Formula Satisfiability (SAT)

• Given
  • n boolean variables
  • m boolean operators (and, or, not, if, if and only if)
  • parentheses ()
• Is there a set of the n input values for which the expression will produce a true?
• Much like CIRCUIT_SAT, SAT can be verified in polynomial time.
• Reduce CIRCUIT_SAT ≤_PSAT
  • Brute force approach will blow up.
    • 
    • If the output of this were split three ways, the last expression would be copied three times...
      • So the trick is
        • Number each wire that is not a primary input
        • Represent each gate as an iff.
        • Represent the entire circuit as a and of these iff.
        • 

        • T = w9 ∧ (w1 ⇔ (a ∧ b))
                 ∧ (w2 ⇔ (c ∧ d))
                 ∧ (w3 ⇔ (¬ w1 ∨ ¬ w2))
                 ∧ (w4 ⇔ (¬ w1 ∨ w2))
                 ∧ (w5 ⇔ (w1 ∨ ¬ w2))
                 ∧ (w6 ⇔ (w1 ∨ w2))
                 ∧ (w7 ⇔ (w3 ∧ w4))
                 ∧ (w8 ⇔ (w5 ∧ w6))
                 ∧ (w9 ⇔ (w7 ∨ w8))
          	     

        • Just a reminder, the truth table for ⇔

        • p	q	p⇔q
          t	t	t
          t	f	f
          f	t	f
          f	f	t

        • So w1 ⇔ expr will return true if and only if the wire and the expression produce the same results.
      • This construction can be done in polynomial time (on the number of gates plus inputs)
      • So any circuit can be reduced to a boolean expression in polynomial time.
      • And if we satisfy the expression, we can satisfy the circuit.
      • So SAT ∈ NC

    3-CNF

    • Given a boolean expression which
      • Has n variables
      • Each expression is the or of three variables
      • Expressions are connected with ands
      • (x₁ ∨ ¬ x₂ ∨ x₃) ∧ (x₃ ∨ x₂ ∨ x4) ∧ ... ∧(x_j ∨ ¬ x_k ∨ x_n)
      • Is there an input set that will satisfy this expression?
    • To show 3-CNF ∈ NPC
      • Show SAT ≤_P 3-CNF
      • This is a three step process
        1. Convert an arbitrary expression to a parse tree, with the literals as leaves and expressions as internal nodes.
           • This can be done in polynomial time. (Take Compiles)
           • And we can turn the tree into an and expression like in the last construction.
           • Consider ((x₁ → x₂) ∨ ¬((¬ x₁ ⇔ x₃) ∨ x₄)) ∨ ¬ x₂
           • 
           • Becomes

           •  f = y1 ∧ (y1 ⇔ (y2 ∧ y3))
                     ∧ (y2 ⇔ (y4 ∨ y5))
             	∧ (y3 ⇔ (¬ y2))
             	∧ (y4 ⇔ (x1 → x2))
             	...
             		  

           • Note that each sub expression has at most three literals
      • Convert this expression to an or of ands.
        • Consider: y1 ⇔ (y2 ∧ y3)

        • y1	y2	y3	y1 ⇔ (y2 ∧ y3)
          t	t	t	t
          t	t	f	t
          t	f	t	t
          t	f	f	f
          f	t	t	f
          f	t	f	f
          f	f	t	f
          f	f	f	t

        • From this we can construct the corresponding expression

        • 	       (y1 ∧ y2 ∧ y3) ∨
          	       (y1 ∧ y2 ∧ ¬  y3) ∨
          	       (y1 ∧ ¬  y2 ∧ y3) ∨
          	       (¬ y1 ∧ ¬  y2 ∧ ¬ y3)
          	       

        • Note that each term in the expression can be replace by at most 8 terms.
        • Which is constant!
      • So now we can apply DeMorgan's to the entire expression
        • Replace all ∧ with ∨ and vice versa
        • Not all individual terms.
        • And add in extra factors if a term does not include three factors.
        • But again, this is a polynomial time transformation.
    • And 3-CNF ∈ NPC

    Some More Reductions

    • CLRS continues in this way for quite a while
    • This is a picture from the book.
    • 
    • CLIQUE is a graph problem
      • Given an undirected graph G=(V,E) does the graph have a clique of size k?
      • A clique C ⊂ V such that for each par u∈ C, v∈ C, the edge (u,v)∈ E
      • The construction is cool, they turn a 3-CNF expression into a graph
      • Each term is a node
      • Edges are between terms in different factors which do not negate each other.
      • This moves the problems into graphs.
    • VERTEX-COVER is a graph problem
      • Given a graph G=(V,E) find C ⊂ V such that for all edges (u,v) ∈ E then u ∈ C or v ∈ C.
      • The optimization problem is to find the minimum vertex cover
      • The decision problem is does a cover of size k exist?