Satisfiability

In mathematical logic, satisfiability and validity are elementary concepts of semantics. A formula is satisfiable if it is possible to find an interpretation (model) that makes the formula true. A formula is valid if all interpretations make the formula true. The opposites of these concepts are unsatisfiability and invalidity, that is, a formula is unsatisfiable if none of the interpretations make the formula true, and invalid if some such interpretation makes the formula false. These four concepts are related to each other in a manner exactly analogous to Aristotle's square of opposition.

The four concepts can be raised to apply to whole theories: a theory is satisfiable (valid) if one (all) of the interpretations make(s) each of the axioms of the theory true, and a theory is unsatisfiable (invalid) if all (one) of the interpretations make(s) each of the axioms of the theory false.

It is also possible to consider only interpretations that make all of the axioms of a second theory true. This generalization is commonly called satisfiability modulo theories.

Boolean satisfiability problem

In computer science, the Boolean Satisfiability Problem (sometimes called Propositional Satisfiability Problem and abbreviated as SATISFIABILITY or SAT) is the problem of determining if there exists an interpretation that satisfies a given Boolean formula. In other words, it asks whether the variables of a given Boolean formula can be consistently replaced by the values TRUE or FALSE in such a way that the formula evaluates to TRUE. If this is the case, the formula is called satisfiable. On the other hand, if no such assignment exists, the function expressed by the formula is identically FALSE for all possible variable assignments and the formula is unsatisfiable. For example, the formula "a AND NOT b" is satisfiable because one can find the values a = TRUE and b = FALSE, which make (a AND NOT b) = TRUE. In contrast, "a AND NOT a" is unsatisfiable.

SAT is one of the first problems that was proven to be NP-complete. This means that all problems in the complexity class NP, which includes a wide range of natural decision and optimization problems, are at most as difficult to solve as SAT. There is no known algorithm that efficiently solves SAT, and it is generally believed that no such algorithm exists; yet this belief has not been proven mathematically, and resolving the question whether SAT has an efficient algorithm is equivalent to the P versus NP problem, which is the most famous open problem in the theory of computing.

2-satisfiability

In computer science, 2-satisfiability (abbreviated as 2-SAT or just 2SAT) is the problem of determining whether a collection of two-valued (Boolean or binary) variables with constraints on pairs of variables can be assigned values satisfying all the constraints. It is a special case of the general Boolean satisfiability problem, which can involve constraints on more than two variables, and of constraint satisfaction problems, which can allow more than two choices for the value of each variable. But in contrast to those problems, which are NP-complete, it has a known polynomial time solution. Instances of the 2-satisfiability problem are typically expressed as 2-CNF or Krom formulas.

Problem representations

A 2-SAT problem may be described using a Boolean expression with a special restricted form: a conjunction of disjunctions (and of ors), where each disjunction (or operation) has two arguments that may either be variables or the negations of variables. The variables or their negations appearing in this formula are known as terms and the disjunctions of pairs of terms are known as clauses. For example, the following formula is in conjunctive normal form, with seven variables and eleven clauses: