On languages accepted with simultaneous complexity bounds and their ranking problem

In this paper, we consider Turing machines having simultaneous bounds on working space s(n), input head inversions i(n) and ambiguity degree d(n). Besides the standard notion of space complexity (Type 1), we discuss a stronger notion (Type 2). In the deterministic case, we show an optimal Ω ∞(log n/log log n) bound on input head inversions for recognizing nonregular languages on Type 1 machines that does not hold for Type 2 machines. Subsequently, the parallel complexity of the ranking problem is studied. We prove that any language accepted on Type 2 machines within s(n), i(n), d(n) simultaneous bounds such that s(n) · i(n) · d(n)=O(log n) can be ranked in DET.