Conclusive Algorithm

Definition

Conclusive Algorithm

An algorithm $A:X\to Y\cup \{\perp \}$ is conclusive on input $x$ when it returns an ordinary result rather than $\perp$:

$$A(x)\in Y.$$

It is a conclusive algorithm when this holds for every input:

$$A\text{ is conclusive}⟺\forall x\in X:A(x)\in Y⟺\forall x\in X:A(x)\ne \perp .$$

Conclusiveness says that the algorithm always gives an answer; it does not say that the answer is correct. Correctness of conclusive results is the separate property of soundness.