Päätösongelmissa

Päätösongelmissa is a Finnish term that translates to "decision problems" in English. In mathematics and computer science, a decision problem is a question with a yes/no answer. More formally, it is a problem that can be represented as a string and whose solution is either "yes" or "no". The set of all strings for which the answer is "yes" forms a language, and the decision problem is equivalent to recognizing this language.

The study of decision problems is fundamental to computability theory and complexity theory. Computability theory deals with whether a problem can be solved by an algorithm at all, regardless of how much time or memory it takes. Problems that cannot be solved by any algorithm are called undecidable. The most famous example of an undecidable problem is the Halting Problem, which asks whether a given program will eventually stop or run forever.

Complexity theory, on the other hand, focuses on classifying solvable problems based on the resources (such as time or space) required to solve them. Decision problems are often used as a basis for this classification. For instance, the complexity class P consists of decision problems that can be solved by a deterministic Turing machine in polynomial time. The complexity class NP consists of decision problems for which a proposed solution can be verified in polynomial time by a deterministic Turing machine. A major open question in computer science is whether P equals NP.

The concept of decision problems is crucial for understanding the limits of computation and for analyzing the efficiency of algorithms. Many important problems in areas like cryptography, optimization, and artificial intelligence can be formulated as decision problems.