AbstractParallel communicating Watson–Crick automata systems were introduced in [E. Czeizler, E. Czeizler, Parallel communicating Watson–Crick automata systems, in: Z. Ésik, Z. Fülöp (Eds.), Proc. Automata and Formal Languages, Dobogókő, Hungary, 2005, pp. 83–96] as possible models of DNA computations. This combination of Watson–Crick automata and parallel communicating systems comes as a natural extension due to the new developments in DNA manipulation techniques. It is already known, see [D. Kuske, P. Weigel, The Role of the Complementarity Relation in Watson–Crick Automata and Sticker Systems, DLT 2004, Lecture Notes in Computer Science, Vol. 3340, Auckland, New Zealand, 2004, pp. 272–283], that for Watson–Crick finite automata, the complementarity relation plays no active role. However, this is not the case when considering parallel communicating Watson–Crick automata systems. In this paper we prove that non-injective complementarity relations increase the accepting power of these systems. We also prove that although Watson–Crick automata are equivalent to two-head finite automata, this equivalence is not preserved when comparing parallel communicating Watson–Crick automata systems and multi-head finite automata

Czeizler, Elena

Czeizler, Eugen

Elsevier - Publisher Connector 

Theoretical Computer Science 358 (2006) 142–147
www.elsevier.com/locate/tcs
Note
On the power of parallel communicating Watson–Crick
automata systems
Elena Czeizlera,b,∗, Eugen Czeizlera,b
aDepartment of Mathematics, FIN-20014 University of Turku, Finland
bTurku Centre for Computer Science, FIN-20520 Turku, Finland
Received 23 November 2005; received in revised form 9 December 2005; accepted 22 January 2006
Communicated by Gh. Paun
Abstract
Parallel communicating Watson–Crick automata systems were introduced in [E. Czeizler, E. Czeizler, Parallel communicating
Watson–Crick automata systems, in: Z. Ésik, Z. Fülöp (Eds.), Proc. Automata and Formal Languages, Dobogóko˝, Hungary, 2005,
pp. 83–96] as possible models of DNA computations. This combination of Watson–Crick automata and parallel communicating
systems comes as a natural extension due to the new developments in DNA manipulation techniques. It is already known, see
[D. Kuske, P.Weigel, The Role of the Complementarity Relation inWatson–CrickAutomata and Sticker Systems, DLT 2004, Lecture
Notes in Computer Science, Vol. 3340, Auckland, New Zealand, 2004, pp. 272–283], that for Watson–Crick ﬁnite automata, the
complementarity relation plays no active role. However, this is not the case when considering parallel communicatingWatson–Crick
automata systems. In this paper we prove that non-injective complementarity relations increase the accepting power of these systems.
We also prove that although Watson–Crick automata are equivalent to two-head ﬁnite automata, this equivalence is not preserved
when comparing parallel communicating Watson–Crick automata systems and multi-head ﬁnite automata.
© 2006 Elsevier B.V. All rights reserved.
Keywords: Watson–Crick; Parallel communicating automata systems; Complementarity relation
1. Introduction
Watson–Crick ﬁnite automata are a counterpart of ﬁnite automata working on double stranded sequences. They
were introduced in [4] and, as suggested by the name, they are intended as a formalization of DNA manipulation.
One of the main features of these automata is that characters on corresponding positions from the two strands of
the input are related by a complementarity relation similarly with the Watson–Crick complementarity of DNA nu-
cleotides. Several variants of these automata were investigated in [7–9], and [11]; see also [10] for a comprehensive
presentation.
When considering DNA molecules as a possible support for computations, we may exploit two key features: the
Watson–Crick complementarity and the massive parallelism. While Watson–Crick automata make use only of the ﬁrst
∗ Corresponding author. Turku Centre for Computer Science, FIN-20520 Turku, Finland. Tel.: +35823338793.
E-mail addresses: elenac@it.utu.ﬁ (E. Czeizler), eugenc@it.utu.ﬁ (E. Czeizler).
0304-3975/$ - see front matter © 2006 Elsevier B.V. All rights reserved.
doi:10.1016/j.tcs.2006.01.049
E. Czeizler, E. Czeizler / Theoretical Computer Science 358 (2006) 142–147 143
one, parallel communicating Watson–Crick automata systems, introduced in [2], come as a possible answer to the
problem formulated in [10] of combining the two features into a model of DNA computations.
A parallel communicating Watson–Crick automata system, PCWKS for short, consists of a set of Watson–Crick
ﬁnite automata working independently on their own input tapes and communicating states on request. Although every
component has its own double-stranded tape, the input is the same on all of them. At the beginning, all components
are in their initial states and start parsing synchronously the input from left to right. As in the case of other parallel
communicating automata systems, see for example [6], the communication between components is done using special
query states, each of them pointing to exactly one component of the system. When component i reaches a query state
Kj , the current state of the component j will be communicated to i and the computation continues. We refer to [1] and
[3] for different paradigms of parallelism and communication in grammar systems.
Another question from [10] is about the role of the complementarity relation regarding the expressive power of
Watson–Crick automata. A ﬁrst answer is given in [5], where it is proved that the complementarity relation plays no
actual role for Watson–Crick automata, i.e., any language accepted by a Watson–Crick automaton is also accepted
by one with a one-to-one complementarity relation. This result is also extended for the Watson–Crick -automata
introduced in [11].
In this paper we prove that for PCWKS, the complementarity relation plays an active role. In [2] it was shown that
systems with injective complementarity relation accept only regular one-letter languages. Here we prove that if the
relation is not injective, then we can accept also some non-regular one-letter languages.
Since in [10] it is proved thatWatson–Crick automata are equivalent with two-head ﬁnite automata, a natural question
is whether this equivalence is preserved when considering PCWKS and multi-head ﬁnite automata. It is already known,
see [6], that n-head ﬁnite automata are equivalent with parallel communicating ﬁnite automata systems (communicating
by states) with n components and they recognize only semilinear languages. The main result of this paper proves that
PCWKS are more powerful: they recognize all languages accepted by parallel communicating ﬁnite automata systems
and also some non-semilinear languages such as {an2 | n2}.
2. Deﬁnitions
We assume that the reader is familiar with the fundamental concepts from formal languages and automata theory;
for more details we refer to [12].
For a ﬁnite alphabet V we denote by V ∗ the set of all ﬁnite words over V and by  the empty word.
Let now  ⊆ V × V be a symmetric relation, called the Watson–Crick complementarity relation on V , inspired by
the Watson–Crick complementarity of nucleotides in the double stranded DNA molecule. We say that  is injective
if any a ∈ V has a unique complementary symbol b ∈ V with (a, b) ∈ . In accordance with the representation of
DNA molecules, viewed as two strings written one over the other, we write
(
V ∗
V ∗
)
instead of V ∗ × V ∗ and an element
(w1, w2) ∈ V ∗ × V ∗ as
(
w1
w2
)
. We denote
[
V
V
]

=
{[a
b
]
| a, b ∈ V, (a, b) ∈ 
}
and WK(V ) =
[
V
V
]∗

. The
set WK(V ) is called the Watson–Crick domain associated to V and . An element
[
a1
b1
] [
a2
b2
]
. . .
[
an
bn
]
∈ WK(V )
can be also written in a more compact form as
[
w1
w2
]
, where w1 = a1a2 . . . an and w2 = b1b2 . . . bn.
The essential difference between
(
w1
w2
)
and
[
w1
w2
]
is that the former is just an alternative notation for the pair
(
w1
w2
)
,
whereas the latter implies that the words w1 and w2 have the same length and the corresponding letters are connected by
the complementarity relation.
AWatson–Crick ﬁnite automaton is a 6-tuple M = (V , ,Q, q0, F, ), where V is the (input) alphabet,  ⊆ V ×V
is the complementarity relation, Q is a ﬁnite set of states, q0 ∈ Q is the initial state, F ⊆ Q is the set of ﬁnal states,
 : Q ×
(
V ∗
V ∗
)
→ 2Q is a mapping, called the transition function, such that 
(
q,
(
w1
w2
))
= ∅ only for ﬁnitely many
triples (q,w1, w2) ∈ Q × V ∗ × V ∗. We can replace the transition function with rewriting rules, by using
s
(
w1
w2
)
→
(
w1
w2
)
s′instead of s′ ∈ 
(
s,
(
w1
w2
))
.
144 E. Czeizler, E. Czeizler / Theoretical Computer Science 358 (2006) 142–147
For more details on Watson–Crick automata we refer to [10].
A parallel communicating Watson–Crick automata system of degree n, for short PCWKS(n), is an (n + 3)-tuple
A = (V , , A1, A2, . . . , An,K),
where
• V is the input alphabet;
•  is the complementarity relation;
• Ai = (V , ,Qi, qi, Fi, i ), 1 in, are Watson–Crick ﬁnite automata, where the sets Qi are not necessarily
disjoint;
• K = {K1,K2, . . . , Kn} ⊆ ⋃ni=1 Qi is the set of query states.
The automata A1, A2, . . . , An are called the components of the system A.
These systems were introduced in [2] where their accepting power and some closure properties were investigated.
They are composed of several Watson–Crick automata working independently on their tapes and communicating on
request by use of query states. Each of these states points to exactly one component of the system such that, when a
component Ai reaches a query state Kj , the current state of the component Aj will be communicated to Ai and the
computation continues.
A conﬁguration of a PCWKS(n) is a 2n-tuple
(
s1,
(
u1
v1
)
, s2,
(
u2
v2
)
, . . . , sn,
(
un
vn
))
where si is the current state of the component Ai and
(
ui
vi
)
is the part of the input which has not been read yet by the
component Ai , for all 1 in. We deﬁne a binary relation  on the set of all conﬁgurations by setting(
s1,
(
u1
v1
)
, s2,
(
u2
v2
)
, . . . , sn,
(
un
vn
))

(
r1,
(
u′1
v′1
)
, r2,
(
u′2
v′2
)
, . . . , rn,
(
u′n
v′n
))
if and only if one of the following two conditions holds:
• K ∩ {s1, s2, . . . , sn} = ∅,
(
ui
vi
)
=
(
xi
yi
) (
u′i
v′i
)
and ri ∈ i
(
si,
(
xi
yi
))
, for all 1 in;
• for all 1 in such that si = Kji and sji /∈ K we have ri = sji , whereas for all the other 1 ln we have rl = sl .
In this case
(
u′i
v′i
)
=
(
ui
vi
)
, for all 1 in.
If we denote by ∗ the reﬂexive and transitive closure of , then the language recognized by a PCWKS is deﬁned as
L(A) =
{
w1 ∈ V ∗
∣∣∣∣
(
q1,
[
w1
w2
]
, q2,
[
w1
w2
]
, . . . , qn,
[
w1
w2
])
∗
(
s1,
(


)
, s2,
(


)
, . . . , sn,
(


))
, si ∈ Fi, 1 in
}
.
Intuitively, the language accepted by such a system consists of all words w1 such that in every component we reach a
ﬁnal state after reading the input
[
w1
w2
]
.
3. Main result
In [2] it was proved that PCWKS with injective complementarity relations accept only regular one-letter lan-
guages. In this section we prove that by using a non-injective complementarity relation we increase the power of these
systems.
Theorem 1. The language {an2 |n2} can be accepted by a parallel communicating Watson–Crick automata system
with three components and a non-injective complementarity relation.
E. Czeizler, E. Czeizler / Theoretical Computer Science 358 (2006) 142–147 145
Table 1
The transition rules of systemA
lll Component A1 Component A2 Component A3
lll q1
(a
b
)
→
( a
b
)
r1 q2
(


)
→
(


)
K1 q3
(a
b
)
→
(a
b
)
K1
r1
(a
b
)
→
( a
b
)
r1 r1
(


)
→
(


)
K1 r1
(


)
→
(


)
K1
r1
(a
c
)
→
( a
c
)
sbc sbc
(a
b
)
→
(a
b
)
K1 sbc
(a
b
)
→
(a
b
)
K1
sbc
(a
c
)
→
( a
c
)
sc sc
(a
b
)
→
(a
b
)
K1 sc
(


)
→
(


)
K1
sc
(a
c
)
→
( a
c
)
sc scb
(a
c
)
→
(a
c
)
K1 scb
(a
b
)
→
(a
b
)
K1
sc
(a
b
)
→
( a
b
)
scb sb
(a
c
)
→
(a
c
)
K1 sb
(


)
→
(


)
K1
scb
(a
b
)
→
( a
b
)
sb f
c
1
(a
c
)
→
(a
c
)
f2 f
b
1
(a
c
)
→
(a
c
)
f3
sb
(a
b
)
→
( a
b
)
sb f
b
1
(a
b
)
→
(a
b
)
f2 f
c
1
(a
c
)
→
(a
c
)
f3
sb
(a
c
)
→
( a
c
)
sbc f2
(a
c
)
→
(a
c
)
f2 f3
(a
c
)
→
(a
c
)
f3
sb
(


)
→
(


)
f b1 f2
(a
b
)
→
(a
b
)
f2 f3
(a
b
)
→
(a
b
)
f3
sc
(


)
→
(


)
f c1 f2
(


)
→
(


)
f2
f b1
(


)
→
(


)
f b1
f c1
(


)
→
(


)
f c1
Proof. The proof is based on the following observation. A word w ∈ {a}∗ is of the form an2 for some n2 if and
only if:
(1) we can divide w into blocks of equal length and
(2) the number of such blocks is equal to their length.
Hence, we build a system accepting a word w = an2 only when its complement is of the form bncnbncn . . . with exactly
n− 1 alternations between blocks of b’s and c’s. The system is composed of three components; the ﬁrst two verify that
the complement has alternating blocks of b’s and c’s of equal length, while the ﬁrst and the third components verify
that the number of such blocks is equal to the length of the ﬁrst block of b’s.
Formally, we construct a PCWKS A = ({a, b, c}, , A1, A2, A3,K), where
•  = {(a, b), (a, c)},
• A1 = ({a, b, c}, , {q1, r1, sb, sc, sbc, scb, f c1 , f b1 }, q1, {f c1 , f b1 }, 1),
• A2 = ({a, b, c}, , {q2, r1, sb, sc, sbc, scb, f c1 , f b1 , f2,K1}, q2, {f2}, 2),
• A3 = ({a, b, c}, , {q3, r1, sb, sc, sbc, scb, f c1 , f b1 , f3,K1}, q3, {f3}, 3).
The transition functions of the components are given in Table 1.
146 E. Czeizler, E. Czeizler / Theoretical Computer Science 358 (2006) 142–147
For an input of the form[aaaaaaaaaaaaa . . .
bb . . . bcc . . . cbb . . .
]
the system evolves as follows. The ﬁrst component reads the ﬁrst block of b’s transmitting at each step its state to the
other two components. Meanwhile, the third component counts the ﬁrst block by reading
(
a
b
)
and then waits for the
signal of the next alternation of letters. The second component also waits for the same signal but without reading any
input.
When component A1 reads the ﬁrst letter of the next block, it enters a special state sbc signaling both the change
between blocks and its type, in this case from b’s to c’s. From now on, the second component starts reading the input
and by communicating with the ﬁrst component checks that any two consecutive one-letter blocks have equal length.
Also, when receives the signal of the alternation, the third component counts this next block by reading the next
(
a
b
)
.
In general, component A3 reads one
(
a
b
)
each time it receives from A1 a signal of alternation, i.e., state sbc or scb.
The computation evolves similarly, until the ﬁrst component reads all the input. Then, it enters one of the ﬁnal states
f b1 or f
c
1 depending on the type of the last block. Both A2 and A3 acknowledge this signal and react as follows. The
second component enters its ﬁnal state f2 only if it reads on the input tape exactly the letter indicated by the signal.
In this ﬁnal state it ﬁnishes reading the input. The third component also enters its ﬁnal state only if it reads
(
a
c
)
on the
input tape, verifying that the number of blocks equals the length of the ﬁrst block; then it ﬁnishes reading the input.
If there exist two consecutive blocks of b’s and c’s of different lengths, then the second component enters a deadlock.
On the other hand, if the number of blocks of b’s and c’s is not equal to the size of the ﬁrst block, then the third
component enters a deadlock. On both cases the input is not accepted by the system. 
As an immediate consequence we have the following result.
Corollary 2. There exist PCWKS with non-injective complementarity relation accepting non-regular one-letter lan-
guages.
Moreover, any language accepted by a PCWKS with injective complementarity relation can be accepted also by a
system with non-injective complementarity relation. Thus, we have the following result.
Corollary 3. Parallel communicating Watson–Crick automata systems with non-injective complementarity relation
are more powerful than systems with injective complementarity relation.
Although Watson–Crick automata are equivalent to two-head ﬁnite automata, see [10], this is not true anymore when
considering PCWKS and multi-head automata. In order to prove this, let us ﬁrst recall the following result from [6].
Theorem 4. A language is accepted by an n-head ﬁnite automaton if and only if it is accepted by a parallel commu-
nicating ﬁnite automata system with n components.
However, it is proved in [13] that the languages accepted by multihead nondeterministic pushdown automata satisfy
the semilinearity property. Hence, parallel ﬁnite automata system communicating by states accept only semilinear
languages.
Moreover, it is easy to prove that for any parallel communicating ﬁnite automata system with n components we can
construct an equivalent Watson–Crick automata system with n components and the identity complementarity relation.
Since the language {an2 |n2} is not semilinear we obtain the following result.
Corollary 5. Parallel communicating Watson–Crick automata systems are more powerful than multi-head automata
and parallel communicating ﬁnite automata systems respectively.
References
[1] E. Csuhaj-Varjú, J. Dassow, J. Kelemen, Gh. Pa˘un, Grammar Systems: A Grammatical Approach to Distribution and Cooperation, Gordon and
Breach, London, 1994.
E. Czeizler, E. Czeizler / Theoretical Computer Science 358 (2006) 142–147 147
[2] E. Czeizler, E. Czeizler, Parallel communicating Watson–Crick automata systems, in: Z. Ésik, Z. Fülöp (Eds.), Proc. Automata and Formal
Languages, Dobogóko˝, Hungary, 2005, pp. 83–96.
[3] J. Dassow, Gh. Pa˘un, G. Rozenberg, Grammar systems, in: G. Rozenberg, A. Salomaa (Eds.), The Handbook of Formal Languages, Vol. 2,
Springer, Berlin, 1997, pp. 155–213.
[4] R. Freund, Gh. Pa˘un, G. Rozenberg,A. Salomaa,Watson–Crick ﬁnite automata, in: Proc. Third DIMACSWorkshop on DNA Based Computers,
Philadelphia, PA, 1997, pp. 297–328.
[5] D. Kuske, P. Weigel, The Role of the Complementarity Relation in Watson–Crick Automata and Sticker Systems, DLT 2004, Lecture Notes in
Computer Science, Vol. 3340, Auckland, New Zealand, 2004, pp. 272–283.
[6] C. Martín-Vide, A. Mateescu, V. Mitrana, Parallel communicating ﬁnite automata systems communicating by states, Internat. J. Found. Comp.
Sci. 13 (5) (2002) 733–749.
[7] C. Martín-Vide, Gh. Pa˘un, Normal forms for Watson–Crick ﬁnite automata, in: F. Cavoto (Ed.), The Complete Linguist: A Collection of Papers
in Honour of Alexis Manaster Ramer, Lincom Europa, Munich, 2000, pp. 281–296.
[8] V. Mihalache, A. Salomaa, Lindenmayer and DNA: Watson–Crick DOL systems, Current Trends in Theoretical Computer Science, World
Scientiﬁc, Singapore, 2001 pp. 740–751.
[9] A. Pa˘un, M. Pa˘un, State and Transition Complexity of Watson–Crick Finite Automata, FCT’99 Lecture Notes in Computer Science, Vol. 1684,
Iasi, Romania, 1999 pp. 409–420.
[10] Gh. Pa˘un, G. Rozenberg, A. Salomaa, DNA Computing. New Computing Paradigms, Springer, Berlin, 1998.
[11] E. Petre, Watson–Crick -automata, J. Autom. Lang. Comb. 8 (1) (2003) 59–70.
[12] A. Salomaa, Formal Languages, Academic Press, NewYork, 1973 (Revised edition in the series Computer Science Classics, Academic Press,
1987).
[13] O.H. Ibarra, A note on semilinear sets and bounded-reversal multihead pushdown automata, Inform. Process. Lett. 3 (1974) 25–28.


On the power of parallel communicating Watson–Crick automata systems 

file:///data/remote/core/dit/data/elsevier/pdf/886/aHR0cDovL2FwaS5lbHNldmllci5jb20vY29udGVudC9hcnRpY2xlL3BpaS9zMDMwNDM5NzUwNjAwMTI0MQ==.pdf

On the power of parallel communicating Watson–Crick automata systems

Abstract

Similar works

Full text

Available Versions

Elsevier - Publisher Connector

Elsevier - Publisher Connector