Whar do you mean by computational Complexity? Explian about the time and space complexity of a Turing machine.

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Computational complexity is a computer science concept that focuses on the amount of computing resources needed for particular kinds of tasks. In computational complexity theory, researchers assess the kinds of resources that will be needed for a given type or class of task in order to classify different kinds of tasks into various levels of complexity

A Turing Machine (TM) is a mathematical model which consists of an infinite length tape divided into cells on which input is given. It consists of a head which reads the input tape. A state register stores the state of the Turing machine. After reading an input symbol, it is replaced with another symbol, its internal state is changed, and it moves from one cell to the right or left. If the TM reaches the final state, the input string is accepted, otherwise rejected.

A TM can be formally described as a 7-tuple (Q, X, ∑, δ, q0, B, F) where −

  • Q is a finite set of states
  • X is the tape alphabet
  •  is the input alphabet
  • δ is a transition function; δ : Q × X → Q × X × {Left_shift, Right_shift}.
  • q0 is the initial state
  • B is the blank symbol
  • F is the set of final states

Time and Space Complexity of a Turing Machine

For a Turing machine, the time complexity refers to the measure of the number of times the tape moves when the machine is initialized for some input symbols and the space complexity is the number of cells of the tape written.

Time complexity all reasonable functions −

T(n) = O(n log n)

TM’s space complexity −

S(n) = O(n)

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