## Combinatorial lower bound arguments for deterministic and nondeterministic Turing machines

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- by Wolfgang Maass PDF
- Trans. Amer. Math. Soc.
**292**(1985), 675-693 Request permission

## Abstract:

We introduce new techniques for proving quadratic lower bounds for deterministic and nondeterministic $1$-tape Turing machines (all considered Turing machines have an additional one-way input tape). In particular, we derive for the simulation of $2$-tape Turing machines by $1$-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 $1$-tape Turing machines.

## References

- Ronald V. Book, Sheila A. Greibach, and Ben Wegbreit,
*Time- and tape-bounded Turing acceptors and $\textrm {AFLs}$*, J. Comput. System Sci.**4**(1970), 606โ621. MR**267993**, DOI 10.1016/S0022-0000(70)80031-9
P. Duris and Z. Galil, - Ofer Gabber and Zvi Galil,
*Explicit constructions of linear size superconcentrators*, 20th Annual Symposium on Foundations of Computer Science (San Juan, Puerto Rico, 1979) IEEE, New York, 1979, pp.ย 364โ370. MR**598118**
Z. Galil, private communication
P. R. Halmos, - J. Hartmanis and R. E. Stearns,
*On the computational complexity of algorithms*, Trans. Amer. Math. Soc.**117**(1965), 285โ306. MR**170805**, DOI 10.1090/S0002-9947-1965-0170805-7 - F. C. Hennie,
*One-tape, off-line Turing machine computations*, Information and Control**8**(1965), 553โ578. MR**191769** - F. C. Hennie and R. E. Stearns,
*Two-tape simulation of multitape Turing machines*, J. Assoc. Comput. Mach.**13**(1966), 533โ546. MR**210521**, DOI 10.1145/321356.321362 - John E. Hopcroft and Jeffrey D. Ullman,
*Introduction to automata theory, languages, and computation*, Addison-Wesley Series in Computer Science, Addison-Wesley Publishing Co., Reading, Mass., 1979. MR**645539**
R. Kannan, - W. Paul,
*On-line simulation of $k+1$ tapes by $k$ tapes requires nonlinear time*, 23rd annual symposium on foundations of computer science (Chicago, Ill., 1982) IEEE, New York, 1982, pp.ย 53โ56. MR**780380**
W. J. Paul, N. Pippenger, E. Szemeredi and W. Trotter, - Michael O. Rabin,
*Real time computation*, Israel J. Math.**1**(1963), 203โ211. MR**163849**, DOI 10.1007/BF02759719
P. M. B. Vitรกnyi,

*Two tapes are better than one for nondeterministic machines*, Proc. 14th ACM STOC (1982), 1-7. P. Duris, Z. Galil, W. J. Paul and R. Reischuk,

*Two nonlinear lower bounds*, Proc. 15th ACM STOC, 1983, pp. 127-132.

*Ergodic theory*, Lecture Notes, University of Chicago, 1955.

*Alternation and the power of nondeterminism*, Proc. 15th ACM STOC, 1983, pp. 344-346. M. Klawe,

*Non-existence of one-dimensional expanding graphs*, Proc. 22th IEEE FOCS, 1981, pp. 109-113. M. Li,

*On one tape versus two stacks*, preprint (February 1984). W. Maass,

*Simulation of two tapes by one tape requires quadratic time*, abstract (August 1983). โ,

*Quadratic lower bounds for deterministic and nondeterministic Turing machines*, abstract (September 1983). โ,

*Are recursion theoretic arguments useful in complexity theory*(Proc. Internat. Conf. on Logic, Methodology and Philosophy of Science, Salzburg, 1983), North-Holland, Amsterdam, 1983. โ,

*Quadratic lower bounds for deterministic and nondeterministic one-tape Turing machines*, Proc. 16th ACM STOC, 1984, pp. 401-408. โ,

*An optimal lower bound for random access machines and other applications of Ramseyโs theorem*, abstract (June 1984). โ, Meanders,

*Ramseyโs theorem and lower bound arguments*(in preparation). W. Maass and A. Schorr,

*Speed-up of one-tape Turing machines by bounded alternation*(in preparation).

*On determinism versus nondeterminism and related problems*, Proc. 24th IEEE FOCS, 1983, pp. 429-438.

*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.

## Additional Information

- © Copyright 1985 American Mathematical Society
- 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