What is the complement of a language?












5












$begingroup$


If given any language L, how do I find the complement of said language?



I lack the basic understanding required to determine if a language is co-recognizable. I understand that a language, $L$, is co-recognizable if the complement of that language, $overline{L}$, is recognizable.



But given a specific language, I have a hard time figuring out what the actual complement is, and can thus not figure out if that complement is recognizable. An example problem is:




Is the following language co-recognizable?
$L$ = {$<M>$ | $M$ is a
turing machine, and $1010 notin{L(M)}$}







computability







share|cite|improve this question









$endgroup$

















    5












    $begingroup$


    If given any language L, how do I find the complement of said language?



    I lack the basic understanding required to determine if a language is co-recognizable. I understand that a language, $L$, is co-recognizable if the complement of that language, $overline{L}$, is recognizable.



    But given a specific language, I have a hard time figuring out what the actual complement is, and can thus not figure out if that complement is recognizable. An example problem is:




    Is the following language co-recognizable?
    $L$ = {$<M>$ | $M$ is a
    turing machine, and $1010 notin{L(M)}$}







    computability







    share|cite|improve this question









    $endgroup$















      5












      5








      5





      $begingroup$


      If given any language L, how do I find the complement of said language?



      I lack the basic understanding required to determine if a language is co-recognizable. I understand that a language, $L$, is co-recognizable if the complement of that language, $overline{L}$, is recognizable.



      But given a specific language, I have a hard time figuring out what the actual complement is, and can thus not figure out if that complement is recognizable. An example problem is:




      Is the following language co-recognizable?
      $L$ = {$<M>$ | $M$ is a
      turing machine, and $1010 notin{L(M)}$}







      computability







      share|cite|improve this question









      $endgroup$




      If given any language L, how do I find the complement of said language?



      I lack the basic understanding required to determine if a language is co-recognizable. I understand that a language, $L$, is co-recognizable if the complement of that language, $overline{L}$, is recognizable.



      But given a specific language, I have a hard time figuring out what the actual complement is, and can thus not figure out if that complement is recognizable. An example problem is:




      Is the following language co-recognizable?
      $L$ = {$<M>$ | $M$ is a
      turing machine, and $1010 notin{L(M)}$}







      computability




      computability






      share|cite|improve this question













      share|cite|improve this question











      share|cite|improve this question




      share|cite|improve this question










      asked Jan 13 at 17:21









      JakirJakir

      261




      261






















          1 Answer
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          11












          $begingroup$

          Remember that a language is defined as a set of strings. The complement of a language is thus the complement of that set, defined in the usual way: everything not in the set.



          In practice, when talking about the complement of a language, there's usually a particular alphabet you're interested in (which you can infer from context). If all else fails, assume ${0,1}$.



          So in this case, the complement of that language is:




          The set of all binary strings $s$, such that either $s$ isn't a valid encoded Turing machine, or the machine encoded by $s$ accepts $1010$.




          Hint: the problem of whether a string $s$ is a valid encoded Turing machine or not is known to be decidable. So you only need to worry about the second clause.






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            Finding the complement is particularly easy (read: mechanizable) if the the language is given as a predicate subset: Rewrite $L = {<M>|M text{is a TM and} 1010 notin L(M)}$ as $L = {w in {0,1}^* | w = <M>, M text{is a TM and} 1010 notin L(M)} = {w in {0,1}^* | P(w)}$ with $P$ being the predicate in $w$. Now the complement simply is ${w in {0,1}^* | neg P(w)}$. And $neg P(w) equiv w neq <M> lor (w = <M> wedge M text{ TM} wedge 1010 in L(M))$.
            $endgroup$
            – ComFreek
            Jan 14 at 7:59













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          1 Answer
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          active

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          1 Answer
          1






          active

          oldest

          votes









          active

          oldest

          votes






          active

          oldest

          votes









          11












          $begingroup$

          Remember that a language is defined as a set of strings. The complement of a language is thus the complement of that set, defined in the usual way: everything not in the set.



          In practice, when talking about the complement of a language, there's usually a particular alphabet you're interested in (which you can infer from context). If all else fails, assume ${0,1}$.



          So in this case, the complement of that language is:




          The set of all binary strings $s$, such that either $s$ isn't a valid encoded Turing machine, or the machine encoded by $s$ accepts $1010$.




          Hint: the problem of whether a string $s$ is a valid encoded Turing machine or not is known to be decidable. So you only need to worry about the second clause.






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            Finding the complement is particularly easy (read: mechanizable) if the the language is given as a predicate subset: Rewrite $L = {<M>|M text{is a TM and} 1010 notin L(M)}$ as $L = {w in {0,1}^* | w = <M>, M text{is a TM and} 1010 notin L(M)} = {w in {0,1}^* | P(w)}$ with $P$ being the predicate in $w$. Now the complement simply is ${w in {0,1}^* | neg P(w)}$. And $neg P(w) equiv w neq <M> lor (w = <M> wedge M text{ TM} wedge 1010 in L(M))$.
            $endgroup$
            – ComFreek
            Jan 14 at 7:59


















          11












          $begingroup$

          Remember that a language is defined as a set of strings. The complement of a language is thus the complement of that set, defined in the usual way: everything not in the set.



          In practice, when talking about the complement of a language, there's usually a particular alphabet you're interested in (which you can infer from context). If all else fails, assume ${0,1}$.



          So in this case, the complement of that language is:




          The set of all binary strings $s$, such that either $s$ isn't a valid encoded Turing machine, or the machine encoded by $s$ accepts $1010$.




          Hint: the problem of whether a string $s$ is a valid encoded Turing machine or not is known to be decidable. So you only need to worry about the second clause.






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            Finding the complement is particularly easy (read: mechanizable) if the the language is given as a predicate subset: Rewrite $L = {<M>|M text{is a TM and} 1010 notin L(M)}$ as $L = {w in {0,1}^* | w = <M>, M text{is a TM and} 1010 notin L(M)} = {w in {0,1}^* | P(w)}$ with $P$ being the predicate in $w$. Now the complement simply is ${w in {0,1}^* | neg P(w)}$. And $neg P(w) equiv w neq <M> lor (w = <M> wedge M text{ TM} wedge 1010 in L(M))$.
            $endgroup$
            – ComFreek
            Jan 14 at 7:59
















          11












          11








          11





          $begingroup$

          Remember that a language is defined as a set of strings. The complement of a language is thus the complement of that set, defined in the usual way: everything not in the set.



          In practice, when talking about the complement of a language, there's usually a particular alphabet you're interested in (which you can infer from context). If all else fails, assume ${0,1}$.



          So in this case, the complement of that language is:




          The set of all binary strings $s$, such that either $s$ isn't a valid encoded Turing machine, or the machine encoded by $s$ accepts $1010$.




          Hint: the problem of whether a string $s$ is a valid encoded Turing machine or not is known to be decidable. So you only need to worry about the second clause.






          share|cite|improve this answer









          $endgroup$



          Remember that a language is defined as a set of strings. The complement of a language is thus the complement of that set, defined in the usual way: everything not in the set.



          In practice, when talking about the complement of a language, there's usually a particular alphabet you're interested in (which you can infer from context). If all else fails, assume ${0,1}$.



          So in this case, the complement of that language is:




          The set of all binary strings $s$, such that either $s$ isn't a valid encoded Turing machine, or the machine encoded by $s$ accepts $1010$.




          Hint: the problem of whether a string $s$ is a valid encoded Turing machine or not is known to be decidable. So you only need to worry about the second clause.







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered Jan 13 at 17:54









          DraconisDraconis

          4,395718




          4,395718












          • $begingroup$
            Finding the complement is particularly easy (read: mechanizable) if the the language is given as a predicate subset: Rewrite $L = {<M>|M text{is a TM and} 1010 notin L(M)}$ as $L = {w in {0,1}^* | w = <M>, M text{is a TM and} 1010 notin L(M)} = {w in {0,1}^* | P(w)}$ with $P$ being the predicate in $w$. Now the complement simply is ${w in {0,1}^* | neg P(w)}$. And $neg P(w) equiv w neq <M> lor (w = <M> wedge M text{ TM} wedge 1010 in L(M))$.
            $endgroup$
            – ComFreek
            Jan 14 at 7:59




















          • $begingroup$
            Finding the complement is particularly easy (read: mechanizable) if the the language is given as a predicate subset: Rewrite $L = {<M>|M text{is a TM and} 1010 notin L(M)}$ as $L = {w in {0,1}^* | w = <M>, M text{is a TM and} 1010 notin L(M)} = {w in {0,1}^* | P(w)}$ with $P$ being the predicate in $w$. Now the complement simply is ${w in {0,1}^* | neg P(w)}$. And $neg P(w) equiv w neq <M> lor (w = <M> wedge M text{ TM} wedge 1010 in L(M))$.
            $endgroup$
            – ComFreek
            Jan 14 at 7:59


















          $begingroup$
          Finding the complement is particularly easy (read: mechanizable) if the the language is given as a predicate subset: Rewrite $L = {<M>|M text{is a TM and} 1010 notin L(M)}$ as $L = {w in {0,1}^* | w = <M>, M text{is a TM and} 1010 notin L(M)} = {w in {0,1}^* | P(w)}$ with $P$ being the predicate in $w$. Now the complement simply is ${w in {0,1}^* | neg P(w)}$. And $neg P(w) equiv w neq <M> lor (w = <M> wedge M text{ TM} wedge 1010 in L(M))$.
          $endgroup$
          – ComFreek
          Jan 14 at 7:59






          $begingroup$
          Finding the complement is particularly easy (read: mechanizable) if the the language is given as a predicate subset: Rewrite $L = {<M>|M text{is a TM and} 1010 notin L(M)}$ as $L = {w in {0,1}^* | w = <M>, M text{is a TM and} 1010 notin L(M)} = {w in {0,1}^* | P(w)}$ with $P$ being the predicate in $w$. Now the complement simply is ${w in {0,1}^* | neg P(w)}$. And $neg P(w) equiv w neq <M> lor (w = <M> wedge M text{ TM} wedge 1010 in L(M))$.
          $endgroup$
          – ComFreek
          Jan 14 at 7:59




















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