Question

1-3. Let us consider an automaton that recognizes words that contain at least two copies of the symbol -. Let us only consider the alphabet consisting of three symbols: -, a, and b. This automaton has 3 states: ? the start state s corresponding to no minuses; ? the intermediate state i corresponding to words with exactly one minus, and ? the final state f corresponding to words with two or more minuses. Transitions are as follows: • from the state s, minus leads to i, other symbols lead back to s; • from the state i, minus leads to f, other symbols lead back to i; • from the final state f, every symbol leads back to f. This automaton accepts the word a --. 1. Show how the general algorithm will produce a context-free grammar that generates all the words accepted by this automaton -- and only words generated by this automaton. 2. On the example of the word a -- accepted by this automaton, show how the tracing of acceptance of this word by the finite automaton can be translated into a generation of this same word by your context-free grammar. 3. Show how the word a -- can be represented as uvxyz according to the Pumping Lemma for context-free grammars.

          1-3. Let us consider an automaton that recognizes words that contain at least
two copies of the symbol -. Let us only consider the alphabet consisting of
three symbols: -, a, and b. This automaton has 3 states:
? the start state s corresponding to no minuses;
? the intermediate state i corresponding to words with exactly one minus,
and
? the final state f corresponding to words with two or more minuses.
Transitions are as follows:
• from the state s, minus leads to i, other symbols lead back to s;
• from the state i, minus leads to f, other symbols lead back to i;
• from the final state f, every symbol leads back to f.
This automaton accepts the word a --.
1. Show how the general algorithm will produce a context-free grammar that
generates all the words accepted by this automaton -- and only words
generated by this automaton.
2. On the example of the word a -- accepted by this automaton, show
how the tracing of acceptance of this word by the finite automaton can
be translated into a generation of this same word by your context-free
grammar.
3. Show how the word a -- can be represented as uvxyz according to the
Pumping Lemma for context-free grammars.

1-3. Let us consider an automaton that recognizes words that contain at least
two copies of the symbol -. Let us only consider the alphabet consisting of
three symbols: -, a, and b. This automaton has 3 states:
? the start state s corresponding to no minuses;
? the intermediate state i corresponding to words with exactly one minus,
and
? the final state f corresponding to words with two or more minuses.
Transitions are as follows:
• from the state s, minus leads to i, other symbols lead back to s;
• from the state i, minus leads to f, other symbols lead back to i;
• from the final state f, every symbol leads back to f.
This automaton accepts the word a –.
1. Show how the general algorithm will produce a context-free grammar that
generates all the words accepted by this automaton – and only words
generated by this automaton.
2. On the example of the word a – accepted by this automaton, show
how the tracing of acceptance of this word by the finite automaton can
be translated into a generation of this same word by your context-free
grammar.
3. Show how the word a – can be represented as uvxyz according to the
Pumping Lemma for context-free grammars.

Added by Robert B.

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