DPLL algorithm

In logic and computer science, the Davis–Putnam–Logemann–Loveland (DPLL) algorithm is a complete, backtracking-based search algorithm for deciding the satisfiability of propositional logic formulae in conjunctive normal form, i.e. for solving the CNF-SAT problem.

It was introduced in 1961 by Martin Davis, George Logemann and Donald W. Loveland and is a refinement of the earlier Davis–Putnam algorithm, which is a resolution-based procedure developed by Davis and Hilary Putnam in 1960. Especially in older publications, the Davis–Logemann–Loveland algorithm is often referred to as the "Davis–Putnam method" or the "DP algorithm". Other common names that maintain the distinction are DLL and DPLL.

Implementations and applications
The SAT problem is important both from theoretical and practical points of view. In complexity theory it was the first problem proved to be NP-complete, and can appear in a broad variety of applications such as model checking, automated planning and scheduling, and diagnosis in artificial intelligence.

As such, writing efficient SAT solvers has been a research topic for many years. GRASP (1996-1999) was an early implementation using DPLL. In the international SAT competitions, implementations based around DPLL such as zChaff and MiniSat were in the first places of the competitions in 2004 and 2005.

Another application that often involves DPLL is automated theorem proving or satisfiability modulo theories (SMT), which is a SAT problem in which propositional variables are replaced with formulas of another mathematical theory.

The algorithm
The basic backtracking algorithm runs by choosing a literal, assigning a truth value to it, simplifying the formula and then recursively checking if the simplified formula is satisfiable; if this is the case, the original formula is satisfiable; otherwise, the same recursive check is done assuming the opposite truth value. This is known as the splitting rule, as it splits the problem into two simpler sub-problems. The simplification step essentially removes all clauses that become true under the assignment from the formula, and all literals that become false from the remaining clauses.

The DPLL algorithm enhances over the backtracking algorithm by the eager use of the following rules at each step:


 * Unit propagation : If a clause is a unit clause, i.e. it contains only a single unassigned literal, this clause can only be satisfied by assigning the necessary value to make this literal true. Thus, no choice is necessary. Unit propagation consists in removing every clause containing a unit clause's literal and in discarding the complement of a unit clause's literal from every clause containing that complement. In practice, this often leads to deterministic cascades of units, thus avoiding a large part of the naive search space.


 * Pure literal elimination : If a propositional variable occurs with only one polarity in the formula, it is called pure. A pure literal can always be assigned in a way that makes all clauses containing it true. Thus, when it is assigned in such a way, these clauses do not constrain the search anymore, and can be deleted.

Unsatisfiability of a given partial assignment is detected if one clause becomes empty, i.e. if all its variables have been assigned in a way that makes the corresponding literals false. Satisfiability of the formula is detected either when all variables are assigned without generating the empty clause, or, in modern implementations, if all clauses are satisfied. Unsatisfiability of the complete formula can only be detected after exhaustive search.

The DPLL algorithm can be summarized in the following pseudocode, where Φ is the CNF formula:

Input: A set of clauses Φ. Output: A truth value indicating whether Φ is satisfiable.

function DPLL(Φ) // unit propagation: while there is a unit clause {l} in Φ do Φ ← unit-propagate(l, Φ); // pure literal elimination: while there is a literal l that occurs pure in Φ do Φ ← pure-literal-assign(l, Φ); // stopping conditions: if Φ is empty then return true; if Φ contains an empty clause then return false; // DPLL procedure: l ← choose-literal(Φ); return DPLL(Φ ∧ {l}) or DPLL(Φ ∧ {¬l});

In this pseudocode,  and   are functions that return the result of applying unit propagation and the pure literal rule, respectively, to the literal   and the formula. In other words, they replace every occurrence of  with "true" and every occurrence of   with "false" in the formula , and simplify the resulting formula. The  in the   statement is a short-circuiting operator. denotes the simplified result of substituting "true" for  in.

The algorithm terminates in one of two cases. Either the CNF formula  is empty, i.e., it contains no clause. Then it is satisfied by any assignment, as all its clauses are vacuously true. Otherwise, when the formula contains an empty clause, the clause is vacuously false because a disjunction requires at least one member that is true for the overall set to be true. In this case, the existence of such a clause implies that the formula (evaluated as a conjunction of all clauses) cannot evaluate to true and must be unsatisfiable.

The pseudocode DPLL function only returns whether the final assignment satisfies the formula or not. In a real implementation, the partial satisfying assignment typically is also returned on success; this can be derived by keeping track of branching literals and of the literal assignments made during unit propagation and pure literal elimination.

The Davis–Logemann–Loveland algorithm depends on the choice of branching literal, which is the literal considered in the backtracking step. As a result, this is not exactly an algorithm, but rather a family of algorithms, one for each possible way of choosing the branching literal. Efficiency is strongly affected by the choice of the branching literal: there exist instances for which the running time is constant or exponential depending on the choice of the branching literals. Such choice functions are also called heuristic functions or branching heuristics.

Visualization
Davis, Logemann, Loveland (1961) had developed this algorithm. Some properties of this original algorithm are: An example with visualization of a DPLL algorithm having chronological backtracking:
 * It is based on search.
 * It is the basis for almost all modern SAT solvers.
 * It does not use learning or non-chronological backtracking (introduced in 1996).

Related algorithms
Since 1986, (Reduced ordered) binary decision diagrams have also been used for SAT solving.

In 1989-1990, Stålmarck's method for formula verification was presented and patented. It has found some use in industrial applications.

DPLL has been extended for automated theorem proving for fragments of first-order logic by way of the DPLL(T) algorithm.

In the 2010-2019 decade, work on improving the algorithm has found better policies for choosing the branching literals and new data structures to make the algorithm faster, especially the part on unit propagation. However, the main improvement has been a more powerful algorithm, Conflict-Driven Clause Learning (CDCL), which is similar to DPLL but after reaching a conflict "learns" the root causes (assignments to variables) of the conflict, and uses this information to perform non-chronological backtracking (aka backjumping) in order to avoid reaching the same conflict again. Most state-of-the-art SAT solvers are based on the CDCL framework as of 2019.

Relation to other notions
Runs of DPLL-based algorithms on unsatisfiable instances correspond to tree resolution refutation proofs.