An Introduction to Formal Language Theory by Robert N. Moll

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 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 evidently to the next ques­ tions. How could a grammar be written for any such language? How may we payment even if a textual content have been or weren't a well-formed application generated by means of that grammar? How might we parse a software to supply the structural research wanted by means of a compiler? How may possibly we fee for ambiguity to en­ yes software has a distinct research to be handed to the pc? This concentrate on programming languages has now been broadened through the in­ creasing drawback of desktop scientists with designing interfaces which permit people to speak with desktops in a usual language, at the very least touching on difficulties in a few well-delimited area of discourse. the required paintings in computational linguistics attracts on reports either inside of linguistics (the research of human languages) and inside synthetic intelligence. the current quantity is the 1st textbook to mix the themes of formal language thought characteristically taught within the context of software­ ming languages with an advent to concerns in computational linguistics. it really is one in all a sequence, The AKM sequence in Theoretical computing device technology, designed to make key mathematical advancements in machine technology quite simply available to undergraduate and starting graduate students.

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Suppose, then, that we know Rh to be regular for all i', j' and for some k S; n. We must prove that each Rt+1 is also regular. Now, consider (referring to Figure 1) some win Rt+1. Either the path taken when following w never reaches qk' or else it can be broken up into segments W 1 W 2 ... wm , m ~ 2, where W1 takes us from qi to qk via {q1, ... ,qk-d, and Wm takes us from qk to qj via {q1, ... 3 Regular and Finite-State Languages I I J I I ~qJ ----_/ "- States qo through q'-l Figure 1 In the first case, we see that wERt· In the second case, we see that WE Thus R7k .

A can be replaced by w in the context y - z); or of the form (2) S -+ A, provided S does not appear on the right-hand side of any production. A language is context-sensitive if it can be generated by a context-sensitive grammar. Strictly speaking, not every context-free grammar is context-sensItIve, because context-free productions may be of the form A -+ A. However, we shall soon see that every context-free language is indeed a context-sensitive language. In this chapter, we will also study a very useful subclass of the context-free languages, the right-linear languages.

Let L = L(G) where G = (V, X, S, P) is a right-linear grammar. , T(M) = L. The NDA M = (Q, S, Qo, F) is defined by: Q = Vu {q} where q ¢ V, Qo = {S}, F = {q}, and for all A E V and x E X, o(A, x) = {BIB E V and A -+ xB is in P} u {qlA -+ x is in P}. We leave as an exercise the proof that indeed T(M) = L. By Proposition 16, there is a FSA M' such that T(M') = T(M) = L. D Putting together Proposition 2, Theorem 12, and Theorem 18, we conclude that the classes of right-linear languages, regular languages, and finite-state languages coincide.

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