Abstract

We consider the problem of computing the costs---{ in terms of states---of optimal simulations between different kinds of finite automata recognizing unary languages. Our main result is a tight simulation of unary n-state two-way nondeterministic automata by $O({{\rm e}^{\sqrt{{n}\ln{n}}}})$-state one-way deterministic automata. In addition, we show that, given a unary n-state two-way nondeterministic automaton, one can construct an equivalent O(n2 )-state two-way nondeterministic automaton performing both input head reversals and nondeterministic choices only at the ends of the input tape. Further results on simulating unary one-way alternating finite automata are also discussed.

MSC codes

  1. 68Q10
  2. 68Q45
  3. 68Q68

Keywords

  1. formal languages; deterministic
  2. nondeterministic
  3. and alternating finite state automata; unary languages

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References

1.
C. Berge, Graphs and Hypergraphs, North‐Holland, Amsterdam, 1985.
2.
P. Berman and A. Lingas, On the Complexity of Regular Languages in Terms of Finite Automata, Tech. Report 304, Polish Academy of Sciences, 1977.
3.
Jean‐Camille Birget, State‐complexity of finite‐state devices, state compressibility and incompressibility, Math. Systems Theory, 26 (1993), 237–269
4.
Alfred Brauer, On a problem of partitions, Amer. J. Math., 64 (1942), 299–312
5.
Alfred Brauer, James Shockley, On a problem of Frobenius, J. Reine Angew. Math., 211 (1962), 215–220
6.
J. Brzozowski, E. Leiss, On equations for regular languages, finite automata, and sequential networks, Theoret. Comput. Sci., 10 (1980), 19–35
7.
Ashok Chandra, Dexter Kozen, Larry Stockmeyer, Alternation, J. Assoc. Comput. Mach., 28 (1981), 114–133
8.
Marek Chrobak, Finite automata and unary languages, Theoret. Comput. Sci., 47 (1986), 149–158
9.
P. Erdös and R. L. Graham, On a linear diophantine problem of Frobenius, Acta Arith., 37 (1980), pp. 321–331.
10.
A. Fellah, H. Jürgensen, and S. Yu, Constructions for alternating finite automata, Internat. J. Comput. Math., 35 (1990), pp. 117–132.
11.
Viliam Geffert, Nondeterministic computations in sublogarithmic space and space constructibility, SIAM J. Comput., 20 (1991), 484–498
12.
Viliam Geffert, Tally versions of the Savitch and Immerman‐Szelepcsényi theorems for sublogarithmic space, SIAM J. Comput., 22 (1993), 102–113
13.
V. Geffert, private communication, 1997.
14.
John Hopcroft, Jeffrey Ullman, Wprowadzenie do teorii automatów, języków i obliczeń, Wydawnictwo Naukowe PWN, Warsaw, 1994, 480–0, Translated from the 1979 English original by Beata Konikowska
15.
E. Landau, Uber die maximalordung der permutationen gegebenen grades, Archiv. der Math. und Phys., 3 (1903), pp. 92–103.
16.
E. Landau, Handbuch der lehre von der verteilung der primzahlen. I, Teubner, Leipzig, Berlin, 1909.
17.
Ernst Leiss, Succinct representation of regular languages by Boolean automata, Theoret. Comput. Sci., 13 (1981), 323–330
18.
P. Lewis II, R. Stearns, and J. Hartmanis, Memory bounds for recognition of context free and context sensitive languages, in IEEE Conference Record on Switching Circuit Theory and Logical Design, 1965, pp. 191–202.
19.
U. Liubicz, Bounds for the optimal determinization of nondeterministicautonomic automata, Sibirskii Matemat. Journal, 2 (1964), pp. 337–355 (in Russian).
20.
Carlo Mereghetti, Giovanni Pighizzini, Optimal simulations between unary automata, Lecture Notes in Comput. Sci., Vol. 1373, Springer, Berlin, 1998, 139–149
21.
Ivan Niven, Herbert Zuckerman, Hugh Montgomery, An introduction to the theory of numbers, John Wiley & Sons Inc., 1991xiv+529
22.
M. Rabin and D. Scott, Finite automata and their decision problems, IBM J. Res. Develop, 3 (1959), pp. 114–125.
Also in E. F. Moore, Sequential Machines, Addison–Wesley, Reading, MA, 1964, pp. 63–91.
23.
William Sakoda, Michael Sipser, Nondeterminism and the size of two‐way finite automata, ACM, New York, 1978, 275–286
24.
J. C. Shepherdson, The reduction of two‐way automata to one‐way automata, IBM J. Res. Develop, 3 (1959), pp. 198–200.
Also in E. F. Moore, Sequential Machines, Addison–Wesley, Reading, MA, 1964, pp. 92–97.
25.
Michael Sipser, Lower bounds on the size of sweeping automata, J. Comput. System Sci., 21 (1980), 195–202
26.
M. Szalay, On the maximal order in Sn and Sn*, Acta Arith., 37 (1980), pp. 321–331.
27.
Andrzej Szepietowski, Turing machines with sublogarithmic space, Lecture Notes in Computer Science, Vol. 843, Springer‐Verlag, 1994vi+115
28.
Paul Turán, Combinatorics, partitions, group theory, Accad. Naz. Lincei, Rome, 1976, 0–0, 181–200. Atti dei Convegni Lincei, No. 17

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cover image SIAM Journal on Computing
SIAM Journal on Computing
Pages: 1976 - 1992
ISSN (online): 1095-7111

History

Published online: 27 July 2006

MSC codes

  1. 68Q10
  2. 68Q45
  3. 68Q68

Keywords

  1. formal languages; deterministic
  2. nondeterministic
  3. and alternating finite state automata; unary languages

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