hw4sol - CS 181 - Winter 2005 Formal Languages and Automata...

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CS 181 — Winter 2005 Formal Languages and Automata Theory Problem Set #4 Solutions Problem 4.1. (10 points) 1. Show that if all productions of a CFG G are of the form A wB or A w for variables A,B V and strings of terminals w Σ * , then L ( G ) is regular. Solution Given a context-free grammar G = ( V, Σ ,R,S ) where each production R has either form A wB or A w , we will construct a GNFA for L ( G ). Here is the intuition: Our GNFA will have a state for each variable in V an initial state q 0 and a final state q f . Our initial state q 0 has a transition into S on ε . For each production of the form A wB we have a transition from the state A to the state B over regular expression w . For each production of the form A w we have a transition from the state A to the state q f over w . Formally, our GNFA is defined as M = ( V ∪ { q 0 ,q f } , Σ ,δ,q 0 , { q f } ) where the transition function is δ ( q,p ) = ε, if q = q 0 and p = S S w :( q w ) R w, if q V and p = q f S w :( q wp ) R w, if q,p V , otherwise Notice that to handle cases where our grammar has productions A w 1 B and A w 2 B , we have a transition from A to B over w 1 w 2 . Let us now show that in fact L ( M ) = L ( G ). Suppose w L ( G ) then there is some derivation S = w 1 V 1 = w 1 w 2 V 2 = ⇒ ··· = w 1 w 2 ··· w k - 1 V k - 1 = w 1 w 2 ··· w k = w Then clearly the run of M on w with state sequence
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hw4sol - CS 181 - Winter 2005 Formal Languages and Automata...

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