Question

Determine the relationship between NSPACE(n^3) and SPACE((n^6) *

log n) and give justification. If one class is contained in the
other, give justification as to why the containment is proper or
not.

Answer 1

NSPACE(n^3) is contained within SPACE((n^6) * log n) as non-deterministic Turing machines frequently use less space than deterministic ones. However, the containment is not necessarily proper because there is no proof of an explicit separation between these classes.

Step-by-step explanation:

The relationship between NSPACE(n^3) and SPACE((n^6) * log n) involves understanding the complexity classes of space requirements for Turing machines to solve decision problems. NSPACE refers to the space complexity of non-deterministic Turing machines, while SPACE refers to that of deterministic Turing machines.

Generally, a non-deterministic Turing machine can solve problems using less space than a deterministic one would require, which suggests that NSPACE(n^3) is contained within SPACE((n^6) * log n) because, in the worst case, a deterministic Turing machine may simulate a non-deterministic one by exploring all possible branches in its computation, multiplying its space requirements. However, this containment is not necessarily proper because we currently do not have a proof that shows an explicit separation between these two complexity classes, meaning that it is not proven that problems exist which are in SPACE((n^6) * log n) but not in NSPACE(n^3).