Hw8-solutions - CS 381 HW 8 Solutions Dan FitzGerald [email protected] November 1 2006 1 Convert the following CFG to a one state PDA S bA S aB A a

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CS 381 HW 8 Solutions Dan FitzGerald, [email protected] November 1, 2006 1. Convert the following CFG to a one state PDA: S -→ bA S -→ aB A -→ a A -→ bAA A -→ aS B -→ b B -→ bS B -→ aBB The main idea of this problem is summarized on page 238 of the text. Basically, we plan to expand the grammar from left to right on the stack. Whenever we see terminals (a and b in this case), we will compare them to the input tape and erase the symbol from the stack if they match. When we encounter a production (S,A, or B), we will guess which production to follow and expand it onto the stack. We continue this process of eating the terminals at the top of the stack and expanding productions until no productions remain, and we will continue comparing terminals to the in- put and erasing them if they match. In the end, we will reach an empty stack and accept if the string is in the grammar, and thus have a PDA for the grammar. More formally, our PDA P = ( Q, Σ , Γ , δ, q 0 , Z 0 , F ) is de ned as: Q = { 1 } , Σ = { a, b } , δ, q 0 = 1 , Z 0 = Z 0 , S = 1 , F = K where δ is de ned as δ (1 , ±, S ) (1 , bA ) , { (1 , aB ) } δ (1 , ±, A ) → { (1 , a ) , (1 , bAA ) , (1 , aS ) } δ (1 , ±, B ) → { (1 , b ) , (1 , bS ) , (1 , aBB ) } δ (1 , a, a ) → { (1 , ± ) } δ (1 , b, b ) → { (1 , ± ) } All other transitions are empty. ¥ 1
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2. Problem 6.3.3: Convert the PDA P = ( { p, q } , { 0 , 1 } , { X, Z 0 } , δ, q, Z 0 ) to a CFG, if δ is given by δ ( q, 1 , Z 0 ) = { ( q, XZ 0 ) } δ ( q, 1 , X ) = { ( q, XX ) } δ ( q, 0 , X ) = { ( p, X ) } δ ( q, ±, X ) = { ( q, ± ) } δ ( p, 1 , X ) = { ( p, ± ) } δ ( p, 0 , Z 0 ) = { ( q, Z 0 ) } Solution: we seek to construct G = ( V, Σ , R, S ) where L ( G ) = N ( P ) as seen on page 242 of the text. The set of variables, V , consists of (1) the start symbol S and (2) all symbols of the form [ pXq ] where p and q are states in Q and X is a stack symbol in Γ . Thus, our variables are as follows S, [ qXq ] , [ qZ 0 q ] , [ qXp ] , [ qZ 0 p ] , [ pXq ] , [ pZ 0 q ] , [ pXp ] , [ pZ 0 p ] This is simply all combinations of states and stack symbols Now let's create the productions. First we will start with the special start production. We know q is our start symbol and we could go to either p or stay at q
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This note was uploaded on 09/04/2008 for the course CS 381 taught by Professor Hopcroft during the Fall '06 term at Cornell University (Engineering School).

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Hw8-solutions - CS 381 HW 8 Solutions Dan FitzGerald [email protected] November 1 2006 1 Convert the following CFG to a one state PDA S bA S aB A a

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