How to guess the end of pushdown automata by emptying the stack?
$begingroup$
Let language $L={sigma_1w_1csigma_2w_2c...sigma_nw_nc}$ where:
$$
1le n\
sigma_iin {a,b}\
w_iin {a,b}^+\
exists i:ile|w_i|
$$
and at least one condition from the two conditions below holds:
the first $i$ letters of $w_i$ form a different word from $(sigma_1sigma_2...sigma_i)^R$ OR $|w_i|=n$.
Build a pushdown automaton which receives $L$ by emptying its stack.
The solution of these two conditions separately is here: https://imgur.com/a/T9PiPgT.
As far as I understand in the automata for both condition 1 and condion 2 ($|w_i|=n$) essentially:
- We build an automaton which simply describes reading $sigma, w_i, c$ first.
- Then we guess the $w_i$ which is according to condition 1 or 2.
- Then we read the $sigma, c$ surrounding the guessed $w_i$ and the $w_i$ itself.
I'm new to pushdown automata, but it looks like a lot of automata could be solved in such a generic way:
- Read the words according to some logic described in the problem.
- guess some "good" word (by using $epsilon$ connection) and delete it from the stack.
Am I missing something?
Also I don't understand why we need $2$ cases in the solution for each condition. The solutions for each condition look almost identical.
Lastly, I don't understand why in the automaton for the condition $1$ in state $q_5$ we need to read the end of $w_i$ first, then read the "good" $w_i$. Why can't we read the "good" $w_i$ straight after the guess?
proof-explanation formal-languages automata
asked Dec 15 '18 at 8:41
YosYos
1,1631823
$endgroup$
add a comment |
$begingroup$
Let language $L={sigma_1w_1csigma_2w_2c...sigma_nw_nc}$ where:
$$
1le n\
sigma_iin {a,b}\
w_iin {a,b}^+\
exists i:ile|w_i|
$$
and at least one condition from the two conditions below holds:
the first $i$ letters of $w_i$ form a different word from $(sigma_1sigma_2...sigma_i)^R$ OR $|w_i|=n$.
Build a pushdown automaton which receives $L$ by emptying its stack.
The solution of these two conditions separately is here: https://imgur.com/a/T9PiPgT.
As far as I understand in the automata for both condition 1 and condion 2 ($|w_i|=n$) essentially:
- We build an automaton which simply describes reading $sigma, w_i, c$ first.
- Then we guess the $w_i$ which is according to condition 1 or 2.
- Then we read the $sigma, c$ surrounding the guessed $w_i$ and the $w_i$ itself.
I'm new to pushdown automata, but it looks like a lot of automata could be solved in such a generic way:
- Read the words according to some logic described in the problem.
- guess some "good" word (by using $epsilon$ connection) and delete it from the stack.
Am I missing something?
Also I don't understand why we need $2$ cases in the solution for each condition. The solutions for each condition look almost identical.
Lastly, I don't understand why in the automaton for the condition $1$ in state $q_5$ we need to read the end of $w_i$ first, then read the "good" $w_i$. Why can't we read the "good" $w_i$ straight after the guess?
proof-explanation formal-languages automata
asked Dec 15 '18 at 8:41
YosYos
1,1631823
$endgroup$
add a comment |
$begingroup$
Let language $L={sigma_1w_1csigma_2w_2c...sigma_nw_nc}$ where:
$$
1le n\
sigma_iin {a,b}\
w_iin {a,b}^+\
exists i:ile|w_i|
$$
and at least one condition from the two conditions below holds:
the first $i$ letters of $w_i$ form a different word from $(sigma_1sigma_2...sigma_i)^R$ OR $|w_i|=n$.
Build a pushdown automaton which receives $L$ by emptying its stack.
The solution of these two conditions separately is here: https://imgur.com/a/T9PiPgT.
As far as I understand in the automata for both condition 1 and condion 2 ($|w_i|=n$) essentially:
- We build an automaton which simply describes reading $sigma, w_i, c$ first.
- Then we guess the $w_i$ which is according to condition 1 or 2.
- Then we read the $sigma, c$ surrounding the guessed $w_i$ and the $w_i$ itself.
I'm new to pushdown automata, but it looks like a lot of automata could be solved in such a generic way:
- Read the words according to some logic described in the problem.
- guess some "good" word (by using $epsilon$ connection) and delete it from the stack.
Am I missing something?
Also I don't understand why we need $2$ cases in the solution for each condition. The solutions for each condition look almost identical.
Lastly, I don't understand why in the automaton for the condition $1$ in state $q_5$ we need to read the end of $w_i$ first, then read the "good" $w_i$. Why can't we read the "good" $w_i$ straight after the guess?
proof-explanation formal-languages automata
asked Dec 15 '18 at 8:41
YosYos
1,1631823
$endgroup$
Let language $L={sigma_1w_1csigma_2w_2c...sigma_nw_nc}$ where:
$$
1le n\
sigma_iin {a,b}\
w_iin {a,b}^+\
exists i:ile|w_i|
$$
and at least one condition from the two conditions below holds:
the first $i$ letters of $w_i$ form a different word from $(sigma_1sigma_2...sigma_i)^R$ OR $|w_i|=n$.
Build a pushdown automaton which receives $L$ by emptying its stack.
The solution of these two conditions separately is here: https://imgur.com/a/T9PiPgT.
As far as I understand in the automata for both condition 1 and condion 2 ($|w_i|=n$) essentially:
- We build an automaton which simply describes reading $sigma, w_i, c$ first.
- Then we guess the $w_i$ which is according to condition 1 or 2.
- Then we read the $sigma, c$ surrounding the guessed $w_i$ and the $w_i$ itself.
I'm new to pushdown automata, but it looks like a lot of automata could be solved in such a generic way:
- Read the words according to some logic described in the problem.
- guess some "good" word (by using $epsilon$ connection) and delete it from the stack.
Am I missing something?
Also I don't understand why we need $2$ cases in the solution for each condition. The solutions for each condition look almost identical.
Lastly, I don't understand why in the automaton for the condition $1$ in state $q_5$ we need to read the end of $w_i$ first, then read the "good" $w_i$. Why can't we read the "good" $w_i$ straight after the guess?
proof-explanation formal-languages automata
proof-explanation formal-languages automata
asked Dec 15 '18 at 8:41
YosYos
1,1631823
asked Dec 15 '18 at 8:41
YosYos
1,1631823
asked Dec 15 '18 at 8:41
YosYos
1,1631823
asked Dec 15 '18 at 8:41
YosYos
1,1631823
asked Dec 15 '18 at 8:41
YosYos
1,1631823
1,1631823
add a comment |
add a comment |
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