P versus NP is considered as one of the most important open problems in computer science. This consists knowing answer following question: Is equal to NP? It was essentially mentioned 1955 from a letter written by John Nash United States National Security Agency. However, precise statement problem introduced independently Stephen Cook and Leonid Levin. Since that date, all efforts find proof fo...

In English the NP1 V NP2 construction typically involves arguments that are construed as Agent and Patient, or Subject Object. It is associated with notion of transitivity analyzed accordingly, even when it exhibits only syntactic properties but not its semantic characteristics. This phenomenon well-known has been accounted for by linguists (Lakoff 1977; Hopper Thompson 1980, among others) a re...

It is a well-known result of Fagin that the complexity class NP coincides with the class of problems expressible in existential second-order logic (1 ;). Monadic NP is the class of problems expressible in monadic Z;, i.e., 1: with the restriction that the second-order quantifiers range only over sets (as opposed to ranging over, say, binary relations). We prove that connectivity of finite graph...

I assume that most students have encountered Turing machines before. (Students who have not may want to look at Sipser’s book [3].) A Turing machine is defined by an integer k ≥ 1, a finite set of states Q, an alphabet Γ, and a transition function δ : Q×Γk → Q×Γk−1×{L, S,R}k where: • k is the number of (infinite, one-dimensional) tapes used by the machine. In the general case we have k ≥ 3 and ...

• INDEPENDENT -SET = {[G, k] : G is an undirected graph with an independent set of vertices I ⊆ V (G) of size ≥ k} where a subset I is independent iff it contains no edges of G. – Guess a set I of size k. Verify that it contains no edges of G. • CLIQUE = {[G, k] : G is an undirected graph with an independent set of vertices C ⊆ V (G) of size ≥ k} where a subset C is a clique iff every edge on t...