foreword
The SAT problem is an important computer science and artificial intelligence problem that involves, given a set of Boolean variables and a set of clauses, determining whether there exists a variable assignment that makes the entire conjunction paradigm true. This problem has a wide range of uses in practical applications, including hardware design, security protocol verification, etc.
What do you think of CNF
The cnf file essentially provides a rule. We need to construct a CNF formula according to the rule, and judge whether there is a set of variable assignments that can satisfy the CNF formula. So what is this rule, that is,
each cnf sub-formula must be TRUE
. Let's take an example to understand
p cnf 4 3
1 -2 -3 0
-4 2 1 0
1 2 3 4 0
Here the formula consists of 3 clauses. Each clause ends with a 0, indicating the end of the clause. Each number represents a literal, positive numbers indicate the positive of the variable, and negative numbers indicate the negation of the variable.
The parsed CNF formula is as follows:
The first clause: 1 -2 -3 0
This clause can be interpreted as: x1 ∨ ¬x2 ∨ ¬x3
The second clause: -4 2 1 0
This clause can be interpreted as: ¬x4 ∨ x2 ∨ x1
The third clause: 1 2 3 4 0
This clause can be interpreted as: x1 ∨ x2 ∨ x3 ∨ x4
Combined, the entire CNF formula is:
F = (x1 ∨ ¬x2 ∨ ¬x3) ∧ (¬x4 ∨ x2 ∨ x1) ∧ (x1 ∨ x2 ∨ x3 ∨ x4)
This is a conjunctive normal form consisting of three clauses, each of which requires at least one literal to be true for the entire conjunctive normal form to be true . This is a typical SAT problem that requires determining whether there exists a set of variable assignments such that this conjunctive normal form is true. You can choose x1 = 1, x2 = 0, x3 = 0, x4 = 1 as a set of solutions.
How to solve the cnf formula
Determining whether a given CNF formula is satisfiable involves using a SAT solver to try to find variable assignments that satisfy the condition. A SAT solver is a tool specifically designed to solve SAT problems by searching for different combinations of variable assignments to determine if there exists a solution that makes the entire conjunctive normal form true.
Following are the general steps to use a SAT solver to solve a given CNF formulation:
Convert to CNF format: If you already have a problem description in CNF format, then you can skip this step. Otherwise, convert the problem description into CNF format, ensuring that each logical expression is represented as a conjunction of clauses.
Call SAT Solver: Use an appropriate SAT solver to solve the problem. There are many open source and commercial SAT solvers to choose from, such as MiniSat, Z3, CryptoMiniSat, etc.
Provide CNF formulas: Enter CNF formulas into the solver in a specific format. This usually involves passing variables, clauses, and information from the CNF file to the solver.
Waiting for solution results: The SAT solver will try different combinations of variable assignments to find a solution that satisfies the entire CNF formula. It either finds a satisfying solution (the assignment makes the entire formula true), or it determines that the formula is unsatisfiable.
Interpreting results: If the solver finds a satisfying solution, it returns the corresponding variable assignments. You can verify the satisfiability of formulas based on these assignments. If the solver determines that the formula is unsatisfiable, then this means that there is no set of variable assignments that can make the entire formula true.
Note that the SAT problem is NP-complete, which means that in the general case, the solution time of the problem grows exponentially as the number of variables increases. Therefore, for large-scale problems, efficient heuristics or optimization techniques may need to be considered to reduce solution time.
In summary, using a SAT solver to solve a CNF formulation involves converting the problem description into CNF format, invoking the solver to solve it, and interpreting the solution results to determine the satisfiability of the formulation.