Abstract
In this paper we consider two-way counter machines, i.e., two-way finite automata with counters whose contents have no effect on transitions except that an attempt to decrement an empty counter will abort the computation. We show that the deterministic machines have an unsolvable emptiness problem, but that their universe problem is solvable because they accept languages whose complements are context free. In the nondeterministic case, we show that these machines are equivalent to two-way nondeterministic logspace Turing machines, and establish an infinite hierarchy based on the number of weak counters. Finally, we disprove two conjectures concerning the nondeterministic machines.
Similar content being viewed by others
References
B. S. Baker and R. V. Book, Reversal-bounded multipushdown machines,J. Comput. System Sci.,3 (1974), 315–332.
T.-H. Chan, Reversal-bounded computations, Ph.D. Thesis, Department of Computer Science, Cornell University, 1980.
T.-H. Chan, Reversal complexity of counter machines,Proceedings of the Thirteenth Annual ACM Symposium on Theory of Computing, Milwaukee, WI, (1981), pp. 146–157.
P. C. Fischer, A. R. Meyer, and A. L. Rosenberg, Counter machines and counter languages,Math. Systems Theory,2 (1968), 265–283.
S. A. Greibach, A note on the recognition of one counter languages,Rev. Française Autom. Inform. Recherche Opérationelle, R-2,9 (1975), 5–12.
S. A. Greibach, Remarks on the complexity of nondeterministic counter languages,Theoret. Comput. Sci.,1 (1976), 269–288.
S. A. Greibach, Remarks on blind and partially blind one-way multicounter machines,Theoret. Comput. Sci.,7 (1978), 311–324.
E. M. Gurari and O. H. Ibarra, The complexity of decision problems for finite-turn multicounter machines,J. Comput. System Sci.,22 (1981), 220–229.
J. Hartmanis, On nondeterminacy in simple computing devices,Acta Inform.,1 (1972), 336–344.
J. E. Hopcroft and J. D. Ullman,Introduction to Automata Theory, Languages, and Computation, Addison-Wesley, Reading, MA, 1979.
O. H. Ibarra, Reversal-bounded multicounter machines and their decision problem,J. Assoc. Comput. Mach.,25 (1978), 116–133.
O. H. Ibarra, Restricted one-counter machines with undecidable universe problems,Math. Systems Theory,13 (1979), 181–186.
R. M. Karp and R. E. Miller, Parallel program schemata,J. Comput. System Sci.,3 (1969), 147–195.
Yu. Matiyasevich, Enumerable sets are Diophantine,Dokl. Akad. Nauk SSSR,191 (1970), 279–282.
M. Minsky, Recursive unsolvability of Post's problem of Tag and other topics in the theory of Turing machines,Ann. of Math.,74 (1961), 437–455.
S. Miyano, Two-way deterministic multi-weak-counter machines,Theoret. Comput. Sci.,21 (1982), 27–37.
S. Miyano, Remarks on two-way automata with weak-counters,Inform. Process. Lett.,18 (1984), 105–107.
C. Rackoff, The covering and boundedness problems for vector addition systems,Theoret. Comput. Sci.,6 (1978), 223–231.
J. I. Seiferas, Relating refined space complexity classes,J. Comput. System Sci.,14 (1977), 100–129.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Chan, Th. On two-way weak counter machines. Math. Systems Theory 20, 31–41 (1987). https://doi.org/10.1007/BF01692057
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01692057