By Walter J Savitch
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PROOF Given a cfg G0 , we will obtain an equivalent cfg G' in Chomsky normal form. First find a grammar G 1 such that G 1 is nonerasing, has no rules of the form A - B where A and B are nonterminals, and is equivalent to G 0 . 13. Next, obtain a grammar G 2 such that G 2 is nonerasing, is equivalent to Gh and has only three types of productions: 1. Possibly S - A where S is the start symbol 2. A - a, where a is a terminal 3. A - B 1B 2... B n where the B; are nonterminals and n ~ 2 CHAPTER 2. CONTEXT-FREE GRAMMARS 21 To get G 2 from Gh proceed as follows.
Every time we change the program we get a new finite-state model. We are really emphasizing pro- 30 CHAPTER 3. FINITE-STATE ACCEPTORS 31 grams at least as much as hardware. In fact, this text is only concerned with programming, but the mathematics does in some gross sense model the hardware as well. Many people find it a help to their intuition to consider finite-state acceptors as hardware devices. For this reason, we sometimes refer to a finite-state acceptor as a finite-state machine. A finite-state acceptor is neither hardware nor exactly a program but a formal mathematical construct that can model both of these.
Consider a fixed parse tree for z. By Claim lb, some path, from S to a terminal, has n + 2 or more nodes. Consider the longest such path. Going from S to a terminal along this path, we pass through all the nodes of this path. Consider the last n + 2 such nodes. One is labeled with a terminal and n + 1 are labeled with nonterminals. There are only n nonterminals; therefore, two such nodes must be labeled with the same nonterminal. Call this nonterminal A. 5(a). One way to write a derivation corresponding to this tree is 4.