Lecture 7 Push Down Automata

1. Lecture 7Push Down Automata • Parsers based on Push-Down Automata (PDA) • Abstract machine containing : states ,transitions , and a push – down stack • The stack distinguishes it from a FSA and allow it to recognize CFG , not just REs • Two type of parsers: • Top –down parser builds parse tree from start non-terminal • Alt each stage , parser guesses which production to apply to extend the partially built tree towards leaves. • Bottom –up parser builds parse tree by starting at the terminal and working up the tree.

2. Push Down Automata ,cont’d • FSA # Stack • Instead of only looking at the next token ,it can use the stack to choose nest state • Formally ,PDA=(K,A,H,D,q0,z0,F) • K is finite set of states • A is finite input alphabet (tokens) • H is finite stack alphabet • q0 E K is the start state • Z0 E H is initial symbol on stack • F is set of final states • D is transition function mapping (q ,a, z) - ((p1,h1).(p2,h2),….) New state New TOS State Input TOS

3. PDA Example • Recognize balance parentheses • K = (1,2) , A = [( , ) , eof ] , H= { ( , start }, q0 = 1 , z0 = start , F = 2 input = eof , TOS = Start D = [ (1 , eof , start)] -> (2, ). pop (1 , ‘(‘ , λ )---> (1 , ‘(‘ ) push (1 , ‘)’ , ‘(‘)---> (1 , λ ) input = ‘(‘ push ‘ (‘ State Input TOS New State New TOS input = ‘)’ TOS = ‘(‘ pop D(1,eof,start) = (2, λ) D(1 , ‘(‘ , λ ) = (1 , ‘(‘) D(1 , ‘)’ , ‘(‘ ) = (1 , λ ) 2 1

4. PDA programming • Common Idioms • D (q , λ , z ) = (p ,h) • When in state q with TOS z ,change to state p replace z with h without consuming input • D (q , a , z) = (p, λ ) • When in state q with TOS z and input a , read the input , pop the stack , and goto state p • A PDA accepts when moves into a final state • PDA can be non-deterministic • Accept if there exists a sequence of moves that load to a final state (with empty stack )

5. Equivalence Theorem • For every CFG G, it is possible to build a NDPDA M such that μ(G) = μ(M) • PDA will perform a leftmost derivation • Each time it processes a non- terminal , PDA non deterministically chooses a production to apply • Stack holds partially derived string • If TOS is non-terminal X • Ignore input token • Pop stack and push a , where X  a is a production • If TOS is terminals • If current input token is c , then pop stack • Otherwise , reject input

6. Parsing Example • Grammar for balanced parentheses : • S  () | (S) | SS • Input Stack Action ()(()) S pop. push SS ()(()) SS pop. push () ()(()) ()S pop. & read )(()) )S pop. & read (()) S pop. Push (S) (()) (S) pop. & read ()) S) pop. Push () ()) ()) pop & read )) )) pop & read ) ) pop & read

7. Construction of PDRAM • M has single state q • A is set of terminals of grammars • H = A U N (non – terminals of G ) • Q0 = q and z0 = S (start symbol) • F = (q) • Construct a transition for each terminal of G • D (q , c , c)  (q , λ) • Construct a transition for each production X  a of G • D (q , λ , X)  (q , a) • M is non deterministic if anon terminal occurs on the LHS of more than one production • If deterministic the grammar is finite

8. Example • S  () | (S) | SS • 1 D (q, ‘(‘ , ‘( )  (q, λ ) 2 D (q, ‘)’,’)’)  (q, λ ) 3 D(q, λ , S)  (q, ‘()’) 4 D (q, λ , S)  (q, ‘(S)’) 5 D (q, λ, S)  (q, ‘SS’) • Input can be rejected because of bad choice • Input Stack Action ()() S rule 4 ()() (S) rule 1 )() S) rule 3 )() ()) ??? • Input can be rejected because it is not in μ(G) • ) S ????

9. Building Parsers • Unfortunately , NDPDA are more powerful than DPDA • Cannot mechanically transform NDPDA into a machine that can be efficiently simulated • However , another algorithm (Early ) can parse an arbitrary CFG in O(n2) • But, for certain classes of CFG, can build deterministic parsers that run in O(n) time