如何将常规语法转换为有限自动机？

Question

How does one convert a regular grammar into a finite automaton (FA)? For instance, what would a finite automaton corresponding to the following regular grammar look like?

VN = {S, B, D} (nonterminals)
VT = {a, b, c} (terminals)
 P = {S -> aB, S -> bB, B -> bD, D -> b, D -> aD, B -> cB, B -> aS} (productions)

Answer 1

The good news is that this is not too hard. The idea is that each of the nonterminals will become a state in a nondeterministic finite automaton accepting the language of the grammar, and productions will become transitions. Our NFA will have states S, B and D, and will transition among those states according to the production rules. Our NFA looks like this:

       ___a__      _a_
      /      \    /   \
      |      |    \   /
      V      |     \ /
----->S-a,b->B--b-->D
            / \
           /   \
           \_c_/

There was one dangling production D -> b which we haven't added yet. We need to introduce another state, not corresponding to an nonterminal symbol, to allow us to transition from D on b and accept some strings:

       ___a__      _a_
      /      \    /   \
      |      |    \   /
      V      |     \ /
----->S-a,b->B--b-->D--b-->Q
            / \
           /   \
           \_c_/

Now if we make S the initial state and Q the accepting state, we have an NFA that works. If we want a DFA, we might notice that this FA is only nondeterministic because we are lacking required transitions from states S, D and Q. We can add the missing transitions by introducing a new dead state X which will keep track of the NFA we just derived having crashed at some point during its processing:

       ___a__      _a_
      /      \    /   \
      |      |    \   /
      V      |     \ /
----->S-a,b->B--b-->D--b-->Q
      |     / \     |      |
      |    /   \    |      |    a,b,c
      c    \_c_/    c    a,b,c  /   \
      |             |      |    \   /
      V             V      V     \ /
      +-------------+------+----->X