comput complexity 3 (1993), 392-401 1016-3328/93/040392-10 $1o50+0.20 @1993 Birkhiiuser Verlag, Basel TWO TAPES VERSUS ONE FOR OFF-LINE TURING MACHINES WOLFGANG MAASS, GEORG SCHNITGER, ENDRE SZEMERI~DI AND GYIDRGY TURAN Abstract. We prove the first superlinear lower bound for a concrete, polynomial time recognizable decision problem on a Taring machine with one work tape and a two-way input tape (also called off-line 1-tape Turing machine). In particular, for offline Turing machines we show that two tapes are better than one and that three pushdown stores are better than two (both in the deterministic and in the nondeterministic case). Key words, off-line 1-tape Turing machines; two tapes; lower bounds; time; nondeterminism. Subject classifications. 68Q05, 68Q25. 1. Introduction A 1-tape off-line Turing machine (see Hennie 1965, p.166) is a Turing machine (TM) with one work tape and an additional two-way input tape, i.e., an input tape with end markers on which the associated read-only input head can move without restriction in both directions. These TM's are used as the standard model for the analysis of the space complexity of TM-computations. In addi- tion, they are of interest as an intermediate model between the relatively slow 1-tape TM without input tape and the relatively powerful 2-tape TM. No non-trivial lower bounds are known for the recognition of polynomial time computable languages on 2-tape Turing machines. On the other hand, lower bound arguments for concrete languages on restricted TM's have pro- gressed from 1-tape TM's without input tape (Hennie 1965, Rabin 1963) to