View source for Boolean satisfiability problem

{{short description|Problem of determining if a Boolean formula could be made true}}
{{Redirect|3SAT|the Central European television network|3sat}}

In [[logic]] and [[computer science]], the '''Boolean satisfiability problem''' (sometimes called '''propositional satisfiability problem''' and abbreviated '''SATISFIABILITY''', '''SAT''' or '''B-SAT''') is the problem of determining if there exists an [[Interpretation (logic)|interpretation]] that [[Satisfiability|satisfies]] a given [[Boolean logic|Boolean]] [[Formula (mathematical logic)|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 [[Contradiction#Contradiction in formal logic|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''&nbsp;=&nbsp;TRUE and ''b''&nbsp;=&nbsp;FALSE, which make (''a'' AND NOT ''b'')&nbsp;=&nbsp;TRUE. In contrast, "''a'' AND NOT ''a''" is unsatisfiable.

SAT is the first problem that was proven to be [[NP-complete]]—this is the [[Cook–Levin theorem]]. This means that all problems in the complexity class [[NP (complexity)|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 each SAT problem, and it is generally believed that no such algorithm exists, but this belief has not been proven mathematically, and resolving the question of whether SAT has a [[polynomial-time]] algorithm is equivalent to the [[P versus NP problem]], which is a famous open problem in the theory of computing.

Nevertheless, as of 2007, heuristic SAT-algorithms are able to solve problem instances involving tens of thousands of variables and <!---"clauses" hasn't been introduced here:--->formulas consisting of millions of symbols,<ref name="Codish.Ohrimenko.Stuckey.2007"/> which is sufficient for many practical SAT problems from, e.g., [[artificial intelligence]], [[circuit design]],<ref>{{Cite book|last1=Hong|first1=Ted|last2=Li|first2=Yanjing|last3=Park|first3=Sung-Boem|last4=Mui|first4=Diana|last5=Lin|first5=David|last6=Kaleq|first6=Ziyad Abdel|last7=Hakim|first7=Nagib|last8=Naeimi|first8=Helia|last9=Gardner|first9=Donald S.|last10=Mitra|first10=Subhasish|title=2010 IEEE International Test Conference |chapter=QED: Quick Error Detection tests for effective post-silicon validation |date=November 2010|pages=1–10|doi=10.1109/TEST.2010.5699215|isbn=978-1-4244-7206-2|s2cid=7909084}}</ref> and [[automatic theorem proving]].

==Definitions==
A ''[[propositional logic]] formula'', also called ''Boolean expression'', is built from [[Variable (mathematics)|variables]], operators AND ([[Logical conjunction|conjunction]], also denoted by ∧), OR ([[logical disjunction|disjunction]], ∨), NOT ([[negation]], ¬), and parentheses. A formula is said to be ''satisfiable'' if it can be made TRUE by assigning appropriate [[logical value]]s (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]], [[computational complexity theory|complexity theory]],<ref>{{cite book | last = Karp | first = Richard M. | author-link = Richard Karp | chapter = Reducibility Among Combinatorial Problems | title = Complexity of Computer Computations | editor = Raymond E. Miller | editor2 = James W. Thatcher | publisher = Plenum | location = New York | pages = 85–103 | year = 1972 | chapter-url = http://www.cs.berkeley.edu/~luca/cs172/karp.pdf | isbn = 0-306-30707-3 | access-date = 2020-05-07 | archive-date = 2011-06-29 | archive-url = https://web.archive.org/web/20110629023717/http://www.cs.berkeley.edu/~luca/cs172/karp.pdf | url-status = dead }} Here: p.86</ref><ref>{{cite book | isbn=0-201-00029-6 |first1=Alfred V. |last1=Aho |first2=John E. |last2=Hopcroft |first3=Jeffrey D. |last3=Ullman | title=The Design and Analysis of Computer Algorithms | publisher=Addison-Wesley | year=1974 |page=403}}</ref> [[algorithmics]], [[cryptography]]<ref>{{Cite journal|last1=Massacci|first1=Fabio|last2=Marraro|first2=Laura|date=2000-02-01|title=Logical Cryptanalysis as a SAT Problem |journal=Journal of Automated Reasoning |volume=24|issue=1|pages=165–203|doi=10.1023/A:1006326723002|s2cid=3114247 }}</ref><ref>{{Cite book|last1=Mironov|first1=Ilya|last2=Zhang|first2=Lintao|title=Theory and Applications of Satisfiability Testing - SAT 2006 |chapter=Applications of SAT Solvers to Cryptanalysis of Hash Functions |date=2006|editor-last=Biere|editor-first=Armin|editor2-last=Gomes|editor2-first=Carla P.|chapter-url=https://link.springer.com/chapter/10.1007%2F11814948_13|series=Lecture Notes in Computer Science|volume=4121 |publisher=Springer|pages=102–115|doi=10.1007/11814948_13|isbn=978-3-540-37207-3}}</ref> and [[artificial intelligence]].<ref>{{Cite journal | last1 = Vizel | first1 = Y. | last2 = Weissenbacher | first2 = G. | last3 = Malik | first3 = S. | journal = Proceedings of the IEEE | volume = 103 | issue = 11 | pages = 2021–2035 | year = 2015 | doi = 10.1109/JPROC.2015.2455034|title=Boolean Satisfiability Solvers and Their Applications in Model Checking| s2cid = 10190144 }}</ref>{{additional citation needed|date=May 2020}}

===Conjunctive normal form===

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, {{math|size=100%|''x''<sub>1</sub>}} is a positive literal, {{math|size=100%|¬''x''<sub>2</sub>}} is a negative literal, and {{math|size=100%|''x''<sub>1</sub> ∨ ¬''x''<sub>2</sub>}} is a clause. The formula {{math|size=100%|(''x''<sub>1</sub> ∨ ¬''x''<sub>2</sub>) ∧ (¬''x''<sub>1</sub> ∨ ''x''<sub>2</sub> ∨ ''x''<sub>3</sub>) ∧ ¬''x''<sub>1</sub>}} 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 ''x''<sub>1</sub>&nbsp;=&nbsp;FALSE, ''x''<sub>2</sub>&nbsp;=&nbsp;FALSE, and ''x''<sub>3</sub> arbitrarily, since (FALSE ∨ ¬FALSE) ∧ (¬FALSE ∨ FALSE ∨ ''x''<sub>3</sub>) ∧ ¬FALSE evaluates to (FALSE ∨ TRUE) ∧ (TRUE ∨ FALSE ∨ ''x''<sub>3</sub>) ∧ 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,<!---need not list them in detail here---(viz. [[#Exactly-1 3-satisfiability|Exactly-1 3-satisfiability]], [[#XOR-satisfiability|XOR-satisfiability]], and, more general, [[#Schaefer's dichotomy theorem|Schaefer's dichotomy theorem]], discussed below),---> 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 {{math|''R''(''l''<sub>1</sub>,...,''l''<sub>''n''</sub>)}} for some [[Boolean function]] ''R'' and (ordinary) literals {{mvar|''l''<sub>''i''</sub>}}. Different sets of allowed Boolean functions lead to different problem versions.<!---, see [[#Complexity and restricted versions|below]].---> 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 [[#Exactly-1 3-satisfiability|below]], with ''R'' being the ternary operator that is TRUE just when exactly one of its arguments is.
<!---need not explain that already here---If ''R'' is the ternary operator that is TRUE just if exactly one of its arguments is, then a satisfying assignment for the latter formula can be found starting from every possible combination of truth values for ''x'', ''y'', ''z'', except ''x''&nbsp;=&nbsp;''y''&nbsp;=&nbsp;''z''&nbsp;=&nbsp;FALSE, and choosing the values of ''a'', ''b'', ''c'', ''d'' appropriately; cf. the left table under [[#Exactly-1 3-satisfiability|Exactly-1 3-satisfiability]] below.--->

Using the laws of [[Boolean algebra (structure)|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 (''x''<sub>1</sub>∧''y''<sub>1</sub>) ∨ (''x''<sub>2</sub>∧''y''<sub>2</sub>) ∨ ... ∨ (''x''<sub>''n''</sub>∧''y''<sub>''n''</sub>) into conjunctive normal form yields
:{{math|(''x''<sub>1</sub>&nbsp;∨&nbsp;''x''<sub>2</sub>&nbsp;∨&nbsp;…&nbsp;∨&nbsp;''x''<sub>''n''</sub>) ∧}}
:{{math|(''y''<sub>1</sub>&nbsp;∨&nbsp;''x''<sub>2</sub>&nbsp;∨&nbsp;…&nbsp;∨&nbsp;''x''<sub>''n''</sub>) ∧}}
:{{math|(''x''<sub>1</sub>&nbsp;∨&nbsp;''y''<sub>2</sub>&nbsp;∨&nbsp;…&nbsp;∨&nbsp;''x''<sub>''n''</sub>) ∧}}
:{{math|(''y''<sub>1</sub>&nbsp;∨&nbsp;''y''<sub>2</sub>&nbsp;∨&nbsp;…&nbsp;∨&nbsp;''x''<sub>''n''</sub>) ∧ ... ∧}}
:{{math|(''x''<sub>1</sub>&nbsp;∨&nbsp;''x''<sub>2</sub>&nbsp;∨&nbsp;…&nbsp;∨&nbsp;''y''<sub>''n''</sub>) ∧}}
:{{math|(''y''<sub>1</sub>&nbsp;∨&nbsp;''x''<sub>2</sub>&nbsp;∨&nbsp;…&nbsp;∨&nbsp;''y''<sub>''n''</sub>) ∧}}
:{{math|(''x''<sub>1</sub>&nbsp;∨&nbsp;''y''<sub>2</sub>&nbsp;∨&nbsp;…&nbsp;∨&nbsp;''y''<sub>''n''</sub>) ∧}}
:{{math|(''y''<sub>1</sub>&nbsp;∨&nbsp;''y''<sub>2</sub>&nbsp;∨&nbsp;…&nbsp;∨&nbsp;''y''<sub>''n''</sub>)}};
while the former is a disjunction of ''n'' conjunctions of 2 variables, the latter consists of 2<sup>''n''</sup> 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.

==Complexity==

{{Main|Cook–Levin theorem}}

SAT was the first problem known to be [[NP-complete]], as proved by [[Stephen Cook]] at the [[University of Toronto]] in 1971<ref>{{Cite book | last1 = Cook | first1 = Stephen A. | title = Proceedings of the third annual ACM symposium on Theory of computing - STOC '71 | chapter = The complexity of theorem-proving procedures | author-link1=Stephen Cook| pages = 151–158 | year = 1971 | chapter-url=http://www.cs.toronto.edu/~sacook/homepage/1971.pdf |archive-url=https://ghostarchive.org/archive/20221009/http://www.cs.toronto.edu/~sacook/homepage/1971.pdf |archive-date=2022-10-09 |url-status=live| doi = 10.1145/800157.805047| citeseerx = 10.1.1.406.395 | s2cid = 7573663 }}</ref> and independently by [[Leonid Levin]] at the [[Russian Academy of Sciences#The Academy of Sciences of the USSR|Russian Academy of Sciences]] in 1973.<ref>{{cite journal|last=Levin|first=Leonid|author-link=Leonid Levin|title = Universal search problems (Russian: Универсальные задачи перебора, Universal'nye perebornye zadachi)|journal = Problems of Information Transmission (Russian: Проблемы передачи информа́ции, Problemy Peredachi Informatsii)|volume = 9|issue = 3|pages = 115–116|year = 1973}} [http://www.mathnet.ru/php/getFT.phtml?jrnid=ppi&paperid=914&volume=9&year=1973&issue=3&fpage=115&what=fullt&option_lang=eng (pdf)] {{in lang|ru}}, translated into English by {{cite journal|last=Trakhtenbrot|first=B. A.|title = A survey of Russian approaches to ''perebor'' (brute-force searches) algorithms|journal = Annals of the History of Computing |volume = 6|issue = 4|pages = 384–400|year = 1984|doi=10.1109/MAHC.1984.10036|s2cid=950581}}</ref> 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 (complexity)|NP]] can be [[reduction (complexity)|reduced]] to the SAT problem for CNF<ref group=note>The SAT problem for ''arbitrary'' formulas is NP-complete, too, since it is easily shown to be in NP, and it cannot be easier than SAT for CNF formulas.</ref> 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 (discrete mathematics)|graph]] has a [[Graph coloring#Vertex coloring|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|§Algorithms for solving SAT]] below.

===3-satisfiability===
[[File:Sat reduced to Clique from Sipser.svg|thumb|upright=1.25|The 3-SAT instance {{math|(''x'' ∨ ''x'' ∨ ''y'') ∧ (¬''x'' ∨ ¬''y'' ∨ ¬''y'') ∧ (¬''x'' ∨ ''y'' ∨ ''y'')}} reduced to a [[clique problem]]. The green vertices form a 3-clique and correspond to the satisfying assignment ''x''=FALSE, ''y''=TRUE.]]
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 {{math|''l''<sub>1</sub> ∨ ⋯ ∨ ''l''<sub>''n''</sub>}} to a conjunction of {{math|''n'' - 2}} clauses
:{{math|(''l''<sub>1</sub> ∨ ''l''<sub>2</sub> ∨ ''x''<sub>2</sub>) ∧ }}
:{{math|(¬''x''<sub>2</sub> ∨ ''l''<sub>3</sub> ∨ ''x''<sub>3</sub>) ∧ }}
:{{math|(¬''x''<sub>3</sub> ∨ ''l''<sub>4</sub> ∨ ''x''<sub>4</sub>) ∧ ⋯ ∧ }}
:{{math|(¬''x''<sub>''n''−3</sub> ∨ ''l''<sub>''n''−2</sub> ∨ ''x''<sub>''n''−2</sub>) ∧ }}
:{{math|(¬''x''<sub>''n''−2</sub> ∨ ''l''<sub>''n''−1</sub> ∨ ''l''<sub>''n''</sub>)}}
where {{math|''x''<sub>2</sub>,&thinsp;⋯&thinsp;,&thinsp;''x''<sub>''n''−2</sub>}} are [[fresh variable]]s 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.{{sfnp|Aho|Hopcroft|Ullman|1974|loc=Theorem 10.4}}

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]].<ref group=note>i.e. at least as hard as every other problem in NP. A decision problem is NP-complete if and only if it is in NP and is NP-hard.</ref> 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 (discrete mathematics)|graph]] consists of a vertex for each literal, and an edge between each two non-contradicting<ref group=note>i.e. such that one literal is not the negation of the other</ref> literals from different clauses; see the picture. The graph has a ''c''-clique if and only if the formula is satisfiable.{{sfnp|Aho|Hopcroft|Ullman|1974|loc=Theorem 10.5}}

There is a simple randomized algorithm due to Schöning (1999) that runs in time (4/3)<sup>''n''</sup> where ''n'' is the number of variables in the 3-SAT proposition, and succeeds with high probability to correctly decide 3-SAT.<ref name="Schoning.1999">{{cite book| last1=Schöning| first1=Uwe| title=40th Annual Symposium on Foundations of Computer Science (Cat. No.99CB37039)| chapter=A probabilistic algorithm for k-SAT and constraint satisfaction problems| pages=410–414| date=Oct 1999| isbn=0-7695-0409-4| chapter-url=http://homepages.cwi.nl/~rdewolf/schoning99.pdf |archive-url=https://ghostarchive.org/archive/20221009/http://homepages.cwi.nl/~rdewolf/schoning99.pdf |archive-date=2022-10-09 |url-status=live| doi=10.1109/SFFCS.1999.814612| s2cid=123177576}}</ref>

The [[exponential time hypothesis]] asserts that no algorithm can solve 3-SAT (or indeed ''k''-SAT for any {{Math|''k'' > 2}}) in {{math|exp([[Small o notation#Little-o notation|''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.<ref>{{cite journal|first1=Bart |last1=Selman |first2=David |last2=Mitchell |first3=Hector |last3=Levesque | title=Generating Hard Satisfiability Problems| journal=Artificial Intelligence| year=1996| volume=81|issue=1–2 | pages=17–29| doi=10.1016/0004-3702(95)00045-3|citeseerx=10.1.1.37.7362 }}</ref>

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|date=May 2020}} 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|date=May 2020}} This does not lead to a different complexity class either, as each clause {{math|''l''<sub>1</sub> ∨ ⋯ ∨ ''l''<sub>''j''</sub>}} with ''j'' < ''k'' literals can be padded with fixed dummy variables to {{math|''l''<sub>1</sub> ∨ ⋯ ∨ ''l''<sub>''j''</sub> ∨ ''d''<sub>''j''+1</sub> ∨ ⋯ ∨ ''d''<sub>''k''</sub>}}. After padding all clauses, 2<sup>''k''</sup>–1 extra clauses<ref group=note>viz. all [[Canonical form (Boolean algebra)#Maxterms|maxterms]] that can be built with {{math|''d''<sub>1</sub>,⋯,''d''<sub>''k''</sub>}}, except {{math|''d''<sub>1</sub>∨⋯∨''d''<sub>''k''</sub>}}</ref> must be appended to ensure that only {{math|1=''d''<sub>1</sub> = ⋯ = ''d''<sub>''k''</sub> = 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 {{math|¬''x'' ∨ ¬''y'' ∨ ¬''y''}}.

==Special cases of SAT==

===Conjunctive normal form===
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.

===Disjunctive normal form===
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 [[#Definitions|above]] exponential blow-up example for conjunctive normal forms.

===Exactly-1 3-satisfiability===
[[File:Schaefer's 3-SAT to 1-in-3-SAT reduction.gif|thumb|x100px|'''Left:''' Schaefer's reduction of a 3-SAT clause ''x'' ∨ ''y'' ∨ ''z''. The result of ''R'' is {{fontcolor|#00a000|TRUE (1)}} if exactly one of its arguments is TRUE, and {{fontcolor|#a00000|FALSE (0)}} otherwise. All 8 combinations of values for ''x'',''y'',''z'' are examined, one per line. The fresh variables ''a'',...,''f'' can be chosen to satisfy all clauses (exactly one {{fontcolor|#00a000|green}} argument for each ''R'') in all lines except the first, where ''x'' ∨ ''y'' ∨ ''z'' is FALSE. '''Right:''' A simpler reduction with the same properties.]]
A 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). In contrast, ordinary 3-SAT requires that every clause has ''at least'' one TRUE literal. Formally, a one-in-three 3-SAT problem is given as a generalized conjunctive normal form with all generalized clauses using a ternary operator ''R'' that is TRUE just if exactly one of its arguments is. When all literals of a one-in-three 3-SAT formula are positive, the satisfiability problem is called '''one-in-three positive 3-SAT'''.

One-in-three 3-SAT, together with its positive case, is listed as NP-complete problem "LO4" in the standard reference ''Computers and Intractability: A Guide to the Theory of NP-Completeness'' by [[Michael R. Garey]] and [[David S. Johnson]].  One-in-three 3-SAT was proved to be NP-complete by [[Thomas Jerome Schaefer]] as a special case of [[Schaefer's dichotomy theorem]], which asserts that any problem generalizing Boolean satisfiability in a certain way is either in the class P or is NP-complete.<ref name="schaefer">{{Cite conference | last1 = Schaefer | first1 = Thomas J. | year = 1978 | title = The complexity of satisfiability problems | book-title = Proceedings of the 10th Annual ACM Symposium on Theory of Computing | place = San Diego, California | pages = 216–226 | url = http://www.ccs.neu.edu/home/lieber/courses/csg260/f06/materials/papers/max-sat/p216-schaefer.pdf |doi=10.1145/800133.804350 |citeseerx=10.1.1.393.8951 }}</ref>

Schaefer gives a construction allowing an easy polynomial-time reduction from 3-SAT to one-in-three 3-SAT.  Let "(''x'' or ''y'' or ''z'')" be a clause in a 3CNF formula.  Add six fresh Boolean variables ''a'', ''b'', ''c'', ''d'', ''e'', and ''f'', to be used to simulate this clause and no other. <!---now introduced already above---Let ''R''(''u'',''v'',''w'') be a predicate that is TRUE if and only if exactly one of the Booleans ''u'', ''v'', and ''w''
is TRUE.--->Then the formula ''R''(''x'',''a'',''d'') ∧ ''R''(''y'',''b'',''d'') ∧ ''R''(''a'',''b'',''e'') ∧ ''R''(''c'',''d'',''f'') ∧ ''R''(''z'',''c'',FALSE) is satisfiable by some setting of the fresh variables if and only if at least one of ''x'', ''y'', or ''z'' is TRUE, see picture (left).  Thus any 3-SAT instance with ''m'' clauses and ''n'' variables may be converted into an [[equisatisfiable]] one-in-three 3-SAT instance with 5''m'' clauses and ''n'' + 6''m'' variables.{{sfnp|Schaefer|1978|p=222, Lemma 3.5}} Another reduction involves only four fresh variables and three clauses: ''R''(¬''x'',''a'',''b'') ∧ ''R''(''b'',''y'',''c'') ∧ R(''c'',''d'',¬''z''), see picture (right).

===Not-all-equal 3-satisfiability===
{{main|Not-all-equal 3-satisfiability}}
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.<ref name="schaefer"/>

=== Linear SAT ===
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.<ref>{{Cite journal|last1=Arkin|first1=Esther M.|last2=Banik|first2=Aritra|last3=Carmi|first3=Paz|last4=Citovsky|first4=Gui|last5=Katz|first5=Matthew J.|last6=Mitchell|first6=Joseph S. B.|last7=Simakov|first7=Marina|date=2018-12-11|title=Selecting and covering colored points|journal=Discrete Applied Mathematics|language=en|volume=250|pages=75–86|doi=10.1016/j.dam.2018.05.011|issn=0166-218X|doi-access=free}}</ref>

===2-satisfiability===
{{Main|2-satisfiability}}

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 [[NL-complete|complete]] for the complexity class [[NL (complexity)|NL]]. If additionally all OR operations in literals are changed to [[Exclusive or|XOR]] operations, then the result is called '''exclusive-or 2-satisfiability''', which is a problem complete for the complexity class [[SL (complexity)|SL]] = [[L (complexity)|L]].

===Horn-satisfiability===
{{Main|Horn-satisfiability}}

The problem of deciding the satisfiability of a given conjunction of [[Horn clause]]s 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 (complexity)|P's]] version of the Boolean satisfiability problem. Also, deciding the truth of quantified Horn formulas can be done in polynomial time.<ref name="buningkarpinski">{{Cite journal | last1 = Buning | first1 = H.K. | last2 = Karpinski| first2 =  Marek| last3=Flogel|first3=A.|year = 1995 | title = Resolution for Quantified Boolean Formulas | journal = Information and Computation | volume = 117 | issue = 1 | pages = 12–18 | publisher = Elsevier | doi= 10.1006/inco.1995.1025| doi-access = free }}</ref>

Horn clauses are of interest because they are able to express [[Entailment|implication]] of one variable from a set of other variables. Indeed, one such clause ¬''x''<sub>1</sub> ∨ ... ∨ ¬''x''<sub>''n''</sub> ∨ ''y'' can be rewritten as ''x''<sub>1</sub> ∧ ... ∧ ''x''<sub>''n''</sub> → ''y''; that is, if ''x''<sub>1</sub>,...,''x''<sub>''n''</sub> 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, (''x''<sub>1</sub> ∨ ¬''x''<sub>2</sub>) ∧ (¬''x''<sub>1</sub> ∨ ''x''<sub>2</sub> ∨ ''x''<sub>3</sub>) ∧ ¬''x''<sub>1</sub> is not a Horn formula, but can be renamed to the Horn formula (''x''<sub>1</sub> ∨ ¬''x''<sub>2</sub>) ∧ (¬''x''<sub>1</sub> ∨ ''x''<sub>2</sub> ∨ ¬''y''<sub>3</sub>) ∧ ¬''x''<sub>1</sub> by introducing ''y''<sub>3</sub> as negation of ''x''<sub>3</sub>. In contrast, no renaming of (''x''<sub>1</sub> ∨ ¬''x''<sub>2</sub> ∨ ¬''x''<sub>3</sub>) ∧ (¬''x''<sub>1</sub> ∨ ''x''<sub>2</sub> ∨ ''x''<sub>3</sub>) ∧ ¬''x''<sub>1</sub> 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.

{| style="float:right"
| [[File:Boolean satisfiability vs true literal counts.png|thumb|x200px|A formula with 2 clauses may be unsatisfied (red), 3-satisfied (green), xor-3-satisfied (blue), or/and 1-in-3-satisfied (yellow), depending on the TRUE-literal count in the 1st (hor) and 2nd (vert) clause.]]
|}

===XOR-satisfiability===

{| align="right" class="wikitable collapsible collapsed" style="text-align:left; margin: 1em"
|-
! Solving an XOR-SAT example<BR>by [[Gaussian elimination]]
|-
|
{| align="left" class="collapsible collapsed" style="text-align:left"
|-
! Given formula
|-
| ("⊕" means XOR, the {{color|#ff8080|red clause}} is optional)
|-
| (''a''⊕''c''⊕''d'') ∧ (''b''⊕¬''c''⊕''d'') ∧ (''a''⊕''b''⊕¬''d'') ∧ (''a''⊕¬''b''⊕¬''c'') {{color|#ff8080|∧ (¬''a''⊕''b''⊕''c'')}}
|}
|-
|
{| align="left" class="collapsible collapsed" style="text-align:left"
|-
! colspan="9" | Equation system
|-
| colspan="9" | ("1" means TRUE, "0" means FALSE)
|-
| colspan="9" | Each clause leads to one equation.
|-
|  || ''a'' || ⊕ ||   || ''c'' || ⊕ ||   || ''d'' || = 1
|-
|  || ''b'' || ⊕ || ¬ || ''c'' || ⊕ ||   || ''d'' || = 1
|-
|  || ''a'' || ⊕ ||   || ''b'' || ⊕ || ¬ || ''d'' || = 1
|-
|  || ''a'' || ⊕ || ¬ || ''b'' || ⊕ || ¬ || ''c'' || = 1
|-
|  {{color|#ff8080|¬}} || {{color|#ff8080|''a''}} || {{color|#ff8080|⊕}} || || {{color|#ff8080|''b''}} || {{color|#ff8080|⊕}} || || {{color|#ff8080|''c''}} || {{color|#ff8080| ≃ 1}}
|}
|-
|
{| align="left" class="collapsible collapsed" style="text-align:left"
|-
! colspan="6" | Normalized equation system
|-
| colspan="6" | using properties of [[Boolean rings]] (¬''x''=1⊕''x'', ''x''⊕''x''=0)
|-
| ''a'' || ⊕ || ''c'' || ⊕ || ''d'' || = '''1'''
|-
| ''b'' || ⊕ || ''c'' || ⊕ || ''d'' || = '''0'''
|-
| ''a'' || ⊕ || ''b'' || ⊕ || ''d'' || = '''0'''
|-
| ''a'' || ⊕ || ''b'' || ⊕ || ''c'' || = '''1'''
|-
| {{color|#ff8080|''a''}} || {{color|#ff8080|⊕}}  || {{color|#ff8080|''b''}} || {{color|#ff8080|⊕}} || {{color|#ff8080|''c''}} || {{color|#ff8080| ≃ '''0'''}}
|-
| colspan="6" | (If the {{color|#ff8080|red equation}} is present, {{color|#ff8080|it}} contradicts
|-
| colspan="6" | the last black one, so the system is unsolvable.
|-
| colspan="6" | Therefore, Gauss' algorithm is
|-
| colspan="6" | used only for the black equations.)
|}
|-
|
{| align="left" class="collapsible collapsed" style="text-align:left"
|-
! colspan="6" | Associated coefficient matrix
|-
| &nbsp;
|-
! ''a'' !! ''b'' !! ''c'' !! ''d'' !!   !! line
|-
| &nbsp;
|-
| 1 || 0 || 1 || 1
! 1
| A
|-
| 0 || 1 || 1 || 1
! 0
| B
|-
| 1 || 1 || 0 || 1
! 0
| C
|-
| 1 || 1 || 1 || 0
! 1
| D
|}
|-
|
{| align="left" class="collapsible collapsed" style="text-align:left"
|-
! colspan="6" |Transforming to echelon form
|-
| &nbsp;
|-
! ''a'' !! ''b'' !! ''c'' !! ''d'' !!   !! operation
|-
| &nbsp;
|-
| 1 || 0 || 1 || 1
! 1
| A
|-
| 1 || 1 || 0 || 1
! 0
| C
|-
| 1 || 1 || 1 || 0
! 1
| D
|-
| 0 || 1 || 1 || 1
! 0
| B (swapped)
|-
| &nbsp;
|-
| 1 || 0 || 1 || 1
! 1
| A
|-
| 0 || 1 || 1 || 0
! 1
| E = C⊕A
|-
| 0 || 1 || 0 || 1
! 0
| F = D⊕A
|-
| 0 || 1 || 1 || 1
! 0
| B
|-
| &nbsp;
|-
| 1 || 0 || 1 || 1
! 1
| A
|-
| 0 || 1 || 1 || 0
! 1
| E
|-
| 0 || 0 || 1 || 1
! 1
| G = F⊕E
|-
| 0 || 0 || 0 || 1
! 1
| H = B⊕E
|}
|-
|
{| align="left" class="collapsible collapsed" style="text-align:left"
|-
! colspan="6" | Transforming to diagonal form
|-
| &nbsp;
|-
! ''a'' !! ''b'' !! ''c'' !! ''d'' !!   !! operation
|-
| &nbsp;
|-
| 1 || 0 || 1 || 0
! 0
| I = A⊕H
|-
| 0 || 1 || 1 || 0
! 1
| E
|-
| 0 || 0 || 1 || 0
! 0
| J = G⊕H
|-
| 0 || 0 || 0 || 1
! 1
| H
|-
| &nbsp;
|-
| 1 || 0 || 0 || 0
! 0
| K = I⊕J
|-
| 0 || 1 || 0 || 0
! 1
| L = E⊕J
|-
| 0 || 0 || 1 || 0
! 0
| J
|-
| 0 || 0 || 0 || 1
! 1
| H
|-
|}
|-
|
{| align="left" class="collapsible collapsed" style="text-align:left"
|-
! Solution:
|-
| If the {{color|#ff8080|red clause}} is present: || Unsolvable
|-
| Else: || ''a'' = 0 = FALSE
|-
| || ''b'' = 1 = TRUE
|-
| || ''c'' = 0 = FALSE
|-
| || ''d'' = 1 = TRUE
|-
| colspan="2" | '''As a consequence:'''
|-
| colspan="2" | ''R''(''a'',''c'',''d'') ∧ ''R''(''b'',¬''c'',''d'') ∧ ''R''(''a'',''b'',¬''d'') ∧ ''R''(''a'',¬''b'',¬''c'') {{color|#ff8080|∧ ''R''(¬''a'',''b'',''c'')}}
|-
| colspan="2" | is not 1-in-3-satisfiable,
|-
| colspan="2" | while (''a'' ∨ ''c'' ∨ ''d'') ∧ (''b'' ∨ ¬''c'' ∨ ''d'') ∧ (''a'' ∨ ''b'' ∨ ¬''d'') ∧ (''a'' ∨ ¬''b'' ∨ ¬''c'')
|-
| colspan="2" |  is 3-satisfiable with ''a''=''c''=FALSE and ''b''=''d''=TRUE.
|}
|}

Another special case is the class of problems where each clause contains XOR (i.e. [[exclusive or]]) rather than (plain) OR operators.<ref group=note>Formally, generalized conjunctive normal forms with a ternary Boolean function ''R'' are employed, which is TRUE just if 1 or 3 of its arguments is. An input clause with more than 3 literals can be transformed into an equisatisfiable conjunction of clauses á 3 literals similar to [[#3-satisfiability|above]]; i.e. XOR-SAT can be reduced to XOR-3-SAT.</ref> This is in [[P (complexity class)|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]];<ref>{{citation|title=The Nature of Computation|first1=Cristopher|last1=Moore|author1-link=Cristopher Moore|first2=Stephan|last2=Mertens|publisher=Oxford University Press|year=2011|isbn=9780199233212|page=366|url=https://books.google.com/books?id=z4zMiZyAE1kC&pg=PA366}}.</ref> see the box for an example. This recast is based on the [[Boolean algebra (structure)#Boolean rings|kinship between Boolean algebras and Boolean rings]], and the fact that arithmetic modulo two forms a [[finite field]]. Since ''a'' XOR ''b'' XOR ''c'' evaluates to TRUE if and only if exactly 1 or 3 members of {''a'',''b'',''c''} are TRUE, each solution of the 1-in-3-SAT problem for a given CNF formula is also a solution of the XOR-3-SAT problem, and in turn each solution of XOR-3-SAT is a solution of 3-SAT; see the picture. As a consequence, for each CNF formula, it is possible to solve the XOR-3-SAT problem defined by the formula, and based on the result infer either that the 3-SAT problem is solvable or that the 1-in-3-SAT problem is unsolvable.

Provided that the [[P = NP problem|complexity classes P and NP are not equal]], neither 2-, nor Horn-, nor XOR-satisfiability is NP-complete, unlike SAT.

===Schaefer's dichotomy theorem===
{{Main|Schaefer's dichotomy theorem}}
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.<ref name="schaefer"/>

The following table summarizes some common variants of SAT.
{| class="wikitable sortable"
|+
!Code
!Name
!Restrictions
!Requirements
!Class
|-
|3SAT
|3-satisfiability
|Each clause contains 3 literals.
|At least one literal must be true.
|NP-c
|-
|2SAT
|2-satisfiability
|Each clause contains 2 literals.
|At least one literal must be true.
|NL-c
|-
|1-in-3-SAT
|Exactly-1 3-SAT
|Each clause contains 3 literals.
|Exactly one literal must be true.
|NP-c
|-
|1-in-3-SAT+
|Exactly-1 Positive 3-SAT
|Each clause contains 3 positive literals.
|Exactly one literal must be true.
|NP-c
|-
|NAE3SAT
|Not-all-equal 3-satisfiability
|Each clause contains 3 literals.
|Either one or two literals must be true.
|NP-c
|-
|NAE3SAT+
|Not-all-equal positive 3-SAT
|Each clause contains 3 positive literals.
|Either one or two literals must be true.
|NP-c
|-
|PL-SAT
|[[Planar SAT]]
|The incidence graph (clause-variable graph) is [[Planar graph|planar]].
|At least one literal must be true.
|NP-c
|-
|LSAT
|Linear SAT
|Each clause contains 3 literals, intersects at most one other clause, and the intersection is exactly one literal.
|At least one literal must be true.
|NP-c
|-
|HORN-SAT
|Horn satisfiability
|Horn clauses (at most one positive literal).
|At least one literal must be true.
|P-c
|-
|XOR-SAT
|Xor satisfiability
|Each clause contains XOR operations rather than OR.
|The XOR of all literals must be true.
|P
|}

==Extensions 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 function]]s,<ref name="Bryant.German.Velev.1999">R. E. Bryant, S. M. German, and M. N. Velev, [http://portal.acm.org/citation.cfm?id=709275 Microprocessor Verification Using Efficient Decision Procedures for a Logic of Equality with Uninterpreted Functions], in Analytic Tableaux and Related Methods, pp.&nbsp;1–13, 1999.</ref> 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" ([[∃]]) [[Quantifier (logic)|quantifier]]s are allowed to bind the Boolean variables. An example of such an expression would be {{math|size=100%|∀''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 (logic)|tautology]] problem''' is obtained, which is [[co-NP-complete]]. If 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. Using highly parallel ''[[P system]]s'', QBF-SAT problems can be solved in linear time.<ref>{{Cite journal | last1 = Alhazov | first1 = Artiom | last2 = Martín-Vide | first2 = Carlos | last3 = Pan | first3 = Linqiang | title = Solving a PSPACE-Complete Problem by Recognizing P Systems with Restricted Active Membranes |  url=https://www.researchgate.net/publication/220444503 | journal = Fundamenta Informaticae | volume = 58 | pages = 67–77 | year = 2003 }} Here: Sect.3, Thm.3.1</ref>

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 the majority of all assignments make the formula TRUE. It is known to be complete for [[PP (complexity)|PP]], a probabilistic class.
* '''[[Sharp-SAT|#SAT]]''', the problem of counting how many variable assignments satisfy a formula, is a counting problem, not a decision problem, and is [[Sharp-P-complete|#P-complete]].
* '''UNIQUE SAT'''<ref>{{Cite journal|last1=Blass|first1=Andreas|last2=Gurevich|first2=Yuri|date=1982-10-01|title=On the unique satisfiability problem|journal=Information and Control|volume=55|issue=1|pages=80–88|doi=10.1016/S0019-9958(82)90439-9|issn=0019-9958|doi-access=free|hdl=2027.42/23842|hdl-access=free}}</ref> is the problem of determining whether a formula has exactly one assignment. It is complete for [[US (complexity)|US]],<ref>{{Cite web|url=https://complexityzoo.uwaterloo.ca/Complexity_Zoo:U#US|title=Complexity Zoo:U - Complexity Zoo|website=complexityzoo.uwaterloo.ca|access-date=2019-12-05|archive-date=2019-07-09|archive-url=https://web.archive.org/web/20190709142353/https://complexityzoo.uwaterloo.ca/Complexity_Zoo:U#US|url-status=dead}}</ref> 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 is [[Promise problem|restricted]] to formulas having at most one satisfying assignment. The problem is also called '''USAT'''.<ref>{{Cite book |chapter-url=https://www.springer.com/gp/book/9781846282973 |chapter=Supplementary Lecture F: Unique Satisfiability |title=Theory of Computation |last=Kozen |first=Dexter C. |date=2006 |publisher=Springer  |isbn=9781846282973 |series=Texts in Computer Science |page=180 |language=en}}</ref> 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 [[Valiant-Vazirani theorem|shown]]<ref>{{Cite journal | last1 = Valiant | first1 = L. | last2 = Vazirani | first2 = V.| doi = 10.1016/0304-3975(86)90135-0 | title = NP is as easy as detecting unique solutions | url = http://www.cs.princeton.edu/courses/archive/fall05/cos528/handouts/NP_is_as.pdf| journal = Theoretical Computer Science | volume = 47 | pages = 85–93 | year = 1986 | doi-access = free }}</ref> that if there is a practical (i.e. [[Bounded-error probabilistic polynomial|randomized polynomial-time]]) algorithm to solve it, then all problems in [[NP (complexity class)|NP]] can be solved just as easily.
* '''MAX-SAT''', the [[maximum satisfiability problem]], is an [[FNP (complexity)|FNP]] generalization of SAT. It asks for the maximum number of clauses which can be satisfied by any assignment. It has efficient [[approximation algorithm]]s, 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 <ref>{{Cite book|last1=Buldas|first1=Ahto|last2=Lenin|first2=Aleksandr|last3=Willemson|first3=Jan|last4=Charnamord|first4=Anton|title=Advances in Information and Computer Security |chapter=Simple Infeasibility Certificates for Attack Trees |date=2017|editor-last=Obana|editor-first=Satoshi|editor2-last=Chida|editor2-first=Koji|volume=10418|series=Lecture Notes in Computer Science|language=en|publisher=Springer International Publishing|pages=39–55|doi=10.1007/978-3-319-64200-0_3|isbn=9783319642000}}</ref>).

Other generalizations include satisfiability for [[first-order predicate calculus|first]]- and [[second-order logic]], [[constraint satisfaction problem]]s, [[0-1 integer programming]].

==Finding a satisfying assignment==
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[[substitution (logic)|{''x''<sub>1</sub>=TRUE}]], that is, Φ with the first variable ''x''<sub>1</sub> replaced by TRUE, and simplified accordingly. If the answer is "yes", then ''x''<sub>1</sub>=TRUE, otherwise ''x''<sub>1</sub>=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 (complexity)|NP]] ⊆ [[P/poly]] ⇒ [[PH (complexity)|PH]] = [[Polynomial hierarchy#Definitions|Σ<sub>2</sub>]] &nbsp; ([[Karp–Lipton theorem]])
: [[NP (complexity)|NP]] ⊆ [[BPP (complexity)|BPP]] ⇒ [[NP (complexity)|NP]] = [[RP (complexity)|RP]]
: [[P (complexity)|P]] = [[NP (complexity)|NP]] ⇒ [[FP (complexity)|FP]] = [[FNP (complexity)|FNP]]

==Algorithms for solving SAT==
{{main|SAT solver}}

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).<ref name="Codish.Ohrimenko.Stuckey.2007">{{citation |title=Principles and Practice of Constraint Programming – CP 2007|series=Lecture Notes in Computer Science|volume=4741|year=2007|pages=544–558|contribution=Propagation = Lazy Clause Generation|first1=Olga|last1=Ohrimenko |first2=Peter J.|last2=Stuckey|first3=Michael|last3=Codish|doi=10.1007/978-3-540-74970-7_39|isbn=978-3-540-74969-1 |quote=modern SAT solvers can often handle problems with millions of constraints and hundreds of thousands of variables |citeseerx=10.1.1.70.5471}}.</ref> Examples of such problems in [[electronic design automation]] (EDA) include [[formal equivalence checking]], [[model checking]], [[formal verification]] of [[microprocessor|pipelined microprocessors]],<ref name="Bryant.German.Velev.1999"/> [[automatic test pattern generation]], [[routing (electronic design automation)|routing]] of [[FPGA]]s,<ref>{{Cite journal |last1=Gi-Joon Nam |last2=Sakallah |first2=K. A. |last3=Rutenbar |first3=R. A. |title=A new FPGA detailed routing approach via search-based Boolean satisfiability |journal=IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems |volume=21 |issue=6 |pages=674 |year=2002 |url=http://cs-rutenbar.web.engr.illinois.edu/wp-content/uploads/2012/10/rutenbar-sattranscad02.pdf |doi=10.1109/TCAD.2002.1004311 |access-date=2015-09-04 |archive-date=2016-03-15 |archive-url=https://web.archive.org/web/20160315003856/http://cs-rutenbar.web.engr.illinois.edu/wp-content/uploads/2012/10/rutenbar-sattranscad02.pdf |url-status=dead }}</ref> [[Automated planning and scheduling|planning]], and [[Scheduling algorithm|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 (constraint satisfaction)|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{{When|date=June 2024}} attempts have been made to learn an instance's satisfiability using deep learning techniques.<ref>{{cite arXiv |last1=Selsam |first1=Daniel |last2=Lamm |first2=Matthew |last3=Bünz |first3=Benedikt |last4=Liang |first4=Percy |last5=de Moura |first5=Leonardo |last6=Dill |first6=David L. |title=Learning a SAT Solver from Single-Bit Supervision |eprint=1802.03685 |date=11 March 2019|class=cs.AI }}</ref>

SAT solvers are developed and compared in SAT-solving contests.<ref>{{cite web|url=http://www.satcompetition.org/ |title=The international SAT Competitions web page|access-date=2007-11-15}}</ref> Modern SAT solvers are also having significant impact on the fields of software verification, constraint solving in artificial intelligence, and operations research, among others.

==See also==
*[[Unsatisfiable core]]
*[[Satisfiability modulo theories]]
*[[Sharp-SAT|Counting SAT]]
*[[Planar SAT]]
*[[Karloff–Zwick algorithm]]
*[[Circuit satisfiability]]

==Notes==
{{Reflist|group=note}}

== External links ==
{{commons category}}

* [http://www.cril.univ-artois.fr/~roussel/satgame/satgame.php?lang=eng SAT Game]: try solving a Boolean satisfiability problem yourself
* [http://www.satcompetition.org/ The international SAT competition website]
* [http://www.satisfiability.org/ International Conference on Theory and Applications of Satisfiability Testing]
* [https://web.archive.org/web/20060219180520/http://jsat.ewi.tudelft.nl/ Journal on Satisfiability, Boolean Modeling and Computation]
* [http://www.satlive.org SAT Live, an aggregate website for research on the satisfiability problem]
* [http://www.maxsat.udl.cat/ Yearly evaluation of MaxSAT solvers]

==References==
{{Reflist|30em}}

==Sources==
* 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]].

==Further reading==
(by date of publication)
{{refbegin|colwidth=30em}}
* {{cite book|first1=Michael R. |last1=Garey |first2=David S. |last2=Johnson|author-link=Michael R. Garey |author2-link=David S. Johnson |year=1979 |title=Computers and Intractability: A Guide to the Theory of NP-Completeness |publisher=W. H. Freeman |isbn=0-7167-1045-5 |title-link=List of important publications in theoretical computer science#Computers and Intractability: A Guide to the Theory of NP-Completeness |pages=A9.1: LO1–LO7, pp.&nbsp;259–260}}
* {{Cite book |last1=Marques-Silva |first1=J. |last2=Glass |first2=T. |chapter=Combinational equivalence checking using satisfiability and recursive learning |title=Design, Automation and Test in Europe Conference and Exhibition, 1999. Proceedings (Cat. No. PR00078) |pages=145 |year=1999 |isbn=0-7695-0078-1 |doi=10.1109/DATE.1999.761110 |url=http://eprints.soton.ac.uk/265003/1/jpms-date99a.pdf |archive-url=https://ghostarchive.org/archive/20221009/http://eprints.soton.ac.uk/265003/1/jpms-date99a.pdf |archive-date=2022-10-09 |url-status=live}}
* {{Cite journal |last1=Clarke |first1=E. |last2=Biere |first2=A. |last3=Raimi |first3=R. |last4=Zhu |first4=Y. |journal=Formal Methods in System Design |volume=19 |pages=7–34 |year=2001 |title=Bounded Model Checking Using Satisfiability Solving |doi=10.1023/A:1011276507260 |s2cid=2484208}}
* {{Cite book | last1 = Giunchiglia | first1 = E. | last2 = Tacchella | first2 = A. | editor1-last = Giunchiglia | editor1-first = Enrico | editor2-first = Armando | title = Theory and Applications of Satisfiability Testing | series = Lecture Notes in Computer Science | volume = 2919 | year = 2004 | isbn = 978-3-540-20851-8 | doi = 10.1007/b95238 | s2cid = 31129008 | editor2-last = Tacchella}}
* {{Cite journal | last1 = Babic | first1 = D. | last2 = Bingham | first2 = J. | last3 = Hu | first3 = A. J. | title = B-Cubing: New Possibilities for Efficient SAT-Solving | journal = IEEE Transactions on Computers | volume = 55 | issue = 11 | pages = 1315 | year = 2006 | url = http://www.domagoj-babic.com/uploads/Pubs/TCOM06/tcom06.pdf| doi = 10.1109/TC.2006.175| s2cid = 14819050 }}
* {{Cite book | last1 = Rodriguez | first1 = C. | last2 = Villagra | first2 = M. | last3 = Baran | first3 = B. | chapter = Asynchronous team algorithms for Boolean Satisfiability | title = 2007 2nd Bio-Inspired Models of Network, Information and Computing Systems | pages = 66–69 | year = 2007 | chapter-url = http://www.cc.pol.una.py/lcca/publicaciones/optimizacion/2007/Asynchronous%20Team%20Algorithms%20for%20Boolean%20Satisfiability.pdf| doi = 10.1109/BIMNICS.2007.4610083| s2cid = 15185219 }}
* {{cite book |editor1-first=Frank Van |editor1-last=Harmelen |editor2-first=Vladimir |editor2-last=Lifschitz |editor3-first=Bruce |editor3-last=Porter |title=Handbook of knowledge representation|year=2008|publisher=Elsevier|isbn=978-0-444-52211-5|pages=89–134|first1=Carla P. |last1=Gomes|author1-link=Carla Gomes |first2=Henry |last2=Kautz |first3=Ashish |last3=Sabharwal |first4=Bart |last4=Selman |chapter=Satisfiability Solvers|doi=10.1016/S1574-6526(07)03002-7|series=Foundations of Artificial Intelligence|volume=3}}
* {{Cite journal | last1 = Vizel | first1 = Y. | last2 = Weissenbacher | first2 = G. | last3 = Malik | first3 = S. | journal = Proceedings of the IEEE | volume = 103 | issue = 11 | year = 2015 | doi = 10.1109/JPROC.2015.2455034|title=Boolean Satisfiability Solvers and Their Applications in Model Checking| pages = 2021–2035 | s2cid = 10190144 }}
* {{Cite book |first=Donald E. |last=Knuth |author-link=Donald Knuth|title=The Art of Computer Programming |volume=4B: Combinatorial Algorithms, Part 2 |pages=185–369 |chapter=Chapter 7.2.2.2: Satifiability|year=2022 |publisher=Addison-Wesley Professional |isbn=978-0-201-03806-4|title-link=The Art of Computer Programming }}
{{refend}}

{{Logic}}

{{DEFAULTSORT:Boolean Satisfiability Problem}}
[[Category:Boolean algebra]]
[[Category:Electronic design automation]]
[[Category:Formal methods]]
[[Category:Logic in computer science]]
[[Category:NP-complete problems]]
[[Category:Satisfiability problems]]