Download presentation

Presentation is loading. Please wait.

Published byMia miah Teer Modified about 1 year ago

1
Parosh Aziz Abdulla Pritha Mahata Aletta Nyl é n Uppsala University Downward Closed Language Generators

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
Transition Systems Systems and properties (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

4
Reachability Analysis Forward Reachability Backward Reachability Reachability Approaches Forward Reachability Bad states Initial state Post* Backward Reachability Initial states Bad state Pre*

5
Reachability Approaches (contd.) - Backward reachability set is sometimes computable, e.g LCS[AJ96b]. Still, Forward Reachability is an appealing approach. Why ? - Forward Reachability set is usually not computable, e.g LCS[CFI96].

6
Forward Reachability Set of reachable states of a system – R Computability of R Symbolic graph G (V, E) l v 1 v2v2 l V = partitions of R wrt some criterion E : v 1 v 2 iff (e.g control states) (finite state) abstraction

7
Forward Reachability Set of reachable states of a system – R Computability of R Symbolic graph G (V, E) l v 1 v2v2 f h l V = partitions of R wrt some criterion E : v 1 v 2 iff (e.g control states) (finite state) abstraction

8
Forward Reachability (contd.) If G satisfies a safetyproperty G simulates the transition system. 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] !!

9
- finite alphabet - substring relation on * L - a language over * If x L and y x => y L, then L is downward closed. y x L Downward Closed Languages x - downward closed set x - upward closed set

10
Why downward closed languages ? TPN - TPN has monotonicity wrt a preorder on markings. M1M1 M2M2 and M 1 M 3 M 2 M 4 M3M3 M4M4 LCS – Channel Language is downward closed. A channel can always lose messages and become empty. Reachability set is downward-closed for LCS.

11
Why downward closed languages ? Note : Considering safety properties only, markings can be made downward-closed in TPN. Timed Petri Net, N Lossy TPN, N’ Set of Bad States, Bad (upward closed) Initial states, I M MlMl M MlMl M and M l B loss Bad B’ B M B

12
Is R recognizable ? Question : Can we find some generator such that R = L ( ) ? R is upward closed. If a language R A* is downward closed, then R is characterized by finite set of minimal elements {w 1,….,w m }. [Higman] R = w 1 U …. U w m R = w 1 …. w m UU If (A, ) is wqo, (A*, *) is a wqo. (Higman) A (A, ) is wqo if for each a 1,a 2,…. A, there is i,j such that i < j and a i a j

13
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. Question : Can we find s such that w 1, w 2, e are expressed by s ? 1.Let A = {a,b,c} and w 1 = ab, w 2 = bc, then w 1 = A* a A* b A*, w 2 = A* b A* c A* and w 1 = (A\a)*(a+ ) (A\b)* w 2 = (A\b)*(b+ ) (A\c)* = (b+c)*(a+ )(c+a)* = (c+a)*(b a+b)* 2.e = w 1 w 2 = c* a* + c* (b + ) b* (a + ) a* + c* (a + ) (a + c)* a* U

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 (a 1 + a 2 + …. +a m )*, where a 1,a 2,….,a m M A product p over M - a concatenation (possibly empty) (e 1 e 2 e n ), where e 1,e 2,….,e n are atomic expressions over M. Simple regular expression over M - has the form p 1 + p 2 + …. + p n, where p 1,p 2,….,p n are products over M.

15
R is recognizable ! e = c* a* + c* (b + ) b* (a + ) a* + c* (a + ) (a + c)* a* Products of atomic expressions e = sum of products – an SRE w 1 = (b+c)*(a+ )(c+a)* w 2 = (c+a)*(b+ )(a+b)* atomic expressions

16
Lossy Channel System 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. c?m c!n Channel Control ( LTS)

17
Well Quasi Ordering N (N, )is wqo x 1,x 2 ……natural numbers, there is i,j such that i < j and x i x j Natural numbers A (A, = )is wqo, if A is finite, a 1,a 2, a 3,a 4,b, a 5,a 6, a 7,a 8,b, a 9 …. Finite sets N (N *, * )is wqo w 1 * w 2 w 1 = w 2 = * Strings

18
SRE Downward Closed Language Generators (M, =), M : finite alphabet A wqo (A, ) (M*, =*), =* : substring N e.g Let A = N, B = {3} and L(~B) = {0,1,2} U { } A (A *, * ) is wqo (a 1 + a 2 + …. +a m )* s.t a 1,a 2,….,a m M ~B * N e.g Let A = N, B = {3} and L(~B) = {0,1,2}* = (L(~B))* * 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}

19
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(e 1 e n ) = {w 1 ….. w n | w 1 L (e 1 ), ….., w n L (e n )} where e 1,e 2,….,e n are atomic expressions over A. DLG over A – L(p 1 + p 2 + …. + p n ) = L(p 1 ) U ….. U L(p n ), where p 1,p 2,….,p n are products over A.

20
DLG Answer : For a downward closed language R, we can find some generator such that R = L ( ) if 1. for a word w in A*, w = L ( ) and 2. dlgs are closed under intersection. N 1.Let (N, ) be the wqo. and w 1 = 2 3, w 2 = 1 2, then w 1 = N* 2 N* 3 N* and w 2 = N* 1 N* 2 N* w 1 = {0,1}*(N U 0,1,2}* w 2 = 0*(N U { }) {0,1}* ~ = L( 2) L(~ ø) L( 3) ~ ~ = L( 1) L(~ ø) L( 2) ~ 2 = L( 2 3) ~ ~ = L( 1 2) ~ ~ 1

21
DLG (contd.) 2. R = w 1 w 2 = {0}* (N U {0,1}* + {0}* {0,1, } {0,1}* {0,1, } { 0,1 }* + {0}* {0, } { 0}* {0,1,2, } {0,1}* = L( {1} ( {2}) ) + L(………………) + L(……………..) = L( {1}) L(~ ø) L( {2}) + ……………… + ……………… ~ ~ ~ = L( {1} ( {2}) + ……………… + ……………..) ~ ~ ~

22
Bags (A B, B )is wqo Application : Markings of a Petri Net are represented by bags. (A, ) is wqo and is equality. B 1 B B 2 B1B1 B2B2 N N B 1, B 2 : N N

23
Dlg for bags L L( ) A bag dlg, - ~{3} ~ {1} * = {0,1,2} 0* L( ) DLGs for bags DLGs for words with operator both associative and commutative

24
String of Bags S1S1 S2S2 S 1 * S 2 ((A B )*, * ) is wqo

25
Dlg for String of Bags A dlg for string of bags, s = ~{bag} ~ {bag} * = * e.g ~~~~ + ~ = ~ ~ 6 * ~4 ~7 ~3 * + ~4 ~4 ~6 * ~~ + ~ 3 * ~4 ~2 * + * are in language of s. Bag dlg Bag dlg*

26
Dlg for String of Bags(contd.) A dlg for string of bags, s = = a 2 b a e.g ~~~~ + ~ ~ {a,b} * ~{b,c} ~{b,c} ~b * + ~{b,c} ~{a,c} ~a * are in language of s. Bag dlg a b b b c c c c A = {a,b,c} : a finite alphabet a b a c a a c c c c c c c

27
Hierarchical DLGs (A, ) (A*, *)is wqo impliesis a wqo ( Higman’s Theorem). If L A* is downward closed, then L is recognizable by some dlg . We can hierarchically define dlgs over A. Example : (A, ) (A B, B ) ((A B )*, * ) L dc A B is recognizable by a dlg. Strings of Bags(A) Bags(A) A wqo Atomic expressions are dlgs for bag. L’ dc (A B ) * is recognizable by a dlg.

28
Timed Petri Net P1P1 P3P3 P2P2 P4P4 [1:3][2:4] [4:5][1:6] [4:5] [0:1][2:5] [4:5] 2.0 Tokens have “ages” : Real numbers. Conditions on “ages” : Intervals. 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(P 3,4.0) = 2, M(P 1, 2.0) = 1

29
Timed Transitions P1P1 P3P3 P2P2 P4P4 [1:3][2:4] [0:0] P1P1 P3P3 P2P2 P4P4 [2:4] [0:0] [1:3] t t Increase of time by 1.0 [4:5] [0:1] [2:5] [4:5] [0:1][2:5] [4:5] T

30
Discrete Transitions P1P1 P3P3 P2P2 P4P4 [1:3][2:4] [0:0] 0. 0 P1P1 P3P3 P2P2 P4P4 [2:4] [0:0] [1:3] t t Firing t [0:1] [2:5] [4:5] [0:1][4:5] [2:5] [4:5] D

31
Transitions = T D U M1M1 M2M2 If M 1 T M2M2 or M 1 D M2M2 Remark : A TPN can have unbounded number of tokens !! Additionally, there are some lossy transitions in lossy TPN.

32
Ordering on Marking P1P1 P3P3 P2P2 P4P4 [1:3][2:4] [0:0] P1P1 P3P3 P2P2 P4P4 [2:4] [0:0] [1:3] t t [0:1] [2:5] [4:5] [0:1][4:5] [2:5] [4:5] 6.2 M1M1 M2M2 P 1,2. 0 P 2,3. 7 P 2,3. 5 P 1,2.2 P 1,2. 0 P 4,max frac = 0 Increasing fractional parts age >= 5 M1M1 M2M2

33
Finite no. of clocks (e.g Timed Automata) x y Two clocks x,y and c max = 3 Clock values are equivalent in timed automata if they have same integral parts same ordering of fractional parts clock values beyond c max are equivalent 0 Regions

34
Region R : x y V(x) = 0.6, V(y) = 0.5 V € R Not Powerful for Timed Petri Nets…… Regions(Example)

35
Dlgs for LTPN P1P1 P3P3 P2P2 P4P4 [1:3][2:4] [4:5][1:5] [4:5] [0:1][2:5] [4:5] c max = 5 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 c max are equivalent. Unboundedness in two directions : number of tokens age of tokens Abstraction of ages to express sets of markings :

36
Dlgs for LTPN Markings are downward closed for LTPN Constraints = strings of bags over a finite alphabet of (Places x {0,..max}) Sets of markings and Constraints are dlgs for strings of bags over a finite set !!!

37
Universal Regions ! P1P1 P3P3 P2P2 P4P4 [1:3][2:4] [4:5][1:5] [4:5] [0:1][2:5] [4:5] 2.0 Note : M can have at most same number of tokens as R. If M’ < M, then M’ R M = 2 0 4* 5 3 P1P1 P2P2 P3P3 P4P4 R = frac = 0Increasing fracage >= 5 * 3.75 P2P2

38
Universal Regions (contd.) P1P1 P3P3 P2P2 P4P4 [1:3)[2:4) [0:5)[1:3) [4:5) [0:1)[2:5) [4:5) t Let Universal Region R = c max = 5 T dlg Generates O((max-1)*2 + sizeof(product) + 1) new regions by timed transition. 2 3 Max bagZero bag max

39
Universal Regions (contd.) Lot of universal regions !!!Solution : Universal Zones !! t x3x3 x4x4 0 T followed by At most one token in P3 and one token in P4 with ages as follows : 2 3

40
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

41
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.

42
Ongoing Work Compute Post*(R,t) for all transitions t. Apply forward reachability algorithm. Define universal zones.

Similar presentations

© 2017 SlidePlayer.com Inc.

All rights reserved.

Ads by Google