Hw8 - Homework 8  CISC 303 Timo Kötzing([email protected]

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Unformatted text preview: Homework 8  CISC 303 Timo Kötzing ([email protected]) Monday, April 20. Friday, April 24. Handed out: Due Date: Problem 1. (8 points) (i) Design a total acceptor-TM M such that L(M) = {an bn cn | n ≥ 0}. (ii) Design a total function-TM f such that, for any words w0 , w1 , . . . , wn , f ( w0 |w1 | . . . |wn ) = w1 . Problem 2. (8 points) (i) Consider the grammar G0 = ({a, b}, {S, X, Y, Z, V, W }, P, S ) with P as follows. S → V aa | ε (1) X → bS | aba (2) Y → W V W | bba | S (3) Z V → aX | ε → X|Y (4) W → S|Y (6) (5) Compute, iteration by iteration, the reachable symbols of G0 , analogous to Example 2.5.4 in the Lecture Notes. (ii) Consider the grammar G1 = ({a, b}, {S, X, Y, Z, V }, P, S ) with P as follows. S → aY | X (7) X → bZ | aSZ (8) Y → bV b | S (9) Z → aaZaa | ZY a | Xa (10) V → X|ε (11) Compute, iteration by iteration, the productive symbols of G1 , analogous to Example 2.5.6 in the Lecture Notes. (iii) Consider the grammar G2 = ({a, b}, {S, X, Y, Z, V, W }, P, S ) with P as follows. S → bXY a | ZaW (12) X → bX | aX (13) Y → bW b | ε (14) Z → Xb | Za | W a (15) V → aW W a | b (16) W → aba | aSa | ε (17) Apply removeUselessSymbolsCFG, as given in Algorithm 2.5.7 in the Lecture Notes, on G2 . Problem 3. (8 points) Consider the grammar G3 = ({a, b}, {S, X, Y }, P, S ) with P as follows. S → bXY a | a (18) X → bX | aX | SS (19) Y → bXbSSa | ab (20) Apply CFGtoCNF, as given in Algorithm 2.5.13 in the Lecture Notes, on G3 (see Examples 2.5.14 and 2.5.15. Problem 4. (8 points) This is an extra credit problem. 1 Homework 8  CISC 303 Timo Kötzing ([email protected]) (i) Let G = (A, N, P, S ) be a grammar, such that, for all X → α ∈ P , either α = ε, or α = xY for some x ∈ A and Y ∈ N . Construct an NFA M such that L(M) = L(G). Conclude that G accepts a regular language. (ii) Let M be an NFA. Construct a grammar G = (A, N, P, S ) such that L(M) = L(G) and, for all productions X → α ∈ P , either α = ε, or α = xY for some x ∈ A and Y ∈ N . 2 ...
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Hw8 - Homework 8  CISC 303 Timo Kötzing([email protected]

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