Hardwareand Petri nets

Symbolic methods foranalysis and verification

Outline

• Representing Boolean functions with BDDs

• Symbolic traversal for reachability set calculation

• State encoding

• Structural methods for efficient encoding

Representing Boolean functions

a

b

c

b

c c c

1 0 0 1 1 0 1 0

1

1

1 1 1 1

1

0

0

0 0 0 0

0

Representing Boolean functions

a

b b

c c

0 1 1 0

1

1

1 1

1

0

0

0 0

0

Representing Boolean functions

a

b

c c

0 1 1 0

1

1

1 1

0

0

0 0

Representing Boolean functions

a

b

c c

0 1

1

1

1 1

0

0

0 0

Binary Decision Diagrams

a

b

c c

0 1

1

1

1 1

0

0

0 0

Reduced OrderedBinary Decision Diagram

• All variables appear in the same order

• No isomorphic subgraphs

• Canonical form• Efficient form for many

functions• [Bryant,

ACM Comp. Surveys,September 1992]

Reachable markings

p1

p6 p7

p5p3

t2

t5 t6

t1

t4t3

t7

p2 p3

t1

p4 p5

t2

t3 t5t6

p1

t5

p6 p5

t6

p4 p7

t4

p6 p7

t3

p6 p3

t4

p2 p7

p2 p4 t7

Boolean encoding

p1

p6 p7

p5p3

t2

t5 t6

t1

t4t3

t7

p2 p4

0110000

t1

0001100

t2

t3 t5t6

1000000

t5

0000110

t6

0001001

t4

0000011

t3

0010010

t4

0100001

t7

Seven variables:p1 p2 p3 p4 p5 p6 p7

Boolean encoding

0110000

t1

0001100

t2

t3 t5t6

1000000

t5

0000110

t6

0001001

t4

0000011

t3

0010010

t4

0100001

t7

Seven variables:p1 p2 p3 p4 p5 p6 p7

p1 p2 p3 (p4 p6) (p5 p7)

Enabled(t7) = p6 p7

p1 p2 p3 p4 p5 p6 p7

(toggle p1, p6 , p7)

p1 p2 p3 p4 p5 p6 p7

Symbolic Traversal (BFS algorithm)

Reached = From = {m0 }; repeat From [ T To; New = To \ Reached; From = New; Reached = Reached New; until New = Ø;

# iterations: sequential depth of the net

Reachability Set computation

• Based on BFS Image computation

S0 = M0

Si+1 = Si Image (Si)

S0

S1

S2

S3

Monotonic increase until fix point Si+1 = Si

#iteration

Newmarkings

0110000 0001100

1000000

0000110 0001001

0000011

0010010 0100001

001 100

000

110 010

111

011 101

Boolean encoding

t1 t2

t3 t5t6

t5 t6

t4

t3 t4 t7

Seven variables: p1 p2 p3 p4 p5 p6 p7

Sparse encoding:

Optimal encoding:

Three variables log2 |RG| : v1 v2 v3But the reachability graphshould be known a priori ...

Very easy to derive and useLess efficient in terms of BDDs

Encoding for safe PNs

• Not all combinations of tokens are possible.

• Find relations among places to reduce the number of variables!!!

p4p2p1 p3 p3 p4

p1 p2

Ø

Encoding for safe PNs

PN structure: Place Invariant• Set of places with a constant weighted sum of tokens

• Specially efficient for safe PNs (State Machines)

k1p1+ k2p2+…..+ knpn = B

p1+ p2+…..+ pn = 1

• Computed by linear programming techniques

p1

p6 p7

p5p3

t2

t5 t6

t1

t4t3

t7

p2 p4

p1

p6 p7

p5p3

t2

t5 t6

t1

t4t3

t7

p2 p4

State Machine Components

p1

p6 p7

p5p3

t2

t5 t6

t1

t4t3

t7

p2 p4

p1

p6

t2

t5

t1

t3

t7

p2 p4

p1

p7

p5p3

t2

t6

t1

t4

t7

Encoding for safe PNs

p1

p6

t2

t5

t1

t3

t7

p2 p4

p1

p7

p5p3

t2

t6

t1

t4

t7

Two additional variables: v3 v4Two variables: v1 v2

00

0110

11

00

11

0110

Encoding for safe PNs

p1

p6 p7

p5p3

t2

t5 t6

t1

t4t3

t7

p2 p4

Four variables: v1 v2 v3 v4

10 - - - - 10

01 - -

0000

- - 01

- - 1111 - -

1010 0101

0000

1101 0111

1111

1110 1011

t1 t2

t3 t5t6

t5 t6

t4

t3 t4 t7

2010 3002

120611120120 0214

2104

Sparse encoding: 10 variablesDense encoding: 3 variables

010 000

011100111 110

001

Encoding for bounded PNs

p1

p3

t2 t1

p4 t3

p2

2

2

2

max(p2)=3max(p1)=3

max(p4)=8 max(p3)=2

t2t1t1t1 t3t3

t3

t3

t3t2

t2

Invariants of the PN:

I1: 2p1 + 4p2 - p4 = 4

I2: p1 + p2 + p3 = 3

2010 3002

120611120120 0214

2104

Dense encoding: 3 variables

Encoding for bounded PNs

Invariants of the PN:

I1: 2p1 + 4p2 - p4 = 4

I2: p1 + p2 + p3 = 3

Dense encoding: 3 variables

t2t1t1t1 t3t3

t3

t3

t3t2

t20308

t2

Invariants characterizethe set of allpotentially reachablemarkings(an overestimationof the reachablemarkings)

Encoding for bounded PNs

2p1 + 4p2 - p4 = 4

p1

p2 p2p2p2

p4 p4 p4 p4 p4p4p4p4

0 1 3 5 6 7

p4 p4

2 984

01 2 3

212 31 0210 1

0 464 28 0 28 6

p1 + p2 + p3 = 3

p1

p2

p3p3

5 6

2

10

01

p2

7

p3

0

3

0

3 variables4 variables

(8 vars)

Encoding for bounded PNs

p1 + p2 + p3 = 3

p1

p2 p2p2

p3 p3 p3 p3 p3

0 1 3

p3

2 74

0 1 2 3

212 31 0

2 01 10 0

p2

p3 p3

3 4

10

01

M(p1)+M(p2)

p3 p3 p3

0 1 2

2 31

2 1 0

Encoding for bounded PNs

x1

x2x2

x3 x3x3x3

0

T F

FT FT

T

0

F

0

T

1

F

1

T

2

F

2

T

3

F

1 2 3 1 2 0 1 0

p1

p2

2p1 + 4p2 - p4 = 4 p3 = 3 - p1 - p2

x1

x2x2

x3 x3x3x3

2

T F

FT FT

T

1

F

0

T

1

F

0

T

1

F

0

T

0

F

p3

Upper bounds for the state space

F F F F F F

x2

x3x3

x4 x4x4x4

T

T F

FT FT

T

T

F

T

T

T

F

T

T

T

F

T

T

T

F

x2

x3x3

x4 x4x4x4

T

T F

FT FT

T

T

F T F T F T F

x1

T F

2p1 + 4p2 - p4 = 4x1

x2

x3

T

F

FT

T

T

F

F

Characteristic function for potentially reachable states

Experimental results

PN States Vars |BDD| Ni Nnmuller10 4,2 x 102 40 / 20 770 / 189 10 40 1 1

muller20 2,5 x 105 80 / 40 3188 / 668 20 80 9 3

muller30 6,0 x 107 120 / 60 6694 / 1390 30 120 51 13

phil5 8,5 x 104 65 / 25 639 / 158 15 125 2 2

phil10 7,4 x 109 130 / 50 7805 / 433 30 250 40 24

phil15 6,4 x 1014 195 / 75 87419 / 708 45 375 700 124

slot5 1,7 x 106 50 / 25 673 / 129 10 50 14 5

slot10 3,8 x 1011 100 / 50 2516 / 460 20 100 1006 309

CPU

Conclusions

• Formal verification and synthesis oftensuffer from the state explosion problem

• Symbolic techniques can be used to efficientlyrepresent the state space

• Structural techniques are crucial to overestimateand encode the state space

• Try to resist the temptation of using BDDs fromthe very beginning. Use them rationally and onlyif desperate.