Decision problem

A decision problem has only two possible outputs, yes or no (or alternately 1 or 0) on any input.

In computability theory and computational complexity theory, a decision problem is a question in some formal system with a yes-or-no answer, depending on the values of some input parameters. For example, the problem "given two numbers x and y, does x evenly divide y?" is a decision problem. The answer can be either 'yes' or 'no', and depends upon the values of x and y.

Decision problems typically appear in mathematical questions of decidability, that is, the question of the existence of an effective method to determine the existence of some object or its membership in a set; many of the important problems in mathematics are undecidable.

Decision problems are closely related to function problems, which can have answers that are more complex than a simple 'yes' or 'no'. A corresponding function problem is "given two numbers x and y, what is x divided by y?". They are also related to optimization problems, which are concerned with finding the best answer to a particular problem.

A method for solving a decision problem given in the form of an algorithm is called a decision procedure for that problem. A decision procedure for the decision problem "given two numbers x and y, does x evenly divide y?" would give the steps for determining whether x evenly divides y, given x and y. One such algorithm is long division, taught to many school children. If the remainder is zero the answer produced is 'yes', otherwise it is 'no'. A decision problem which can be solved by an algorithm, such as this example, is called decidable.

The field of computational complexity categorizes decidable decision problems by how difficult they are to solve. "Difficult", in this sense, is described in terms of the computational resources needed by the most efficient algorithm for a certain problem. The field of recursion theory, meanwhile, categorizes undecidable decision problems by Turing degree, which is a measure of the noncomputability inherent in any solution.

Research in computability theory has typically focused on decision problems. As explained in the section Equivalence with function problems below, there is no loss of generality.

Definition

A decision problem is any arbitrary yes-or-no question on an infinite set of inputs. Because of this, it is traditional to define the decision problem equivalently as: the set of inputs for which the problem returns yes.

These inputs can be natural numbers, but may also be values of some other kind, such as strings over the binary alphabet {0,1} or over some other finite set of symbols. The subset of strings for which the problem returns "yes" is a formal language, and often decision problems are defined in this way as formal languages.

Alternatively, using an encoding such as Gödel numberings, any string can be encoded as a natural number, via which a decision problem can be defined as a subset of the natural numbers.

Examples

A classic example of a decidable decision problem is the set of prime numbers. It is possible to effectively decide whether a given natural number is prime by testing every possible nontrivial factor. Although much more efficient methods of primality testing are known, the existence of any effective method is enough to establish decidability.

Decidability

A decision problem A is called decidable or effectively solvable if A is a recursive set. A problem is called partially decidable, semidecidable, solvable, or provable if A is a recursively enumerable set. Problems that are not decidable are called undecidable.

The halting problem is an important undecidable decision problem; for more examples, see list of undecidable problems.

Complete problems

Decision problems can be ordered according to many-one reducibility and related feasible reductions such as polynomial-time reductions. A decision problem P is said to be complete for a set of decision problems S if P is a member of S and every problem in S can be reduced to P. Complete decision problems are used in computational complexity to characterize complexity classes of decision problems. For example, the Boolean satisfiability problem is complete for the class NP of decision problems under polynomial-time reducibility.

History

The Entscheidungsproblem, German for "Decision-problem", is attributed to David Hilbert: "At [the] 1928 conference Hilbert made his questions quite precise. First, was mathematics complete... Second, was mathematics consistent... And thirdly, was mathematics decidable? By this he meant, did there exist a definite method which could, in principle be applied to any assertion, and which was guaranteed to produce a correct decision on whether that assertion was true" (Hodges, p. 91). Hilbert believed that "in mathematics there is no ignorabimus' (Hodges, p. 91ff) meaning 'there is no limit to what can be known'. See David Hilbert and Halting Problem for more.

Equivalence with function problems

A function problem consists of a partial function f; the informal "problem" is to compute the values of f on the inputs for which it is defined.

Every function problem can be turned into a decision problem; the decision problem is just the graph of the associated function. (The graph of a function f is the set of pairs (x,y) such that f(x) = y.) If this decision problem were effectively solvable then the function problem would be as well. This reduction does not respect computational complexity, however. For example, it is possible for the graph of a function to be decidable in polynomial time (in which case running time is computed as a function of the pair (x,y) ) when the function is not computable in polynomial time (in which case running time is computed as a function of x alone). The function f(x) = 2x has this property.

Every decision problem can be converted into the function problem of computing the characteristic function of the set associated to the decision problem. If this function is computable then the associated decision problem is decidable. However, this reduction is more liberal than the standard reduction used in computational complexity (sometimes called polynomial-time many-one reduction); for example, the complexity of the characteristic functions of an NP-complete problem and its co-NP-complete complement is exactly the same even though the underlying decision problems may not be considered equivalent in some typical models of computation.

Practical decision

Having practical decision procedures for classes of logical formulas is of considerable interest for program verification and circuit verification. Pure Boolean logical formulas are usually decided using SAT-solving techniques based on the DPLL algorithm. Conjunctive formulas over linear real or rational arithmetic can be decided using the simplex algorithm, formulas in linear integer arithmetic (Presburger arithmetic) can be decided using Cooper's algorithm or William Pugh's Omega test. Formulas with negations, conjunctions and disjunctions combine the difficulties of satisfiability testing with that of decision of conjunctions; they are generally decided nowadays using SMT-solving technique, which combine SAT-solving with decision procedures for conjunctions and propagation techniques. Real polynomial arithmetic, also known as the theory of real closed fields, is decidable, for instance using the cylindrical algebraic decomposition; unfortunately the complexity of that algorithm is excessive for most practical uses.