1. Lecture # 28 Theory Of Automata By Dr. MM Alam 1
2. Lecture 27 recap • Chomsky Normal Form conversion in JFLAP • Push Down Automata Definition • PDA Symbols 2
3. Adding A Pushdown Stack • A PUSHDOWN STACK is a place where input letters can be stored until we want to refer to them again. • It holds the letters it has been fed in a long line. The operation PUSH adds a new letter to the line. • The new letter is placed on top of the STACK, and all the other letters are pushed back (or down) accordingly. • Before the machine begins to process an input string the STACK is presumed to be empty, which means 3 that every storage location in it initially contains a
4. Adding A Pushdown Stack • If the STACK is then fed the letters a, b, c, d by this sequence of instructions: PUSH a PUSH b PUSH c PUSH d • Then top letter in the STACK is d, the second is c, the third is b, and the fourth is a. • If we now execute the instruction: • PUSH b the letter b will be added to the STACK on the top. The d will be pushed down to 4position 2,
5. Adding A Pushdown Stack • One pictorial representation of a STACK with these letters in it is shown below. • Beneath the bottom a we presume that the rest of the STACK, which, like the INPUT TAPE, has infinitely many storage locations, holds only blanks.  b  d  c  b  a  Δ   5
6. Adding A Pushdown Stack • How the following PDA is working: a b 6
7. Adding A Pushdown Stack • Its operation on the • input string aaabbb. • We begin by assuming • that this string has • been put on the TAPE. STACK Δ TAPE a a a b b b Δ 7
8. Adding A Pushdown Stack • Its operation on the • input string aaabbb. • We begin by assuming • that this string has been • put on the TAPE. STACK a Δ TAPE a a a b b b Δ 8
9. Adding A Pushdown Stack • We now read another a • and proceed as before along • the a edge to push it into • the STACK. • Again we are returned • to the READ box. STACK • Again we read an a (our third), • and again this a is pushed onto the STACK.a a TAPE a a a b b b Δ a Δ 9
10. Adding A Pushdown Stack • After the third PUSH a, • we are routed back to • the same READ state again. • This time, we read the letter b. • This means that we take the • b edge out of this state down STACK • to the lower left POP. a a TAPE a a a b b b Δ Δ 10
11. Adding A Pushdown Stack • The b road from the second • READ state now takes us • back to the edge feeding • into the POP state. • So we pop the STACK • again and get another a. • The STACK is now • down to only one a. a • The a line from POP Δ • takes TAPE a to us again a this a bsameb READ. b Δ • There is only one letter left on the input TAPE, a b. STACK 11
12. Adding A Pushdown Stack • We read it and • leave the TAPE empty, • that is, all blanks. • However, the machine does • not yet know that the TAPE • is empty. • It will discover this only when • it next tries to read the TAPE Δ • and TAPE finds Δ. a a a b b b Δ 12
13. Adding A Pushdown Stack • Let 13
14. Adding A Pushdown Stack 14
15. Example Example • The PALINDROMEX, language of all words of the form s X reverse(s) where s is any string in (a + b)*. • The words in this language are { X aXa bXb aaXaa abXba baXab bbXbb aaaXaaa aabXbaa ...} • They all contain exactly one X, and this X marks the middle ofthe word. • We can build a deterministic PDA that accepts the language PALINDROMEX. • It has the same basic structure as the PDA we had for the language {anbn}. 15
16. Adding A Pushdown Stack Example • In the first part of the machine the STACK is loaded with the letters from the input string just as the initial a's from anbn were pushed onto the STACK. • The letters go into the STACK first letter on the bottom, second letter on top of it, and so on till the last letter pushed in ends up on top. • When we read the X we know we have reached the middle of the input. 16
17. Adding A Pushdown Stack Example • We can then begin to compare the front half of the word (which is reversed in the STACK) with the back half (still on the TAPE) to see that they match. • We begin by storing the front half of the input string in the STACK with this part of the machine. 17
18. Adding A Pushdown Stack Example • If we READ an a, we PUSH an a. If we READ a b, we PUSH a b, and on and on until we encounter the X on the TAPE. • After we take the first half of the word and stick it into the STACK, we have reversed the order of the letters and it looks exactly like the second half of the word. • For example, if we begin with the input string abbXbba 18
19. Adding A Pushdown Stack Example • At the moment we are just about to read the X we have: b b a b b X b b a Δ a Δ 19
20. Adding A Pushdown Stack Example • When we read the X we do not put it into the STACK. It is used up the process of transferring us to phase two. • In order to reach ACCEP these two should be the same letter for letter, down to the blanks.: 20