Combinatorial lower bound arguments for deterministic and nondeterministic Turing machines

Author:
Wolfgang Maass

Journal:
Trans. Amer. Math. Soc. **292** (1985), 675-693

MSC:
Primary 03D15; Secondary 03D10, 68Q15, 94A17

DOI:
https://doi.org/10.1090/S0002-9947-1985-0808746-4

MathSciNet review:
808746

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Abstract: We introduce new techniques for proving quadratic lower bounds for deterministic and nondeterministic -tape Turing machines (all considered Turing machines have an additional one-way input tape). In particular, we derive for the simulation of -tape Turing machines by -tape Turing machines an optimal quadratic lower bound in the deterministic case and a nearly optimal lower bound in the nondeterministic case. This answers the rather old question whether the computing power of the considered types of Turing machines is significantly increased when more than one tape is used (problem Nos. 1 and 7 in the list of Duris, Galil, Paul, Reischuk [**3**]). Further, we demonstrate a substantial superiority of nondeterminism over determinism and of co-nondeterminism over nondeterminism for -tape Turing machines.

**[1]**R. V. Book, S. A. Greibach and B. Wegbreit,*Time and tape bounded Turing acceptors and*'*s*, J. Comput. System Sci.**4**(1970), 606-621. MR**0267993 (42:2893)****[2]**P. Duris and Z. Galil,*Two tapes are better than one for nondeterministic machines*, Proc. 14th ACM STOC (1982), 1-7.**[3]**P. Duris, Z. Galil, W. J. Paul and R. Reischuk,*Two nonlinear lower bounds*, Proc. 15th ACM STOC, 1983, pp. 127-132.**[4]**O. Gabber and Z. Galil,*Explicit constructions of linear size superconcentrators*, Proc. 20th IEEE FOCS, 1979, pp. 364-370. MR**598118 (83g:68097)****[5]**Z. Galil, private communication**[6]**P. R. Halmos,*Ergodic theory*, Lecture Notes, University of Chicago, 1955.**[7]**J. Hartmanis and R. E. Stearns,*On the computational and complexity of algorithms*, Trans. Amer. Math. Soc.**117**(1965), 285-306. MR**0170805 (30:1040)****[8]**F. C. Hennie,*One-tape off-line Turing machine computations*, Inform. and Control**8**(1965), 553-578. MR**0191769 (32:9171)****[9]**F. C. Hennie and R. E. Stearns,*Two-tape simulation of multitape Turing machines*, J. Assoc. Comput. Mach.**13**(1966), 533-546. MR**0210521 (35:1413)****[10]**J. E. Hopcroft and J. D. Ullman,*Introduction to automata theory, languages and computation*, Addison-Wesley, Reading, Mass., 1979. MR**645539 (83j:68002)****[11]**R. Kannan,*Alternation and the power of nondeterminism*, Proc. 15th ACM STOC, 1983, pp. 344-346.**[12]**M. Klawe,*Non-existence of one-dimensional expanding graphs*, Proc. 22th IEEE FOCS, 1981, pp. 109-113.**[13]**M. Li,*On one tape versus two stacks*, preprint (February 1984).**[14]**W. Maass,*Simulation of two tapes by one tape requires quadratic time*, abstract (August 1983).**[15]**-,*Quadratic lower bounds for deterministic and nondeterministic Turing machines*, abstract (September 1983).**[16]**-,*Are recursion theoretic arguments useful in complexity theory*(Proc. Internat. Conf. on Logic, Methodology and Philosophy of Science, Salzburg, 1983), North-Holland, Amsterdam, 1983.**[17]**-,*Quadratic lower bounds for deterministic and nondeterministic one-tape Turing machines*, Proc. 16th ACM STOC, 1984, pp. 401-408.**[18]**-,*An optimal lower bound for random access machines and other applications of Ramsey's theorem*, abstract (June 1984).**[19]**-, Meanders,*Ramsey's theorem and lower bound arguments*(in preparation).**[20]**W. Maass and A. Schorr,*Speed-up of one-tape Turing machines by bounded alternation*(in preparation).**[21]**W. J. Paul.*On-line simulation of**tapes by**tapes requires nonlinear time*, Proc. 23rd IEEE FOCS, 1982, pp. 53-56. MR**780380****[22]**W. J. Paul, N. Pippenger, E. Szemeredi and W. Trotter,*On determinism versus nondeterminism and related problems*, Proc. 24th IEEE FOCS, 1983, pp. 429-438.**[23]**M. O. Rabin,*Real time computation*, Israel J. Math.**1**(1963), 203-211. MR**0163849 (29:1148)****[24]**P. M. B. Vitányi,*One queue or two pushdown stores take square time on a one-head tape unit*, Report CS-R8406 of the Centre for Mathematics and Computer Science, Amsterdam, 1984.

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DOI:
https://doi.org/10.1090/S0002-9947-1985-0808746-4

Article copyright:
© Copyright 1985
American Mathematical Society