Relativizing time and space

The problem of finding the precise relationship between computation time afldspa-ce is very important ·in complexity theory. Because this relationship remains unknown, there is an exponential discrepancy when upper and lower bounds are both expressed in tenns of time or space alone.· For example, not only is it not known whether a linear upper bound f~r space implies simultaneous upper bounds of linear space and polynomial time but also it is not known whether a linear upper bound for space implies a polynomial upper bound for time regardless of how much space is used. One method of approaching questions regarding the relationship between complexity classes is to study relativized complexity classes. One might attempt to prove that NP =PSPACE if and only if for every oracle set A, NP(A) = PSPACE"(A); if one could prove this, then one could conclude that NP , PSPACE since it is known that there exists a set A such that NP(A), PSPACE{A) [2,.3,13], and knowing that NP ~ PSPACE would allow one to conclude that there exists a problem solvable in linear space but not solvable in polynomial time. However this equivalence has not been established. Different notions of controlled relativizations have been studied, on the one hand by controlling the resources bounding the query tape of