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aa:ct:models [2019/10/07 21:02] costin.lupu.almighty |
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| - | Let $math[M] be a ''TM'' and $math[w = c_1c_2 ... c_n] be a word which is built from the alphabet of $math[M]. We build the Universal Turing Machine $math[U], as follows: | + | Let $math[M] be a ''TM'' and $math[w = c_1c_2 ... c_n] be a word which is built from the alphabet of $math[M]. We build the Universal Turing Machine $math[U] as follows: |
| - | * The input of $math[U] is $math[enc(M)\#enc(s_0)\#c_1\#c_2 ... c_n]. Note that $math[enc(s_0)] encodes the initial state of $math[M] while $math[c_1] is the fi rst symbol from $math[w]. The portion of the tape $math[enc(s_0)\#c_1\#c_2 ... c_n] will be used to mark the current con guration of $math[M], namely the current state of $math[M] (initially $math[s_0]), the contents of $math[M]'s tape, and $math[M]'s current head position. More generally, this portion of the tape is of the form $math[enc(s_i)\#u\#v], with $math[u, v \in \Sigma_b^*] and $math[s_i] being the current state of $math[M]. The last symbol of $math[u] marks the current symbol, while $math[v] is the word which is to the left of the head. Initially, the current symbol is the first one, namely $math[c_1]. | + | * The input of $math[U] is $math[enc(M)\#enc(s_0)\#c_1\#c_2 ... c_n]. Note that $math[enc(s_0)] encodes the initial state of $math[M] while $math[c_1] is the first symbol from $math[w]. The portion of the tape $math[enc(s_0)\#c_1\#c_2 ... c_n] will be used to mark the current configuration of $math[M], namely the current state of $math[M] (initially $math[s_0]), the contents of $math[M]'s tape, and $math[M]'s current head position. More generally, this portion of the tape is of the form $math[enc(s_i)\#u\#v], with $math[u, v \in \Sigma_b^*] and $math[s_i] being the current state of $math[M]. The last symbol of $math[u] marks the current symbol, while $math[v] is the word which is to the left of the head. Initially, the current symbol is the first one, namely $math[c_1]. |
| - | * $math[U] will scan the initial state of $math[M], then it will move on the initial symbol from $math[w], and finally will move on the portion of $math[enc(M)] were transitions are encoded. Once a valid transition is found, it will execute it: | + | * $math[U] will scan the initial state of $math[M], then it will move on the initial symbol from $math[w] and finally will move on the portion of $math[enc(M)] were transitions are encoded. Once a valid transition is found, it will execute it: |
| - $math[U] will change the initial state to the current one, according to the transition; | - $math[U] will change the initial state to the current one, according to the transition; | ||
| - $math[U] will change the original symbol in $math[w] according to the transition; | - $math[U] will change the original symbol in $math[w] according to the transition; | ||
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| In what follows, we examine the **Church-Turing Thesis** which is an **argument** for using the Turing Machine as a computational model: | In what follows, we examine the **Church-Turing Thesis** which is an **argument** for using the Turing Machine as a computational model: | ||
| - | //Any problem which can be solved byh the Turing Machine is //"universally solvable"//.// | + | //Any problem which can be solved by the Turing Machine is //"universally solvable"//.// |
| The term //"universally solvable"// cannot be given a precise mathematical definition. We only know solvability w.r.t. a given model of computation (abstract or concrete). | The term //"universally solvable"// cannot be given a precise mathematical definition. We only know solvability w.r.t. a given model of computation (abstract or concrete). | ||