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A representation of the relation among complexity classes. This is a list of complexity classes in computational complexity theory. For other computational and complexity subjects, see list of computability and complexity topics. Many of these classes have a 'co' partner which consists of the complements of all languages in the original class ...
A decision problem is EXPSPACE-complete if it is in EXPSPACE, and every problem in EXPSPACE has a polynomial-time many-one reduction to it. In other words, there is a polynomial-time algorithm that transforms instances of one to instances of the other with the same answer. EXPSPACE-complete problems might be thought of as the hardest problems ...
A reduction is a transformation of one problem into another problem, i.e. a reduction takes inputs from one problem and transforms them into inputs of another problem. For instance, you can reduce ordinary base-10 addition x + y {\displaystyle x+y} to base-2 addition by transforming x {\displaystyle x} and y {\displaystyle y} to their base-2 ...
Two notable examples in mathematics that have been solved and closed by researchers in the late twentieth century are Fermat's Last Theorem [1] and the four-color theorem. [2] [3] An important open mathematics problem solved in the early 21st century is the Poincaré conjecture. Open problems exist in all scientific fields.
An example of a decision problem is deciding with the help of an algorithm whether a given natural number is prime. Another example is the problem, "given two numbers x and y, does x evenly divide y?" A method for solving a decision problem, given in the form of an algorithm, is called a decision procedure for that problem.
Other examples of EXPTIME-complete problems include the problem of evaluating a position in generalized chess, [5] checkers, [6] or Go (with Japanese ko rules). [7] These games have a chance of being EXPTIME-complete because games can last for a number of moves that is exponential in the size of the board.
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By the definition of NP, each problem instance that is answered 'yes' has a polynomial-size certificate which serves as a proof for the 'yes' answer. Thus, the set of these tuples ( x , y ) {\displaystyle (x,y)} forms a relation, representing the function problem "given x {\displaystyle x} in L {\displaystyle L} , find a certificate y ...