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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 NLcomplete. 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. ImmermanSzelepcs´ 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 NLhard. 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 ....
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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.
 Spring '08
 Sturtivant,C

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