Push-down Automata

1. Push-down Automata Section 3.3 Fri, Oct 21, 2005

2. Push-Down Automata • Recall that DFAs accept regular languages. • We want to design machines similar to DFAs that will accept context-free languages. • These machines will need to be more powerful. • To handle a language like {anbn | n 0}, the machine needs to “remember” the number of as. • To do this, we use a stack. • A push-down automaton (PDA) is essentially an NFA with a stack.

3. Definition of a Push-Down Automaton • A push-down automaton (PDA) is a sextuple (K, Σ, Γ, Δ, s, F) where • K is a finite set of states. • Σ is a finite alphabet of tape symbols. • Γ is a finite alphabet of stack symbols. • sK is the start state. • FK is the set of final states. • Δ is the transition relation, i.e., Δ is a finite subset of (K (Σ  {e})  Γ*)  (K Γ*).

4. Transitions • Let ((p, a, β), (q, γ))  Δ be a transition. • It means that we • Move from state p. • Read a from the tape, • Pop the string β from the stack, • Move to state q, • Push string γ onto the stack. • The first three (p, a, β), are “input.” • The last two (q, γ) are “output.”

5. a, ;  p q Transitions • We will draw it as

6. Pushing and Popping • When we push β, we push the symbols of β as we read them right to left. • When we push the string abc, we push c, then push b, then push a. • When we pop γ, we pop the symbols of γ as we read them from left to right (reverse order). • When we pop the string abc, we pop a, then pop b, then pop c.

7. Pushing and Popping • Thus, if we push the string abc and then pop it, we will get back abc, not cba. • If we wanted to reverse the order, we would use three separate transitions: • Push a • Push b • Push c

8. Configurations • A configuration fully describes the current “state” of the PDA. • The current state p. • The remaining input w. • The current stack contents . • Thus, a configuration is a triple (p, w, )  (K, *, *).

9. Computations • A configuration (p, w, ) yields a configuration (p', w', ') in one step, denoted (p, w, ) (p', w', '), if there is a transition ((p, a, ), (p', )) Δ such that w = aw',  = , and ' =  for some  *. • The reflexive, transitive closure of  is denoted *.

10. Accepting Strings • After processing the string on the tape, • The PDA is in either a final or a nonfinal state, and • The stack is either empty or not empty. • The input string is accepted if • The ending state is a final state, and • The stack is empty. • That is, the string w* is accepted if (s, w, e) * (f, e, e) for some fF.

11. Accepting Strings • One may define acceptance “by final state” only. • The input is accepted if and only if the last state is a final state, regardless of whether the stack is empty. • One may define acceptance “by empty stack” only. • The input is accepted if and only if the stack is empty once the input is processed, regardless of which state the PDA is in.

12. a, e; a b, a; e a, e; a s p q b, a; e Example of a PDA • Run the following PDA on the input string aaabbb.

13. Example of a PDA • The steps in the processing are • (s, aaabbb, e)  (p, aabbb, a)  (p, abbb, aa)  (p, bbb, aaa)  (p, bb, aa)  (p, b, a)  (q, e, e).

14. The Language of a PDA • The language of a PDA A is L(A) = {w  *A accepts w}. • What is the language of the PDA in the previous example?

15. a, e; a b, a; e e, e; e s p Example of a PDA • What is the language of the following PDA?

16. Examples of PDAs • Let  = {a, b}. • Design a PDA that accepts the language {wcwR | w *}. • Design a PDA that accepts the language {wwR | w *}.

17. Examples of PDAs • Design a PDA whose language is {ambn 0 m < n}. • Design a PDA whose language is {ambn 0 n < m}.

18. Examples of PDAs • Design a PDA whose language is {ambncndm | m  0, n  0}. • Design a PDA whose language is {ambmcndn | m  0, n  0}. • Design a PDA whose language is {ambncpdq | m + n = p + q}. • Design a PDA whose language is {ambnck | m = n or m = k}.