The Boolean SATisfiability Problem and the orthogonal group O(n)O(n)

We explore the relations between the Boolean Satisfiability Problem with $n$ literals and the orthogonal group $O(n)$ and show that all solutions lie in the compact and disconnected real manifold of dimension $n (n-1)/2$ of this group.Comment: 11 pages, no figures, 6 reference

Polynomial Time Algorithm for Boolean Satisfiability Problem

This is the latest in a series of articles aimed at exploring the relationship between the complexity classes of P and NP. In the previous papers, we have proved that the sat CNF problem is polynomially reduced to the problem of finding a special covering for a set under the special decomposition of this set and vice versa. That is, these problems are polinomially equivalent. This means that the problem of finding a special covering for a set under a special decomposition of this set, is an NP-complete problem. We also described algorithmic procedures that determine whether there is a special covering for a set under a special decomposition of this set. In this article we prove that all these algorithmic procedures have polynomial time complexity with respect to the length of input data. In addition, we will describe an algorithm that, given any Boolean function in conjunctive normal form (CNF), determines in polynomial time whether this function is satisfiable. We will prove that the time complexity of this algorithm is bounded by the cube of the length of the input data. Also, if the function is not satisfiable, the algorithm deduces this result noting the reason for this result. We have implemented an algorithm in Python, and successfully tested it on Boolean functions represented in CNF with tens of thousands of variables and tens of thousands of clauses

Solving the boolean satisfiability problem using multilevel techniques

There are many complex problems in computer science that occur in knowledge-representation (artificial thinking), artificial learning, Very Large Scale Integration (VLSI) design, security protocols and other areas. These complex problems may be deduced into satisfiability problems where the Boolean Satisfiability Problem (SAT) may be applied. This deduction is made in order to simplify complex problems into a specific propositional logic problem. The SAT problem is the most well-known nondeterministic polynomial time (NP) complete problem in computer science. It is a Boolean expression which is composed of a specific amount of variables (literals), clauses that contain disjunctions of the literals and conjunctions of the clauses. The literals have the logical values TRUE and FALSE, the task is to find a truth assignment that makes the entire expression TRUE. The main goal of the thesis is to solve the SAT problem using a clustering technique - Multilevel - combined first with Tabu Search and combined thereafter with finite Learning Automata. Tabu Search and finite Learning Automata are two very efficient approaches that have been used to solve SAT. Benchmark experiments are conducted in order to disclose whether combining Multilevel with existing solutions to solve SAT will provide better results - than the two mentioned approaches alone - mainly in terms of computational efficienc