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Computational Complexity (Definition)

The field classifying computational problems according to their inherent difficulty and resource requirements.
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The statement of the theorem

Let TM\text{TM} denote the set of all deterministic Turing Machines. For any M and input xM \text{ and input } x, let TM(x)T_M(x) be the time complexity function, and SM(x)S_M(x) be the space complexity function. A language L is decidable in time O(f(n))L \text{ is decidable in time } O(f(n)) if there exists a TM M_L \text{ such that } L(M_L) = L \text{ and } T_{M_L}(x) \restrict{x \text{ length } n} \text{ is bounded by } c \text{ for some constant } c \text{ and } n \text{ large enough}.\n\nWe define the complexity class P\text{P} (Polynomial Time) as:\n\nP=⋃k is a positive integerPk\text{P} = \bigcup_{k \text{ is a positive integer}} \text{P}_k \n\nwhere Pk\text{P}_k is the set of all languages LL such that LL is decided by a TM MM whose running time TM(n)T_M(n) is bounded by O(nk)O(n^k). Formally, P=DTIME(poly(n))\text{P} = \text{DTIME}(\text{poly}(n)), where DTIME(f(n))=Alg(L) s.t. L is decided by a TM M with TM(n) bounded by f(n)\text{DTIME}(f(n)) = \text{Alg}(\text{L}) \text{ s.t. } \text{L} \text{ is decided by a TM } M \text{ with } T_M(n) \text{ bounded by } f(n).\n\nMore generally, the computational complexity of a problem A\text{A} is characterized by the minimal resource function f(n)f(n) such that $\text{A} \text{ belongs to } \text{DTIME}(f(n)) \text{ or } \text{DSPACE}(f(n)).