Boolean satisfiability problem

A propositional logic formula, also called Boolean expression, is built from variables, operators AND (conjunction, also denoted by ∧), OR (disjunction, ∨), NOT (negation, ¬), and parentheses. A formula is said to be satisfiable if it can be made TRUE by assigning appropriate logical values (i.e. TRUE, FALSE) to its variables. The Boolean satisfiability problem (SAT) is, given a formula, to check whether it is satisfiable. This decision problem is of central importance in many areas of computer science, including theoretical computer science, complexity theory,^[5][6] algorithmics, cryptography^[7][8] and artificial intelligence.^{[9][additional citation(s) needed]}

A literal is either a variable (in which case it is called a positive literal) or the negation of a variable (called a negative literal). A clause is a disjunction of literals (or a single literal). A clause is called a Horn clause if it contains at most one positive literal. A formula is in conjunctive normal form (CNF) if it is a conjunction of clauses (or a single clause).

For example, x₁ is a positive literal, ¬x₂ is a negative literal, and x₁ ∨ ¬x₂ is a clause. The formula (x₁ ∨ ¬x₂) ∧ (¬x₁ ∨ x₂ ∨ x₃) ∧ ¬x₁ is in conjunctive normal form; its first and third clauses are Horn clauses, but its second clause is not. The formula is satisfiable, by choosing x₁ = FALSE, x₂ = FALSE, and x₃ arbitrarily, since (FALSE ∨ ¬FALSE) ∧ (¬FALSE ∨ FALSE ∨ x₃) ∧ ¬FALSE evaluates to (FALSE ∨ TRUE) ∧ (TRUE ∨ FALSE ∨ x₃) ∧ TRUE, and in turn to TRUE ∧ TRUE ∧ TRUE (i.e. to TRUE). In contrast, the CNF formula a ∧ ¬a, consisting of two clauses of one literal, is unsatisfiable, since for a=TRUE or a=FALSE it evaluates to TRUE ∧ ¬TRUE (i.e., FALSE) or FALSE ∧ ¬FALSE (i.e., again FALSE), respectively.

For some versions of the SAT problem, it is useful to define the notion of a generalized conjunctive normal form formula, viz. as a conjunction of arbitrarily many generalized clauses, the latter being of the form R(l₁,...,l_n) for some Boolean function R and (ordinary) literals l_i. Different sets of allowed Boolean functions lead to different problem versions. As an example, R(¬x,a,b) is a generalized clause, and R(¬x,a,b) ∧ R(b,y,c) ∧ R(c,d,¬z) is a generalized conjunctive normal form. This formula is used below, with R being the ternary operator that is TRUE just when exactly one of its arguments is.

Using the laws of Boolean algebra, every propositional logic formula can be transformed into an equivalent conjunctive normal form, which may, however, be exponentially longer. For example, transforming the formula (x₁∧y₁) ∨ (x₂∧y₂) ∨ ... ∨ (x_n∧y_n) into conjunctive normal form yields

while the former is a disjunction of n conjunctions of 2 variables, the latter consists of 2ⁿ clauses of n variables.

However, with use of the Tseytin transformation, we may find an equisatisfiable conjunctive normal form formula with length linear in the size of the original propositional logic formula.

SAT was the first problem known to be NP-complete, as proved by Stephen Cook at the University of Toronto in 1971^[10] and independently by Leonid Levin at the Russian Academy of Sciences in 1973.^[11] Until that time, the concept of an NP-complete problem did not even exist. The proof shows how every decision problem in the complexity class NP can be reduced to the SAT problem for CNF^[a] formulas, sometimes called CNFSAT. A useful property of Cook's reduction is that it preserves the number of accepting answers. For example, deciding whether a given graph has a 3-coloring is another problem in NP; if a graph has 17 valid 3-colorings, then the SAT formula produced by the Cook–Levin reduction will have 17 satisfying assignments.

NP-completeness only refers to the run-time of the worst case instances. Many of the instances that occur in practical applications can be solved much more quickly. See §Algorithms for solving SAT below.

Like the satisfiability problem for arbitrary formulas, determining the satisfiability of a formula in conjunctive normal form where each clause is limited to at most three literals is NP-complete also; this problem is called 3-SAT, 3CNFSAT, or 3-satisfiability. To reduce the unrestricted SAT problem to 3-SAT, transform each clause l₁ ∨ ⋯ ∨ l_n to a conjunction of n - 2 clauses

where x₂, ⋯ , x_n−2 are fresh variables not occurring elsewhere. Although the two formulas are not logically equivalent, they are equisatisfiable. The formula resulting from transforming all clauses is at most 3 times as long as its original; that is, the length growth is polynomial.^[12]

3-SAT is one of Karp's 21 NP-complete problems, and it is used as a starting point for proving that other problems are also NP-hard.^[b] This is done by polynomial-time reduction from 3-SAT to the other problem. An example of a problem where this method has been used is the clique problem: given a CNF formula consisting of c clauses, the corresponding graph consists of a vertex for each literal, and an edge between each two non-contradicting^[c] literals from different clauses; see the picture. The graph has a c-clique if and only if the formula is satisfiable.^[13]

There is a simple randomized algorithm due to Schöning (1999) that runs in time (4/3)ⁿ where n is the number of variables in the 3-SAT proposition, and succeeds with high probability to correctly decide 3-SAT.^[14]

The exponential time hypothesis asserts that no algorithm can solve 3-SAT (or indeed k-SAT for any k > 2) in exp(o(n)) time (that is, fundamentally faster than exponential in n).

Selman, Mitchell, and Levesque (1996) give empirical data on the difficulty of randomly generated 3-SAT formulas, depending on their size parameters. Difficulty is measured in number recursive calls made by a DPLL algorithm. They identified a phase transition region from almost-certainly-satisfiable to almost-certainly-unsatisfiable formulas at the clauses-to-variables ratio at about 4.26.^[15]

3-satisfiability can be generalized to k-satisfiability (k-SAT, also k-CNF-SAT), when formulas in CNF are considered with each clause containing up to k literals.^{[citation needed]} However, since for any k ≥ 3, this problem can neither be easier than 3-SAT nor harder than SAT, and the latter two are NP-complete, so must be k-SAT.

Some authors restrict k-SAT to CNF formulas with exactly k literals.^{[citation needed]} This does not lead to a different complexity class either, as each clause l₁ ∨ ⋯ ∨ l_j with j < k literals can be padded with fixed dummy variables to l₁ ∨ ⋯ ∨ l_j ∨ d_j+1 ∨ ⋯ ∨ d_k. After padding all clauses, 2^k–1 extra clauses^[d] must be appended to ensure that only d₁ = ⋯ = d_k = FALSE can lead to a satisfying assignment. Since k does not depend on the formula length, the extra clauses lead to a constant increase in length. For the same reason, it does not matter whether duplicate literals are allowed in clauses, as in ¬x ∨ ¬y ∨ ¬y.

Conjunctive normal form (in particular with 3 literals per clause) is often considered the canonical representation for SAT formulas. As shown above, the general SAT problem reduces to 3-SAT, the problem of determining satisfiability for formulas in this form.

A 3-SAT formula is Linear SAT (LSAT) if each clause (viewed as a set of literals) intersects at most one other clause, and, moreover, if two clauses intersect, then they have exactly one literal in common. An LSAT formula can be depicted as a set of disjoint semi-closed intervals on a line. Deciding whether an LSAT formula is satisfiable is NP-complete.^[16]

SAT is easier if the number of literals in a clause is limited to at most 2, in which case the problem is called 2-SAT. This problem can be solved in polynomial time, and in fact is complete for the complexity class NL. If additionally all OR operations in literals are changed to XOR operations, then the result is called exclusive-or 2-satisfiability, which is a problem complete for the complexity class SL = L.

The problem of deciding the satisfiability of a given conjunction of Horn clauses is called Horn-satisfiability, or HORN-SAT. It can be solved in polynomial time by a single step of the unit propagation algorithm, which produces the single minimal model of the set of Horn clauses (w.r.t. the set of literals assigned to TRUE). Horn-satisfiability is P-complete. It can be seen as P's version of the Boolean satisfiability problem. Also, deciding the truth of quantified Horn formulas can be done in polynomial time.^[17]

Horn clauses are of interest because they are able to express implication of one variable from a set of other variables. Indeed, one such clause ¬x₁ ∨ ... ∨ ¬x_n ∨ y can be rewritten as x₁ ∧ ... ∧ x_n → y; that is, if x₁,...,x_n are all TRUE, then y must be TRUE as well.

A generalization of the class of Horn formulas is that of renameable-Horn formulae, which is the set of formulas that can be placed in Horn form by replacing some variables with their respective negation. For example, (x₁ ∨ ¬x₂) ∧ (¬x₁ ∨ x₂ ∨ x₃) ∧ ¬x₁ is not a Horn formula, but can be renamed to the Horn formula (x₁ ∨ ¬x₂) ∧ (¬x₁ ∨ x₂ ∨ ¬y₃) ∧ ¬x₁ by introducing y₃ as negation of x₃. In contrast, no renaming of (x₁ ∨ ¬x₂ ∨ ¬x₃) ∧ (¬x₁ ∨ x₂ ∨ x₃) ∧ ¬x₁ leads to a Horn formula. Checking the existence of such a replacement can be done in linear time; therefore, the satisfiability of such formulae is in P as it can be solved by first performing this replacement and then checking the satisfiability of the resulting Horn formula.

SAT is trivial if the formulas are restricted to those in disjunctive normal form, that is, they are a disjunction of conjunctions of literals. Such a formula is indeed satisfiable if and only if at least one of its conjunctions is satisfiable, and a conjunction is satisfiable if and only if it does not contain both x and NOT x for some variable x. This can be checked in linear time. Furthermore, if they are restricted to being in full disjunctive normal form, in which every variable appears exactly once in every conjunction, they can be checked in constant time (each conjunction represents one satisfying assignment). But it can take exponential time and space to convert a general SAT problem to disjunctive normal form; to obtain an example, exchange "∧" and "∨" in the above exponential blow-up example for conjunctive normal forms.

Another NP-complete variant of the 3-satisfiability problem is the one-in-three 3-SAT (also known variously as 1-in-3-SAT and exactly-1 3-SAT). Given a conjunctive normal form with three literals per clause, the problem is to determine whether there exists a truth assignment to the variables so that each clause has exactly one TRUE literal (and thus exactly two FALSE literals).

Another variant is the not-all-equal 3-satisfiability problem (also called NAE3SAT). Given a conjunctive normal form with three literals per clause, the problem is to determine if an assignment to the variables exists such that in no clause all three literals have the same truth value. This problem is NP-complete, too, even if no negation symbols are admitted, by Schaefer's dichotomy theorem.^[18]

Another special case is the class of problems where each clause contains XOR (i.e. exclusive or) rather than (plain) OR operators. This is in P, since an XOR-SAT formula can also be viewed as a system of linear equations mod 2, and can be solved in cubic time by Gaussian elimination.^[19]

The restrictions above (CNF, 2CNF, 3CNF, Horn, XOR-SAT) bound the considered formulae to be conjunctions of subformulas; each restriction states a specific form for all subformulas: for example, only binary clauses can be subformulas in 2CNF.

Schaefer's dichotomy theorem states that, for any restriction to Boolean functions that can be used to form these subformulas, the corresponding satisfiability problem is in P or NP-complete. The membership in P of the satisfiability of 2CNF, Horn, and XOR-SAT formulae are special cases of this theorem.^[18]

The following table summarizes some common variants of SAT.

An extension that has gained significant popularity since 2003 is satisfiability modulo theories (SMT) that can enrich CNF formulas with linear constraints, arrays, all-different constraints, uninterpreted functions,^[20] etc. Such extensions typically remain NP-complete, but very efficient solvers are now available that can handle many such kinds of constraints.

The satisfiability problem becomes more difficult if both "for all" (∀) and "there exists" (∃) quantifiers are allowed to bind the Boolean variables. An example of such an expression would be ∀x ∀y ∃z (x ∨ y ∨ z) ∧ (¬x ∨ ¬y ∨ ¬z); it is valid, since for all values of x and y, an appropriate value of z can be found, viz. z=TRUE if both x and y are FALSE, and z=FALSE else. SAT itself (tacitly) uses only ∃ quantifiers. If only ∀ quantifiers are allowed instead, the so-called tautology problem is obtained, which is co-NP-complete. If any number of both quantifiers are allowed, the problem is called the quantified Boolean formula problem (QBF), which can be shown to be PSPACE-complete. It is widely believed that PSPACE-complete problems are strictly harder than any problem in NP, although this has not yet been proved.

Ordinary SAT asks if there is at least one variable assignment that makes the formula true. A variety of variants deal with the number of such assignments:

• MAJ-SAT asks if at least half of all assignments make the formula TRUE. It is known to be complete for PP, a probabilistic class. Surprisingly, MAJ-kSAT is demonstrated to be in P for every finite integer k.^[21]
• #SAT, the problem of counting how many variable assignments satisfy a formula, is a counting problem, not a decision problem, and is #P-complete.
• UNIQUE SAT^[22] is the problem of determining whether a formula has exactly one assignment. It is complete for US,^[23] the complexity class describing problems solvable by a non-deterministic polynomial time Turing machine that accepts when there is exactly one nondeterministic accepting path and rejects otherwise.
• UNAMBIGUOUS-SAT is the name given to the satisfiability problem when the input formula is promised to have at most one satisfying assignment. The problem is also called USAT.^[24] A solving algorithm for UNAMBIGUOUS-SAT is allowed to exhibit any behavior, including endless looping, on a formula having several satisfying assignments. Although this problem seems easier, Valiant and Vazirani have shown^[25] that if there is a practical (i.e. randomized polynomial-time) algorithm to solve it, then all problems in NP can be solved just as easily.
• MAX-SAT, the maximum satisfiability problem, is an FNP generalization of SAT. It asks for the maximum number of clauses which can be satisfied by any assignment. It has efficient approximation algorithms, but is NP-hard to solve exactly. Worse still, it is APX-complete, meaning there is no polynomial-time approximation scheme (PTAS) for this problem unless P=NP.
• WMSAT is the problem of finding an assignment of minimum weight that satisfy a monotone Boolean formula (i.e. a formula without any negation). Weights of propositional variables are given in the input of the problem. The weight of an assignment is the sum of weights of true variables. That problem is NP-complete (see Th. 1 of^[26]).

Other generalizations include satisfiability for first- and second-order logic, constraint satisfaction problems, 0-1 integer programming.

While SAT is a decision problem, the search problem of finding a satisfying assignment reduces to SAT. That is, each algorithm which correctly answers whether an instance of SAT is solvable can be used to find a satisfying assignment. First, the question is asked on the given formula Φ. If the answer is "no", the formula is unsatisfiable. Otherwise, the question is asked on the partly instantiated formula Φ{x₁=TRUE}, that is, Φ with the first variable x₁ replaced by TRUE, and simplified accordingly. If the answer is "yes", then x₁=TRUE, otherwise x₁=FALSE. Values of other variables can be found subsequently in the same way. In total, n+1 runs of the algorithm are required, where n is the number of distinct variables in Φ.

This property is used in several theorems in complexity theory:

• NP ⊆ P/poly ⇒ PH = Σ₂ (Karp–Lipton theorem)
• NP ⊆ BPP ⇒ NP = RP
• P = NP ⇒ FP = FNP

Since the SAT problem is NP-complete, only algorithms with exponential worst-case complexity are known for it. In spite of this, efficient and scalable algorithms for SAT were developed during the 2000s and have contributed to dramatic advances in the ability to automatically solve problem instances involving tens of thousands of variables and millions of constraints (i.e. clauses).^[3] Examples of such problems in electronic design automation (EDA) include formal equivalence checking, model checking, formal verification of pipelined microprocessors,^[20] automatic test pattern generation, routing of FPGAs,^[27] planning, and scheduling problems, and so on. A SAT-solving engine is also considered to be an essential component in the electronic design automation toolbox.

Major techniques used by modern SAT solvers include the Davis–Putnam–Logemann–Loveland algorithm (or DPLL), conflict-driven clause learning (CDCL), and stochastic local search algorithms such as WalkSAT. Almost all SAT solvers include time-outs, so they will terminate in reasonable time even if they cannot find a solution. Different SAT solvers will find different instances easy or hard, and some excel at proving unsatisfiability, and others at finding solutions. Recent attempts have been made to learn an instance's satisfiability using deep learning techniques.^[28]

SAT solvers are developed and compared in SAT-solving contests.^[29] Modern SAT solvers are also having significant impact on the fields of software verification, constraint solving in artificial intelligence, and operations research, among others.

Theoretical algorithms with increasingly better worst-case runtime guarantees have been given in the last decades, including an ${\displaystyle O^{*}(1.0638^{L})}$ algorithm for clause sets of length (total literal count) ${\displaystyle L}$,^[30][31] an ${\displaystyle O^{*}(1.2226^{m})}$ algorithm for sets of ${\displaystyle m}$ clauses,^[32][33] and an ${\displaystyle O^{*}(1.32793^{n})}$ algorithm for 3-SAT with ${\displaystyle n}$ variables.^[34] Here, the notation "${\displaystyle O^{*}(.)}$" means "up to a polynomial factor", i.e. ${\displaystyle O^{*}(f(n))=O(f(n)n^{O(1)})}$. Earlier runtime guarantees are shown in the diagram.

• Unsatisfiable core
• Satisfiability modulo theories
• Counting SAT
• Planar SAT
• Karloff–Zwick algorithm
• Circuit satisfiability

• This article includes material from https://web.archive.org/web/20070708233347/http://www.sigda.org/newsletter/2006/eNews_061201.html by Prof. Karem A. Sakallah.