Storage Requirements for Deterministic Polynomial Time Recognizable Languages

An intriguing question is whether (log n)^2 space is enough to recognize the class 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 and it has been shown that if SP has tape complexity (log n)^k, then every member of 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.