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lect20-SAT,CookThmpnp2

lect20-SAT,CookThmpnp2 - CooksTheorem:AnNP CompleteProblem...

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1 The Satisfiability Problem Cook’s Theorem: An NP- Complete Problem Restricted SAT: CSAT, 3SAT

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2 Boolean Expressions Boolean, or propositional-logic  expressions are built from variables and  constants using the operators AND,  OR, and NOT. Constants are true and false, represented  by 1 and 0, respectively. We’ll use concatenation (juxtaposition) for  AND, + for OR, - for NOT,  unlike the text .
3 Example : Boolean expression (x+y)(-x + -y) is true only when variables  x and y have opposite truth values. Note : parentheses can be used at will,  and are needed to modify the  precedence order NOT (highest), AND,  OR.

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4 The Satisfiability Problem ( SAT  ) Study of boolean functions generally is  concerned with the set of  truth  assignments   (assignments of 0 or 1 to  each of the variables) that make the  function true. NP-completeness needs only a simpler  question (SAT): does there exist a truth  assignment making the function true?
5 Example : SAT (x+y)(-x + -y) is satisfiable. There are, in fact, two satisfying truth  assignments: 1. x=0; y=1. 2. x=1; y=0. x(-x) is not satisfiable.

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6 SAT as a Language/Problem An instance of SAT is a boolean  function. Must be coded in a finite alphabet. Use special symbols (, ), +, - as  themselves. Represent the i-th variable by symbol x  followed by integer i in binary.
7 Example : Encoding for SAT (x+y)(-x + -y) would be encoded by the  string  (x1+x10)(-x1+-x10)

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8 SAT is in  NP There is a multitape NTM that can decide if a  Boolean formula of length n is satisfiable. The NTM takes O(n 2 ) time along any path. Use nondeterminism to guess a truth  assignment on a second tape. Replace all variables by guessed truth  values. Evaluate the formula for this assignment. Accept if true.
9 Cook’s  Theorem SAT is NP-complete. Really a stronger result: formulas may be  in conjunctive normal form (CSAT) – later. To prove, we must show how to  construct a polytime reduction from  each language L in  NP  to SAT. Start by assuming the most resticted  possible form of NTM for L (next slide).

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10 Assumptions About NTM for L 1. One tape only. 2. Head never moves left of the initial  position. 3. States and tape symbols are disjoint. Key Points : States can be named  arbitrarily, and the constructions  many-tapes-to-one  and  two-way- infinite-tape-to-one  at most square the  time.
11 More About the NTM M for L Let p(n) be a polynomial time bound for  M.

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