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aa:decidability [2019/10/14 22:02] costin.lupu.almighty |
aa:decidability [2019/10/15 06:53] (current) costin.lupu.almighty |
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| $def[Decision, acceptance] | $def[Decision, acceptance] | ||
| Let $math[M] be a Turing Machine and $math[f \in Hom(\mathbb{N}, \{0, 1\})]. | Let $math[M] be a Turing Machine and $math[f \in Hom(\mathbb{N}, \{0, 1\})]. | ||
| - | * $math[M] **decides** $math[f], iff for all $math[n \in \mathbb{N}]: $math[M(w) = 1] whenever $math[f(n^w) = 1] and $math[M(w) = 0] whenever $math[f(n^w) = 0]. | + | * $math[M] **decides** $math[f] iff for all $math[n \in \mathbb{N}]: $math[M(w) = 1] whenever $math[f(n^w) = 1] and $math[M(w) = 0] whenever $math[f(n^w) = 0]. |
| - | * $math[M] **accepts** $math[f] iff for all $math[n \in \mathbb{N}]: $math[M(w) = 1] iff $math[f(n^w) = 1], and $math[M(w) = \perp] iff $math[f(n)=0]. | + | * $math[M] **accepts** $math[f] iff for all $math[n \in \mathbb{N}]: $math[M(w) = 1] iff $math[f(n^w) = 1], and $math[M(w) = \perp] iff $math[f(n)=0]. |
| $end | $end | ||
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| Let $math[M] be a Turing Machine and $math[w \in \Sigma^*] be a word. | Let $math[M] be a Turing Machine and $math[w \in \Sigma^*] be a word. | ||
| - | * We build the Turing Machine $math[\Pi_{M,w}(\omega) = ]//"Replace $math[\omega] by $math[w] and then simulate $math[M(w)]"//. | + | * We build the Turing Machine $math[\Pi_{M,w}(\omega) = ]//"Replace $math[\omega] by $math[w] and then simulate $math[M(w)]"//. |
| * The construction of $math[\Pi_{M,w}] depends on $math[M] and $math[w], as suggested by the indices. | * The construction of $math[\Pi_{M,w}] depends on $math[M] and $math[w], as suggested by the indices. | ||
| * Note that building $math[\Pi_{M,w}] can be easily derived from the Universal Turing Machine. | * Note that building $math[\Pi_{M,w}] can be easily derived from the Universal Turing Machine. | ||
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| * run $math[M_{all}] with input $math[enc(\Pi_{M,w})], and return the output. | * run $math[M_{all}] with input $math[enc(\Pi_{M,w})], and return the output. | ||
| * it is easy to see that: | * it is easy to see that: | ||
| - | * $math[(\forall \omega \in \Sigma^* : \Pi_{M,w}(\omega) \mbox{ halts})] iff $math[M(w)] halts. | + | * $math[(\forall \omega \in \Sigma^* : \Pi_{M,w}(\omega) \mbox{ halts})] iff $math[M(w)] halts. |
| === Step 2 (contraposition) === | === Step 2 (contraposition) === | ||
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| The relationship between $math[f_{all}] and $math[f_h] is called **Turing-reducibility**, and is written: | The relationship between $math[f_{all}] and $math[f_h] is called **Turing-reducibility**, and is written: | ||
| - | $math[f_h \leq_T f_{all}] | + | $math[f_h \leq_T f_{all}] |
| This relation indicates a strong bond between the two problems which can be interpreted as //$math[f_{all}] is at least as hard as $math[f_{h}]//, in the sense that if we can decide $math[f_{all}] using a Turing Machine, then it is trivial to decide $math[f_h]. | This relation indicates a strong bond between the two problems which can be interpreted as //$math[f_{all}] is at least as hard as $math[f_{h}]//, in the sense that if we can decide $math[f_{all}] using a Turing Machine, then it is trivial to decide $math[f_h]. | ||
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| $def[Turing-reducibility] | $def[Turing-reducibility] | ||
| - | //Let $math[f_A, f_B \in Hom(\mathbb{N}, \{0, 1\})]. We say $math[f_A] is// Turing-reducible //to $math[f_B], and write $math[f_A \leq_T f_B] iff there exists a// decidable transformation// $math[T \in Hom(\mathbb{N},\mathbb{N})] such that $math[f_A(n) = 1] iff $math[f_B(T(n)) = 1].// | + | //Let $math[f_A, f_B \in Hom(\mathbb{N}, \{0, 1\})]. We say $math[f_A] is// Turing-reducible //to $math[f_B], and write $math[f_A \leq_T f_B] iff there exists a// decidable transformation// $math[T \in Hom(\mathbb{N},\mathbb{N})] such that $math[f_A(n) = 1] iff $math[f_B(T(n)) = 1].// |
| $end | $end | ||