An Introduction to Formal Language Theory by Robert N. Moll

By Robert N. Moll

The research of formal languages and of similar households of automata has lengthy been on the middle of theoretical desktop technology. until eventually lately, the most purposes for this centrality have been hooked up with the specification and analy­ sis of programming languages, which led certainly to the subsequent ques­ tions. How may a grammar be written for this type of language? How may perhaps we payment even if a textual content have been or weren't a well-formed application generated via that grammar? How might we parse a software to supply the structural research wanted through a compiler? How may we cost for ambiguity to en­ certain application has a special research to be handed to the pc? This specialize in programming languages has now been broadened by means of the in­ creasing quandary of computing device scientists with designing interfaces which enable people to speak with desktops in a average language, not less than relating difficulties in a few well-delimited area of discourse. the mandatory paintings in computational linguistics attracts on stories either inside 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 idea characteristically taught within the context of application­ ming languages with an creation to concerns in computational linguistics. it's one in all a sequence, The AKM sequence in Theoretical computing device technological know-how, designed to make key mathematical advancements in laptop technology easily available to undergraduate and starting graduate students.

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PROOF. Let us use the notation a ~ p to indicate that o(p, a) = q in M. Let z behavior on z by writing q = a 1 az ... ak' Then we can trace M's When Izl ~ n, at least n + 1 states must appear in the trace, and so some state, say r, must appear twice. 44 2 Grammars and Machines U a,+! , lead to the same final state of M, and so if uvw E T(M) then we have uvmw E T(M) for all D m~Q 20 Example. The language L = {anbnln ~ O} is not regular. Suppose L were regular. Then L = T(M), where M has k states, and consider akb k.

2. State and prove an empty-string lemma for right-linear grammars. 3. Write a context-sensitive grammar for the language {a"ba"ca"ln > O}. 4. , 1, 100, 1001, etc. 5. , for which A ~ A. Describe carefully an algorithm that will detect all variables A for which A ~ A. (Hint: Proceed inductively. First find all A's for which A b A, then having found those for which A ~ A, find those for which A m,J.! A. ") 6. Let G = (V,X,S,P) where V = {S}, X = {a,b}, and P is the following set of productions: S -+ aSb S -+ aSa S -+ bSa S -+ bSb S-+A Show that L(G) is a right-linear language.

2. 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}.

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