BU CS 332 –Theory of Computation · 2020-02-18 · BU CS 332 –Theory of Computation Lecture 8: • Equivalence between PDAs and CFGs • Closure Properties Reading: Sipser Ch

BU CS 332 – Theory of Computation

Lecture 8:• Equivalence between PDAs and CFGs• Closure Properties

Reading:Sipser Ch 2.2

Mark BunFebruary 18, 2020

Pushdown Automaton (the idea)• Nondeterministic finite automaton + stack• Stack has unlimited size, but machine can only manipulate (push, pop, read) symbol at the top

2/18/2020 CS332 ‐ Theory of Computation 2

Input 𝑎 𝑏 𝑎 𝑎

Finite control

…

𝑥𝑥𝑦 Memory: Infinite Stack

Transitions of the form:𝑎, 𝑥 → 𝑥′

Example: Even Palindromes


𝜀, 𝜀 → $ 𝑎, 𝜀 → 𝑎

𝜀, 𝜀 → 𝜀

𝑎,𝑎 → 𝜀𝜀, $ → 𝜀

0

𝑓

𝑏, 𝜀 → 𝑏

𝑏, 𝑏 → 𝜀

Pushdown Automaton (formal)A PDA is a 6‐tuple • is a finite set of states• is the input alphabet• is the stack alphabet• : is the transition function• is the start state• is the set of final states


accepts a string if, starting from 0 and an empty stack, there exists a path to an accept state that can be followed by reading all of .

Examplehas an equal number of ’s and ’s

Algorithmic description


Examplehas an equal number of ’s and ’s

State diagram


CFGs vs. PDAsThe language of a PDA is the set of all strings it accepts.

Theorem: The class of languages recognized by PDAs is exactly the context‐free languages.

Theorem 1: Every CFG has an equivalent PDATheorem 2: Every PDA has an equivalent CFG


CFG ‐> PDA


CFG ‐> PDA ConversionSuppose language is generated by CFG =

Goal: Construct a PDA recognizing

Idea: will guess the steps of the CFG derivation of its input , and use its stack to check the derivation

A helpful intermediate abstractionGeneralized PDA: Can push an entire string to the stack in one move


Algorithmic Description1. Place and the start variable on the stack2. Repeat forever:

a) If the top of the stack holds variable :Nondeterministically guess a rule Replace with on the stack

b) If the top of the stack holds terminal :Pop and verify that it matches the next input char

c) If the top of the stack holds :Accept if the input is empty


State Diagram


𝜀,𝐴 → 𝑢 [for every rule 𝐴 → 𝑢]𝜎,𝜎 → 𝜀 [for every terminal 𝐴 → 𝜎]

𝜀, 𝜀 → 𝑆$

𝜀, $ → 𝜀

0

𝑓

𝑙𝑜𝑜𝑝

Example


𝜀, 𝜀 → 𝑆$

𝜀, $ → 𝜀

0

𝑓

𝑙𝑜𝑜𝑝

Example


𝜀, $ → 𝜀

0

𝑓

𝑙𝑜𝑜𝑝

PDA ‐> CFG


PDA ‐> CFG ConversionTheorem 2: Every PDA has an equivalent CFGSuppose is recognized by PDAGoal: Construct a CFG = generating

First simplify so that:1. It has a single accept state 𝑓

2. It empties the stack before accepting3. Every transition either pushes a symbol or pops a

symbol (but not both)


Simplification Example


𝜀, 𝜀 → $ 𝑎, 𝜀 → 𝑎

𝜀, 𝜀 → 𝜀

𝑎,𝑎 → 𝜀𝜀, $ → 𝜀

0

𝑓

𝑏, 𝜀 → 𝑏

𝑏, 𝑏 → 𝜀

Conversion IdeaVariables: for every pair of states in PDA Idea: generates all strings that can take from (with an empty stack) to (with an empty stack)


Example


𝜀, 𝜀 → $ 𝑎, 𝜀 → 𝑎

𝜀, 𝜀 → #

𝑎,𝑎 → 𝜀𝜀, $ → 𝜀

0 𝑏, 𝜀 → 𝑏

𝑏, 𝑏 → 𝜀

𝜀, # → 𝜀

𝜀, 𝜀 → #

𝜀, 𝜀 → #

What strings should 𝐴 generate?

1

2

34

5



6

𝜀, # → 𝜀

What rules should define ?

Let be a string generated by

Two cases:1) Stack first empties after reading all of

2) Stack empties before reaching the end of


stackheight

inputstring 𝑝 𝑞

stackheight

𝑝 𝑞inputstring

1. Stack first empties after reading all of


stackheight

inputstring

Add rule

2. Stack empties before reaching the end of


stackheight

inputstring

Add rule

Formal CFG Construction

Three kinds of rules:1. For every :If and ,

include the rule

2. For every include the rule 3. For every include the rule


Sketch of proof that CFG generates Claim: ∗ if and only if takes from to , beginning and ending with empty stackProof idea:

Induction on number of steps of derivation of from

Induction on number of steps of computation taking from to


Closure Properties


Closure Properties

• The class of CFLs is closed underUnionConcatenationStarIntersection with regular languages

• Beware: It is not closed under intersection or complement(Find counterexamples!)


Closure under union (Proof 1)Let be a CFL recognized by PDA and let be a CFL recognized by PDA Goal: Construct a PDA recognizing


Closure under union (Proof 2)Let be a CFL generated by CFG and let be a CFL recognized by CFG Goal: Construct a CFG recognizing

= =

Relabel variables so and are disjoint

Let =


BU CS 332 –Theory of Computation · 2020-02-18 · BU CS 332 –Theory of Computation Lecture 8: • Equivalence between PDAs and CFGs • Closure Properties Reading: Sipser Ch

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