Decision Problems

A decision problem consists of a specification of a subset of the possible instances. Given an instance, one is required to determine whether the instance is in the specified set (e.g., the set of prime numbers, the set of connected graphs, or the set of sorted sequences). For example, consider the problem where one is given a natural number, and is asked to determine whether or not the number is a prime. One important case, which corresponds to the aforementioned search problems, is the case of the set of instances having a solution; that is, for any binary relation R ⊆ {0, 1}∗ × {0, 1}∗ we consider the set {x : R(x) = ∅}. Indeed, being able to determine whether or not a solution exists is a prerequisite to being able to solve the corresponding search problem (as per Definition 1.1). In general, decision problems refer to the natural task of making a binary decision, a task that is not uncommon in daily life (e.g., determining whether a traffic light is red). In any case, in the following definition of solving decision problems, the potential solver is again a function; that is, in this case the solver is a Boolean function, which is supposed to indicate membership in a predetermined set.