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Adding short description: "Check the validity of a logic formula" |
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{{Short description|Check the validity of a logic formula}}
==Overview==
▲The '''Davis–Putnam algorithm''' was developed by [[Martin Davis]] and [[Hilary Putnam]] for checking the validity of a [[first-order logic]] formula using a [[Resolution (logic)|resolution]]-based decision procedure for [[propositional logic]]. Since the set of valid first-order formulas is [[recursively enumerable]] but not [[recursive]], there exists no general algorithm to solve this problem. Therefore, the Davis-Putnam algorithm only terminates on valid formulas. Today, the term "Davis-Putnam algorithm" is often used synonymously with the resolution-based propositional decision procedure that is actually only one of the steps of the original algorithm.
[[File:Resolution.png|thumb|400px|Two runs of the Davis-Putnam procedure on example propositional ground instances.
''Top to bottom, Left:'' Starting from the formula <math>(a \lor b \lor c) \land (b \lor \lnot c \lor \lnot f) \land (\lnot b \lor e)</math>, the algorithm resolves on <math>b</math>, and then on <math>c</math>. Since no further resolution is possible, the algorithm stops; since the [[empty clause]] couldn't be derived, the result is "''satisfiable''".
''Right:'' Resolving the given formula on <math>b</math>, then on <math>a</math>, then on <math>c</math> yields the empty clause; hence the algorithm returns "''unsatisfiable''".]]
The procedure is based on [[Herbrand's theorem]], which implies that an [[satisfiable|unsatisfiable]] formula has an unsatisfiable [[ground instance]], and on the fact that a formula is
The procedure for checking validity of a formula ''φ'' roughly consists of these three parts:▼
* put the formula ''¬φ'' in [[prenex]] form and eliminate quantifiers ▼
** If some instance is unsatisfiable, then return that ''φ'' is valid. Else continue checking.
The last part is a [[SAT solver]] based on [[Resolution (logic)|resolution]] (as seen on the illustration), with an eager use of [[unit propagation]] and pure literal elimination (elimination of clauses with variables that occur only positively or only negatively in the formula).{{clarify|reason=In the algorithm below, adding to and removing from a formula should be defined. I guess the formula has to be in clausal normal form (this should be explained then), and adding and removing is achieved by set operations? Moreover, the 'consistent set of literals' test, and the concepts 'polarity', 'pure literal' and 'unit propagation' should be briefly explained.|date=June 2021}}
▲The procedure roughly consists of these three parts:
▲* put the formula in [[prenex]] form and eliminate quantifiers
▲* generate one by one all propositional ground instances
▲* and check for each if it is satisfiable
{{Algorithm-begin|name=DP SAT solver}}
Input: A set of clauses Φ.
Output: A Truth Value: true if Φ can be satisfied, false otherwise.
'''function''' DP-SAT(Φ)
'''repeat'''
** for every clause <math>c</math> containing the variable and every clause <math>n</math> containing the negation of the variable▼
// ''unit propagation:''
'''while''' Φ contains a unit clause {''l''} '''do'''
'''for every''' clause ''c'' in Φ that contains ''l'' '''do'''
Φ ← ''remove-from-formula''(''c'', Φ);
'''for every''' clause ''c'' in Φ that contains ¬''l'' '''do'''
Φ ← ''remove-from-formula''(''c'', Φ);
Φ ← ''add-to-formula''(''c'' \ {¬''l''}, Φ);
// ''eliminate clauses not in normal form:''
'''for every''' clause ''c'' in Φ that contains both a literal ''l'' and its negation ¬''l'' '''do'''
Φ ← ''remove-from-formula''(''c'', Φ);
// ''pure literal elimination:''
'''while''' there is a literal ''l'' all of which occurrences in Φ have the same polarity '''do'''
'''for every''' clause ''c'' in Φ that contains ''l'' '''do'''
Φ ← ''remove-from-formula''(''c'', Φ);
// ''stopping conditions:''
'''if''' Φ is empty '''then'''
'''return''' true;
'''if''' Φ contains an empty clause '''then'''
'''return''' false;
// ''Davis-Putnam procedure:''
pick a literal ''l'' that occurs with both polarities in Φ
▲
// ''resolve c with n:''
''r'' ← (''c'' \ {''l''}) ∪ (''n'' \ {¬''l''});
Φ ← ''add-to-formula''(''r'', Φ);
'''for every''' clause ''c'' that contains ''l'' or ¬''l'' '''do'''
Φ ← ''remove-from-formula''(''c'', Φ);
{{Algorithm-end}}
At each step of the SAT solver, the intermediate formula generated is [[equisatisfiable]]
The [[Davis–Putnam–Logemann–Loveland algorithm]] is a 1962 refinement of the propositional satisfiability step of the Davis–Putnam procedure which requires only a linear amount of memory in the worst case. It eschews the resolution for ''the splitting rule'': a backtracking algorithm that chooses a literal ''l'', and then recursively checks if a simplified formula with ''l'' assigned a true value is satisfiable or if a simplified formula with ''l'' assigned false is. It still forms the basis for today's (as of 2015) most efficient complete [[SAT solver]]s.
==See also==
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|last=Davis
|first=Martin
|
| title=A Computing Procedure for Quantification Theory
| journal =[[Journal of the ACM]]
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| issue = 3
| pages = 201–215
|
}}
▲| doi=10.1145/321033.321034}}
*{{cite journal
| last=Davis
| first=Martin |author2=Logemann, George |author3=Loveland, Donald
| title=A Machine Program for Theorem Proving
| journal =[[Communications of the ACM]]
Line 47 ⟶ 80:
| issue=7
| pages = 394–397
|
| url=http://portal.acm.org/citation.cfm?doid=368273.368557
| doi=10.1145/368273.368557
| hdl-access=free
}}
* {{cite conference
| author = R. Dechter
|
| editor = J. Doyle and E. Sandewall and P. Torasso
| title = Directional Resolution: The Davis–Putnam Procedure, Revisited
|
▲ | booktitle = Principles of Knowledge Representation and Reasoning: Proc. of the Fourth International Conference (KR'94)
| pages = 134–145
| publisher = Kaufmann
}}
* {{cite book|author=John Harrison|title=Handbook of practical logic and automated reasoning|url=https://archive.org/details/handbookpractica00harr|url-access=limited|year=2009|publisher=Cambridge University Press|isbn=978-0-521-89957-4|pages=[https://archive.org/details/handbookpractica00harr/page/n100 79]–90}}
{{DEFAULTSORT:Davis-Putnam algorithm}}
[[Category:Boolean algebra]]
[[Category:Constraint
[[Category:Automated theorem proving]]
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