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--- aa:pcp [2019/10/17 08:07]
pdmatei
+++ aa:pcp [2021/01/17 22:48] (current)
cata.chiru
@@ Line 24: / Line 24: @@
   * for this proof, we consider that Turing Machines have only two final states $math[F=\{s_{yes},s_{no}\}], which model the output of $math[0/1]. It is straightforward how one can take an arbitrary Turing Machine an turn it into a //yes/no//-Turing Machine.
   * We require that the execution of $math[M] on $math[w] never moves the head to the left of the first symbol of the word. Making sure this happens is a little more technically involved, however algorithmically possible. We shall skip these details.
-  * We require that the word $w\neq \epsilon$, hence we do not allow simulations of the empty word. To accommodate for this, we can easily add a new symbol to the alphabet, which is used specifically for encoding the empty word.
+  * We require that the word $ w \neq \epsilon $, hence we do not allow simulations of the empty word. To accommodate for this, we can easily add a new symbol to the alphabet, which is used specifically for encoding the empty word.
@@ Line 151: / Line 151: @@
 Direction "$math[\Leftarrow]". Suppose we have a matching in our constructed PCP instance. Since $math[*] can only appear between other symbols from $math[\Sigma], it follows that any matching can only start with the pair $math[\left(\begin{array}{l} \lhd \alpha_1 \\ \lhd \beta_1 \rhd \end{array}\right)], and must continue with pairs $math[\left(\begin{array}{l} \lhd \alpha_i \\ \beta_i \rhd \end{array}\right)], from which we can immediately construct a matching in FTPCP.
-===== PCP is undecidable (old) =====
-We prove PCP is undecidable via a reduction: PCP $math[\leq_T] FTPCP.
-Let $math[ \left\{\left(\begin{array}{l} \alpha_i \\ \beta_i \end{array}\right)\right\}_{ 1 \leq i \leq n}] be an instance of PCP. We build an instance of FTPCP as follows.
-First, consider an arbitrary word $math[w=c_1\ldots c_k] over alphabet $math[\Sigma]. We define:
-  * $math[\lhd w = *c_1*c_2* \ldots *c_k]
-  * $math[w \rhd = c_1*c_2* \ldots *c_k*]
-  * $math[\lhd w \rhd = *c_1*c_2* \ldots *c_k*]
-where the symbol $math[*] does not occur in $math[\Sigma].
-Using our word constructions, we can construct the following instance of FTPCP:
-$math[ \left\{\left(\begin{array}{l} \lhd \alpha_i \\ \lhd \beta_i \rhd \end{array}\right)\right\} \cup \left\{\left(\begin{array}{l} \lhd \alpha_i \\ \beta_i \rhd \end{array}\right)\right\}_{ 1 \leq i \leq n} \cup \left\{\left(\begin{array}{l} \lhd @ \\ @ \end{array}\right)\right\}]
-where $math[*] and $math[@] are symbols outside of the alphabet of the PCP instance. The intuition of the construction is as follows:
-  * since $math[*] cannot appear in any word, any matching must start with the first pair (all others are incompatible)
-  * any matching attempt will leave a trailing $math[*], which can be solved by adding the final tile.
-It is straightforward that our instance of FTPCP is //matchable// iff PCP is //matchable//.