CSC 3130: Automata theory and formal languages Andrej Bogdanov The Chinese University of Hong Kong Pushdown automata Fall 2008

Motivation We had two ways to describe regular languages How about context-free languages? regular expression DFANFA syntactic computational CFGpushdown automaton syntactic computational

Pushdown automata versus NFA Since context-free is more powerful than regular, pushdown automata must generalize NFAs state control 0100 input NFA

Pushdown automata A pushdown automaton has access to a stack, which is a potentially infinite supply of memory state control 0100 input pushdown automaton (PDA) … stack

Pushdown automata As the PDA is reading the input, it can push / pop symbols in / out of the stack state control 0100 input pushdown automaton (PDA) Z0Z0 01 stack … 1

Rules for pushdown automata The transitions are nondeterministic Stack is always accessed from the top Each transition can pop a symbol from the stack and / or push another symbol onto the stack Transitions depend on input symbol and on last symbol popped from stack Automaton accepts if after reading whole input, it can reach an accepting state

Example L = {0 n #1 n : n ≥ 0} state control 000# input Z0Z0 aa stack a … 111 read 0 push a read # read 1 pop a pop Z 0

Shorthand notation read 0 push a read # read 1 pop a pop Z 0 0,  / a #,  /  , Z 0 /  1, a /  read, pop / push

Formal definition A pushdown automaton is (Q, , , , q 0, Z 0, F) : –Q is a finite set of states; –  is the input alphabet; –  is the stack alphabet, including a special symbol Z 0 ; –q 0 in Q is the initial state; –Z 0 in  is the start symbol; –F  Q is a set of final states; –  is the transition function  : Q  (   {  })  (   {  }) → subsets of Q  (   {  }) stateinput symbolpop symbolstatepush symbol

Notes on definition We use slightly different definition than textbook Example 0,  / a #,  / , Z 0 /  1, a /   : Q  (   {  })  (   {  }) → subsets of Q  (   {  }) q0q0 q1q1 q2q2  (q 0, 0,  ) = {(q 0, a)}  (q 0, 1,  ) = ∅  (q 0, #,  ) = {(q 1,  )}  (q 0, 0,  ) = ∅...

A convention Sometimes we denote “transitions” by: This will mean: –Intuitively, pop b, then push c 1, c 2, and c 3 a, b  / c 1 c 2 c 3 q0q0 q1q1 ,  / c 2 q0q0 q1q1 a, b  / c 1 ,  / c 3 intermediate states

Examples Describe PDAs for the following languages: –L = {w#w R : w  ∈  *},  = {0, 1, #} –L = {ww R : w  ∈  *},  = {0, 1} –L = {w: w has same number of 0s and 1s},  = {0, 1} –L = {0 i 1 j : i ≤ j ≤ 2i},  = {0, 1}

Main theorem A language L is context-free if and only if it is accepted by some pushdown automaton. context-free grammarpushdown automaton

From CFGs to PDAs Idea: Use PDA to simulate (rightmost) derivations A → 0A1 A → B B → # A  0A1  00A11  00B11  00#11 PDA control: CFG: write start variable stack: Z0AZ0A replace production in reverse Z 0 1A0 pop terminals and match Z 0 1A e, e / A 0, 0 / e e, A / 1A0 input: 00#11 0#11 replace production in reverse Z 0 11A0e, A / 1A0 0#11 pop terminals and match Z 0 11A0, 0 / e #11 replace production in reverse Z 0 11Be, A / B #11

From CFGs to PDAs If, after reading whole input, PDA ends up with an empty stack, derivation must be valid Conversely, if there is no valid derivation, PDA will get stuck somewhere –Either unable to match next input symbol, –Or match whole input but stack non empty

Description of PDA for CFGs Repeat the following steps: –If the top of the stack is a variable A : Choose a rule A →  and substitute A with  –If the top of the stack is a terminal a : Read next input symbol and compare to a If they don’t match, reject (die) –If top of stack is Z 0, go to accept state

Description of PDA for CFGs q0q0 q1q1 q2q2 ,  / S a, a /  for every terminal a , A /  k...  1 for every production A →  1...  k , Z 0 / 

From PDAs to CFGs First, we simplify the PDA: –It has a single accept state q f –Z 0 is always popped exactly before accepting –Each transition is either a push, or a pop, but not both context-free grammarpushdown automaton ✓

From PDAs to CFGs We look at the stack in an accepting computation: a Z0Z0 Z0Z0 a Z0Z0 a Z0Z0 a Z0Z0 a Z0Z0 a Z0Z0 a Z0Z0 a Z0Z0 a Z0Z0 Z0Z0 baccc a portions that preserve the stack q0q0 q1q1 q3q3 q1q1 q7q7 q0q0 q1q1 q2q2 q1q1 q3q3 q7q7 A 03 = {x: x leads from q 0 to q 3 and preserves stack}  11  01  0 00 input state stack qfqf

From PDAs to CFGs a Z0Z0 Z0Z0 a Z0Z0 a Z0Z0 a Z0Z0 a Z0Z0 a Z0Z0 a Z0Z0 a Z0Z0 a Z0Z0 Z0Z0 baccc a q0q0 q1q1 q3q3 q1q1 q7q7 q1q1 q2q2 q1q1 q7q7  11  01  0 00 input state stack A 11 A 03 0  A 11 → 0A 03  q0q0 q3q3 qfqf

From PDAs to CFGs a Z0Z0 Z0Z0 a Z0Z0 a Z0Z0 a Z0Z0 a Z0Z0 a Z0Z0 a Z0Z0 a Z0Z0 a Z0Z0 Z0Z0 baccc a q0q0 q1q1 q3q3 q1q1 q7q7 q1q1 q2q2 q1q1 q7q7  11  01  0 00 input state stack A 03 A 13 A 13 → A 10 A 03 q0q0 q3q3 A 10 qfqf

From PDAs to CFGs qiqi qjqj a,  / t b, t /  q i’ q j’ A ij → aA i’j’ b qiqi qjqj qkqk A ik → A ij A jk qiqi A ii →  variables: A ij qfqf a, Z 0 /  qiqi A 0f → A 0i a start variable: A0fA0f

Example 0,  / a #,  / , Z 0 /  1, a /  q0q0 q1q1 q2q2 start variable: A 02 productions: A 00 → A 00 A 00 A 00 → A 01 A 10 A 00 → A 03 A 30 A 01 → A 01 A 11 A 01 → A 02 A 21 A 00 → ... A 11 →  A 22 →  A 01 → 0A 01 1 A 01 → #A 33 A 33 →  0,  / a #,  / $ , Z 0 /  1, a /  q0q0 q1q1 q2q2 q3q3 , $ /  A 02 → A 01