Storage requirements for deterministic polynomialtime recognizable languages

An intriguing question is whether (log n)² space is enough to recognize the class {A figure is presented} of languages recognizable in deterministic polynomial time. This question has earlier been narrowed down to the storage required to recognize a particular language called SP. SP is clearly in {A figure is presented} and it has been shown that if SP has tape complexity (log n)^k, then every member of {A figure is presented} has tape complexity (log n)^k. This paper presents further evidence in support of the conjecture that SP cannot be recognized using storage (log n)^k for any k. We have no techniques at present for proving such a statement for Turing machines in general; we prove the result for a suitably restricted device.