Reducing Pcp (Post's correspondence problem) to mPcp












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Recently I have been studying Post's correspondence problem ($Pcp$), and I have stumbled upon a problem where I need to find a reduction from $Pcp$ to a modified version, $mPcp$. This modified version requires any matching sequence of "dominoes" $p_{i_1}, p_{i_2}, ..., p_{i_m}$ to start with the first "domino", $p_1$.



I am very well aware of the fact that there exists a match for
an instance $I=langle p_1, ..., p_nrangle$ of $Pcp$ if and only if there exists a match for one of the instances $I_1, I_2, ..., I_n$ of $mPcp$ where $I_j=langle p_j, p_{j+1}, ..., p_n, p_1, ..., p_{j-1}rangle$ for $j=1, ..., n$. In other words, if and only if $I$ is a yes-instance for $Pcp$, there is a way to rotate the list of "dominoes" in a cyclic manner such that the starting piece in the found match for $Pcp$ is put in the first position.



However, it is not clear to me how this observation could lead to a mapping from an instance $I$ of $Pcp$ to the "correct" instance $I_j$ of $mPcp$ that can be carried out in a finite amount of steps for any $I$, thus serving as a suitable reduction.



Could anybody point me in the right direction? Thank you very much.










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$endgroup$












  • $begingroup$
    Given a sequence of dominoes $p_1,...,p_n$, you can "rotate" the sequence by removing the first piece and appending it to the end of the sequence. With n pieces, you have n sequences each beginning with $p_i$ for $i = 1,...,n$. You then call mPcp once for each of the n sequences. Pcp will be a "Yes" instance iff mPcp is a "Yes" instance for at least one of the n instances.
    $endgroup$
    – user137481
    May 9 '16 at 17:42












  • $begingroup$
    The problem is that calling $mPcp$ for all $i$ could result in an infinite loop if $I$ is a no instance of $Pcp$, thus not satisfying the requirement that the reduction be carried out in a finite number of steps.
    $endgroup$
    – Dyon J Don Kiwi van Vreumingen
    May 10 '16 at 9:12










  • $begingroup$
    Pcp is undecidable. I'm assuming that you are trying to prove that mPcp is undecidable. To start, we assume that mPcp is decidable and let M be the decider for mPcp. A decider always halts. So, there is no "infinite loop" even for 'No" instances. However, if mPcp is decidable, M could also be used to decide Pcp which gives us a contradiction. Hence, mPcp must also be undecidable.
    $endgroup$
    – user137481
    May 10 '16 at 15:16
















0












$begingroup$


Recently I have been studying Post's correspondence problem ($Pcp$), and I have stumbled upon a problem where I need to find a reduction from $Pcp$ to a modified version, $mPcp$. This modified version requires any matching sequence of "dominoes" $p_{i_1}, p_{i_2}, ..., p_{i_m}$ to start with the first "domino", $p_1$.



I am very well aware of the fact that there exists a match for
an instance $I=langle p_1, ..., p_nrangle$ of $Pcp$ if and only if there exists a match for one of the instances $I_1, I_2, ..., I_n$ of $mPcp$ where $I_j=langle p_j, p_{j+1}, ..., p_n, p_1, ..., p_{j-1}rangle$ for $j=1, ..., n$. In other words, if and only if $I$ is a yes-instance for $Pcp$, there is a way to rotate the list of "dominoes" in a cyclic manner such that the starting piece in the found match for $Pcp$ is put in the first position.



However, it is not clear to me how this observation could lead to a mapping from an instance $I$ of $Pcp$ to the "correct" instance $I_j$ of $mPcp$ that can be carried out in a finite amount of steps for any $I$, thus serving as a suitable reduction.



Could anybody point me in the right direction? Thank you very much.










share|cite|improve this question











$endgroup$












  • $begingroup$
    Given a sequence of dominoes $p_1,...,p_n$, you can "rotate" the sequence by removing the first piece and appending it to the end of the sequence. With n pieces, you have n sequences each beginning with $p_i$ for $i = 1,...,n$. You then call mPcp once for each of the n sequences. Pcp will be a "Yes" instance iff mPcp is a "Yes" instance for at least one of the n instances.
    $endgroup$
    – user137481
    May 9 '16 at 17:42












  • $begingroup$
    The problem is that calling $mPcp$ for all $i$ could result in an infinite loop if $I$ is a no instance of $Pcp$, thus not satisfying the requirement that the reduction be carried out in a finite number of steps.
    $endgroup$
    – Dyon J Don Kiwi van Vreumingen
    May 10 '16 at 9:12










  • $begingroup$
    Pcp is undecidable. I'm assuming that you are trying to prove that mPcp is undecidable. To start, we assume that mPcp is decidable and let M be the decider for mPcp. A decider always halts. So, there is no "infinite loop" even for 'No" instances. However, if mPcp is decidable, M could also be used to decide Pcp which gives us a contradiction. Hence, mPcp must also be undecidable.
    $endgroup$
    – user137481
    May 10 '16 at 15:16














0












0








0





$begingroup$


Recently I have been studying Post's correspondence problem ($Pcp$), and I have stumbled upon a problem where I need to find a reduction from $Pcp$ to a modified version, $mPcp$. This modified version requires any matching sequence of "dominoes" $p_{i_1}, p_{i_2}, ..., p_{i_m}$ to start with the first "domino", $p_1$.



I am very well aware of the fact that there exists a match for
an instance $I=langle p_1, ..., p_nrangle$ of $Pcp$ if and only if there exists a match for one of the instances $I_1, I_2, ..., I_n$ of $mPcp$ where $I_j=langle p_j, p_{j+1}, ..., p_n, p_1, ..., p_{j-1}rangle$ for $j=1, ..., n$. In other words, if and only if $I$ is a yes-instance for $Pcp$, there is a way to rotate the list of "dominoes" in a cyclic manner such that the starting piece in the found match for $Pcp$ is put in the first position.



However, it is not clear to me how this observation could lead to a mapping from an instance $I$ of $Pcp$ to the "correct" instance $I_j$ of $mPcp$ that can be carried out in a finite amount of steps for any $I$, thus serving as a suitable reduction.



Could anybody point me in the right direction? Thank you very much.










share|cite|improve this question











$endgroup$




Recently I have been studying Post's correspondence problem ($Pcp$), and I have stumbled upon a problem where I need to find a reduction from $Pcp$ to a modified version, $mPcp$. This modified version requires any matching sequence of "dominoes" $p_{i_1}, p_{i_2}, ..., p_{i_m}$ to start with the first "domino", $p_1$.



I am very well aware of the fact that there exists a match for
an instance $I=langle p_1, ..., p_nrangle$ of $Pcp$ if and only if there exists a match for one of the instances $I_1, I_2, ..., I_n$ of $mPcp$ where $I_j=langle p_j, p_{j+1}, ..., p_n, p_1, ..., p_{j-1}rangle$ for $j=1, ..., n$. In other words, if and only if $I$ is a yes-instance for $Pcp$, there is a way to rotate the list of "dominoes" in a cyclic manner such that the starting piece in the found match for $Pcp$ is put in the first position.



However, it is not clear to me how this observation could lead to a mapping from an instance $I$ of $Pcp$ to the "correct" instance $I_j$ of $mPcp$ that can be carried out in a finite amount of steps for any $I$, thus serving as a suitable reduction.



Could anybody point me in the right direction? Thank you very much.







computability






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edited May 9 '16 at 15:04







Dyon J Don Kiwi van Vreumingen

















asked May 9 '16 at 10:48









Dyon J Don Kiwi van VreumingenDyon J Don Kiwi van Vreumingen

1012




1012












  • $begingroup$
    Given a sequence of dominoes $p_1,...,p_n$, you can "rotate" the sequence by removing the first piece and appending it to the end of the sequence. With n pieces, you have n sequences each beginning with $p_i$ for $i = 1,...,n$. You then call mPcp once for each of the n sequences. Pcp will be a "Yes" instance iff mPcp is a "Yes" instance for at least one of the n instances.
    $endgroup$
    – user137481
    May 9 '16 at 17:42












  • $begingroup$
    The problem is that calling $mPcp$ for all $i$ could result in an infinite loop if $I$ is a no instance of $Pcp$, thus not satisfying the requirement that the reduction be carried out in a finite number of steps.
    $endgroup$
    – Dyon J Don Kiwi van Vreumingen
    May 10 '16 at 9:12










  • $begingroup$
    Pcp is undecidable. I'm assuming that you are trying to prove that mPcp is undecidable. To start, we assume that mPcp is decidable and let M be the decider for mPcp. A decider always halts. So, there is no "infinite loop" even for 'No" instances. However, if mPcp is decidable, M could also be used to decide Pcp which gives us a contradiction. Hence, mPcp must also be undecidable.
    $endgroup$
    – user137481
    May 10 '16 at 15:16


















  • $begingroup$
    Given a sequence of dominoes $p_1,...,p_n$, you can "rotate" the sequence by removing the first piece and appending it to the end of the sequence. With n pieces, you have n sequences each beginning with $p_i$ for $i = 1,...,n$. You then call mPcp once for each of the n sequences. Pcp will be a "Yes" instance iff mPcp is a "Yes" instance for at least one of the n instances.
    $endgroup$
    – user137481
    May 9 '16 at 17:42












  • $begingroup$
    The problem is that calling $mPcp$ for all $i$ could result in an infinite loop if $I$ is a no instance of $Pcp$, thus not satisfying the requirement that the reduction be carried out in a finite number of steps.
    $endgroup$
    – Dyon J Don Kiwi van Vreumingen
    May 10 '16 at 9:12










  • $begingroup$
    Pcp is undecidable. I'm assuming that you are trying to prove that mPcp is undecidable. To start, we assume that mPcp is decidable and let M be the decider for mPcp. A decider always halts. So, there is no "infinite loop" even for 'No" instances. However, if mPcp is decidable, M could also be used to decide Pcp which gives us a contradiction. Hence, mPcp must also be undecidable.
    $endgroup$
    – user137481
    May 10 '16 at 15:16
















$begingroup$
Given a sequence of dominoes $p_1,...,p_n$, you can "rotate" the sequence by removing the first piece and appending it to the end of the sequence. With n pieces, you have n sequences each beginning with $p_i$ for $i = 1,...,n$. You then call mPcp once for each of the n sequences. Pcp will be a "Yes" instance iff mPcp is a "Yes" instance for at least one of the n instances.
$endgroup$
– user137481
May 9 '16 at 17:42






$begingroup$
Given a sequence of dominoes $p_1,...,p_n$, you can "rotate" the sequence by removing the first piece and appending it to the end of the sequence. With n pieces, you have n sequences each beginning with $p_i$ for $i = 1,...,n$. You then call mPcp once for each of the n sequences. Pcp will be a "Yes" instance iff mPcp is a "Yes" instance for at least one of the n instances.
$endgroup$
– user137481
May 9 '16 at 17:42














$begingroup$
The problem is that calling $mPcp$ for all $i$ could result in an infinite loop if $I$ is a no instance of $Pcp$, thus not satisfying the requirement that the reduction be carried out in a finite number of steps.
$endgroup$
– Dyon J Don Kiwi van Vreumingen
May 10 '16 at 9:12




$begingroup$
The problem is that calling $mPcp$ for all $i$ could result in an infinite loop if $I$ is a no instance of $Pcp$, thus not satisfying the requirement that the reduction be carried out in a finite number of steps.
$endgroup$
– Dyon J Don Kiwi van Vreumingen
May 10 '16 at 9:12












$begingroup$
Pcp is undecidable. I'm assuming that you are trying to prove that mPcp is undecidable. To start, we assume that mPcp is decidable and let M be the decider for mPcp. A decider always halts. So, there is no "infinite loop" even for 'No" instances. However, if mPcp is decidable, M could also be used to decide Pcp which gives us a contradiction. Hence, mPcp must also be undecidable.
$endgroup$
– user137481
May 10 '16 at 15:16




$begingroup$
Pcp is undecidable. I'm assuming that you are trying to prove that mPcp is undecidable. To start, we assume that mPcp is decidable and let M be the decider for mPcp. A decider always halts. So, there is no "infinite loop" even for 'No" instances. However, if mPcp is decidable, M could also be used to decide Pcp which gives us a contradiction. Hence, mPcp must also be undecidable.
$endgroup$
– user137481
May 10 '16 at 15:16










2 Answers
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http://infolab.stanford.edu/~ullman/ialc/slides/slides14.pdf



To summarise what is said in the attached pdf:



Let the dominos be (u1,v1), (u2,v2),...(un,vn).
(* is some symbol not already in the alphabet) For all the u's, put a * after them, and for all the v's, put a star before them, like such:
(u1*,v1), (u2,v2),...(un,*vn)



Now in the MPCP if we wanna start with the first term, we put a * before u1 like such:
(u1,v1), (u2,v2),...(un,*vn)



So now you the first domino MUST start, as it is the only one which has the first symbol matching. If there is a solution to this, it must start with the first domino.



Now to make the endings match, we add a domino:
(@,*@)
where @ is anything not in the alphabet.



Now MPCP is yes iff PCP is yes.






share|cite|improve this answer











$endgroup$





















    -2












    $begingroup$

    Put the *s before (or after) each symbol, not the words. So for d=(01101,001) add (0*1*1*0*1*,*0*0*1) but no (01101*,*001). If d is a starting domino add (*0*1*1*0*1*,*0*0*1), too.






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      2 Answers
      2






      active

      oldest

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      2 Answers
      2






      active

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      active

      oldest

      votes









      0












      $begingroup$

      http://infolab.stanford.edu/~ullman/ialc/slides/slides14.pdf



      To summarise what is said in the attached pdf:



      Let the dominos be (u1,v1), (u2,v2),...(un,vn).
      (* is some symbol not already in the alphabet) For all the u's, put a * after them, and for all the v's, put a star before them, like such:
      (u1*,v1), (u2,v2),...(un,*vn)



      Now in the MPCP if we wanna start with the first term, we put a * before u1 like such:
      (u1,v1), (u2,v2),...(un,*vn)



      So now you the first domino MUST start, as it is the only one which has the first symbol matching. If there is a solution to this, it must start with the first domino.



      Now to make the endings match, we add a domino:
      (@,*@)
      where @ is anything not in the alphabet.



      Now MPCP is yes iff PCP is yes.






      share|cite|improve this answer











      $endgroup$


















        0












        $begingroup$

        http://infolab.stanford.edu/~ullman/ialc/slides/slides14.pdf



        To summarise what is said in the attached pdf:



        Let the dominos be (u1,v1), (u2,v2),...(un,vn).
        (* is some symbol not already in the alphabet) For all the u's, put a * after them, and for all the v's, put a star before them, like such:
        (u1*,v1), (u2,v2),...(un,*vn)



        Now in the MPCP if we wanna start with the first term, we put a * before u1 like such:
        (u1,v1), (u2,v2),...(un,*vn)



        So now you the first domino MUST start, as it is the only one which has the first symbol matching. If there is a solution to this, it must start with the first domino.



        Now to make the endings match, we add a domino:
        (@,*@)
        where @ is anything not in the alphabet.



        Now MPCP is yes iff PCP is yes.






        share|cite|improve this answer











        $endgroup$
















          0












          0








          0





          $begingroup$

          http://infolab.stanford.edu/~ullman/ialc/slides/slides14.pdf



          To summarise what is said in the attached pdf:



          Let the dominos be (u1,v1), (u2,v2),...(un,vn).
          (* is some symbol not already in the alphabet) For all the u's, put a * after them, and for all the v's, put a star before them, like such:
          (u1*,v1), (u2,v2),...(un,*vn)



          Now in the MPCP if we wanna start with the first term, we put a * before u1 like such:
          (u1,v1), (u2,v2),...(un,*vn)



          So now you the first domino MUST start, as it is the only one which has the first symbol matching. If there is a solution to this, it must start with the first domino.



          Now to make the endings match, we add a domino:
          (@,*@)
          where @ is anything not in the alphabet.



          Now MPCP is yes iff PCP is yes.






          share|cite|improve this answer











          $endgroup$



          http://infolab.stanford.edu/~ullman/ialc/slides/slides14.pdf



          To summarise what is said in the attached pdf:



          Let the dominos be (u1,v1), (u2,v2),...(un,vn).
          (* is some symbol not already in the alphabet) For all the u's, put a * after them, and for all the v's, put a star before them, like such:
          (u1*,v1), (u2,v2),...(un,*vn)



          Now in the MPCP if we wanna start with the first term, we put a * before u1 like such:
          (u1,v1), (u2,v2),...(un,*vn)



          So now you the first domino MUST start, as it is the only one which has the first symbol matching. If there is a solution to this, it must start with the first domino.



          Now to make the endings match, we add a domino:
          (@,*@)
          where @ is anything not in the alphabet.



          Now MPCP is yes iff PCP is yes.







          share|cite|improve this answer














          share|cite|improve this answer



          share|cite|improve this answer








          edited Nov 21 '16 at 6:20

























          answered Nov 20 '16 at 18:28









          PolkaDotPolkaDot

          246




          246























              -2












              $begingroup$

              Put the *s before (or after) each symbol, not the words. So for d=(01101,001) add (0*1*1*0*1*,*0*0*1) but no (01101*,*001). If d is a starting domino add (*0*1*1*0*1*,*0*0*1), too.






              share|cite|improve this answer











              $endgroup$


















                -2












                $begingroup$

                Put the *s before (or after) each symbol, not the words. So for d=(01101,001) add (0*1*1*0*1*,*0*0*1) but no (01101*,*001). If d is a starting domino add (*0*1*1*0*1*,*0*0*1), too.






                share|cite|improve this answer











                $endgroup$
















                  -2












                  -2








                  -2





                  $begingroup$

                  Put the *s before (or after) each symbol, not the words. So for d=(01101,001) add (0*1*1*0*1*,*0*0*1) but no (01101*,*001). If d is a starting domino add (*0*1*1*0*1*,*0*0*1), too.






                  share|cite|improve this answer











                  $endgroup$



                  Put the *s before (or after) each symbol, not the words. So for d=(01101,001) add (0*1*1*0*1*,*0*0*1) but no (01101*,*001). If d is a starting domino add (*0*1*1*0*1*,*0*0*1), too.







                  share|cite|improve this answer














                  share|cite|improve this answer



                  share|cite|improve this answer








                  edited Dec 4 '17 at 15:07









                  Parcly Taxel

                  43.1k1372101




                  43.1k1372101










                  answered Dec 4 '17 at 14:19









                  tkrisztkrisz

                  1




                  1






























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