Course Notes on Formal Languages and Compilers

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If the automaton has an ε - transition such as B ε A ,
then the grammar rule will be A→B, but this must be removed, as already
explained, so that the grammar will be regular.
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In this way we can convert between Regular Grammars and Finite
Automata in both directions.
For example, the grammars
automata are:-
G 1 , G 2 corresponding to the previous two
G 1 = ( { A, B}, {a,b}, P 1 , {A,B} ) where the rules in P 1 are
A→Aa
B→Bb
A→ ε
B→Ab
G 2 = ( {A,B}, {a, b}, P 2 , {B} ) where the rules in P 2 are
A→Aa
B→Bb
A→ε
B→A
<strong>Notes</strong>: 1) G 1 is a backward deterministic regular grammar.
2) Eliminating B→A from G 2 will give us a regular grammar which is
not in backward deterministic form.
More analogies between finite automata and regular grammars:
1) ε - transitions can always be eliminated - similar to removing rules of the
form A→B from grammars.
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2) A deterministic finite automaton has no transitions of the form
a
A C where B ≠ C.
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a
B
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It also has no ε - transitions and has at most one initial state. Otherwise,
the finite automaton is non-deterministic (The first automaton we
presented is deterministic, the second is non-deterministic).
Deterministic finite automata correspond to backward deterministic
regular grammars.