Davis–Putnam algorithm

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-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.

Overview

Two runs of the Davis-Putnam procedure on example propositional ground instances. Top to bottom, Left: Starting from the formula ${\displaystyle (a\lor b\lor c)\land (b\lor \lnot c\lor \lnot f)\land (\lnot b\lor e)}$, the algorithm resolves on ${\displaystyle b}$, and then on ${\displaystyle c}$. 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 ${\displaystyle b}$, then on ${\displaystyle a}$, then on ${\displaystyle c}$ yields the empty clause; hence the algorithm returns "unsatisfiable".

The procedure is based on Herbrand's theorem, which implies that an 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.

The procedure roughly consists of these three parts:

• put the formula in prenex form and eliminate quantifiers
• generate all propositional ground instances, one by one
• check if each instance is satisfiable

The last part is probably the most innovative one, and works as follows (cf. picture):

• for every variable in the formula
  • for every clause ${\displaystyle c}$ containing the variable and every clause ${\displaystyle n}$ containing the negation of the variable
    • resolve c and n and add the resolvent to the formula
  • remove all original clauses containing the variable or its negation

At each step, the intermediate formula generated is equisatisfiable, but possibly not equivalent, to the original formula. The resolution step leads to a worst-case exponential blow-up in the size of the formula.

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 still forms the basis for today's (as of 2015) most efficient complete SAT solvers.

See also

• Herbrandization

References

• Davis, Martin; Putnam, Hilary (1960). "A Computing Procedure for Quantification Theory". Journal of the ACM. 7 (3): 201–215. doi:10.1145/321033.321034.
• Davis, Martin; Logemann, George; Loveland, Donald (1962). "A Machine Program for Theorem Proving". Communications of the ACM. 5 (7): 394–397. doi:10.1145/368273.368557. hdl:2027/mdp.39015095248095.
• R. Dechter; I. Rish. "Directional Resolution: The Davis–Putnam Procedure, Revisited". In J. Doyle and E. Sandewall and P. Torasso (ed.). Principles of Knowledge Representation and Reasoning: Proc. of the Fourth International Conference (KR'94). Kaufmann. pp. 134–145.
• John Harrison (2009). Handbook of practical logic and automated reasoning. Cambridge University Press. pp. 79–90. ISBN 978-0-521-89957-4.