( 1 ( 2) ) L() L(..)
L( 1 ( 2) ..)
?
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Bags
(A, ) is wqo and is equality.
B1, B2 N N
B1
B2
B1 B B2
(AB, B)
is wqo

• Application Markings of a Petri Net are
  represented by bags.

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Dlg for bags
DLGs for bags ? DLGs for words with operator
both associative and
commutative.
A bag dlg, ? - 3 1

0,1,2 0
? L(?)
0 0 0 2
? L(?)
1 0 0
? L(?)
0 0 3
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String of Bags
S1
S2
S1 S2
((AB), ) is wqo
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Dlg for String of Bags
A dlg for string of bags, ?s
bag bag

32 6
2 3


? ?
4 7 3
4 4 6
6
Bag dlg








4 2
3
Bag dlg
?






0 125


9
2 1 0
3
5 8
e.g



16 210
5 3 2 1
21
3 3
are in language of ?s.
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Dlg for String of Bags(contd.)
A a,b,c a finite alphabet
A dlg for string of bags, ? s
a2 b

? ?
b,c b,c b
b,c a,c a
a,b




Bag dlg




a b
c c c c c
e.g
a a
a a c c
a b
a c
b b c c c c
are in language of ?s.
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Hierarchical DLGs

• (A, )

(A, )
is wqo implies
is a wqo ( Higmans Theorem).

• If L ? A is downward closed, then L is
  recognizable by some dlg ?.

• We can hierarchically define dlgs over A.

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Timed Petri Net
P1
P2
3.0
2.0
13
24
Tokens have ages Real numbers.
01
25
Conditions on ages Intervals.
45
45
45
16
4.0
0.0
P3
P4
Extended bags of Real Numbers
Mapping from real numbers to natural numbers N U
?. B 4.0, 4.0, 2.0 B(4.0)
2
Marking M A Ebag over (Places x Reals).
M(P3,4.0) 2, M(P1, 2.0) 1
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Timed Transitions
T
P1
P2
P1
P2
3.0
2.0
3.0
4.0
13
24
24
13
25
01
25
01
Increase of time by 1.0
t
t
45
45
45
45
00
00
00
00
P3
P4
P3
P4
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Discrete Transitions
D
P1
P2
P1
P2
3.0
2.0
13
24
24
13
Firing t
01
45
01
45
t
t
45
45
25
25
00
00
00
00
0.0
0.0
P3
P4
P3
P4
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Transitions
U

D
T
M2
If M1
M2
M1
T
or M1
M2
D
Additionally, there are some lossy transitions in
lossy TPN.
Remark A TPN can have unbounded number of
tokens !!
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Ordering on Marking
P1
P2
P1
P2
3.7
2.0
2.2 2.0
3.5
13
24
24
13
01
45
01
45
t
t
45
45
25
25
00
00
00
00
4.0
6.2
P3
P4
P3
P4
M1
M2
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Regions

• Finite no. of clocks (e.g Timed Automata)

y
3
Two clocks x,y and cmax 3
2
1
0
x
0
1
2
3

• Clock values are equivalent in timed automata if
  they have
• same integral parts
• same ordering of fractional parts
• clock values beyond cmax are equivalent

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Regions(Example)

• Region R

y

• V(x) 0.6, V(y) 0.5
• V R

1
0
x
1
0
Not Powerful for Timed Petri Nets
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Dlgs for LTPN
P1
P2
13
24
01
25

• Unboundedness in two directions
• number of tokens
• age of tokens

45
45
45
15
P3
P4
cmax 5
Abstraction of ages to express sets of markings

• Tokens with same fractional parts are in the
  same ebag.
• Ordering of ebags is according to the ordering
  of
• fractional parts of ages.
• Ages of tokens beyond cmax are equivalent.

36
Dlgs for LTPN
Constraints strings of bags over
a finite alphabet of (Places x
0,..max)
Sets of markings
and
Markings are downward closed for LTPN
Constraints are dlgs for strings of bags over a
finite set !!!
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Universal Regions !
P2
P1
3.5 3.75
2.0
13
24
Note M can have at most same number
of tokens as R.
01
25
45
45
45
15
4.2
P3
P4
frac 0
Increasing frac
age gt 5

4 5
3
2 0
R
If M lt M, then M ? R
4.2 4.2
2.0
3.5
3.75
M
P1
P2
P3
P4
P2
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Universal Regions (contd.)
3
2
Let Universal Region R
T
dlg
Max bag
Zero bag
cmax 5
2
3
4
2


P1
P2
4
2
3
4


13)
24)
3
4
3
max


01)
25)
t
45)
45)
05)
13)
4
max

4
max

max max
P3
P4
Generates O((max-1)2 sizeof(product) 1)
new regions by timed transition.
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Universal Regions (contd.)
3
2
followed by
t
T
x4
5
At most one token in P3 and one token in P4 with
ages as follows
4
3
2
1
0
0
1
2
3
x3
4
5
Lot of universal regions !!!
Solution Universal Zones !!
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Acceleration

• Compute Post
• Acceleration - a sequence of transitions at
  each step
• Lossy Channel system - accelerate by
  arbitrary iteration of control loops
• Lossy TPN - accelerate by
• arbitrary firing of enabled transitions
  followed by
• timed transitions and
• combine atomic expressions of the universal
  regions

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Comparison with earlier TPN work

• Forward Reachability
  Backward Reachability
• Compute Post
  Compute Pre
• Markings are downward closed(lossy TPN).
  Markings are upward closed.
• Universal region.
  Existential region.
• Maximal number of tokens in a
  Minimal number of tokens
• universal region.
  in an
  existential region.

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Ongoing Work

• Compute Post(R,t) for all transitions t.

• Define universal zones.

• Apply forward reachability algorithm.