A subfield of theoretical computer science one of whose primary goals is to classify and compare the practical difficulty of solving problems about finite combinatorial objects
Computational complexity theory is a subfield of theoretical computer science one of whose primary goals is to classify and compare the practical difficulty of solving problems about finite combinatorial objects – e.g. given two natural numbers n and m, are they relatively prime? Given a propositional formula $\phi$, does it have a satisfying assignment? If we were to play chess on a board of size $n×n$, does white have a winning strategy from a given initial position? These problems are equally difficult from the standpoint of classical computability theory in the sense that they are all effectively decidable. Yet they still appear to differ significantly in practical difficulty. For having been supplied with a pair of numbers $m>n>0$, it is possible to determine their relative primality by a method (Euclid’s algorithm) which requires a number of steps proportional to $\log(n)$. On the other hand, all known methods for solving the latter two problems require a ‘brute force’ search through a large class of cases which increase at least exponentially in the size of the problem instance.
Complexity theory attempts to make such distinctions precise by proposing a formal criterion for what it means for a mathematical problem to be feasibly decidable – i.e. that it can be solved by a conventional Turing machine in a number of steps which is proportional to a polynomial function of the size of its input.