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| {{Confusing|article|date=February 2009}}
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| 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 set|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.
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| 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 valid if and only if its negation is unsatisfiable. Taken together, these facts imply that to prove the validity of ''φ'' it is enough to prove that a ground instance of ''¬φ'' is unsatisfiable. If ''φ'' is not valid, then the search for an unsatisfiable ground instance will not terminate.
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| The procedure roughly consists of these three parts:
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| * put the formula in [[prenex]] form and eliminate quantifiers
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| * generate all propositional ground instances, one by one
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| * check if each instance is satisfiable
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| The last part is probably the most innovative one, and works as follows:
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| * for every variable in the formula
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| ** for every clause <math>c</math> containing the variable and every clause <math>n</math> containing the negation of the variable
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| *** [[Resolution (logic)|resolve]] ''c'' and ''n'' and add the resolvent to the formula
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| ** remove all original clauses containing the variable or its negation
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| At each step, the intermediate formula generated is [[equisatisfiable]] to the original formula, but it does not retain [[Logical equivalence|equivalence]]. The resolution step leads to a worst-case exponential blow-up in the size of the formula. The [[DPLL algorithm]] is a refinement of the propositional satisfiability step of the Davis–Putnam procedure, that requires only a linear amount of memory in the worst case.
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| ==See also==
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| *[[Herbrandization]]
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| ==References==
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| * {{cite journal
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| |last=Davis
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| |first=Martin
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| | coauthors= Putnam, Hilary
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| | title=A Computing Procedure for Quantification Theory
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| | journal =[[Journal of the ACM]]
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| | volume = 7
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| | issue = 3
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| | pages = 201–215
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| | year = 1960
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| | url = http://portal.acm.org/citation.cfm?coll=GUIDE&dl=GUIDE&id=321034
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| | doi=10.1145/321033.321034}}
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| *{{cite journal
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| | last=Beckford
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| | first=Jahbrill
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| | coauthors=Logemann, George, and Loveland, Donald
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| | title=A Machine Program for Theorem Proving
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| | journal =[[Communications of the ACM]]
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| | volume=5
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| | issue=7
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| | pages = 394–397
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| | year=1962
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| | url=http://portal.acm.org/citation.cfm?doid=368273.368557
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| | doi=10.1145/368273.368557}}
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| * {{cite conference
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| | author = R. Dechter
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| | coauthors = I. Rish
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| | editor = J. Doyle and E. Sandewall and P. Torasso
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| | year =
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| | title = Directional Resolution: The Davis–Putnam Procedure, Revisited
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| | conference =
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| | booktitle = Principles of Knowledge Representation and Reasoning: Proc. of the Fourth International Conference (KR'94)
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| | pages = 134–145
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| | publisher = Starswager18
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| | url =
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| | conferenceurl =
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| }}
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| * {{cite book|author=John Harrison|title=Handbook of practical logic and automated reasoning|year=2009|publisher=Cambridge University Press|isbn=978-0-521-89957-4|pages=79–90}}
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| {{DEFAULTSORT:Davis-Putnam algorithm}}
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| [[Category:Boolean algebra]]
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| [[Category:Constraint programming]]
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| [[Category:Automated theorem proving]]
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| {{formalmethods-stub}}
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