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Downward Closed Language Generators
Parosh Aziz Abdulla
Pritha Mahata
Aletta Nylén
Uppsala University
Outline
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Reachability Approaches
Downward-closed languages
Recognizability of Reachable sets
Simple Regular Expressions
Downward closed language generators
Hierarchical dlgs
Timed Petri Net
Ongoing Work
Systems and properties
 Transition Systems
(Set of states,
set of initial states,
alphabet, transition rules)
 Safety Properties ( Nothing bad will ever happen)
Verification of Safety property
Reachability of a bad state
in the system
Reachability Approaches
Forward Reachability
Reachability Analysis
Backward Reachability
Forward Reachability
Backward Reachability
Initial state
Post*
Initial
states
Bad states
Pre*
Bad state
Reachability Approaches(contd.)
- Forward Reachability set is usually not
computable , e.g LCS[CFI96].
- Backward reachability set is sometimes computable,
e.g LCS[AJ96b].
Still, Forward Reachability is an appealing approach.
Why ?
Forward Reachability
Set of reachable states of a system – R
Computability of R
V = partitions of R wrt some criterion
• (finite state) abstraction
• Symbolic graph G (V, E)
E : v1
l
v2
iff
(e.g control states)
l
v1
v2
Forward Reachability
Set of reachable states of a system – R
Computability of R
V = partitions of R wrt some criterion
• (finite state) abstraction
• Symbolic graph G (V, E)
E : v1
l
v2
iff
(e.g control states)
f
v1
l
h
v2
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 !
Yes, if R is downward-closed [ABJ98] !!
Downward Closed Languages
 L -
finite alphabet
substring relation on *
a language over *
If x L and y x => y  L,
then L is downward closed.
y
•
x
•
L
x
- downward closed set
x
- upward closed set
Why downward closed languages ?
LCS
– Channel Language is downward closed. A channel can
always lose messages and become empty.
Reachability set is downward-closed for LCS .
TPN
-
TPN has monotonicity wrt a preorder
M1
M3
M2
M4
and M1
M2
on markings.
M3
M4
Why downward closed languages ?
Timed Petri Net, N
Lossy TPN, N’
Set of Bad States, Bad (upward closed)
Initial states, I
Initial states, I
loss
M
Bad
M
B’
Ml
B
Bad
Ml
Note : Considering safety
properties only, markings can be
made downward-closed in TPN.
and Ml
M
B
B
M
B’
B’
Is R recognizable ?
for each a1,a2,…. A, there is i,j such that
(A, ) is wqo if
i < j and ai
If (A,
) is wqo, (A*,
aj
*) is a wqo. (Higman)
If a language R  A* is downward closed, then
R is upward closed.
R is characterized by finite set of minimal elements {w1,….,wm}. [Higman]
R = w1 U …. U wm
….
U
U
R = w1
wm
Question : Can we find some generator  such that R = L() ?
Is R recognizable ? (contd.)
Answer : We can find some generator  such that R = L() if
 for a word w in A*, w
= L() and
 generators are closed under intersection.
Let A = {a,b,c} and w1 = ab, w2 = bc,
then w1
w1
= A* a A* b A* ,
w2
= A* b A* c A* and
= (A\a)*(a+)(A\b)*
w2
= (A\b)*(b+)(A\c)*
=
2.
e = w1
(b+c)*(a+)(c+a)*
U
1.
w2
=
(c+a)*(b+)(a+b)*
= c* a* + c* (b + ) b* (a +  ) a* +
c* (a + ) (a + c)* a*
Question : Can we find  s such that w1 , w2 , e are expressed by  s ?
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
A product p over M - a concatenation (possibly empty)
 (e1 • e2 ••••• en ), where e1,e2,….,en are atomic expressions over M.
Simple regular expression over M - has the form

p1 + p2 + …. + pn , where p1,p2,….,pn are products over M.
R is recognizable !
w1 =
(b+c)*(a+)(c+a)*
atomic expressions
w2 =
(c+a)*(b+)(a+b)*
e = c* a* + c* (b + ) b* (a +  ) a* + c* (a + ) (a + c)* a*
Products of atomic expressions
e = sum of products – an SRE
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.
Well Quasi Ordering
Natural numbers
is wqo x1,x2……natural numbers, there is i,j such that
(N , )
i < j and xi
xj
Finite sets
(A , = ) is wqo, if A is finite,
a1,a2, a3,a4,b, a5,a6, a7,a8,b, a9….
Strings
w1 =
*
w2
= 1.
2 .
3 . 4
w1
*
w2
(N*, *) is wqo
3 . 2.
5 . 3. 7 . 1. 1
SRE
(M, =) ,
Downward Closed Language Generators
M : finite alphabet
A wqo (A , )
(M*, =*) , =* : substring
(A*,
Atomic expressions :
Let B  A.
(a +  ) s.t
a M
*)
is wqo
~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 {}
(a1 + a2 + …. +am )*
s.t a1,a2,….,am  M
*~B
e.g Let A = N, B = {3} and
* = {0,1,2}* = (L(~B))*
L(~B)
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.
DLG
Answer : For a downward closed language R, we
can find some generator  such that R = L() if
= L() and
1.
for a word w in A*, w
2.
dlgs are closed under intersection.
1.
Let (N,
) be the wqo. and w1 = 2 • 3, w2 = 1 • 2,
then w1
= N* 2 N* 3 N*
w1
and
= {0,1}*(N U {}){0,1,2}*
=
L( ~ 2) L(~ ø) L(~ 3)
=
L( ~ 2 • ~3)
1
w2 = N* 1 N* 2 N*
w2
= 0*(N U {}){0,1}*
=
=
L( ~ 1) L(~ ø) L( ~2)
L( ~ 1 • ~ 2)
2
DLG (contd.)
R = w1

2.
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}) + ……………… + ……………..)

Bags
(A, ) is wqo and
is equality.
B1, B2 : N
B1
B2
B1
B
B2
(AB, B) is wqo

Application : Markings of a Petri Net are represented by bags.
N
Dlg for bags
DLGs for bags  DLGs for words with operator • both associative and
commutative.
A bag dlg,  -
~{3}
~* {1}
= {0,1,2} 0*
0 0 02

L()
1 0 0

L()

L()
0 0 3
String of Bags
S1
S2
S1
* S2
((AB)*, *) is wqo
Dlg for String of Bags
A dlg for string of bags, s =
=
~
~
+ ~* 6
+
~* {bag}
~{bag}
=~
32
~4
~7
~*
6
~4
+
*
~3
2
~4
3
Bag dlg
*
~6
*
~
e.g
3
3
~4
*
~2
+
0
125
5 8
5 3
2 1
+ ~* 3
21
Bag dlg*
9
3
16
210
are in language of s.
2 1 0
Dlg for String of Bags(contd.)
A = {a,b,c} : a finite alphabet
A dlg for string of bags,  s =
=
e.g
~
~
{a,b}
+ *~
~
+
a2
b
~{b,c}
~{b,c}
*~b
+
ccccc
~{b,c}
~{a,c}
*~a
Bag dlg
ab
aacc
bb
cc
cc
aa
a
b
are in language of s.
ac
Hierarchical DLGs

(A, ) is wqo implies

If L  A* is downward closed, then L is recognizable by some dlg .

We can hierarchically define dlgs over A.
(A*, *) is a wqo ( Higman’s Theorem).
Example :
A wqo
(A, )
Bags(A)
(AB, B)
Ldc  AB is recognizable by a dlg.
((AB)*, *)
Atomic expressions are dlgs for bag.
Strings of Bags(A)
L’dc  (AB)* is recognizable by a dlg.
Timed Petri Net
P
P
1
2
2.0
[1:3]
[2:4]
3.
0
[0:1]
[2:5]
[4:5]
[4:5]
[4:5]
[1:6]
4.
0
0.
0
P3
P
Tokens have “ages” : Real numbers.
Conditions on “ages” : Intervals.
4
Extended bags of Real Numbers :
Mapping from real numbers to natural numbers N U {ω}.
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
Timed Transitions
P
1 2.
0
P
P
P
2
1 3.
24.
3.0
[1:3]
[2:4]
[0:1]
[2:5]
t
[4:5]
P3
[0:0]
[0:0]
[4:5]
P
4
0
[0:1]
Increase of time by 1.0
[4:5]
P3
[1:3]
[2:4]
0
[2:5]
t
[0:0]
[0:0]
[4:5]
P
4
T
Discrete Transitions
P
1 2.
0
P
P
P
2
1
2
3.0
[1:3]
[2:4]
[0:1]
[1:3]
[4:5]
Firing t
[0:1]
[4:5]
t
[2:5]
P3
[0:0]
[0:0]
[2:4]
t
[4:5]
P
4
[2:5]
0.
0
P3
[0:0]
[0:0]
[4:5]
0.
0
P
4
D
Transitions
=
M1
T
M2
U
If M1
or M1
D
M2
T
D
M2
Additionally, there are some lossy transitions in lossy TPN.
Remark : A TPN can have unbounded number of tokens !!
Ordering on Marking
P
1 2.
0
P
P
P
2
1 2.2
23.
3.7
[1:3]
2.0
[1:3]
[2:4]
[0:1]
[4:5]
[2:4]
[0:1]
[4:5]
t
[2:5]
[0:0]
[0:0]
P3
t
[4:5]
P
M1
5
[2:5]
4.
0
[0:0]
[0:0]
6.2
P3
P
4
M2
P2,3.
7
M1
P1,2.
0
M2
P2,3.
P1,2.
P1,2.2
5
0
frac = 0 Increasing fractional parts
[4:5]
P4,max
age >= 5
4
Regions
• Finite no. of clocks (e.g Timed Automata)
y
3
Two clocks x,y and cmax = 3
2
1
00
1
2
3
x
Clock values are equivalent in timed automata if they have
 same integral parts
 same ordering of fractional parts
 clock values beyond cmax are equivalent
Regions(Example)
• Region R :
y
 V(x) = 0.6, V(y) = 0.5
1
0
0
V€R
1
x
Not Powerful for Timed Petri Nets……
Dlgs for LTPN
P
P
1
2
[1:3]
[2:4]
[0:1]
[2:5]
[4:5]
[4:5]
[4:5]
[1:5]
P3
Unboundedness in two directions :
• number of tokens
• age of tokens
P
cmax = 5
4
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.
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 !!!
Universal Regions !
P
P
1
2
2.0
3.5 3.75
[1:3]
[2:4]
[0:1]
Note : M can have at most same number
of tokens as R.
[2:5]
[4:5]
[4:5]
[1:5]
[4:5]
4.
2
P3
P
4
frac = 0
R =
M=
2
Increasing frac
4*
5
0
2.0
3.5
3.75
P
P
P
age >= 5
3
*
If M’ < M,
then M’  R
4.2 4.2
P
P
Universal Regions (contd.)
Let Universal Region R =
3
2
Zero bag
cmax = 5
dlg
2
P
P
1
2
[1:3)
T
Max bag
3
4
2
3
4
+
+
4
2
+
3
4
+
[2:4)
[0:1)
[2:5)
+
3
max
+
t
[4:5)
[0:5)
P3
[1:3)
[4:5)
4
max
P
4
Generates O((max-1)*2 + sizeof(product) + 1)
new regions by timed transition.
+
4
max
max
max
+
Universal Regions (contd.)
3
2
t
followed by
T
x4
5
At most one token in P3 and one token in P4
with ages as follows :
4
3
2
1
00 1
2
3
4
5
x3
Lot of universal regions !!! Solution : Universal Zones !!
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
Comparison with earlier TPN work

Forward 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
universal region.
Minimal number of tokens
in an existential region.
Backward Reachability
Ongoing Work
• Compute Post*(R,t) for all transitions t.
• Define universal zones.
• Apply forward reachability algorithm.