11 Facts About Undecidable problem

In computability theory and computational complexity theory, an undecidable problem is a decision problem for which it is proved to be impossible to construct an algorithm that always leads to a correct yes-or-no answer.

The halting problem is an example: it can be proven that there is no algorithm that correctly determines whether arbitrary programs eventually halt when run.

Decision Undecidable problem is any arbitrary yes-or-no question on an infinite set of inputs.

Formally, a decision Undecidable problem is a subset of the natural numbers.

The corresponding informal Undecidable problem is that of deciding whether a given number is in the set.

In computability theory, the halting Undecidable problem is a decision Undecidable problem which can be stated as follows:.

Such a problem is said to be undecidable if there is no computable function that correctly answers every question in the problem set.

The connection between these two is that if a decision problem is undecidable then there is no consistent, effective formal system which proves for every question A in the problem either "the answer to A is yes" or "the answer to A is no".

One of the first problems suspected to be undecidable, in the second sense of the term, was the word problem for groups, first posed by Max Dehn in 1911, which asks if there is a finitely presented group for which no algorithm exists to determine whether two words are equivalent.

Since we have only one equation but n variables, infinitely many solutions exist in the complex plane; however, the Undecidable problem becomes impossible if solutions are constrained to integer values only.

Matiyasevich showed this Undecidable problem to be unsolvable by mapping a Diophantine equation to a recursively enumerable set and invoking Godel's Incompleteness Theorem.