Transparency No. 1 Formal Language and Automata Theory Homework 6

Homework Transparency No. 2 PDA 1.Design a PDA (accepting by empty stack) to accept the language L 1 = { x  {a,b}* | x contains the same number of a's and b's. 1.1 Sketch your pseudo code 1.2 Draw its transition diagram 1.3 Give a formal definition of the machine. Namely, suppose the machine is M = (Q, , , ,s, , F), then you should list the content of each component of the machine.

Homework Transparency No. 3 PDA 2. Design a PDA to accept the language L 2 = { xy  {a,b}* | |x| = |y| but x  y }. 1.1 Sketch your pseudo code 1.2 Draw its transition diagram 1.3 Give a formal definition of the machine. Namely, suppose the machine is M = (Q, , , ,s, , F), then you should list the content of each component of the machine. Also remember to specify that your PDA accepts either by empty stack or by final state.

Homework Transparency No. 4 Equivalence of CFG and PDA 3.Find a single-state PDA M = (Q, , , ,s, , { }) which accepts by empty stack and is equivalent to the following CFG: S  ABS | AB A  aA | a B  bA 4.Let PDA M2 = ({p,q}, {0,1}, {A, X}, , p, X, {q}) where X is the bottom symbol, and  is given as follows: (p, 0, X)  (p,AX), (q,1, A)  (q,  ) (p, 0, A)  (p, AA), (q, , A)  (q,  ) (p. 1, A)  (q,  ),(q, , X)  (q,  ). Suppose M2 accepts by final state and empty stack, i.e., x is accepted iff (p, x, X)  * (q, ,  ). Find a CFG equivalent to M2.

Homework Transparency No. 5 STM programming 5. Design a standard Turing machine (TM) M to accept the language of all palindromes P = { x ∈ {0,1}* | x = reverse(x) }. On input x ∈ {0,1}* of length n, the machine will start its execution on the initial configuration (s, [x, 0) and ends on the accept state t with the final configuration (t,[  …  …, -). The following is an informal description for M: 1.R (i.e., Move right ) 2.case(input) of { 3. ‘  ’(blank) => goto_t (accept) 4. ‘0’ => 5. ‘  ’ (i.e., write  ) R Move_right_until_  L(i.e., move left) 6. case (input) of { 7. ‘  ’(blank) => goto t (accept) 8. ‘0’ => ‘  ’ L (Move_left_until_  ) goto_1 9. ‘1’, ‘[‘ => goto_r (reject) } 10. ‘1’ => ‘  ’ R Move_right_until_  L 11. case (input) of { 12. ‘  ’(blank) => goto_t (accept) 13. ‘1’ => ‘  ’ L Move_left_until_  goto _1 14. ‘0’, ‘[‘ => goto_r (reject) } 15.} t : halt r : halt

Homework Transparency No translate the above description into (flow-chart-like) state-transition diagram 5.2 Write your formal result using the following table: state\  \ symbol [01  ss tFhalt rFhalt

Homework Transparency No. 7 CSL and CSG 6. The language A = {xx | x  {a,b}* } is not context-free but context-sensitive. Design a context-sensitive grammar(CSG) for it.