We state in this abstract a machine-independent characterization of the class of functions computable by log-space bounded deterministic Turi~ g machines. The characterization is similar to RW Ritchie's characterization of the linear space computable functions as equal to Grzegorczyk's class 2 [1, 2].

Both Jones [3] and Stockmeyer and Meyer [4] observed that most, if not all, of the polynomial-time reducibilities of Cook [5], and Karp [6], and others are actually log-space reducibilities. A computation which runs in log-space must halt in polynomial time, so log-space reducibility implies polynomial-time reducibility.~ Whether the converse implication holds is an important open problem. Indeed, a positive solution to this problem would solve the wellknown problem of equivalence between deterministic and nondeterministic linear Bounded automata.) Our theorem below emphasizes the fact that the concept of log-space bounded computations is relatively independent of the detailed conventions of the definition of log-space bounded machines. The theorem also provides what is essentially a high-level programmir~ language for defining and proving that various reductions or computations can be carried out in log-space.