Formally, to say that a problem can be solved in polynomial time is to say that there exists an algorithm that, given an n-bit instance of the problem as input, can produce a solution in time O(nc), where c is a constant that depends on the problem but not the particular instance of the problem.
Alan Cobham's 1965 paper entitled "The intrinsic computational difficulty of functions" is one of the earliest mentions of the concept of the complexity classP, consisting of problems decidable in polynomial time. Cobham theorized that this complexity class was a good way to describe the set of feasibly computable problems. Any problem that cannot be contained in P is not feasible, but if a real-world problem can be solved by an algorithm existing in P, generally such an algorithm will eventually be discovered.
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