DPLL: All-Time Hit SAT-Solver

Enter DPLL! - An Introduction

Ever wondered how computers solve those tricky SAT problems? Meet the DPLL algorithm, a cornerstone in the world of logic puzzles. What does DPLL stand for? Its very creators Davis, Putnam, Logemann, and Loveland! It's a classic in the field of computer science, simplifying complex problems by systematically exploring possible solutions.

What is DPLL??

DPLL algorithm is designed to determine the satisfiability of logical formulas. It is essentially a "Trial and Error" method, but more sophisticated. The algorithm works by systematically exploring possible assignments and simplifying the formula through techniques like unit propagation and pure literal elimination. This process reduces the complexity of the problem, making it easier to find a solution or prove that no solution exists.

So, how does it work?

• 
  Unit Propagation: Unit literals are assigned TRUE (Because if they are not, the clause and in turn the whole sentence would become FALSE). When a literal (a unit) is assigned, the effects fall upon the clauses containing either the literal or its negated version. There are two points to be noted:

1. clause contains literal

A clause is true if any literal is true, even if the other literals do not yet have truth values

Hence, the clause can be removed from the list of clauses that make the problem (a sentence).

For example, the sentence (A ∨ B ∨ C) is true if A is true, regardless of the values of B and C.

2. clause contains -literal

A sentence is false if any clause is false, which occurs when each of its literals is false. This can occur long before the model is complete.

So, the literals that become FALSE because of the new assignment can be removed from corresponding clauses.

• Pure Literal Elimination: Next, it is about literals that only appear in one form (either always positive or always negative) also called "pure literals". These are assigned TRUE (For obvious reasons - A literal that is never negated cannot be the cause of a failure when it is assigned and only aids in satisfying the clause and the sentence).




• Splitting: Finally, the remaining subproblems are tackled by choosing a variable and assigning it a value. If the assigned value leads to FAILURE, the alternate value is assigned after backtracking. To improve the algorithm further, a heuristic can be employed to choose the variable.




• Success or Failure: This is determined by examining the list of clauses. Because of unit propagation done after each assignment, clauses might get removed or shortened.
  
  No clauses in the list would mean that every clause was deemed TRUE and got removed from the list. So, the assignments done so far are returned, indicating success.
  
  An empty clause in the list suggests that every literal belonging to the clause was deemed FALSE and was removed from the clause. This means that the whole sentence is FALSE and FAILURE (None) is returned.

Pseudocode

Artificial Intelligence - A Modern Approach

(Third Edition) by Stuart J. Russell and Peter Norvig

(Sure, you did your research, but hey, there's always more to discover!)

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