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--- aa:intro:complexity_theory [2016/08/03 11:29]
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+++ aa:intro:complexity_theory [2016/08/03 11:34] (current)
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 ==== 1.6.1 Remark (Practical applications of reductions) ====
-//As illustrated before, reductions of the type $math[\leq_p] are a theoretical tool which is useful for providing $math[NP]-hardness. Reductions also have practical applications. For instance, most $math[NP]-complete problems are solved by employing $math[SAT] solvers, which, as discussed in the former chapters, may be quite fast in general case. Thus, a specific problem instance is cast (via an appropriate transformation) into a formula $math[\varphi], such that $math[varphi] is satisfiable iff the answer to the instance is// yes.
+//As illustrated before, reductions of the type $math[\leq_p] are a theoretical tool which is useful for providing $math[NP]-hardness. Reductions also have practical applications. For instance, most $math[NP]-complete problems are solved by employing $math[SAT] solvers, which, as discussed in the former chapters, may be quite fast in general case. Thus, a specific problem instance is cast (via an appropriate transformation) into a formula $math[\varphi], such that $math[\varphi] is satisfiable iff the answer to the instance is// yes.
+==== $math[SAT]. The first $math[NP]-complete problem. ====
+We observe that the "//hen and eggs//" issue still holds in our scenario. To apply our technique, we need an initial $math[NP]-hard problem in the first place. This problem is provided by Cook's Theorem, which proves that $math[SAT] is $math[NP]-complete. The technique for the proof relies on building, for each $math[NTM \mbox{ } M] (and hence, for each problem in $math[NP]), a formula $math[\varphi_M] such that it is satisfiable iff there exists a sequence in the computation tree leading to **success**.