hwsol2.2 - Fedor Korsakov CSci 5403 Spring 2010 Revised...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Fedor Korsakov CSci 5403, Spring 2010 Revised Solution 2.2 (a) Define the language 2sat = { φ | φ is a satisfiable 2cnf formula } . Prove that 2sat is NL-complete. Hint : A 2cnf clause ( a ∨ b ) is equivalent to a → b , and a 2cnf formula is unsatisfiable iff it has a chain of inferences that (by transitivity) can be simplified to x → x → x . Solution: 2 SAT ∈ NL . With the knowledge that 2cnf clauses can be expressed through implication relationships, we are able to construct a directed graph, such that each variable x corresponds to vertices x and ¬ x , and the edges correspond to implica- tion relationships. Immerman-Szelepcs´ enyi Theorem demonstrates that NOPATH ∈ NL . This can be used for a logspace reduction of 2 SAT to NOPATH , since using NOPATH twice can demonstrate the nonexistence of a path from x to ¬ x and from ¬ x to x . If such chain is found, reject. 2 SAT is NL-hard. NOPATH can be logspace reduced to 2 SAT . The reduction is as follows: every edge ( a,b ) becomes 2 SAT clause ¬ a ∨ b . Additionally, we need the clauses s ∨ s and ¬ t ∨ ¬ t for starting and ending vertices, respectively. The resulting boolean formula can be thought of as containing clauses that give true values to reachable vertices. ¬ s ∨ t sets t to true, and prevents ¬ t ∨ ¬ t from being satisfied, therefore the entire formula does not belong to 2 SAT ....
View Full Document

This note was uploaded on 10/21/2011 for the course CSCI 5403 taught by Professor Sturtivant,c during the Spring '08 term at Minnesota.

Page1 / 3

hwsol2.2 - Fedor Korsakov CSci 5403 Spring 2010 Revised...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online