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Wednesday, 12 August 2015

DSPACE


Image result for https://en.wikipedia.org/wiki/DSPACE



Image result for https://en.wikipedia.org/wiki/DSPACE


In computational complexity theoryDSPACE or SPACE is the computational resource describing the resource of memory space for a deterministic Turing machine. It represents the total amount of memory space that a "normal" physical computer would need to solve a given computational problem with a given algorithm. It is one of the most well-studied complexity measures, because it corresponds so closely to an important real-world resource: the amount of physical computer memory needed to run a given program.
The measure DSPACE is used to define complexity classes, sets of all of the decision problems which can be solved using a certain amount of memory space. For each functionf(n), there is a complexity class SPACE(f(n)), the set of decision problems which can be solved by a deterministic Turing machine using space O(f(n)). There is no restriction on the amount of computation time which can be used, though there may be restrictions on some other complexity measures (like alternation).
Several important complexity classes are defined in terms of DSPACE. These include:
  • REG = DSPACE(O(1)), where REG is the class of regular languages. In fact, REG = DSPACE(o(log log n)) (that is, Ω(log log n) space is required to recognize any non-regular language).[1] [2]
Proof: Suppose that there exists a non-regular language L ∈ DSPACE(s(n)), for s(n) = o(log log n). Let M be a Turing machine deciding L in space s(n). By our assumption M ∉DSPACE(O(1)); thus, for any arbitrary k ∈ \mathbb{N}, there exists an input of M requiring more space than k.
Let x be an input of smallest size, denoted by n, that requires more space than k, and \mathcal{C} be the set of all configurations of M on input x. Because M ∈ DSPACE(s(n)), then |\mathcal{C}| \le 2^{c.s(n)} = o(log n), where c is a constant depending on M.
Let S denote the set of all possible crossing sequences of M on x. Note that the length of a crossing sequence of M on x is at most |\mathcal{C}|: if it is longer than that, then some configuration will repeat, and M will go into an infinite loop. There are also at most |\mathcal{C}| possibilities for every element of a crossing sequence, so the number of different crossing sequences of M on x is
|S|\le|\mathcal{C}|^{|\mathcal{C}|} \le (2^{c.s(n)})^{2^{c.s(n)}}= 2^{c.s(n).2^{c.s(n)}}< 2^{2^{2c.s(n)}}=2^{2^{o(\log \log n)}} = o(n)

See more  https://en.wikipedia.org/wiki/DSPACE