$A,C$ are recursive and $Acdot B = C$, Is $B$ is recursive?
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Past year paper question.
Prove or disprove: Suppose $A,C$ are recursive and $Acdot B = C$ (The dot denotes concatenation). Then $B$ is recursive.
I think it is false.
Let $B$ be any non-recursive language containing the empty string $epsilon$.
Let $A=C=Sigma^*$.
Then $A cdot B = C$.
Question is how can I construct $B$?
formal-languages
asked Nov 21 at 8:29
eatfood
1827
add a comment |
up vote
0
down vote
favorite
Past year paper question.
Prove or disprove: Suppose $A,C$ are recursive and $Acdot B = C$ (The dot denotes concatenation). Then $B$ is recursive.
I think it is false.
Let $B$ be any non-recursive language containing the empty string $epsilon$.
Let $A=C=Sigma^*$.
Then $A cdot B = C$.
Question is how can I construct $B$?
formal-languages
asked Nov 21 at 8:29
eatfood
1827
Why do you mean by constructing a non-recursive language?
– J.-E. Pin
Nov 21 at 8:35
I mean is there a concrete example of a non-recursive language, that contains the empty string.
– eatfood
Nov 21 at 8:57
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Past year paper question.
Prove or disprove: Suppose $A,C$ are recursive and $Acdot B = C$ (The dot denotes concatenation). Then $B$ is recursive.
I think it is false.
Let $B$ be any non-recursive language containing the empty string $epsilon$.
Let $A=C=Sigma^*$.
Then $A cdot B = C$.
Question is how can I construct $B$?
formal-languages
asked Nov 21 at 8:29
eatfood
1827
Past year paper question.
Prove or disprove: Suppose $A,C$ are recursive and $Acdot B = C$ (The dot denotes concatenation). Then $B$ is recursive.
I think it is false.
Let $B$ be any non-recursive language containing the empty string $epsilon$.
Let $A=C=Sigma^*$.
Then $A cdot B = C$.
Question is how can I construct $B$?
formal-languages
formal-languages
asked Nov 21 at 8:29
eatfood
1827
asked Nov 21 at 8:29
eatfood
1827
asked Nov 21 at 8:29
eatfood
1827
asked Nov 21 at 8:29
eatfood
1827
asked Nov 21 at 8:29
eatfood
1827
1827
Why do you mean by constructing a non-recursive language?
– J.-E. Pin
Nov 21 at 8:35
I mean is there a concrete example of a non-recursive language, that contains the empty string.
– eatfood
Nov 21 at 8:57
add a comment |
Why do you mean by constructing a non-recursive language?
– J.-E. Pin
Nov 21 at 8:35
I mean is there a concrete example of a non-recursive language, that contains the empty string.
– eatfood
Nov 21 at 8:57
Why do you mean by constructing a non-recursive language?
– J.-E. Pin
Nov 21 at 8:35
Why do you mean by constructing a non-recursive language?
– J.-E. Pin
Nov 21 at 8:35
I mean is there a concrete example of a non-recursive language, that contains the empty string.
– eatfood
Nov 21 at 8:57
I mean is there a concrete example of a non-recursive language, that contains the empty string.
– eatfood
Nov 21 at 8:57
add a comment |
1 Answer
1
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oldest
votes
up vote
1
down vote
Probably you can just suppose that there exists a non-recursive language that contains the empty string, and nobody will demand a proof of this.
Adding or removing a finite number of strings will not change the recursiveness or non-recursiveness of a language. In particular, adding the empty string will not change the complexity (take a TM for the language, add by a non-deterministic choice the possibility to accept the empty string instead of starting the original computation; even if the original TM rejects the empty string, the new one will accept it). So if you are allowed to suppose without proof that there exist non-recursive languages, then you can just take any of them and add the empty string and it will still be non-recursive.
answered Nov 21 at 9:49
Peter Leupold
56826
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
Probably you can just suppose that there exists a non-recursive language that contains the empty string, and nobody will demand a proof of this.
Adding or removing a finite number of strings will not change the recursiveness or non-recursiveness of a language. In particular, adding the empty string will not change the complexity (take a TM for the language, add by a non-deterministic choice the possibility to accept the empty string instead of starting the original computation; even if the original TM rejects the empty string, the new one will accept it). So if you are allowed to suppose without proof that there exist non-recursive languages, then you can just take any of them and add the empty string and it will still be non-recursive.
answered Nov 21 at 9:49
Peter Leupold
56826
add a comment |
up vote
1
down vote
Probably you can just suppose that there exists a non-recursive language that contains the empty string, and nobody will demand a proof of this.
Adding or removing a finite number of strings will not change the recursiveness or non-recursiveness of a language. In particular, adding the empty string will not change the complexity (take a TM for the language, add by a non-deterministic choice the possibility to accept the empty string instead of starting the original computation; even if the original TM rejects the empty string, the new one will accept it). So if you are allowed to suppose without proof that there exist non-recursive languages, then you can just take any of them and add the empty string and it will still be non-recursive.
answered Nov 21 at 9:49
Peter Leupold
56826
add a comment |
up vote
1
down vote
up vote
1
down vote
Probably you can just suppose that there exists a non-recursive language that contains the empty string, and nobody will demand a proof of this.
Adding or removing a finite number of strings will not change the recursiveness or non-recursiveness of a language. In particular, adding the empty string will not change the complexity (take a TM for the language, add by a non-deterministic choice the possibility to accept the empty string instead of starting the original computation; even if the original TM rejects the empty string, the new one will accept it). So if you are allowed to suppose without proof that there exist non-recursive languages, then you can just take any of them and add the empty string and it will still be non-recursive.
answered Nov 21 at 9:49
Peter Leupold
56826
Probably you can just suppose that there exists a non-recursive language that contains the empty string, and nobody will demand a proof of this.
Adding or removing a finite number of strings will not change the recursiveness or non-recursiveness of a language. In particular, adding the empty string will not change the complexity (take a TM for the language, add by a non-deterministic choice the possibility to accept the empty string instead of starting the original computation; even if the original TM rejects the empty string, the new one will accept it). So if you are allowed to suppose without proof that there exist non-recursive languages, then you can just take any of them and add the empty string and it will still be non-recursive.
answered Nov 21 at 9:49
Peter Leupold
56826
answered Nov 21 at 9:49
Peter Leupold
56826
answered Nov 21 at 9:49
Peter Leupold
56826
answered Nov 21 at 9:49
Peter Leupold
56826
56826
add a comment |
add a comment |
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Why do you mean by constructing a non-recursive language?
– J.-E. Pin
Nov 21 at 8:35
I mean is there a concrete example of a non-recursive language, that contains the empty string.
– eatfood
Nov 21 at 8:57