Beslissbaarheid

Beslissbaarheid is a concept in theoretical computer science and mathematical logic that characterizes whether a given decision problem can be resolved by an algorithm that always terminates with a correct yes‑or‑no answer. A problem is said to be beslisbaar (decidable) when there exists a Turing machine, or an equivalent computational model, which halts on every input and yields the appropriate outcome. The notion originates from early work on the Entscheidungsproblem by Alonzo Church and Alan Turing in the 1930s, where they proved that a general algorithm for first‑order logic does not exist, thereby establishing the existence of undecidable problems.

In formal language theory, a language is decidable if its membership problem can be decided by a deterministic Turing machine. Typical examples of decidable problems include regular language recognition, context‑free grammar emptiness, and the halting problem restricted to specific program classes such as linear bounded automata. Conversely, classic undecidable problems encompass the general halting problem, Hilbert’s tenth problem, and the Post correspondence problem.

Decidability is closely linked to computability and complexity. While decidability concerns the existence of any terminating algorithm, complexity theory refines this by measuring the resources required, such as time or space. A problem may be decidable yet intractable if the fastest known algorithm requires super‑polynomial resources.

Research in decidability explores the boundaries between decidable and undecidable domains, often by restricting problem parameters or considering alternative computational models. The study informs practical fields such as program verification, automated theorem proving, and formal methods, where establishing decidability guarantees the reliability of automated analysis tools.