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Another definition is "a problem whose social complexity means that it has no determinable stopping point". [2] Moreover, because of complex interdependencies, the effort to solve one aspect of a wicked problem may reveal or create other problems. Due to their complexity, wicked problems are often characterized by organized irresponsibility.
Many, if not most, undecidable problems in mathematics can be posed as word problems: determining when two distinct strings of symbols (encoding some mathematical concept or object) represent the same object or not. For undecidability in axiomatic mathematics, see List of statements undecidable in ZFC.
A decision problem is a question which, for every input in some infinite set of inputs, answers "yes" or "no". [2] Those inputs can be numbers (for example, the decision problem "is the input a prime number?") or values of some other kind, such as strings of a formal language.
It is also very possible that a proof would not lead to practical algorithms for NP-complete problems. The formulation of the problem does not require that the bounding polynomial be small or even specifically known. A non-constructive proof might show a solution exists without specifying either an algorithm to obtain it or a specific bound ...
Experiments in the 1960s and early 1970s asked participants to solve relatively simple, well-defined, but not previously seen laboratory tasks. [ 17 ] [ 18 ] These simple problems, such as the Tower of Hanoi , admitted optimal solutions that could be found quickly, allowing researchers to observe the full problem-solving process.
The definition: A real number is algebraic if it’s the root of some polynomial with integer coefficients. For example, x²-6 is a polynomial with integer coefficients, since 1 and -6 are integers.
Many mathematical problems have been stated but not yet solved. These problems come from many areas of mathematics, such as theoretical physics, computer science, algebra, analysis, combinatorics, algebraic, differential, discrete and Euclidean geometries, graph theory, group theory, model theory, number theory, set theory, Ramsey theory, dynamical systems, and partial differential equations.
This definition is equivalent to the verifier-based definition because a nondeterministic Turing machine could solve an NP problem in polynomial time by nondeterministically selecting a certificate and running the verifier on the certificate. Similarly, if such a machine exists, then a polynomial time verifier can naturally be constructed from it.