By Robert N. Moll

The examine of formal languages and of comparable households of automata has lengthy been on the center of theoretical computing device technological know-how. until eventually lately, the most purposes for this centrality have been hooked up with the specification and analy sis of programming languages, which led evidently to the next ques tions. How may a grammar be written for this kind of language? How may perhaps we fee no matter if a textual content have been or weren't a well-formed software generated by means of that grammar? How may well we parse a software to supply the structural research wanted via a compiler? How may we cost for ambiguity to en definite software has a special research to be handed to the pc? This specialise in programming languages has now been broadened by way of the in creasing quandary of laptop scientists with designing interfaces which permit people to speak with desktops in a ordinary language, at the very least referring to difficulties in a few well-delimited area of discourse. the mandatory paintings in computational linguistics attracts on reports either inside of linguistics (the research of human languages) and inside of synthetic intelligence. the current quantity is the 1st textbook to mix the themes of formal language thought normally taught within the context of software ming languages with an creation to matters in computational linguistics. it really is certainly one of a sequence, The AKM sequence in Theoretical machine technology, designed to make key mathematical advancements in machine technological know-how conveniently obtainable to undergraduate and starting graduate students.

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Prove that the language {anbncmln, m > O} is context-free. 3. Prove Theorem 9. 4. For L c X*, define Init(L) = {z E X*lthere is aWE X* and zw E L}. (a) What is Init(L), where L = {a"b"ln > O}? (b) Prove that if L is right-linear, then Init(L) is right-linear. (c) Prove that if L is context-free, then Init(L) is context-free. (d) Does Init(L) = Init(lnit(L»? Prove this claim or give a counterexample. 5. Use closure under union to show that each of the following language is contextfree: (a) {a%ili #- j} (b) {a,b}* - {aibili ~ O} (c) {w E {a,b}*lw = WR}.

3 Theorem. Let G be an arbitrary context-free grammar. Then there is an equivalent grammar G', L(G'), in Chomsky normal form. 2 Normal Forms for Context-Free Grammars 55 PROOF. We assume that G satisfies (ii), (iii), and (iv) of the preceding lemma. For each production A -+ Bl B2 ... Bm in P, where m ;::: 2, we may also assume that all the B's are variables. For suppose Bj is a terminal symbol, then we can replace it in the production with a new variable not in V, Bj, and we add the additional production Bj -+ Bj • The set of productions P' of G' is obtained by first including in it all the productions in P which are of the form A -+ w where Iwl ~ 2.

Then the preceding informal description may be rewritten as follows: (1)