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aa:classes [2016/11/21 09:48] ccalin [Completeness with respect to a class] |
aa:classes [2016/11/25 16:15] (current) pdmatei |
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| $proof | $proof | ||
| - | Suppose $math[f \in P]. Hence there exists a TM $math[M_f] which decides $math[f] in polynomial time $math[S_f(n)]. Since $math[f \leq_p f'], there exists a TM $math[T] which transforms the input of $math[f] into one of $math[f'], such that: $math[\forall w: f(w) = 1 \iff f'(T(w)) = 1]. Furthermore, the execution time of $math[T] is a polynomial function $math[S_T(n)]. We construct the Turing Machine $math[M^*] as follows: | + | Suppose $math[f \in P]. Hence there exists a TM $math[M_f] which decides $math[f] in polynomial time $math[S_f(n)]. Since $math[f' \leq_p f], there exists a TM $math[T] which transforms the input of $math[f'] into one of $math[f], such that: $math[\forall w: f'(w) = 1 \iff f(T(w)) = 1]. Furthermore, the execution time of $math[T] is a polynomial function $math[S_T(n)]. We construct the Turing Machine $math[M^*] as follows: |
| * read the input $math[w] | * read the input $math[w] | ||
| * compute $math[w' = T(w)] | * compute $math[w' = T(w)] | ||
| Line 92: | Line 92: | ||
| Note that the above proposition establishes an implication ($math[A \implies B]). We cannot practically benefit from this proposition unless we know $math[A] is true (which, as already said, has not been proved for any problem). | Note that the above proposition establishes an implication ($math[A \implies B]). We cannot practically benefit from this proposition unless we know $math[A] is true (which, as already said, has not been proved for any problem). | ||
| - | However, this proposition is still true, because it validates our intuition regarding the concept of **polynomial reduction** - it clearly reflects our intuition regarding **hardness**. | + | However, this proposition is still true, because it validates our intuition regarding the concept of **polynomial reduction** - it clearly |
| + | reflects our intuition regarding **hardness**. | ||
| ==== Hardness with respect to a class ==== | ==== Hardness with respect to a class ==== | ||
| Line 127: | Line 128: | ||
| $end | $end | ||
| - | ==== Complet | + | ==== Completeness with respect to a class ==== |
| - | eness with respect to a class ==== | + | |
| $def[Completeness w.r.t. a class] | $def[Completeness w.r.t. a class] | ||