Communication Aspects of Computation of Systems of
Tomasz Jurdzi´ nski1, Mirosław Kutyłowski13, and Krzysztof Lory´ s1
(some results in cooperation with PavolˇDuriˇ s2)
1Computer Science Institute, University of Wrocław
2Institute of Informatics, Comenius University, Bratislava
3Department of Mathematics and Computer Science, Pozna´ n University
Many computing systems can be modeled by systems of cooperating finite au-
tomata. In fact, any existing physical device is finite, even though we often think in
terms of models with infinite memory.
Here, we consider systems of synchronized finite automata performing together
computation on an input string. Each automaton has its own read-only head that moves
Computational power of such devices depends on the number of states of automata, the
numberof automata,the amount of communicationbetween automataand the way they
cooperate. We concentrate on the communication and cooperation aspects of the com-
putation of the systems.
Today, communication capacity is one of the most important bottlenecks of computer
systems. For this reason, there is growing interest in communication complexity of
problems. From practical point of view, corresponding complexity measures are often
more important than traditional measures such as time and space complexities.
Automata of the considered systems may communicate by broadcasting messages.
We introduce the notion of message complexity as a measure of amount of communi-
cation exchanged. Each automaton in any step of the system can send a message. We
say that a language L has message complexity f
which recognizes L and on each input word the automata of the system send messages
in at most f
Somewhat related to our consideration are papers on the trade-offs between com-
putation time and space of multihead systems [3,6] (here space denotes logarithm of
the number of internal states). However, these results are based on the fact that in or-
der to examine distant parts of the input the heads must spend quite a long time to get
there. Connected to our results is also research on trade-offs between communication
and space, started in last years. Perhaps the most prominent results of this kind [1,8]
concern matrix multiplication problem. In the strongest version, the computing model
assumed is very general (i.e. branching programs).
We show a number of hierarchy results for message complexity measure. For each
constant k there is a language, which may be recognized with k
not be recognized with k messages. We give an example of a language that requires
a language requires more than a constant number of messages. We present a language
that requires Θ
? if there exists a multihead system
? 1 messages and can-
? messages and we conjecture that Ω
? messages are necessary, if
? messages. For a large family of functions f, f
? , we prove that there is a language which requires Θ
??? messages. Fi-
nally,we present a hierarchywith respect to message complexityformany-valuedfunc- Download full-text
tionswhichrequiresuperlinearnumberofmessages.We givealso analmosttightbound
for message complexity of matrix multiplication. This last result can be also derived
from . However, we give a relatively simple and direct proof making intensive use of
Over the years researchers studied different extensions and restrictions of the model of
systems. In this modeleach automaton,called also a processor,hasno knowledgeofthe
states of other automata. Automata may not send messages. Merely, there is a central
processing unit that may freeze any automaton or let it proceed its work.
The restrictions given on multiprocessor systems are so severe that one may expect
them to be muchweaker thanmultiheadsystems. We provethat this intuitiveconjecture
is false for nondeterministic case. However, it seems to be not the case for determinis-
tic systems. We showed, by Kolmogorov complexity analysis, the separation between
classes of languagesrecognizedbydeterministic multiheadand multiprocessorsystems
which consist of two one-way devices. On the other hand we show that it is possible to
overcome power of multihead systems by adding two extra processors.
The main effort of presented results is concentrated on lower bounds. The most power-
ful technique we use is an incompressibility method, based on Kolmogorovcomplexity
analysis. We apply also counting and pumping arguments. In analysis of low message
complexity classes we found some relations between the computations of multihead
systems and solutions of some systems of diophantine equations and inequalities.
Further research. It would be interesting to investigate other versions versions of sys-
tems of finite automata (e.g. probabilistic). Some research in this subject was started by
1. P. Beame, M. Tompa, P. Yan, Communication-Space Tradeoffs for Unrestricted Protocols,
SICOMP 23 (1994), 652–661.
2. A.O. Buda, Multiprocessor automata, IPL 25 (1987), 257-161.
3. P. Duriˇ s, Z. Galil, A Time-Space Tradeoff for Language Recognition, MST 17 (1984), 3–12.
4. P.Duriˇ s,T. Jurdzi´ nski, M. Kutyłowski, K. Lory´ s, Powerof Cooperation and Multihead Finite
Systems, in Proc. ICALP’98, 896–907.
5. T. Jurdzi´ nski, M. Kutyłowski, K. Lory´ s, Multiparty Finite Computations, in Proc. CO-
6. M. Karchmer, Two Time-Space Tradeoffsfor Element Distinctness, TCS47 (1986), 237–246.
7. M. Li, P. Vitanyi, An Introduction to Kolmogorov Complexity and its Applications, Springer-
8. T. Lam, P. Tiwari, M. Tompa, Trade-offs between Communication and Space, JCSS 45
9. Ioan I. Macarie, Multihead Two-Way Probabilistic Finite Automata, Theory of Computing
Systems 30(1) (1997), 91–109.