77-I62 661 SIMULATIONS AMONG MULTIDIMENSIONAL ITERATIVE ARRAYS / ITERATIVE TREE AUTOMA (U) ILLINOIS UNIV AT URBANA COORDINATED SCIENCE LAS J L TRAHAN JAN 86 M UNCLASSIFIED UILU-ENG-86-2202 F/G 1211 N I EEBhEEBhEEBE Ifllllllflllll IIIIIIIIIIIIIIl..flf, HIIIIE
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IIIIIIIIIIIIIIl..flf, HIIIIE and alternating Turing machines are important models of parallel computation. A d ... A deterministic Turing machine ... who established that DTM(t) ...
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77-I62 661 SIMULATIONS AMONG MULTIDIMENSIONAL ITERATIVE ARRAYS
/ITERATIVE TREE AUTOMA (U) ILLINOIS UNIV AT URBANACOORDINATED SCIENCE LAS J L TRAHAN JAN 86 M
UNCLASSIFIED UILU-ENG-86-2202 F/G 1211 N
I EEBhEEBhEEBEIfllllllflllllIIIIIIIIIIIIIIl..flf,HIIIIE
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MICROCOPY RESOLUTION TEST CHARTNATIONAL BUREAU OF STANOARDS-I963-A
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January 1986 UILU-ENG-86-2202ACT-66
COORDINATED SCIENCE LABORATORYCollege of Engineering
ID
SIMULATIONS AMONG MULTI-DIMENSIONAL ITERATIVE ARRAYS,ITERATIVE TREE AUTOMATA, ANDALTERNATING TURING MACHINES
BJerry Lee Trahan
DTICSIELECTED
DEC 2 7 9856S
UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN
Approved for Public Release. Distribution Unlimited. 85 12 27 028
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a.. NAME OP PERFORMING ORGANIZATION IS. OFPFICE SYMBOL 7& NAME OF MONITORING ORGANIZATIONCoordinated Science Laboratory (f.,asbw,University of Illinois N/A Office of Naval ResearchSc. ADDRESS Cty. S&ate and ZIP Code) 7b. ADDRESS (City. State .nd ZIP Code)
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IIIIELO GROUP 7 Uu s. an. I ;Iterative array, alternting turing machine, parallelcomputation, sinulation, computational complexity theory
IC5 I i_ c 2 FeTs#~SS. A63TRACT (Continue on nevrse if nacewMry and iden if'y by MEock number)---We present three simulations: a simulation of an Alternating Turing machine (ATM)
operating in time T(n) by an iterative tree automation (tTA) int. -.... -:, a simulation
of a d-d mensional iterative array (dIA) operating in time T(n) by an ATM 4-_.
-O((T(-4. , and a simulation of an ITA operating in time T(n) by an ATM.±Lt tme-O((T(.)}n-) The first two improve previously known results. The first implies the
simulation of a nondeterministic Turing machine by an ITA in time O(T(n)) of Culik and
Yu (19844i The second is stronger than the simulation of a dlA by an ATM in ti"_0((T(n)) /logT(n)) of Seiferas (1977) and Dymond and Tompa (1985). 7 . #.
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20. 0ISTRI E.TION,AVA!LABIL.TY OF ASSTRACT 121. ABSTRACT SECURIrY :LASSIFICA TIO.'
UNCLASSiFiED/UNL;MIE. SAME AS APT. "Z
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11. tree automata, and alternating turing machines
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SIMULATIONS AMONG MULTIDIMENSIONAL ITERATIVE ARRAYS.ITERATIVE TREE AUTOMATA.
AND ALTERNATING TURING MACIlINES
BY
JERRY LEE TRAHAN
B.S.. Louisiana State University and A. & M. College. 1983
THESIS
Submitted in partial fulfillment of the requirementsfor the degree of Master of Science in Electrical Engineering
in the Graduate College of theUniversity of Illinois at Urbana-Champaign. 1986
Accession For
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ABSTRACTU
We present three simulations: a simulation of an alternating Turing machine (ATM)
operating in time T(n ) by an iterative tree automaton (ITA) in time O(T(n )). a simulation
of a d-dimensional iterative array (dIA) operating in time T(n) by an ATM in time
O((T(n )Yd). and a simulation of an ITA operating in time T(n) by an ATM in time
SO((T(n ))). The first two improve previously known results. The first implies the simula-
tion of a nondeterministic Turing machine by an ITA in time O(T(n )) of Culik and Yu
(1984). The second is stronger than the simulation of a dIA by an ATM in time
0 ((T(n ))d+ /logT(n )) of Seiferas (1977) and Dymond and Tompa (1985).
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ACKNOWLEDGEMENTS
I wish to thank my thesis advisor. Dr. Michael C. Loui. for his encouragement. guidance. and
constructive suggestions for this thesis. I also wish to thank my parents for their constant
support throughout my education. This work was supported in part by the Office of Naval
Research under contract N00014-85-K-0570.
,
2
TABLE OF CONTENTS
Chapter Page
1. INTRODUCTION ....................................................... I
accepting. A configuration with a universal state is accepting if every successor configuration
is accepting. An ATM accepts an input string if its initial configuration is accepting. An
ATM has a two-way read-only input tape with endmarkers and k worktapes. which are ini-
tially blank. A step of an ATM consists of reading one symbol from each worktape and
reading an input symbol. then writing a symbol on each of the worktapes, moving each of
the heads left or right one tape square or not moving the tape heads. and choosing a new state
-from the set specified by the transition function.
One can describe all possible computations of an ATM on some input string as a compu-
tation tree. All nodes are configurations. the root is the initial configuration. and the children
of any configuration c are exactly those configurations that can be reached from c in one step
according to the transition rules of the ATM. The leaves of the tree are the final
configurations and may be accepting or rejecting. A branch of the computation tree is a
downward directed path from the root: in other words, a branch is a sequence of
configurations starting with the initial configuration. Assume that for an ATM to run in time
T(n). all branches terminate in at most T(n) steps.
Formally. an ATM AM is a septuple AM - (k .Q .Z.r.8.q,,.g ) where
k is the number of wotk~apes.
Q is a finite set of states.
Z is a finite input alphabet (S kZ is an endmarker).
r is a finite worktape alphabet (B E r is the blank symbol).
s: Q xf x(z U i$ )"P(Q x(F -B I)' xleft . right .stationary 1h is the transition
function, where P(S) is he power set of S. that is. the collection of subsets of S.
q, E Q is the initiai sate. and
g: Q -- universal . existential , accept reject is a mapping identifying each state as a
universal, existential, accepting, or rejecting state.
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Chapter 3
LITERATURE REVIEW
This chapter reviews the literature related to the research reported in this thesis.
Chandra. Kozen. and Stockmeyer (1981) present the concept of alternation. (The same tauthors originally presented the concept in Chandra and Stockmeyer (1976) and Kozen
(1976).) This thesis uses their ATM model. They derive significant relationships between
classes of languages accepted by time and space bounded ATMs and those accepted by time
and space bounded DTMs. In particular. logarithmic alternating space is equivalent to poly-
nomial deterministic time. and polynomial alternating time is equivalent to polynomial
deterministic space. Paul. Prauss. and Reischuk (1980) demonstrate that an ATM with a sin-
gle tape can simulate an ATM with multiple tapes in linear time.
Dymond and Tompa (1985) prove another result related to this thesis. They establish
that DTM(t) Q ATM(t/log t). Their proof associates the computation of the DTM with an
acyclic directed graph. They use a two-person pebbling game to pebble the graph within a
time bound of O(n/log n) for a graph with n vertices. Next, the ATM steps simulate the
two-person pebbling of the graph. In the pebbling game. one person's moves correspond to
existential choices of the ATM. and the other person's moves correspond to universal choices
of the ATM.
Paterson (1972) represents a TM computation as a two-dimensional diagram of succes-
sive tape configurations. He employs divide-and-conquer in both time and space dimensions.
This method is generalized in this thesis in the simulation of a dIA by an ATM.. Loui (1981)
establishes a space bound for a DTM to accept the same language as a d-dimensional NTM
with one worktape head. The proof utilizes a generalization of crossing sequences across the,boundaries of d-dimensional boxes of the worktape. He uses a divide-and-conquer method to
recursively partition the boxes.
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Rosenfeld (1979) presents a good review of iterative automata.
Cole (1969) formally presents the d-dimensional iterative array of finite state
machines. He establishes that the computing speed of a dIA can be increased by a constant
factor by enlarging the set-of states of each machine. He proves that the class of context-free
languages does not contain all the sets of strings accepted by a dIA. nor do the sets of strings
accepted by a dIA contain all context-free languages. He proves that computing capability
increases as the number of dimensions increases.
Seiferas (1977a) extends Cole's work on deterministic dIAs to nondeterministic dlAs
(NdIAs). He derives that
NTM(td ) Q NdIA(t).
NdIA(t) Q NTM(td ") and
dliA(t) Q DTM(tdO1).
The second result is related to the simulation of an NdIA by an ATM given in this thesis.
His simulation uses about nd steps of a one-dimensional (2d+1) head TM to simulate the nth
step in a computation of a dIA.
Seiferas (1977b) establishes that a dIA with direct central control is no more powerful
than a regular dIA. and that a regular dIA can simulate a dIA with direct central control in
linear time. In this simulation. the finite state machine at the origin of the dIA controls the
dIA indirectly by propagating the value of its state outward using only local commun4cation.
Culik and Yu (1984) construct a language L such that an ITA accepts L in real-time.
but no dA can accept L in real-time. They state that the converse problem. that is, whether
real-time dA languages (d >2) are properly contained in real-time ITA languages. is an open
problem. They establish that an ITA can simulate an NTM in linear time. Their simulation
provides a basis for the simulation of an ATM by an ITA in Chapter 4.
I.*,1
9
Chapter 4
P THE ITA SIMULATION OF THE DIA
This chapter contains a simulation of the ATM by the ITA anid a simulation of the dIA
by the ATM and proofs of the correctness of each simulation.
Lemma 1: Every language recognized in time tby a k -tape ATM can be recognized in time
0 (t ) by a one-tape ATM.
This result is from Paul. Prauss. and Reischuk (1980). They specify that the one-tape
ATM does not have separate input and output tapes.
Lemma 2: Every t steps of an ATM with at most c >.3 choices at each step can be simulated
by (c -1 )t steps of an ATM with at most 2 choices at each step.
Proof: Let Ml = (k .Q .L..q,.g) be an ATM with at most c choices at each step. We
define an ATM Al' =(k .Q,.z.r.S*.q, g) with at most 2 choices at each step such that M'
polynomial time; hence. the computational complexity class P would equal the computational
cbmplexity class NP. The equality P = NP is widely believed to be unlikely; hence,a polyno-
mial time simulation of an NTM by a dA is unlikely. Therefore. a polynomial time simula-
tion of an ATM by a dIA is unlikely.
Proposition 8: For all T(n). every language recognized in time T(n) by an ATM can be recog-
nized in time 0(22T' )) by a dIA.
Briefly, a (deterministic) dA can simulate an ATM in exponential time as follows. The
dlA can simply compute each branch of the computation tree of the ATMvl in turn. Each
branch corresponds to the actions of a DTM. and a dIA is able to simulate a DTM in linear
time according to Seiferas (1977b).
Theorem 7 and Proposition 8 together yield an exponential time bound for a dIA simu-
lation of an ITA. One would expect this bound because the number of finite state machines
potentially involved in a computation grows polynomially with time for a dIA. but grows
exponentially with time for an ITA.
4i..
.44"
-71,
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Chapter 6
CONCLUSIONS AND OPEN PROBLEMS
This thesis has presented three simulations and discussed a fourth. When combined.
Theorems 5 and 6 imply that an ITA can simulate a dIA in time O (t d ). and Theorem 7 and
Proposition 8 imply that a dIA can simulate an ITA in exponential time.
These results and the work done in obtaining them suggest several open problems.
1. Can the time bounds of Theorems 5. 6. and 7 be improved?
2. Is there a language L such that some dIA recognizes L in linear time, but every ITA
requires superlinear time to recognize L? Culik and Yu (1984) pose this question. but for
real-time. One candidate considered for L is a string of the form
;"L =(x I# #..X.. ,,, ##y: ...#y,, )
where x l.x x,,, is an unordered list of values, and y I.y . y,, is a sorted list of the same
values. This candidate fails because an ITA can sort in time O(n) according to Browning
(1979). and. though a 21A can sort in time O(vr'J logn ) according to Thompson and Kung
(1977). Nassimi and Sahni (1979). and Stout (1982). the dIA must write down the output.
leading to a total time requirement of O(n). A second possible candidate is a language of
binary strings that represent connected d-dimensional figures.
3. How much time is required for an X-tree array to simulate an ATM? An X-tree is a
binary tree with additional edges connecting all nodes at the same level in the tree. An X-
tree array is an iterative array of finite state machines organized into an X-tree.
S.4. How much time is required for an A'rM to simulate an X-tree array?
5. How can an ITA with depth as a function of the length of the input string simulate
an ATM or an NTM?
35
6. How much time and space are required for an ATM with a limit on the number of
its alternations to simulate either the dIA or the ITA?
A further possible area for future research is the alternating iterative array. such as
either an alternating dA or an alternating ITA. The addition of universal choices to non-
deterministic dIAs or ITAs adds a second kind of parallelism. Initial work could be done in
relating such a model to other models of computation.
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REFERENCES
S. A. Browning (1979). "Computations on a Tree of Processors." Caltech Conjerence on VLSI.. pp. 453-478. January 1979.
A. K. Chandra. D. C. Kozen. and L. J. Stockmeyer (1981). "Alternation. 0 J. of the Associationfor Computing Machinery, vol. 28. no. 1. pp. 114-133. January 1981.
A. K. Chandra and L. J. Stockmeyer (1976). "Alternation." Proceedings o1 the 17th IEEESymposium on Foundations of Computer Science, pp. 98-108. October 1976.
S. N. Cole (1969). "Real-Time Computation by n-Dimensional Iterative Arrays of Finite-State Machines." IEEE Trans. Comput., vol. C-18. no. 4. pp. 349-365. April 1969.
K. Culik 11 and S. Yu (1984). "Iterative Tree Automata." Theoretical Computer Science. vol.32. no. 3, pp. 227-247. August 1984.
P. W. Dymond and S. A. Cook (1980). "Hardware Complexity and Parallel Computation."Proceedings oj the 21st IE'. Symposium on Foundations of Computer Science, pp.360-372. October 1980.
P. W. Dymond and M. Tompa (1985). "Speedups of Deterministic Machines by SynchronousParallel Machines." J. of Computer and System Sciences, vol. 30. no. 2. pp. 149-161.April 1985.
S. Fortune and J. Wyllie (1978). "Parallelism in Random Access Machines." Proccedirg ofthe Tenth Annual ACM Symposium on Theory of Computing, pp. 114-118. May 1978.
J. E. Hopcroft and J. D. Ullman (1979). Introduction to Automata Theory. Languages. andComputation, Reading. MA: Addison-wesley. 1979.
D. Kozen (1976), "On Parallelism in Turing Machines." Proceedings of the 17th IEEE Sympo-sium on Foundations of Computer Science, pp. 89-97. October 1976.
M. C. Loui (1981). "A Space Bound for One-Tape Multidimensional Turing Machines -zot ), -:Theoretica( Computer Science. vol. 15, no. 3. pp. 311-320. September 1981.
D. Nassimi and S. Sahni (1979). "Bitonic Sort on a Mesh-Connected Parallel Computer." IEEETrans. Comput., vol. C-27. no. 1. January 1979.
M. S. Paterson (1972). "Tape Bounds for Time-Bounded Turing Machines." J. of Corru;lerand SYstem Sciences, vol. 6. no. 2. pp. 116-124. April 1972.
W. .. Paul, W. J. Prauss. and R. Reischuk (1980)). "On Alternation." Acta InlOrmaticc. -ol.14. no. 3. pp. 243-255. September 1980.
A. Rosenfeld (1979). Picture Languages. New York. NY: Academic Press. 1979.
W. J. Savitch (1970). "Relationships Between Nondeterministic and Deterministic Tape Con-piexities.' J. ,f Computer and Syzem Sciences. vol. 4. no. 2. pp. 177-192. April 1970.
J. I. Seiferas (19 7 7 a). "Linear-Time Computation by Nondeterministic Multidimensional
AL,
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Iterative Arrays." SIAM 1. Comput.. vol. 6. no. 3. pp. 487-504. September 1977.
J. I. Seiferas (1977b). "Iterative Arrays with Direct Central Control." Acta Inf(w'maica. vol.8. no. 2. pp. 177-192. 1977.
SQ. F. Stout (1982). "Using Clerks in Parallel Processing." Proceedings of the 23rd IEEE Sym-posium on Foundations of Computer Science, pp. 272-279. November 1982.
C. D. Thompson and H. T. Kung (1977). "Sorting on a Mesh-Connected Parallel Computer."Communications of the ACM, vol. 20, no. 4. pp. 263-271. April 1977.