[17] R. van Glabbeek, The linear time - branching time spectrum, in: Proc.

CONCUR'90, Lecture Notes in Computer Science, Vol. 458, (Springer, Berlin,

1990) 278-297.

22

References

[1] S. Christensen, Decidability and decomposition in process algebras. Ph.D.

Thesis, Univ. of Edinburgh, U.K., CST-105-93, 1993.

[2] S. Christensen, Y. Hirshfeld and F. Moller, Bisimulation equivalence is decidable

for basic parallel processes, in: Proc. CONCUR'93, Lecture Notes in Computer

Science, Vol. 715 (Springer, Berlin, 1993) 143-157.

[3] S. Christensen and H. H�uttel, Decidability issues for in�nite-state processes { a

survey, Bulletin of the EATCS Vol. 51 (1993) 156-166.

[4] J. Esparza, On the uniform word problem for commutative context-free

grammars. Technical Report, Lab. of Foundations of Computer Science, Univ.

of Edinburgh, U.K., 1993.

[5] J. Esparza and M. Nielsen, Decidability issues for Petri nets { a survey, Bulletin

of the EATCS Vol. 52 (1993) 245-262.

[6] Y. Hirshfeld, Petri nets and the equivalence problem, in: Proc. CSL'93, 1993

Conf. of European Association for Computer Science Logic, Lecture Notes in

Computer Science, Vol. 832 (Springer, Berlin, 1993).

[7] Y. Hirshfeld, M. Jerrum and F. Moller, A polynomial time algorithm for deciding

bisimulation equivalence of normed basic parallel processes, Tech. Report, Dept.

of Computer Science, Univ. of Edinburgh, U.K., 1994.

[8] H. H�uttel, Undecidable equivalences for basic parallel processes, Lecture Notes

in Computer Science, Vol. 789, (Springer, Berlin, 1994) 454-464.

[9] D. Huynh, The complexity of semilinear sets, Inform. Process. Cybernet. (EIK)

18 (1982) 291-338.

[10] D. Huynh, Commutative grammars: the complexity of uniform word problems,

Inform. and Comput. 57 (1983) 21-39.

[11] D. Huynh, The complexity of equivalence problems for commutative grammars,

Inform. and Comput. 66 (1985) 103-121.

[12] D. Huynh, A simple proof for the �

P

2

upper bound of the inequivalence problem

for semilinear sets, Inform. Process. Cybernet. (EIK) 22 (1986) 147-156.

[13] J. Peterson, Petri Net Theory and the Modeling of Systems (Prentice-Hall,

Englewood Cli�s, NJ, 1981).

[14] W. Reisig, Petri Nets: An Introduction, EATCS Monographs in Computer

Science, Vol. 4 (Springer-Verlag, Berlin, 1985).

[15] L. Stockmeyer, The polynomial-time hierarchy, Theoret. Comput. Sci. 3 (1977)

1-22.

[16] H. Yamasaki, Normal Petri nets, Theoret. Comput. Sci. 31 (1984) 307-315.

21

Now we show the size of the semilinear set representation as well as the

time required for generating such a representation. From Condition 1, each

component of � is bounded by 2

d

1

s

2

; hence jBj (i.e., the number of distinct

bases of the semilinear set) is bounded by (2

d

1

s

2

)

k

� 2

d

1

s

3

(k(� s) is the

dimension of �). As a result, the size of B is bounded by k � (log

2

(2

d

1

s

2

)) �

(2

d

1

s

3

). Likewise, from Condition 2 the size of each period �

�

is bounded by

k � (log

2

(2

d

2

s

2

)) � (2

d

2

s

3

). In summary, the size of

S

�2B

L(�; �

�

) is (the size of

B) +

X

�2B

(the size of �

�

), which is bounded by O(2

c

1

s

3

), for some constant c

1

.

As for the amount of time needed to generate the semilinear set, �rst recall

from Theorem 4 that the reachability problem for BPP-nets in NP. From

our earlier discussion, each base vector � is of size k � (log

2

(2

d

2

s

2

)), which is

polynomial in O(s

3

). Hence, the reachability of � from �

0

can be checked in

DTIME(2

O(s

3

)

). Similarly, checking the existence of a positive loop satisfying

Conditions 2 (a) and (b) can also be done in DTIME(2

O(s

3

)

). By exhaustive

search, the desired semilinear set can be constructed in DTIME(2

c

2

s

3

), for

some constant c

2

. 2

To show our main result, we also require the following known result con-

cerning the complexity of the equivalence problem for semilinear sets (see [9]

and [12]):

Lemma 6 (from Corollary 5.2 in [9]) The equivalence problem for semilinear

sets is in �

P

2

.

From Theorem 5 and Lemma 6, we immediately have:

Corollary 7 The equivalence problem for BPP-nets is solvable in DTIME(2

2

d�s

3

),

where s is the size of the PN, and d is some �xed constant.

As for the lower bound, it is known that the equivalence problem for

commutative context-free grammars is �

P

2

-hard (see [11]). Since commutative

context-free grammars are a special case of BPP-nets, the following lower

bound for BPP-nets follows immediately.

Theorem 8 The equivalence problem for BPP-nets is �

P

2

� hard.

Acknowledgement

The author thanks the anonymous referees for their comments and suggestions

which improved the presentation as well as the correctness of this paper.

20

maximize

m

X

i=1

x

i

subject to

8

>

<

>

:

�(�) + A � (x

1

; :::; x

m

)

T

� 0

(x

1

; :::; x

m

) � #

�

In words, solution X=(x

1

; :::; x

m

) represents the transition count vector of the

maximum �rable sequence contained in � using only tokens accumulated as a

result of �ring circuit �. Consider two cases:

(Case 1): X 6= 0, for some � in C. Since � is circuit-free, guaranteed by

Lemma 3 there exists a �

0

such that #

�

0= (x

1

; :::; x

m

) and ��

0

is enabled

in some �� in �

0

�

00

z }| {

�

7�! ��

�

7�!�

00

. Let � = �(��

0

) = �(�) + A � X

T

� 0. Since

(�(�)(p

i

) � mn;8p

i

2 P), the maximum number of tokens that can pile up

in a place using a circuit-free transition sequence is bounded by (mn)n

l�1

;

where l = minfm;k � 1g. (The worst-case scenario arises when �

0

is a path

p

i

t

g

1

p

g

1

� � � p

g

j�1

t

g

j

p

g

j

� � � t

g

l

p

g

l

along which each transition t

g

j

`ampli�es' the

token count of its input place by a factor of n.) Hence, (�(i) = �(��

0

)(p

i

) �

mn+(mn)n

l�1

� 2mn

l

(� 2

d

2

s

2

, for some constant d

2

), 8p

i

2 P ), and j��

0

j �

�j�j

z}|{

m +

�j�

0

j

z }| {

mn+mn

2

+mn

3

+ � � �+mn

l

� (l + 1)mn

l

. We let D := D [ f�g,

C := C � f�g, �̂ := �̂ + �(�

0

), and � := a �rable sequence (in �̂) using the

transitions from �

.

�

0

. See Figure 5(d). (Notice that �̂+�(� ) = � and � being

circuit-free imply the existence of such a �rable sequence (Lemma 3)). Clearly,

the invariant remains true.

(Case 2): The above iteration ends when X = 0, for every � in C, or C = ;.

We claim that �

00

�

7�! ~�, for some ~�. If this is the case, by letting the base (�)

and the set (�

�

) of the periods of the semilinear set be ~� and f�(�)j� 2 Cg[D,

respectively, we have �

0

�

00

�

7�! ~�, and ~�+(

X

�2C

�(�))+(

X

�2D

�) = �. Furthermore,

in our earlier discussion we know that 8p

i

2 P , �

00

(p

i

) � n + 2m

3

n � 3m

3

n.

Hence, j� j � 3m

3

n+ 3m

3

n

2

+ � � �+ 3m

3

n

l

� 3lm

3

n

l

, and ~�(p

i

) � (3m

3

n)n

l�1

(� 2

d

1

s

2

, for some constant d

1

), since � is circuit-free. See Figure 5(e). Now

we prove the claim (i.e., �

00

�

7�! ~�). If the �ring of � in �

00

fails for some

transition t in � , then it must be the case that some circuit in C provides a

token for t in the original path { contradicting the fact that in (Case 2), either

the solution to the above optimization problem is 0 or C is empty. (Notice

that the token needed by t cannot come from any ��

0

that has already been

added to D; otherwise, �

0

violates the requirement of being the maximum

solution.) By picking a constant d

3

so that j��

0

j � (l + 1)mn

l

� 2

d

3

s

2

and

�

00

� � (n+ 2m

3

n) + 3m

3

n

l

� 2

d

3

s

2

, Condition 2(b)iii holds.

19

Fig. 5. The construction of a semilinear set representation of a BPP-net.

18

into Q. It is important to note that for every � in Q, � is enabled in some �� in

�

0

�

00

z }| {

�

7�! ��

�

7�!�

00

. Now consider the length of �

00

. According to Condition 1(c)

of Lemma 2, the number of distinct circuits `collected' along �

00

is bounded

by m. Furthermore, the above discussion suggests that there are at most m

copies in existence for each such circuit. Hence, the sum of the lengths of all

such circuits is bounded by m

3

(there are at most m

2

circuits, each of which is

of length � m). In addition, the length of �

1

� � ��

h�1

is bounded by (h�1)�m

(� m

2

). As a consequence, j�

00

j � m

3

+m

2

(� 2m

3

); hence, no component in

�

00

exceeds n+ 2m

3

n (i.e., an upper bound on the size of the initial marking

+ an upper bound on the number of tokens that can be deposited into a place

resulting from �ring �

00

) = O(m

3

n).

Now consider �

0

�

7�! �, which covers no circuits. Unlike �

1

; :::; �

h�1

(all

of which are short), in � (=�

h

) the number of times a transition is used may

not be bounded by a polynomial. In our subsequent discussion, we show how

pieces of the su�x � (if it is too long) can be paired with circuits in Q, result-

ing in a su�ciently short su�x. Upon the completion of this `pairing' process,

the base and periods of the semilinear set follow immediately. Such a con-

struction is done in an iterative fashion. Initially, let variables C (a multiset

of circuits), D (a multiset of vectors), �̂ (a marking, which can also be viewed

as a vector), and � (a transition sequence) be Q, ;, �

0

, �, respectively. The key

of our construction relies on proving that the following remains an invariant

as the iteration progresses:

� (�

0

�

00

7�! �

00

) ^ (�

00

+ (

X

�2C

�(�)) + (

X

�2D

�) = �̂) ^ (�̂

�

7�! �).

To illustrate the intuition, consider Figure 5(c). We show that if � (which

equals � initially) is `too long,' then it must contain a short segment �

0

(see

Figure 5(c)) that can be paired with some circuit � in C (see Figure 5(d))

in such a way that �(��

0

) remains nonnegative. (Hence, �(��

0

) can then

be placed into the �nal set of periods.) By repeatedly doing so, � can be

shortened.

Clearly the invariant holds initially. In what follows, we explain in detail

how the above intuition of constructing the semilinear set (in an iterative

fashion) is implemented. Recall that an ordering is assumed on the elements

of P and T , (i.e., P = fp

1

,...,p

k

g and T = ft

1

,...,t

m

g). Let A be a k � m

addition matrix of (P; T; ') so that a

i;j

= '(t

j

,p

i

) - '(p

i

,t

j

). Now let � be a

circuit in C. Consider the following optimization formula (in which x

1

; :::; x

m

are nonnegative variables):

17

O(2

c

1

s

3

)), where

1. B is the set of all reachable markings with no component larger than 2

d

1

s

2

,

and

2. �

�

is the set of all # 2 N

k

such that

(a) # has no component larger than 2

d

2

s

2

, and

(b) 9 �; �

1

; �

2

2 T

�

, 9 marking �

1

,

i. �

0

�

1

7�! �

1

�

2

7�! �,

ii. �

1

�

7�! �

1

+ #,

iii. j�j; j�

1

�

2

j � 2

d

3

s

2

.

Proof. Let P be of k places and m transitions, and n be a number such

that �

0

(p) � n and j'(t; p)j � n, 8t 2 T; p 2 P , i.e., no integer mentioned

in P is larger than n. (Recall that for BPP-nets, it is always the case that

j'(p; t)j � 1, 8t 2 T; p 2 P .) Clearly, m;k � s and n � 2

s

.

[

�2B

L(�; �

�

) � R(P) is obvious, since, according to Condition 2(b)ii, � can

be pumped in marking �

1

for an arbitrary number of times. Therefore, it is

su�cient to show R(P) �

[

�2B

L(�; �

�

). The proof is somewhat involved. To

better explain the details, Figure 5 illustrates the key steps of the proof. The

reader is encouraged to consult Figure 5 as our discussion progresses.

Let � 2 R(P) be a reachablemarking. According to Lemma 2, there exists a

sequence �

1

�

1

�

2

�

2

� � ��

h

�

h

which witnesses �

0

�

7�! �, and satis�es Conditions

(1), (2) and (3) stated in the description of Lemma 2. See Figure 5(a). For

ease of explanation, let �

0

= �

1

�

1

�

2

�

2

� � ��

h

, � = �

h

, and �

0

�

0

7�! �

0

�

7�! �, for

some �

0

. Recall that �(�

i

) =

r

i

X

j=1

a

i

j

�(c

i

j

) for some integers a

i

1

; a

i

2

; :::; a

i

r

i

> 0

(see Lemma 2). (In Figure 5, for example, �

1

consists of three circuits c

1

; c

2

,

and c

3

of multiplicities 60, 50 and 10, respectively.) Consider circuits in �

i

,

1 � i � h. It is clear from Condition 2(b) of Lemma 2 that for any place p, each

of �

i

; :::; �

h�1

consumes at most one token from p; hence, the entire sequence

�

i

; :::; �

h�1

consumes at most h� i tokens from p. Now if a

i

j

> m(� h� i), for

some j, 1 � j � r

i

, then �

0

�

0

.

c

i

j

7�! (�

0

��(c

i

j

)) remains a valid path. In words,

a copy of circuit c

i

j

can be cut without rendering the path invalid. (This is

mainly because if the �ring of �

i

; :::; �

h�1

hinders on circuit c

i

j

, m copies of

c

i

j

s su�ce.) By trimming excess copies of circuits in �

0

repeatedly (called the

resulting sequence �

00

), we have �

0

�

00

7�! �

00

, for some �

00

, such that no circuit in

�

00

appears more than m times, and �

00

+

X

�2Q

�(�) = �

0

, where Q is a multiset

containing those circuits cut in the above trimming process. In Figure 5(b),

m is assumed to be 10; hence, 50 copies of c

1

and 40 copies of c

2

are thrown

16

2. Let �

i

and �

0

i

(� 0) be marking variables, and a

i

j

and b

j

be scalar variables

carrying positive integer values. Set up the following linear inequalities to

capture PN computation �

0

= �

1

�

1

7�! �

0

1

�

1

7�! �

2

�

2

7�! �

0

2

�

2

7�! � � � �

h

�

h

7�!

�

0

h

�

h

7�! �:

8

>

>

>

>

>

>

>

>

>

>

>

>

<

>

>

>

>

>

>

>

>

>

>

>

>

:

�

1

= �

0

�����(1)

�

0

i

= �

i

+

r

i

X

j=1

a

i

j

��(c

i

j

);81 � i � h �����(2)

�

i+1

= �

0

i

+�(�

i

);81 � i � h � 1 �����(3)

� = �

0

h

+

l

X

j=1

b

j

��(t

h

j

) �����(4)

(1) is trivial. For (2), the validity of �

0

i

being reachable from �

i

is guaranteed

by Lemma 1. Since �

i

(1 � i < h) is circuit-free, (3) is su�cient to ensure

the reachability of �

i+1

from �

0

i

through the �ring of �

i

, as Lemma 3 suggests.

Likewise, circuit-freedom of �

h

justi�es (4). (Notice that the need to consider

�

h

separately comes from the fact that �

h

may contain multiple copies of a

transition. See Lemma 2.)

In view of the above, � is reachable from �

0

i� the above system of linear

inequalities has integer solutions with respect to variables �

i

, �

0

i

, a

i

j

and b

j

.

This completes the proof of the theorem. 2

The NP upper bound of testing reachability for BPP-nets was �rst shown

in [4], providing a simpler proof for the NP upper bound of the uniform word

problem for context-free commutative grammars, which was originally shown

in [10]. (The concept of the so-called siphon plays a crucial in the proof of

[4].) By providing a new characterization for paths in BPP-nets, we o�er yet

another approach to solving the reachability problem for BPP-nets.

4 The equivalence problem for BPP-nets

In this section, we investigate the complexity of the equivalence problem for

BPP-nets. Our upper bound is obtained by demonstrating `small' semilinear

set representations for the reachability sets of BPP-nets. More precisely, we

have the following theorem:

Theorem 5 Let P=((P, T, '), �

0

) be a BPP-net of size s. For some �xed

constants c

1

; c

2

; d

1

; d

2

; d

3

independent of s, we can construct in DTIME(2

c

2

s

3

)

a semilinear reachability set R(P) =

[

�2B

L(�; �

�

) (whose size is bounded by

15

one more transition becomes absent in the remaining sequence. Repeat the

above procedure (at most m times) until no more circuit can be added to C

1

.

Following Lemma 1, �

1

, a sequence consisting of a

1

j

copies of c

1

j

(1 � j � i

1

),

can be constructed. Now suppose �

1

= �

0

+�(�

1

), and �

1

= �

.

�

1

. Guaranteed

by the claim stated in the beginning of the proof, there exists a rearrangement

�

0

1

of �

1

such that �

1

�

0

1

7�!. If �

0

1

is circuit-free, we are done; otherwise, let �

00

1

be the shortest pre�x of �

0

1

such that the remaining sequence �

0

1

.

�

00

1

covers

a circuit, say �c, which is not token-free with respect to �

2

, where �

1

�

00

1

7�! �

2

.

(Notice that it is possible for a �c to be marked in �

1

; in this case, �

00

1

is empty.)

Since �

00

1

is circuit-free, and each transition in �

00

1

removes at most one token

from a place, �

00

1

can be rearranged into �

1

�

0

1

in such a way that (8t 2 T,

#

�

1

(t) � 1), (�(�

1

)(p) � 1;8p 2 P ), and (�

0

2

(�c) > 0), where �

1

�

1

7�! �

0

2

.

(Intuitively, �

1

is a simple path (in the graph-theoretic sense) leading to some

place in �c.) The remaining �

i

and �

i

can be constructed similarly. 2

Using Lemma 2, we are able to set up a system of linear inequalities to

capture reachability for BPP-nets, giving rise to an NP upper bound for the

reachability problem since integer linear programming is in NP. Before doing

so, we require the following known result from [16].

Lemma 3 (From Theorem 3.3 in [16]) If a PN P=((P, T, '), �

0

) has no

token-free circuits in every reachable marking, then R(P)=f�j� = �

0

+A� x �

0, for some x 2N

m

g, where m is the number of transitions in T , and A is the

addition matrix. In words, � = �

0

+ A � x � 0 is a su�cient and necessary

condition for reachability provided that no token-free circuit is reachable in the

PN.

As a direct consequence, � = �

0

+A � x � 0 is also a su�cient and necessary

condition for reachability for circuit-free PNs.

Theorem 4 The reachability problem for BPP-nets can be solved in NP.

Proof. As shown in Lemma 2, � is reachable from the initial marking �

0

i�

there exists a sequence �=�

1

�

1

�

2

�

2

� � � �

h

�

h

(�

0

�

7�! �) meeting the three

conditions stated in Lemma 2.

The desired system of linear inequalities can be set up as follows:

1. For 1 � i � h, guess C

i

(=fc

i

1

; :::; c

i

r

i

g) and verify the connectivity condi-

tion; for 1 � i � h� 1, guess the sequence �

i

and check Conditions 2(a),

(b) and (c) of Lemma 2; guess the set ft

h

1

; :::; t

h

l

g of transitions used in

�

h

and verify the circuit-freedom condition. It is not hard to see that

checking each of the above can be done in polynomial time.

14

Fig. 4. A picture describing the concept used in the proof of Lemma 2.

13

�

3

�

0

7�!, it must be the case that j�

0

j � j�j.) Let � = (�

.

�

1

)

.

�. We let X be

fpj

�

t = fpg; t 2 Tr(�)g, i.e.,X consists of all the input places of transitions in

Tr(�). Clearly, �

4

(p) = 0;8p 2 X. See Figure 4. We now make the following

observations:

1. 8p 2 X;9t

0

2 Tr(�); such that p 2 t

0�

. (This is because �

4

(p)+�(�)(p) =

�

2

(p) � 0 and �

4

(p) = 0.)

2. there must be some place r in X such that either (i) �

1

(r) > 0, or (ii)

(9 t

1

2 Tr(�

1

�)) (9t

2

2 Tr(�

1

�)) such that (r 2 t

�

1

) and

�

t

2

= frg. (If

neither (i) nor (ii), then none of the transitions in Tr(�) could be �red

in the original sequence �. The existence of a t

2

results from �

4

(r) = 0.)

Let R be the set of all places r satisfying Observation 2(i) or (ii) above.

What we need next is to show that at least one place in R must be along a

circuit consisting of some places in X and some transitions in Tr(�). Suppose,

to the contrary, that none of R is on a circuit; then there must be an s 2 R

such that s cannot be reached from the remaining places in R through places

in X and transitions in Tr(�). For s, let t

3

be a transition guaranteed by

Observation 1 above. Due to the selection of s, t

3

could never have been �red

in � since its input place would never possess a token (because the input place

of t

3

(i.e.,

�

t

3

) is not in R, and none of R is capable of supplying a token to

�

t

3

directly or indirectly) { a contradiction. Intuitively, one can think of R

as places through which tokens are `pumped' into the sub-PN consisting of

places in X and transitions in Tr(�).

Let r 2 R be a place on a circuit, and t

2

(whose existence is guaranteed

by Observation 2) be a transition in �

1

� removing a token from r. If t

2

is in �

1

(which comprises only circuits from C), then c must have shared some place

with one of the circuits in C { violating Assumption (b) of the claim. If t

2

is

in �, then r is marked during the course of the path �, which implies that c

should have been added to � { violating the assumption about � being the

longest. This completes the proof of the claim.

In what follows, we only show how �

1

and �

1

are constructed; the remaining

sequences can be obtained similarly. Suppose � covers a circuit c which is not

token-free in �

0

(i.e., �

0

(c) > 0). Initially let C

1

=fc

1

1

g = fcg. (If � does not

cover any circuit marked in �

0

, then �

1

is empty.) The associated integer

a

1

1

is the maximum number of occurrences of c in �, i.e., a

1

1

(#

c

) � #

�

but

a(#

c

) 6� #

�

;8a > a

1

1

. Let �

0

be the resulting sequence of removing a

1

1

copies

of c from �. That is, �

0

= �

.

a

1

1

z }| {

c � � � c. It is important to notice that at least one

of c's transitions is no longer in existence in �

0

. The next step is to �nd, if one

exists, a circuit c

0

which shares some place with at least one of C

1

; then add c

0

to C

1

(i.e., c

1

2

= c

0

) and remove a

1

2

copies of c

0

from �

0

, where a

1

2

is the maximum

number of occurrences of c

0

in �

0

. Upon the completion of the above, at least

12

(a) there exists a set C

i

=fc

i

1

; :::; c

i

r

i

g (r

i

� m) of connected circuits such

that

�(�

i

) =

r

i

X

j=1

a

i

j

�(c

i

j

) for some integers a

i

1

; a

i

2

; :::; a

i

r

i

> 0,

(b) the remaining sequence �

i

� � � �

h

�

h

does not cover any circuit which

shares some place with circuits in C

i

, and

(c)

h

X

i=1

jC

i

j � m, i.e., the total number of distinct circuits considered above

is bounded by the number of transitions of the PN.

2. 8i; 1 � i � h� 1,

(a) #

�

i

(t) � 1;8t 2 T (in words, all transitions in �

i

are distinct),

(b) �(�

i

)(p) � 1;8p 2 P (in words, �

i

removes at most one token from

any place), and

(c) �

i

is circuit-free (i.e., it does not cover any circuit).

3. �

h

is circuit-free. Notice that �

h

may contain multiple copies of a transi-

tion.

Proof. We begin by proving the following claim which tells how `cut-and-

paste' technique can be applied to BPP-nets.

� Claim: Consider a path �

1

�

7�! �

2

. Let C=fc

1

; c

2

; :::; c

z

g be a set of con-

nected circuits and a

1

; a

2

; :::; a

z

be positive integers such that

(a) (9i; 1 � i � z) (�

1

(c

i

) > 0) (i.e., c

i

is not token-free in marking �

1

)

(b) �

.

(c

a

1

1

� � � c

a

z

z

) does not cover any circuit that shares some place with

circuits in C, and

(c)

z

X

j=1

a

j

(#

c

j

) � #

�

.

Then there exist �

1

and �

2

such that

(1) #

�

1

=

z

X

j=1

a

j

(#

c

j

),

(2) #

�

2

= #

�

.

�

1

, and

(3) �

1

�

1

7�! �

3

�

2

7�! �

2

, for some �

3

.

(In words, � can be rearranged into �

1

�

2

such that �

1

consists of the largest

collection of connected circuits with at least one of them marked in �

1

.)

To prove our claim, �rst notice that �

1

�

1

7�! �

3

is guaranteed by Lemma 1; it

su�ces to prove that �

3

�

2

7�! �

2

, for some �

2

which is a rearrangement of �

.

�

1

.

Suppose, to the contrary, that none of the permutations of �

.

�

1

is �rable in

�

3

. We let � be a longest sequence such that #

�

< #

�

.

�

1

and �

3

�

7�! �

4

,

for some �

4

. (By `longest' we mean that for all �

0

with #

�

0< #

�

.

�

1

and

11

Fig. 3. Rearranging a path in a BPP-net into a canonical one.

10

Fig. 2. A set of connected circuits.

9

in C. To help explain the proof, see Figure 2.

(Induction Basis): For n = 1, the result is quite obvious. (The sequence t

1

� � � t

r

in Figure 2 can be �red an arbitrary number of times.)

(Induction Hypothesis): Assume that the assertion is true for n � h.

(Induction Step): Consider n = h + 1. Starting from place p

1

, let p

2

; ::; p

r

,

for some r, be places along c

1

. Let C

j

(1 � j � r) be the largest connected

subset of C � fc

1

g � (

[

0�l�j�1

C

l

) in which one of its circuits contains place

p

j

. (Notice that C

j

might be empty.) See Figure 2. By induction hypothesis,

all circuits in C

j

can be �red arbitrarily, provided that p

j

is marked. Let t

i

(1 � i � r) be the transition from places p

i

to p

i+1

along circuit c

1

(assuming

that p

r+1

= p

1

). Then the desired sequence � is the following: (sequence guar-

anteed by induction hypothesis for C

1

) t

1

(sequence guaranteed by induction

hypothesis for C

2

) � � � t

r�1

(sequence guaranteed by induction hypothesis for

C

r

) t

r

(t

1

� � � t

r

)

a

1

�1

. 2

The idea of rearranging an arbitrary path in a BPP-net into a `canonical'

one is as follows. To give the reader a better feel for such a rearrangement,

we accompany our subsequent discussion with Figure 3. Suppose �

�

7�! is a

path, and c is a circuit covered by � such that �(c) > 0. (In Figure 3, circuit

c consists of transitions a; b; d; e, and f .) Then we use c as a `seed' to grow

the largest collection of connected circuits that are covered by � (for example,

circuits c and c

0

in Figure 3). We then follow a `short' circuit-free transition

sequence of the remaining path until reaching a marking in which a non-

token-free circuit (with respect to the current marking) which is covered by

the subsequent path exists. (See marking �

0

and circuit c

00

in Figure 3.) Using

such a newly found circuit as a new seed and repeating the above procedure,

we are able rearrange an arbitrary path of a BPP-net into a `canonical' one as

the following lemma indicates. Notice that the above procedure need not be

repeated for more than m times, because for each of the circuits collected in

a marking, at least one of its transitions must be absent from the remaining

path.

It is important to point out that the above rearrangement procedure is

merely `conceptual.' That is, we do not actually carry out the above procedure

in the derivation of our complexity result. What the rearrangement concept

does is that it suggests the existence of a canonical computation, upon which

our derivation of semilinear set representations for BPP-nets relies.

Lemma 2 Let � be a reachable marking in a BPP-net P = ((P, T, '), �

0

).

Then there exists a sequence �=�

1

�

1

�

2

�

2

� � ��

h

�

h

(1 � h � m, �

i

; �

i

2 T

�

)

which witnesses �

0

�

7�! � and satis�es the following conditions:

1. 8i; 1 � i � h,

8

integer is the number of bits of its binary representation. The size of a set (or

vector) of integers is de�ned to be the sum of the lengths of the components.

The size of a linear set L(�; �) is the size of vector � plus the sum of the

sizes of the vectors in �. The size of a semilinear set is the sum of the sizes of

its constituent linear sets. Consider a Petri net P=((P; T; '); �

0

), where P =

fp

1

; :::; p

k

g and T = ft

1

; :::; t

m

g. Each transition '(p

i

; t

j

) = d ('(t

j

; p

i

) = d)

can be thought of as a four tuple (0; i; j; d) ((1; j; i; d)). (The �rst component

(0 or 1) is to indicate the ow direction (0: from a place to a transition; 1: from

a transition to a place). In this way, ' can be treated as a set of four tuples.

Now the size of Petri net P can be de�ned as dlog

2

ke + dlog

2

me + the sum

of the sizes of elements in ' + the size of �

0

. Since the binary representation

is used, the �ring of a transition may result in removing (or adding) 2

s

tokens

from (to) a place, where s is the size of the Petri net.

For more about Petri nets and their related problems, see [13,14].

3 Canonical paths in BPP-nets

To derive the complexity of the equivalence problem, we begin with a few

lemmas which are important in characterizing computations in BPP-nets. In

the literature, one of the few techniques proven to be useful for analyzing PNs

relies on the ability to rearrange PN paths into some `canonical' form. As one

might expect, the simple structure of circuits in BPP-nets (in particular, the

ability to repeat a circuit for an arbitrary number of times at any marking

at which the circuit is marked) suggests a good starting point for devising

a rearrangement technique. The �rst attempt, perhaps, is to �re a circuit

immediately when one of its transitions becomes enabled, even though the

transitions of the circuit are interleaved with others in the original path. Un-

fortunately, such an attempt does not work as Figure 1 indicates. (In Figure 1,

`acdeb

0

is a legal �ring sequence, whereas `ab(any permutation of cde)

0

is not.)

To circumvent such a di�culty, we �rst present a nice property concerning

any set of connected circuits in BPP-nets.

Lemma 1 Let C=fc

1

; c

2

; :::; c

n

g be a set of connected circuits in a BPP-net

P and � be a marking with �(c

i

) > 0, for some i. For arbitrary integers

a

1

; a

2

; :::; a

n

> 0, there exists a sequence � such that �

�

7�! and#

�

=

n

X

j=1

a

j

(#

c

j

).

(In words, from � there exists a �rable sequence � utilizing circuit c

j

exactly

a

j

times, for every j.)

Proof. Without loss of generality, we assume i = 1, and let p

1

be a place in c

1

such that �(p

1

) > 0. The proof is done by induction on the number of circuits

7

Fig. 1. A BPP-net.

6

� p

�

=ftj'(p; t) � 1; t 2 Tg is the set of output transitions of p;

t

�

=fpj'(t; p) � 1; p 2 Pg is the set of output places of t.

�

�

p=ftj'(t; p) � 1; t 2 Tg is the set of input transitions of p;

�

t=fpj'(p; t) � 1; p 2 Pg is the set of input places of t.

Notice that if �

0

�

7�! �, then �

0

+ A �#

�

= �. (The converse, however, does

not necessarily hold.) Given a path �

�

7�! �

0

, a sequence �

0

is said to be a

rearrangement of � if #

�

= #

�

0and �

�

0

7�! �

0

.

A PN ((P; T; '); �

0

) is said to be a BPP-net [4] if

(1). 8t 2 T; j

�

tj = 1, (i.e., every transition has exactly one input place), and

(2). 8p 2 P; t 2 T; '(p; t) � 1 (i.e., every arc going from a place to a transition

has weight 1).

A circuit of a PN is a `simple' closed path in the PN graph. (By `simple'

we mean all nodes are distinct along the closed path.) It is important to note

that every circuit c = p

1

t

1

p

2

t

2

� � � p

n

t

n

p

1

in a BPP-net must have

�

t

i

= fp

i

g,

for every i; 1 � i � n. See Figure 1 for an example of a BPP-net. (Notice

that the �ring of a transition may deposit more than one token into a place.

In Figure 1, for example, the �ring of transition e adds 10 tokens to place

p

2

.) Given a circuit c = p

1

t

1

p

2

t

2

� � � p

n

t

n

p

1

, let P

c

= fp

1

; p

2

; � � � ; p

n

g denote the

set of places in c. (With a slight abuse of notation, we sometimes use c to

denote transition sequence t

1

t

2

� � � t

n

of circuit p

1

t

1

p

2

t

2

� � � p

n

t

n

p

1

when places

are not important.) We de�ne the token count of circuit c in marking � to be

�(c) =

X

p2P

c

�(p). A circuit c is said to be token-free in � i� �(c) = 0. A set of

circuits C=fc

1

; c

2

; :::; c

n

g is said to be connected i� for every i; j, 1 � i; j � n,

there exist 1 � h

1

; h

2

; :::; h

r

� n, for some r, such that h

1

= i, h

r

= j, and

for every 1 � l < r, P

c

h

l

\ P

c

h

l+1

6= ;. In words, every pair of neighboring

circuits in the sequence c

h

1

; c

h

2

; :::; c

h

r

share at least one place. For a simple

circuit c, we also use #

c

to denote the vector count of transitions used in c,

i.e., #

c

(i) = 1 if t

i

is in c; #

c

(i) = 0, otherwise. A sequence � is said to cover

circuit c if #

c

� #

�

, i.e., every transition of c appears in �.

For a vector � 2 N

k

and a �nite set � (= fv

1

; :::; v

n

g, for some n) � N

k

,

the set L(�; �) = fvj9a

1

:::; a

n

2 N; v = � +

n

X

i=1

a

i

� v

i

g is called the linear set

with base � over the set of periods �. A semilinear set is a �nite union of linear

sets.

To deal with the complexity issue, it is essential to de�ne the sizes of

Petri nets and semilinear sets in a precise manner. Throughout this paper,

each integer will be represented by its binary representation. The length of an

5

string �) using symbols from �. We write �

+

to denote �

�

� f�g.

A Petri net (PN, for short) is a triple (P; T; '), where P is a �nite set of

places, T is a �nite set of transitions, and ' is a ow function ' : (P � T ) [

(T � P ) ! N . In this paper, k and m will be reserved for jPj (the number of

places in P) and jTj (the number of transitions in T), respectively. A marking

is a mapping � : P ! N . A transition t 2 T is enabled at a marking � i�

for every p 2 P , '(p; t) � �(p). A transition t may �re at a marking � if t is

enabled at �. We then write �

t

7�! �

0

, where �

0

(p) = �(p) � '(p; t) + '(t; p)

for all p 2 P . A sequence of transitions � = t

1

:::t

n

is a �ring sequence from

�

0

i� �

0

t

1

7�! �

1

t

2

7�! � � �

t

n

7�! �

n

for some sequence of markings �

1

,...,�

n

.

(We also write `�

0

�

7�! �

n

'.) We write `�

0

�

7�!' to denote that � is enabled

and can be �red from �

0

, i.e., �

0

�

7�! i� there exists a marking � such that

�

0

�

7�! �. The notation �

0

�

7�! � is used to denote the existence of a � such

that �

0

�

7�! �. A marked PN is a pair ((P; T; '); �

0

), where (P; T; ') is a PN,

and �

0

is a marking called the initial marking. Throughout the rest of this

paper, the word `marked' will be omitted if it is clear from the context. By

establishing an ordering on the elements of P and T (i.e., P = fp

1

; :::; p

k

g and

T = ft

1

; :::; t

m

g), we de�ne the k �m addition matrix A of (P; T; ') so that

a

i;j

= '(t

j

; p

i

) � '(p

i

; t

j

). Thus, if we view a marking � as a k-dimensional

column vector in which the ith component is �(p

i

), each column a

j

of A is

then a k-dimensional vector such that if �

t

j

7�! �

0

, then �

0

= � + a

j

. Let

P = ((P; T; '); �

0

) be a PN. The reachability set of P is the set R(P) =

f� j �

0

�

7�! � for some � 2 T

�

g. The reachability equivalence problem (or

simply equivalence problem) is that of determining, given two PNs P

1

and P

2

with the same set of places, whether R(P

1

) = R(P

2

).

For ease of expression, the following notations will be used extensively

throughout the rest of this paper. (Let �; �

0

be transition sequences, p be a

place, and t be a transition.)

� #

�

(t) represents the number of occurrences of t in �. (For convenience, we

sometimes treat #

�

as an m-dimensional vector assuming that an ordering

on T is established (jT j = m).)

� �(�) = A � #

�

de�nes the displacement of �. (Notice that if �

�

7�! �

0

,

then �(�) = �

0

� �.) For a place p 2 P , we write �(�)(p) to denote the

component of �(�) corresponding to place p.

� Tr(�) = ftjt 2 T;#

�

(t) > 0g, denoting the set of transitions used in �.

� j�j is the number of transitions in �, i.e., j�j = n if � = t

1

:::t

n

.

� �

.

�

0

is de�ned inductively as follows. Suppose �

0

= t

1

:::t

n

. Let �

0

be �. If t

i

is in �

i�1

, let �

i

be �

i�1

with the leftmost occurrence of t

i

deleted; otherwise,

let �

i

= �

i�1

. Finally, let �

.

�

0

= �

n

. For example, if � = t

1

t

2

t

3

t

4

t

5

and

�

0

= t

4

t

3

t

1

, then �

.

�

0

= t

2

t

5

. Intuitively, �

.

�

0

represents the transition

sequence resulting from removing each transition of �

0

from the leftmost

occurrence of such a transition in � (if the transition exists).

4

of the reachability equivalence problem for BPP-nets, where s is the size of

the problem instance (when a standard binary encoding scheme is sued) and

d is a �xed constant.

The contributions of this paper include the following. Our DTIME(2

2

d�s

3

)

result improves upon the previous decidability result presented in [4]. (In [4],

the decidability result was obtained by showing the reachability sets of BPP-

nets to be e�ectively semilinear. The work, however, did not reveal any com-

plexity bounds for the reachability equivalence problem.) As for the lower

bound, at this moment we are able to show the problem to be �

P

2

-hard

2

,

directly following a result presented in [11] concerning the complexity of the

equivalence problem for commutative context-free grammars. As a by-product,

our analysis yields yet another proof for the NP upper bound of the reachabil-

ity problem for BPP-nets. (We show that checking the reachability property

for BPP-nets is tantamount to solving an integer linear programming problem.

The approach used in [4], on the other hand, requires that certain structure

(called siphon) of Petri nets be examined.) Finally, we feel that the new char-

acterization for paths in BPP-nets is interesting in its own right, and may

have other applications to the analysis of Petri nets.

The remainder of this paper is structured as follows. In Section 2, we

formally de�ne the model of Petri nets, the reachability equivalence problem,

and the notations used throughout this paper. In Section 3, we show that

BPP-net computations can always be rearranged into some canonical form,

facilitating the use of integer linear programming to solve the reachability

problem. Finally, in Section 4, we derive small semilinear set representations

for BPP-nets, which, in turn, give rise to an upper bound for the reachability

equivalence problem.

2 Preliminaries

Let Z (N) denote the set of (nonnegative) integers, and Z

k

(N

k

) the

set of vectors of k (nonnegative) integers. For a k-dimensional vector v, let

v(i); 1 � i � k, denote the ith component of v. For a k � m matrix A, let a

i;j

,

1 � i � k; 1 � j � m, denote the element in the ith row and the jth column of

A, and let a

j

denote the jth column of A. For a given value of k, let 0 denote

the vector of k zeros (i.e., 0(i) = 0 for i = 1; :::; k). We let jSj be the number of

elements in set S. Given a column vector x, we let x

T

(which is a row vector)

denote the transpose of x. Given an alphabet (i.e., a �nite set of symbols) �,

we write �

�

to denote the set of all �nite-length strings (including the empty

2

�

P

2

denotes the set of all languages whose complements are in the second level

of the polynomial-time hierarchy [15].

3

given for the NP upper bound of the uniform word problem, taking advantage

of the connection between BPP-nets and commutative context-free grammars

as well as the fact that the reachability problem for BPP-nets is solvable in

NP. Second, surprising results have been shown regarding the issue of de-

ciding equivalence for labeled BPP-nets with respect to various equivalence

notions de�ned in the linear time/branching time hierarchy of [17]. Deciding

bisimulation equivalence has been shown to be decidable [2]. However, for all

the equivalences of the linear time/branching time hierarchy below bisimula-

tion equivalence, deciding equivalence turns out to be undecidable [8]. The

undecidability result is somewhat surprising, taking into consideration the

rather limited computational power of BPP-nets. As for reachability equiva-

lence (which coincides with the conventional equivalence of Petri net reach-

ability sets), it has recently been shown in [4] that BPP-nets always exhibit

e�ective semilinear reachability sets, thus yielding a decidability result.

Motivated by the work (in particular, the technique) of [4], in this paper

we develop a new characterization for paths in BPP-nets. As we will see later,

the simple structure of circuits in BPP-nets plays a crucial role in our analysis.

(A circuit of a Petri net is simply a closed path (i.e., a cycle) in the Petri net

graph.) By and large, the presence of complex circuits, in general, is trouble-

some in Petri net analysis. In fact, strong evidence has suggested that circuits

constitute the major stumbling block in the analysis of Petri nets. To get a feel

for why this is the case, it is well known that in a Petri net P with initial mark-

ing �

0

, a marking � is reachable (from �

0

) in P only if there exists a column

vector x 2 N

k

such that �

0

+A � x = �, where k is the number of transitions

in P and A is the addition matrix of P. The converse, however, does not nec-

essarily hold. In fact, lacking a simple necessary and su�cient condition for

reachability in general has been blamed for the high degree of complexity in

the analysis of Petri nets. (Otherwise, one could tie the reachability analysis of

Petri nets to the integer linear programming problem, which is relatively well

understood.) There are restricted classes of Petri nets for which necessary and

su�cient conditions for reachability are available. Most notable, of course, is

the class of circuit-free Petri nets (i.e., Petri nets without circuits) for which

the equation �

0

+A � x = � is su�cient and necessary to capture reachability.

A slight relaxation of the circuit-freedom constraint yields the same necessary

and su�cient condition for the class of Petri nets without token-free circuits

in every reachable marking [16]. By taking advantage of simple circuits o�ered

by BPP-nets, in this paper we show that any path in a BPP-net can be re-

arranged into some canonical form, which, in turn, facilitates the derivation

of `small' semilinear set representations for BPP-nets. This result, in conjunc-

tion with a known result concerning the complexity of deciding equivalence

for semilinear sets presented in [9,12], yields a DTIME(2

2

d�s

3

)

1

upper bound

1

DTIME(f(n)) represents the class of languages accepted by deterministic Turing

machines using at most f(n) time.

2

On reachability equivalence for BPP-nets

Hsu-Chun Yen

Department of Electrical Engineering, National Taiwan University, Taipei,

Taiwan 106, Republic of China

Abstract

In this paper, we study the complexity of the reachability equivalence problem

for BPP-nets. BPP-nets are closely related to Basic Parallel Processes, which form

a subclass of Milner's CCS. We show the reachability equivalence problem for BPP-

nets to be solvable in DTIME(2

2

ds

3

), where d is a constant and s is the size of the

problem instance, when a standard binary encoding scheme is used. To that end,

we provide a new characterization for computations in BPP-nets, which, in turn,

facilitates the derivation of small semilinear set representations for the reachability

sets of BPP-nets. As for the lower bound, the problem is shown to be �

P

2

�hard. Our

results improve upon the previous decidability result of the reachability equivalence

problem for BPP-nets.

1 Introduction

BPP-nets provide a net semantics for Basic Parallel Processes (BPP, for

short), which form a subclass of Milner's CCS (see, e.g., [1,3,7]). Simply speak-

ing, a BPP-net is a Petri net in which each transition has exactly one input

place, and the �ring of a transition removes exactly one token from its input

place [4,5]. It seems, on the surface, that the computational power of BPP-

nets is rather limited. The limitation is a direct consequence of the inability

for BPP-nets to model `synchronization' actions, which require places to syn-

chronize through transition �rings. (This is why such Petri nets are also called

communication-free nets [6].)

What makes BPP-nets theoretically interesting, aside from their close con-

nection to BPPs, includes the following. First, BPP-nets are also computa-

tionally equivalent to the so-called commutative context-free grammars de�ned

and investigated in [10,11]. Of many problems considered in [10], the uniform

word problem was shown to be solvable in NP through a somewhat compli-

cated proof. In a recent article [4], an alternative and simpler proof has been

Preprint submitted to Elsevier Science 27 January 1996