When does the initial state lead to empty string when converting a DFA to context-free grammar?
up vote
0
down vote
favorite
I saw several examples of converting DFA to context-free grammar. Sometimes the grammar output includes the production rule such that the initial state $S$ leads to empty string: $Stoepsilon$, however in some examples the grammar didn't include $Stoepsilon$ rule.
From my observations it seems that if $q_0$ is the initial and accepting state in some DFA then the resulting grammar will include the rule: $q_0toepsilon$, if $q_0$ initial state is not accepting then the rule $q_0toepsilon$ will not be in the context-free grammar production rules.
Is my observation correct?
automata context-free-grammar
asked Dec 2 at 13:45
Yos
1,227623
add a comment |
up vote
0
down vote
favorite
I saw several examples of converting DFA to context-free grammar. Sometimes the grammar output includes the production rule such that the initial state $S$ leads to empty string: $Stoepsilon$, however in some examples the grammar didn't include $Stoepsilon$ rule.
From my observations it seems that if $q_0$ is the initial and accepting state in some DFA then the resulting grammar will include the rule: $q_0toepsilon$, if $q_0$ initial state is not accepting then the rule $q_0toepsilon$ will not be in the context-free grammar production rules.
Is my observation correct?
automata context-free-grammar
asked Dec 2 at 13:45
Yos
1,227623
1
Yes, it's true, and easy to prove. Just prove that $varepsilon$ is in the language recognized by a DFA if and only if the initial state is accepting.
– saulspatz
Dec 2 at 15:22
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
I saw several examples of converting DFA to context-free grammar. Sometimes the grammar output includes the production rule such that the initial state $S$ leads to empty string: $Stoepsilon$, however in some examples the grammar didn't include $Stoepsilon$ rule.
From my observations it seems that if $q_0$ is the initial and accepting state in some DFA then the resulting grammar will include the rule: $q_0toepsilon$, if $q_0$ initial state is not accepting then the rule $q_0toepsilon$ will not be in the context-free grammar production rules.
Is my observation correct?
automata context-free-grammar
asked Dec 2 at 13:45
Yos
1,227623
I saw several examples of converting DFA to context-free grammar. Sometimes the grammar output includes the production rule such that the initial state $S$ leads to empty string: $Stoepsilon$, however in some examples the grammar didn't include $Stoepsilon$ rule.
From my observations it seems that if $q_0$ is the initial and accepting state in some DFA then the resulting grammar will include the rule: $q_0toepsilon$, if $q_0$ initial state is not accepting then the rule $q_0toepsilon$ will not be in the context-free grammar production rules.
Is my observation correct?
automata context-free-grammar
automata context-free-grammar
asked Dec 2 at 13:45
Yos
1,227623
asked Dec 2 at 13:45
Yos
1,227623
asked Dec 2 at 13:45
Yos
1,227623
asked Dec 2 at 13:45
Yos
1,227623
asked Dec 2 at 13:45
Yos
1,227623
1,227623
1
Yes, it's true, and easy to prove. Just prove that $varepsilon$ is in the language recognized by a DFA if and only if the initial state is accepting.
– saulspatz
Dec 2 at 15:22
add a comment |
1
Yes, it's true, and easy to prove. Just prove that $varepsilon$ is in the language recognized by a DFA if and only if the initial state is accepting.
– saulspatz
Dec 2 at 15:22
1
1
Yes, it's true, and easy to prove. Just prove that $varepsilon$ is in the language recognized by a DFA if and only if the initial state is accepting.
– saulspatz
Dec 2 at 15:22
Yes, it's true, and easy to prove. Just prove that $varepsilon$ is in the language recognized by a DFA if and only if the initial state is accepting.
– saulspatz
Dec 2 at 15:22
add a comment |
active
oldest
votes
active
oldest
votes
active
oldest
votes
active
oldest
votes
active
oldest
votes
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Some of your past answers have not been well-received, and you're in danger of being blocked from answering.
Please pay close attention to the following guidance:
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3022654%2fwhen-does-the-initial-state-lead-to-empty-string-when-converting-a-dfa-to-contex%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
1
Yes, it's true, and easy to prove. Just prove that $varepsilon$ is in the language recognized by a DFA if and only if the initial state is accepting.
– saulspatz
Dec 2 at 15:22