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| {{Redirect|3SAT|the Central European television network|3sat}}
| | It is very common to have a dental emergency -- a fractured tooth, an abscess, or severe pain when chewing. Over-the-counter pain medication is just masking the problem. Seeing an emergency dentist is critical to getting the source of the problem diagnosed and corrected as soon as possible.<br><br><br><br>Here are some common dental emergencies:<br>Toothache: The most common dental emergency. This generally means a badly decayed tooth. As the pain affects the tooth's nerve, treatment involves gently removing any debris lodged in the cavity being careful not to poke deep as this will cause severe pain if the nerve is touched. Next rinse vigorously with warm water. Then soak a small piece of cotton in oil of cloves and insert it in the cavity. This will give temporary relief until a dentist can be reached.<br><br>At times the pain may have a more obscure location such as decay under an old filling. As this can be only corrected by a dentist there are two things you can do to help the pain. Administer a pain pill (aspirin or some other analgesic) internally or dissolve a tablet in a half glass (4 oz) of warm water holding it in the mouth for several minutes before spitting it out. DO NOT PLACE A WHOLE TABLET OR ANY PART OF IT IN THE TOOTH OR AGAINST THE SOFT GUM TISSUE AS IT WILL RESULT IN A NASTY BURN.<br><br>Swollen Jaw: This may be caused by several conditions the most probable being an abscessed tooth. In any case the treatment should be to reduce pain and swelling. An ice pack held on the outside of the jaw, (ten minutes on and ten minutes off) will take care of both. If this does not control the pain, an analgesic tablet can be given every four hours.<br><br>Other Oral Injuries: Broken teeth, cut lips, bitten tongue or lips if severe means a trip to a dentist as soon as possible. In the mean time rinse the mouth with warm water and place cold compression the face opposite the injury. If there is a lot of bleeding, apply direct pressure to the bleeding area. If bleeding does not stop get patient to the emergency room of a hospital as stitches may be necessary.<br><br>Prolonged Bleeding Following Extraction: Place a gauze pad or better still a moistened tea bag over the socket and have the patient bite down gently on it for 30 to 45 minutes. The tannic acid in the tea seeps into the tissues and often helps stop the bleeding. If bleeding continues after two hours, call the dentist or take patient to the emergency room of the nearest hospital.<br><br>Broken Jaw: If you suspect the patient's jaw is broken, bring the upper and lower teeth together. Put a necktie, handkerchief or towel under the chin, tying it over the head to immobilize the jaw until you can get the patient to a dentist or the emergency room of a hospital.<br><br>Painful Erupting Tooth: In young children teething pain can come from a loose baby tooth or from an erupting permanent tooth. Some relief can be given by crushing a little ice and wrapping it in gauze or a clean piece of cloth and putting it directly on the tooth or gum tissue where it hurts. The numbing effect of the cold, along with an appropriate dose of aspirin, usually provides temporary relief.<br><br>In young adults, an erupting 3rd molar (Wisdom tooth), especially if it is impacted, can cause the jaw to swell and be quite painful. Often the gum around the tooth will show signs of infection. Temporary relief can be had by giving aspirin or some other painkiller and by dissolving an aspirin in half a glass of warm water and holding this solution in the mouth over the sore gum. AGAIN DO NOT PLACE A TABLET DIRECTLY OVER THE GUM OR CHEEK OR USE THE ASPIRIN SOLUTION ANY STRONGER THAN RECOMMENDED TO PREVENT BURNING THE TISSUE. The swelling of the jaw can be reduced by using an ice pack on the outside of the face at intervals of ten minutes on and ten minutes off.<br><br>If you are you looking for more on [http://www.youtube.com/watch?v=90z1mmiwNS8 Washington DC Dentist] review our own internet site. |
| In [[computer science]], '''Boolean''', or '''propositional''', '''satisfiability''' (often written '''SATISFIABILITY''' or abbreviated '''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 establishes if the variables of a given Boolean formula can be assigned in such a way as to make the formula [[Validity|evaluate to TRUE]]. If no such assignments exist, the function expressed by the formula is [[Contradiction#Contradiction_in_formal_logic|identically FALSE]] for all possible variable assignments. In this latter case, it is called '''unsatisfiable''', otherwise '''satisfiable'''. 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.
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| SAT was the first known example of an [[NP-complete]] problem. That briefly means that there is no known algorithm that efficiently solves all instances of SAT, and it is generally believed (but not proven, see [[P versus NP problem]]) that no such algorithm can exist. Further, a wide range of other naturally occurring decision and optimization problems can be transformed into instances of SAT. A class of algorithms called SAT solvers can efficiently solve a large enough subset of SAT instances to be useful in various practical areas such as [[circuit design]] and [[automatic theorem proving]], by solving SAT instances made by transforming problems that arise in those areas. Extending the capabilities of SAT solving algorithms is an ongoing area of research. However, no current such methods can efficiently solve ''all'' SAT instances.
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| ==Basic definitions and terminology==
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| 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.
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| 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.
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| The '''Boolean satisfiability problem''' ('''SAT''') is, given a formula, to check whether it is satisfiable. | |
| This [[decision problem]] is of central importance in various areas of [[computer science]], including [[theoretical computer science]], [[computational complexity theory|complexity theory]], [[algorithmics]], and [[artificial intelligence]].
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| There are several special cases of the Boolean satisfiability problem in which the formulas are required to have a particular structure.
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| A '''literal''' is either a variable, then called '''positive literal''', or the negation of a variable, then called '''negative literal'''.
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| A '''clause''' is a disjunction of literals (or a single literal).
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| A clause is called '''[[Horn clause]]''' if it contains at most one positive literal.
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| A formula is in '''[[conjunctive normal form]]''' ('''CNF''') if it is a conjunction of clauses (or a single clause).
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| For example, "''x''<sub>1</sub>" is a positive literal, "¬''x''<sub>2</sub>" is a negative literal, "''x''<sub>1</sub> ∨ ¬''x''<sub>2</sub>" is a clause, and "(''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 a formula in conjunctive normal form, its 1st and 3rd clause is a Horn clause, but its 2nd clause is not. The formula is satisfiable, choosing ''x''<sub>1</sub>=FALSE, ''x''<sub>2</sub>=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 and ''a''=FALSE it evaluates to TRUE ∧ ¬TRUE (i.e. to FALSE) and FALSE ∧ ¬FALSE (i.e. again to FALSE), respectively.
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| For some versions of the SAT problem,
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| <!---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),--->
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| 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''(''l''<sub>1</sub>,...,''l''<sub>''n''</sub>) for some boolean operator ''R'' and (ordinary) literals ''l''<sub>''i''</sub>. Different sets of allowed boolean operators lead to different problem versions.
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| <!---, see [[#Complexity and restricted versions|below]].--->
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| 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 if exactly one of its arguments is.
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| <!---need not explain that alread yhere---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''=''y''=''z''=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.--->
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| 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
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| (''x''<sub>1</sub>∧''y''<sub>1</sub>) ∨ (''x''<sub>2</sub>∧''y''<sub>2</sub>) ∨ ... ∨ (''x''<sub>''n''</sub>∧''y''<sub>''n''</sub>)
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| into conjunctive normal form yields
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| (''x''<sub>1</sub>∨''x''<sub>2</sub>∨…∨''x''<sub>''n''</sub>) ∧
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| (''y''<sub>1</sub>∨''x''<sub>2</sub>∨…∨''x''<sub>''n''</sub>) ∧
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| (''x''<sub>1</sub>∨''y''<sub>2</sub>∨…∨''x''<sub>''n''</sub>) ∧
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| (''y''<sub>1</sub>∨''y''<sub>2</sub>∨…∨''x''<sub>''n''</sub>) ∧ ... ∧
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| (''x''<sub>1</sub>∨''x''<sub>2</sub>∨…∨''y''<sub>''n''</sub>) ∧
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| (''y''<sub>1</sub>∨''x''<sub>2</sub>∨…∨''y''<sub>''n''</sub>) ∧
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| (''x''<sub>1</sub>∨''y''<sub>2</sub>∨…∨''y''<sub>''n''</sub>) ∧
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| (''y''<sub>1</sub>∨''y''<sub>2</sub>∨…∨''y''<sub>''n''</sub>);
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| while the former is a disjunction of ''n'' conjunctions of 2 variables, the latter consists of 2<sup>''n''</sup> clauses of ''n'' variables.
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| ==Complexity and restricted versions==
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| ===Unrestricted satisfiability (SAT)===
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| {{Main|Cook–Levin theorem}}
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| SAT was the first known [[NP-complete]] problem, as proved by [[Stephen Cook]] in 1971<ref>{{cite book| author=Stephen A. Cook| chapter=The Complexity of Theorem-Proving Procedures| title=Proc. 3rd ann. Symp. on Theory of Computer| year=1971| pages=151–158| url=http://www.cs.toronto.edu/~sacook/homepage/1971.pdf| doi=10.1145/800157.805047}}</ref> and independently by [[Leonid Levin]] in 1973.<ref>{{cite journal|last=Levin|first=Leonid|authorlink=Leonid Levin|title = Universal search problems ({{lang-ru|Универсальные задачи перебора}}, Universal'nye perebornye zadachi)|journal = Problems of Information Transmission ({{lang-ru|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)] {{ru icon}}, 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}}</ref> Until that time, the concept of an NP-complete problem did not even exist.
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| 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.
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| A useful property of Cook's reduction is that it preserves the number of accepting answers. For example, deciding whether a given [[graph (mathematics)|graph]] has a [[Graph coloring#Vertex_coloring|3-coloring]] is another problem in NP; if a graph has 17 valid 3-colorings, the SAT formula produced by the Cook–Levin reduction will have 17 satisfying assignments.
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| 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.
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| SAT is trivial if the formulas are restricted to those in '''[[disjunctive normal form]]''', that is, they are 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; for an example exchange "∧" and "∨" in the [[#Basic definitions and terminology|above]] exponential blow-up example for conjunctive normal forms.
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| ===3-satisfiability===
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| [[File:Sat reduced to Clique from Sipser.svg|thumb|The 3-SAT instance (''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.]]
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| 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'''.
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| To reduce the unrestricted SAT problem to 3-SAT, transform each clause "''l''<sub>1</sub> ∨ ... ∨ ''l''<sub>''n''</sub>" to a conjunction of ''n''-2 clauses
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| "(''l''<sub>1</sub> ∨ ''l''<sub>2</sub> ∨ ''x''<sub>2</sub>) ∧
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| (¬''x''<sub>2</sub> ∨ ''l''<sub>3</sub> ∨ ''x''<sub>3</sub>) ∧
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| (¬''x''<sub>3</sub> ∨ ''l''<sub>4</sub> ∨ ''x''<sub>4</sub>) ∧ ... ∧
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| (¬''x''<sub>''n''-3</sub> ∨ ''l''<sub>''n''-2</sub> ∨ ''x''<sub>''n''-2</sub>) ∧
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| (¬''x''<sub>''n''-2</sub> ∨ ''l''<sub>''n''-1</sub> ∨ ''l''<sub>n</sub>)", where ''x''<sub>2</sub>,...,''x''<sub>''n''-2</sub> are fresh variables not occurring elsewhere.
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| Although both 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, i.e. the length growth is polynomial.<ref name="Aho.Hopcroft.Ullman.1974">{{cite book| author=Alfred V. Aho, John E. Hopcroft, Jeffrey D. Ullman| title=The Design and Analysis of Computer Algorithms| year=1974| publisher=Addison-Wesley}}; here: Thm.10.4</ref>
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| 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 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 (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 isn't the negation of the other</ref> literals from different clauses, cf. picture. The graph has a ''c''-clique if and only if the formula is satisfiable.{{refn|Aho, Hopcroft, Ullman<ref name="Aho.Hopcroft.Ullman.1974"/> (1974); Thm.10.5}}
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| 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 clauses and succeeds with high probability to correctly decide 3-SAT.<ref name="Schoning.1999">{{cite book| author=Uwe Schöning| chapter=A Probabilistic Algorithm for $k$-SAT and Constraint Satisfaction Problems| title=Proc. 40th Ann. Symp. Foundations of Computer Science|date=Oct 1999| pages=410-414| url=http://homepages.cwi.nl/~rdewolf/schoning99.pdf| doi=10.1109/SFFCS.1999.814612}}</ref>
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| The [[exponential time hypothesis]] is that no algorithm can solve 3-SAT faster than [[Small o notation#Little-o notation|''o'']](''exp''(''n'')).
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| Selman, Mitchell, and Levesque (1996) give empirical data on the difficulty of randomly generated 3-SAT formulas, depending on their size parameters.
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| Difficuly is measured in number recursive calls made by a [[#Algorithms for solving SAT|DPLL algorithm]].<ref>{{cite journal| author=Bart Selman, David Mitchell, Hector Levesque| title=Generating Hard Satisfiability Problems| journal=Artificial Intelligence| year=1996| volume=81| pages=17–29| url=http://citeseerx.ist.psu.edu/viewdoc/download;jsessionid=3CBEAB7E11BF4B2283E9F383810060C1?doi=10.1.1.37.7362&rep=rep1&type=pdf}}</ref>
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| 3-satisfiability can be generalized to '''k-satisfiability''' ('''k-SAT'''), when formulas in CNF are considered with each clause containing up to ''k'' literals.
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| 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.
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| Some authors restrict k-SAT to CNF formulas with '''exactly k literals'''. This doesn't lead to a different complexity class either, as each clause "''l''<sub>1</sub> ∨ ... ∨ ''l''<sub>''j''</sub>" with ''j''<''k'' literals can be padded with fixed dummy variables to
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| "''l''<sub>1</sub> ∨ ... ∨ ''l''<sub>''j''</sub> ∨ ''d''<sub>''j''+1</sub> ∨ ... ∨ ''d''<sub>''k''</sub>".
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| 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 ''d''<sub>1</sub>,...,''d''<sub>''k''</sub>, except "''d''<sub>1</sub>∨...∨''d''<sub>''k''</sub>"</ref> have to be appended to ensure that only ''d''<sub>1</sub>=...=''d''<sub>k</sub>=FALSE can lead to a satisfying assignment. Since ''k'' doesn't 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 (like e.g. "¬''x'' ∨ ¬''y'' ∨ ¬''y''"), or not.
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| ===Exactly-1 3-satisfiability===
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| [[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.]]
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| 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''').
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| Given a conjunctive normal form, 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.
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| 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.
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| One-in-three 3-SAT is listed as NP-complete problem "LO4" in the standard reference, ''Computers and Intractability: A Guide to the Theory of NP-Completeness''
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| by [[Michael R. Garey]] and [[David S. Johnson]]. It was proved to be NP-complete by [[Thomas J. 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">
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| {{cite conference
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| | last = Schaefer
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| | first = Thomas J.
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| | title = The complexity of satisfiability problems
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| | booktitle = Proceedings of the 10th Annual ACM Symposium on Theory of Computing
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| | place = San Diego, California
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| | pages = 216–226
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| | year = 1978
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| | url = http://www.ccs.neu.edu/home/lieber/courses/csg260/f06/materials/papers/max-sat/p216-schaefer.pdf
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| | doi = 10.1145/800133.804350
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| }}</ref>
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| 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.
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| <!---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''
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| is TRUE.--->
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| 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.<ref>(Schaefer, 1978), p.222, Lemma 3.5</ref>
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| 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).
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| The one-in-three 3-SAT problem is often used in the literature as a known NP-complete problem in a reduction to show that other problems are NP-complete.{{citation needed|date=September 2013}}
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| ===2-satisfiability===
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| {{Main|2-satisfiability}}
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| 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, the result is called '''exclusive-or 2-satisfiability''', which is a problem complete for the complexity class [[SL (complexity)|SL]] = [[L (complexity)|L]].
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| ===Horn-satisfiability===
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| {{Main|Horn-satisfiability}}
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| The problem of deciding the satisfiability of a given conjunction of Horn clauses is called '''Horn-satisfiability''', or '''HORN-SAT'''.
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| 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).
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| Horn-satisfiability is [[P-complete]]. It can be seen as [[P (complexity)|P's]] version of the Boolean satisfiability problem.
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| 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'' needs to be TRUE as well.
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| A generalization of the class of Horn formulae is that of renamable-Horn formulae, which is the set of formulae that can be placed in Horn form by replacing some variables with their respective negation.
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| 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>.
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| 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.gif|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"
| |
| |-
| |
| ! 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
| |
| |-
| |
| |
| |
| |-
| |
| ! ''a'' !! ''b'' !! ''c'' !! ''d'' !! !! line
| |
| |-
| |
| |
| |
| |-
| |
| | 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
| |
| |-
| |
| |
| |
| |-
| |
| ! ''a'' !! ''b'' !! ''c'' !! ''d'' !! !! operation
| |
| |-
| |
| |
| |
| |-
| |
| | 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)
| |
| |-
| |
| |
| |
| |-
| |
| | 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
| |
| |-
| |
| |
| |
| |-
| |
| | 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
| |
| |-
| |
| |
| |
| |-
| |
| ! ''a'' !! ''b'' !! ''c'' !! ''d'' !! !! operation
| |
| |-
| |
| |
| |
| |-
| |
| | 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
| |
| |-
| |
| |
| |
| |-
| |
| | 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 operator ''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]], see box for an example.{{citation needed|reason=Although the example computation seems to be ok, it is not immediately obvious why Gauss' algorithm is applicable and its solution(s) are solutions of XOR-SAT. In particular, the algorithm needs a field, not a ring, and its solutions need not be in {0,1} in general.|date=September 2013}} This recast is based on the [[Boolean_algebra_(structure)#Boolean_rings|kinship between Boolean algebras and Boolean rings]]. 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, cf. picture. As a consequence, for each CNF formula, either the 3-SAT problem or the 1-in-3-SAT problem can be decided in cubic time to be solvable or unsolvable, respectively.{{clarify|reason=This consequence seems strange, but valid. Somebody please crosscheck.|date=September 2013}}
| |
| | |
| 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. The assumption that P and NP are not equal is [[Unsolved problems of mathematics#Millennium Prize Problems|currently not proven]].
| |
| | |
| ===Schaefer's dichotomy theorem===
| |
| {{Main|Schaefer's dichotomy theorem}}
| |
| The restrictions above (CNF, 2CNF, 3CNF, Horn, XOR-SAT) bound the considered formulae to be conjunction of subformulae; each restriction states a specific form for all subformulae: for example, only binary clauses can be subformulae in 2CNF.
| |
| | |
| Schaefer's dichotomy theorem states that, for any restriction to Boolean operators that can be used to form these subformulae, 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.
| |
| | |
| ==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. 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]]s 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 (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.
| |
| | |
| A number of variants deal with the number of variable assignments making the formula TRUE. Ordinary SAT asks if there is at least one such assignment. '''MAJ-SAT''', which asks if the majority of all assignments make the formula TRUE, is complete for [[PP (complexity)|PP]], a probabilistic class. The problem of '''how many variable assignments''' satisfy a formula, not a decision problem, is in [[Sharp-P|#P]]. '''UNIQUE-SAT''' is the problem of determining whether a formula has exactly one assignment, it is complete for [[US (complexity)|US]].{{clarify|reason=As long as there is no article available, the complexity class 'US' should be defined here explicitly.|date=September 2013}} When the input is [[Promise problem|restricted]] to formulas having at most one satisfying assignment (or none), the problem is called '''UNAMBIGOUS-SAT'''. A solving algorithm for UNAMBIGOUS-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 doi|10.1016/0304-3975(86)90135-0}}</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.
| |
| | |
| The '''[[maximum satisfiability problem]]''', an [[FNP (complexity)|FNP]] generalization of SAT, 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.
| |
| | |
| Other generalisations include satisfiability for [[first-order predicate calculus|first]]- and [[second-order logic]], [[constraint satisfaction problem]]s, [[0-1 integer programming]].
| |
| | |
| ==Self-reducibility==
| |
| The SAT problem is '''self-reducible''', that is, each algorithm which correctly answers if 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}]], i.e. 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 of self-reducibility is used in several theorems in complexity theory:
| |
| | |
| : [[NP (complexity)|NP]] ⊆ [[P/poly]] ⇒ [[PH (complexity)|PH]] = [[Polynomial_hierarchy#Definitions|Σ<sub>2</sub>]] ([[Karp–Lipton theorem]]){{clarify|reason=When adding the link for Σ2, I guessed from the 'Karp–Lipton theorem' article that the 'Polynomial hierarchy', rather than e.g. the 'Arithmetical hierarchy' was meant. This should be crosschecked by a complexity theory expert.|date=September 2013}}
| |
| : [[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==
| |
| | |
| Since the SAT problem is NP-complete, only algorithms with exponential worst-case complexity are known for it. In spite of this state of the art in complexity theory, efficient and scalable algorithms for SAT were developed over the last decade and have contributed to dramatic advances in our ability to automatically solve problem instances involving tens of thousands of variables and millions of constraints (i.e. clauses).{{citation needed|reason=Until 2013-09-16, this article claimed, without giving a particular source, both 'tens of thousands of variables and millions of constraints' (provisionally kept) and 'thousands of variables and tens of thousands of constraints' (removed). A source for one or the other version (or even for both) should be named.|date=September 2013}} 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 doi|10.1109/TCAD.2002.1004311}}</ref> [[Automated planning and scheduling|planning]], and [[Scheduling algorithm|scheduling problems]], and so on. A SAT-solving engine is now considered to be an essential component in the [[electronic design automation|EDA]] toolbox.
| |
| | |
| There are two classes of high-performance [[algorithms]] for solving instances of SAT in practice: the [[conflict-driven clause learning]] algorithm, which can be viewed as a modern variant of the [[DPLL algorithm]] (well known implementation include [[Chaff algorithm|Chaff]],<ref>{{cite doi|10.1145/378239.379017}}</ref> [[GRASP (SAT solver)|GRASP]])<ref>{{cite doi|10.1109/12.769433}}</ref> and [[stochastic local search algorithm|stochastic]] [[Local search (constraint satisfaction)|local search]] algorithms, such as [[WalkSAT]].
| |
| | |
| A DPLL SAT solver employs a systematic backtracking search procedure to explore the (exponentially sized) space of variable assignments looking for satisfying assignments. The basic search procedure was proposed in two seminal papers in the early 60s (see references below) and is now commonly referred to as the [[Davis–Putnam–Logemann–Loveland algorithm]] ("DPLL" or "DLL").<ref>{{cite doi|10.1145/321033.321034}}</ref><ref>{{cite doi|10.1145/368273.368557}}</ref> Theoretically, exponential lower bounds have been proved for the DPLL family of algorithms.
| |
| | |
| In contrast, randomized algorithms like the PPSZ algorithm by Paturi, Pudlak, Saks, and Zane set variables in a random order according to some heuristics, for example bounded-width [[Resolution (logic)|resolution]]. If the heuristic can't find the correct setting, the variable is assigned randomly. The PPSZ algorithm has a runtime of <math>O(2^{0.386n})</math> for 3-SAT with a single satisfying assignment. Currently this is the best known runtime for this problem. In the setting with many satisfying assignments the randomized algorithm by Schöning has a better bound.<ref name="Schoning.1999"/><ref name="ppsz_algorithm">[http://dl.acm.org/citation.cfm?id=1066101 "An improved exponential-time algorithm for k-SAT"], Paturi, Pudlak, Saks, Zani</ref>
| |
| | |
| Modern SAT solvers (developed in the last ten years) come in two flavors: "conflict-driven" and "look-ahead".{{clarify|reason=3 paragraphs before here, 'two classes', viz. 'conflict-driven clause learning DPLL' and 'stochastic local search' were named. Please indicate if the latter essentially means the same as 'look-ahead' here.|date=September 2013}} Conflict-driven solvers augment the basic DPLL search algorithm with efficient conflict analysis, clause learning, non-[[chronological backtracking]] (aka [[backjumping]]), as well as "two-watched-literals" unit propagation, adaptive branching, and random restarts. These "extras" to the basic systematic search have been empirically shown to be essential for handling the large SAT instances that arise in [[electronic design automation]] (EDA). Look-ahead solvers have especially strengthened reductions (going beyond unit-clause propagation) and the heuristics, and they are generally stronger than conflict-driven solvers on hard instances (while conflict-driven solvers can be much better on large instances which actually have an easy instance inside).
| |
| | |
| Modern SAT solvers are also having significant impact on the fields of software verification, constraint solving in artificial intelligence, and operations research, among others. Powerful solvers are readily available as [[free and open source software]]. In particular, the conflict-driven [http://minisat.se/ MiniSAT], which was relatively successful at the [http://www.satcompetition.org/ 2005 SAT competition], only has about 600 lines of code. A modern Parallel SAT solver is [http://www.cril.univ-artois.fr/~jabbour/manysat.htm ManySAT]. It can achieve super linear speed-ups on important classes of problems. An example for look-ahead solvers is [http://www.st.ewi.tudelft.nl/sat/march_dl.php march_dl], which won a prize at the [http://www.satcompetition.org/ 2007 SAT competition].
| |
| | |
| Certain types of large random satisfiable instances of SAT can be solved by [[survey propagation]] (SP). Particularly in [[hardware design]] and [[hardware verification|verification]] applications, satisfiability and other logical properties of a given propositional formula are sometimes decided based on a representation of the formula as a [[binary decision diagram]] (BDD).
| |
| | |
| 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.
| |
| All of these behaviors can be seen in the SAT solving contests.<ref>{{cite web|url=http://www.satcompetition.org/ |title=The international SAT Competitions web page|accessdate=2007-11-15}}</ref>
| |
| | |
| ==See also==
| |
| *[[Unsatisfiable core]]
| |
| *[[Satisfiability Modulo Theories]]
| |
| *[[Sharp-SAT|Counting SAT]]
| |
| *[[Karloff–Zwick algorithm]]
| |
| | |
| ==Notes==
| |
| {{reflist|group=note}}
| |
| | |
| ==References==
| |
| | |
| {{reflist}}
| |
| References are ordered by date of publication:
| |
| {{refbegin|colwidth=25em}}
| |
| * {{cite book|author = [[Michael R. Garey]] and [[David S. Johnson]] | year = 1979 | title = [[List of important publications in computer science#Computers and Intractability: A Guide to the Theory of NP-Completeness|Computers and Intractability: A Guide to the Theory of NP-Completeness]] | publisher = W.H. Freeman | isbn = 0-7167-1045-5}} A9.1: LO1 – LO7, pp. 259 – 260.
| |
| * {{cite doi|10.1109/DATE.1999.761110}}
| |
| * {{cite doi|10.1023/A:1011276507260}}
| |
| * {{cite doi|10.1007/b95238}}
| |
| * {{cite doi|10.1109/TC.2006.175}}
| |
| * {{cite doi|10.1109/BIMNICS.2007.4610083}}
| |
| * {{cite book|editor=Frank Van Harmelen, Vladimir Lifschitz, Bruce Porter|title=Handbook of knowledge representation|year=2008|publisher=Elsevier|isbn=978-0-444-52211-5|pages=89–134|author=Carla P. Gomes, Henry Kautz, Ashish Sabharwal, Bart Selman|chapter=Satisfiability Solvers|doi=10.1016/S1574-6526(07)03002-7|series=Foundations of Artificial Intelligence|volume=3}}
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| {{refend|colwidth=25em}}
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| == See also ==
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| * [[Circuit satisfiability]]
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| ==External links==
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| === SAT problem format ===
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| A SAT problem is often described in the [http://www.domagoj-babic.com/uploads/ResearchProjects/Spear/dimacs-cnf.pdf DIMACS-CNF] format: an input file in which each line represents a single disjunction. For example, a file with the two lines
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| 1 -5 4 0
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| -1 5 3 4 0
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| represents the formula "(''x''<sub>1</sub> ∨ ¬''x''<sub>5</sub> ∨ ''x''<sub>4</sub>) ∧ (¬''x''<sub>1</sub> ∨ ''x''<sub>5</sub> ∨ ''x''<sub>3</sub> ∨ ''x''<sub>4</sub>)".
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| Another common format for this formula is the 7-bit [[ASCII]] representation "(x1 | ~x5 | x4) & (~x1 | x5 | x3 | x4)".
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| [http://users.ics.aalto.fi/tjunttil/bcsat/ BCSAT] is a tool that converts input files in human-readable format to the DIMACS-CNF format.
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| === Online SAT solvers ===
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| * [http://www.boolsat.com BoolSAT] – Solves formulas in the DIMACS-CNF format or in a more human-friendly format: "a and not b or a". Runs on a server.
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| * [http://www.msoos.org/2013/09/minisat-in-your-browser/ minisat-in-your-browser] – Solves formulas in the DIMACS-CNF format. Runs on the browser.
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| === Offline SAT solvers ===
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| * [http://minisat.se/ MiniSAT] – DIMACS-CNF format.
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| * [http://fmv.jku.at/lingeling/ Lingeling] – won a gold medal in a 2011 SAT competition.
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| ** [http://fmv.jku.at/picosat/ PicoSAT] – an earlier solver from the Lingeling group.
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| * [http://www.sat4j.org/ Sat4j] – DIMACS-CNF format. Java source code available.
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| * [http://www.labri.fr/~lsimon/glucose Glucose] – DIMACS-CNF format.
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| * [http://reasoning.cs.ucla.edu/rsat/home.html RSat] – won a gold medal in a 2007 SAT competition.
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| * [http://ubcsat.dtompkins.com/ UBCSAT]. Supports unweighted and weighted clauses, both in the DIMACS-CNF format. C source code hosted on [https://github.com/dtompkins/ubcsat GitHub].
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| * [http://www.msoos.org/cryptominisat2 CryptoMiniSat] – won a gold medal in a 2011 SAT competition. C++ source code hosted on [https://github.com/msoos/cryptominisat GitHub]. Tries to put many useful features of MiniSat 2.0 core, PrecoSat ver 236, and Glucose into one package, adding many new features
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| * [http://www.domagoj-babic.com/index.php/ResearchProjects/Spear Spear] – Supports bit-vector arithmetic. Can use the DIMACS-CNF format or the [http://www.domagoj-babic.com/uploads/ResearchProjects/Spear/spear-format.pdf Spear format].
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| ** [http://www.domagoj-babic.com/index.php/ResearchProjects/HyperSAT HyperSAT] – Written to experiment with B-cubing search space pruning. Won 3rd place in a 2005 SAT competition. An earlier and slower solver from the developers of Spear.
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| * [http://logic.pdmi.ras.ru/~basolver/ BASolver]
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| * [http://argo.matf.bg.ac.rs/?content=downloads ArgoSAT]
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| * [http://dudka.cz/fss Fast SAT Solver] – based on [[genetic algorithm]]s.
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| * [http://www.princeton.edu/~chaff/zchaff.html zChaff] – not supported anymore.
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| * [http://users.ics.aalto.fi/tjunttil/bcsat/ BCSAT] – human-readable boolean circuit format (also converts this format to the DIMACS-CNF format and automatically links to MiniSAT or zChaff).
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| === SAT applications ===
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| * [http://www.mqasem.net/sat/winsat/ WinSAT v2.04]: A Windows-based SAT application made particularly for researchers.
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| === Conferences ===
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| International Conference on Theory and Applications of Satisfiability Testing:
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| * [http://sat2013.cs.helsinki.fi/ SAT 2013]
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| * [http://sat2012.fbk.eu SAT 2012]
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| * [http://www.lri.fr/SAT2011/ SAT 2011]
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| * [http://ie.technion.ac.il/SAT10/ SAT 2010]
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| * [http://www.cs.swansea.ac.uk/~csoliver/SAT2009/ SAT 2009]
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| * [http://wwwcs.uni-paderborn.de/cs/ag-klbue/en/workshops/sat-08/sat08-main.php SAT 2008]
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| * [http://sat07.ecs.soton.ac.uk/ SAT 2007]
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| === Publications ===
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| * [http://jsat.ewi.tudelft.nl Journal on Satisfiability, Boolean Modeling and Computation]
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| * [http://www.ictp.trieste.it/~zecchina/SP/ Survey Propagation]
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| === Benchmarks ===
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| * [http://www.nlsde.buaa.edu.cn/~kexu/benchmarks/benchmarks.htm Forced Satisfiable SAT Benchmarks]
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| * [http://www.satlib.org SATLIB]
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| * [http://www.cs.ubc.ca/~babic/index_benchmarks.htm Software Verification Benchmarks]
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| * [http://www.aloul.net/benchmarks.html Fadi Aloul SAT Benchmarks]
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| SAT solving in general:
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| * http://www.satlive.org
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| * http://www.satisfiability.org
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| === Evaluation of SAT solvers ===
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| * [http://www.maxsat.udl.cat/ Yearly evaluation of SAT solvers]
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| * [http://www.maxsat.udl.cat/08/ms08.pdf SAT solvers evaluation results for 2008]
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| * [http://www.satcompetition.org International SAT Competitions]
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| * [http://www.satcompetition.org/2002/onlinereport/node2.html History]
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| More information on SAT:
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| * [http://www.mqasem.net/sat/sat SAT and MAX-SAT for the Lay-researcher]
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| ----
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| ''This article includes material from a column in the ACM [http://www.sigda.org SIGDA] [http://www.sigda.org/newsletter/index.html e-newsletter] by [http://www.eecs.umich.edu/~karem Prof. Karem Sakallah] <br />
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| Original text is available [http://www.sigda.org/newsletter/2006/eNews_061201.html here]''
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| {{logic}}
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| {{DEFAULTSORT:Boolean Satisfiability Problem}}
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| [[Category:Boolean algebra]]
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| [[Category:Electronic design automation]]
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| [[Category:Formal methods]]
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| [[Category:Logic in computer science]]
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| [[Category:NP-complete problems]]
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| [[Category:Satisfiability problems]]
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