Question

In a 2cnf-formula, each clause is an OR of at most two terms, and the clauses are combined with ANDs. For example, ( x₁​​ V x​₂ ) ∧ (x₃​ ∨ x₄​ ) ∧ (x₁​ ∨ x₃​ ) is a 2 cnf-formula. Define the language 2SAT of satisfiable 2 cnf-formulas: 2SAT={<ϕ>∣ϕ is a satisfiable 2 cnf-formula }. Show that 2SAT is in P.

Answer 1

The language 2SAT, representing satisfiable 2 cnf-formulas, is in P because the satisfiability of such formulas can be determined in polynomial time using graph theory methods, such as constructing an implication graph and checking for strongly connected components.

Step-by-step explanation:

The question asks to demonstrate that the language 2SAT of satisfiable 2 cnf-formulas is in P, meaning that it can be solved in polynomial time, implying it's not NP-complete like 3SAT. A 2 cnf-formula is a conjunctive normal form with at most two literals per clause. The problem can be solved using graph theory techniques, specifically by constructing an implication graph from the formula where each literal and its negation are vertices, and edges represent the implications of satisfying one literal on the other in a clause. If the graph contains a strongly connected component (SCC) that contains both a literal and its negation, the formula is unsatisfiable; otherwise, it is satisfiable. This SCC check can be done using Tarjan's algorithm, which runs in linear time relative to the number of literals and clauses, thus confirming that 2SAT is in P.

Answer 2

2SAT belongs to the complexity class P because there are known algorithms like resolution and graph-based techniques which can determine the satisfiability of 2-CNF formulas in polynomial time.

Step-by-step explanation:

The question asks to show that 2SAT, the problem of determining whether a 2-CNF formula is satisfiable, is in the complexity class P, which stands for polynomial time. To illustrate this, it's important to understand that a 2-CNF formula is a conjunction (AND) of clauses, where each clause is a disjunction (OR) of at most two literals.

The process to determine satisfiability for such formulas can be accomplished in polynomial time using an algorithm like the resolution method or graph-based methods such as implication graphs and strongly connected components (SCCs).

For example, an implication graph method works by constructing a directed graph of implications and then using Tarjan's algorithm to find SCCs. If a variable and its negation belong to the same SCC, the formula is unsatisfiable; otherwise, it is satisfiable. This process runs in polynomial time, confirming that 2SAT is, indeed, in P.