Download Abstract machines and grammars by Walter J Savitch PDF

By Walter J Savitch

ISBN-10: 0316771619

ISBN-13: 9780316771610

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Extra resources for Abstract machines and grammars

Example text

Finally, it remains only to show that either x or v is nonempty. Again, consider the subtree A ~ vAx ~ vwx. Since the grammar is in Chomsky normal form, it can be written as A ~ BC ~ vAx ~ vwx = z 1z 2, where B and C are nonterminals such that B ~ z 1 and C ~ z 2. Since all Chomsky normal form grammars are nonerasing, it follows that both z 1 and z 2 are nonempty. Now the second A must be derived from either the B or the C. If it is derived from the B, then z 2 lies completely to the right of w.

4 The machine M is said to be a deterministic finite-state acceptor provided 8(q, a) contains at most one element for every q in S and a in r.. If M is deterministic and 8 (q, a) = {p}, then we will usually omit the braces and write 8(q, a)= p. If 8(q, a) is the empty set, we will sometimes indicate this by CHAPTER 3. " There is a technical difference between {p} and p. Sometimes this difference can be very important. However, in this context we can identify the two things without causing any problems.

3. 4. A ~aB. A ~ aC ~ a{3B, for some C inN and some {3 in T*. There is some C such that Cis in 8(A, a) and C ~ {38. (A, a{3) (C, {3) and (C, {3) 1.!... (B, A), for some C. 5. (A, a) r r (B, A). Statements (3) and (4) are equivalent by the induction hypothesis and the definitions involved. All other adjacent statements are equivalent by the definitions involved. Thus Claim 1 is proven. To see that L (G) = L, note that, by Claim 1 and the definitions involved, we can show the following chain of statements to be equivalent: a is (B, A), for in L(G); S ~a; S ~ aB, for some accepting state B; (S, a) some accepting state B; a is in A (M).

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