Sakallah univ of michigan 5 implications in cnf

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Unformatted text preview: a + c + ¬d) (¬b + ¬c + ¬d) b 8 c d b c d c d d d (¬b + ¬c + d) Source: Karem A. Sakallah, Univ. of Michigan 5 Implications in CNF • Implications in a CNF formula are caused by unit clauses – unit clause is a CNF term for which all variables except one are assigned • the value of that clause can be implied immediately Example: (a+^b+c) (a=0)(b=1)⇒(c=1) a • No implications in circuit: c b – All clauses satisfied: 1 1 1 0 x (^a+^b+c)(a+^c)(b+^c) 0 0 1 0 x 0 0 1 0 0 0 0 0 – Not all satisfies (How do we avoid exploring that part of the circuit?) 1 x x x x x x 1 x 6 Example a c b Implications: (^a+^b+ c) 1 1 x 1 1 x x 0 0 (a+^c) 0 1 0 x x 1 x x 1 (b+^c) x x 0 0 x 1 x 0 1 x 1 0 x 1 x 7 Case Splitting with Implications 1 2 3 a (a + b + c) (a + b + ¬c) (¬a + b + ¬c) 4 5 (¬a + c + ¬d) (¬b + ¬c + ¬d) 8 (¬b + ¬c + d) Source: Karem A. Sakallah, Univ. of Michigan b c (a + c + d) (¬a + c + d) 6 7 b a b 4 3 5 7 b c 4 3 5 7 c d c 6 5 8 5 6 8 8 6 6 d 6 6 8 Implementation • • Clauses are stores in array Track sensitivity of clauses for changes: – all literals but one assigned -> implication – all literals but two assigned -> clause is sensitive to a change of either literal – all other clauses are insensitive and do not need to be observed • Learning: – learned implications are added to the CNF formula as additional clauses • limit the size of the clause • limit the “lifetime” of a clause, will be removed after some time • Non-...
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