BU CS 332 Theory of Computation Lecture 7: Reading: More on CFGs - - PowerPoint PPT Presentation

BU CS 332 – Theory of Computation

Lecture 7:

• More on CFGs
• Pushdown Automata

Reading: Sipser Ch 2.1‐2.3

Mark Bun February 12, 2020

Context‐Free Grammar (Formal)

A CFG is a 4‐tuple

• is a finite set of variables
• is a finite set of terminal symbols (disjoint from )
• is a finite set of production rules of the form

, where and

∗

• is the start symbol

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Context‐Free Grammar

Example Grammar Parse Tree | | | Derivation

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Context‐Free Languages

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𝑀 is a context‐free language if it is the language of some CFG

Questions about CFLs

• 1. Which languages are not

context‐free?

• 2. How do we recognize

whether 𝑥 ∈ 𝑀?

• 3. What are the closure

properties of CFLs?

Pumping Lemma for context‐free languages

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Let be a context‐free language. Then there exists a “pumping length” such that 1. 2. 3.

𝑗 𝑗 

for all For every where , can be split into five parts where:

Pumping Lemma example

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Claim:

• is not context‐free

Proof: Assume is context‐free with pumping length

• 1. Find

with

• 2. Show that

cannot be pumped If = with | | , then… Case 1: both contain only one kind of symbol Case 2: Either or contains two kinds of symbols

Pumping Lemma example

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Claim:

• is not context‐free

Proof: Assume is context‐free with pumping length

• 1. Find

with

• 2. Show that

cannot be pumped If = with | | , then… Case 1: both contain only one kind of symbol

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Pumping Lemma example

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Claim:

• is not context‐free

Proof: Assume is context‐free with pumping length

• 1. Find

with

• 2. Show that

cannot be pumped If = with | | , then… Case 2: Either or contains two kinds of symbols

Pumping Lemma: Proof idea

Let be a context‐free language. If is long enough, then every parse tree for has a repeated variable.

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Pumping Lemma Proof

What does “long enough” mean? (How do we choose the pumping length ?)

• Let

be a CFG for

• Suppose the right‐hand side of every rule in

uses at most symbols

• Let

Claim: If with , then the smallest parse tree for has height at least

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Pumping Lemma Proof

Claim: If with , then the smallest parse tree for has height at least

• By the pigeonhole principle, there is a path down the parse tree

with a repeated variable 𝑆

• Choose two such occurrences within the bottom 𝑊 1 levels

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Context‐Free Languages

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𝑀 is a context‐free language if it is the language of some CFG

Questions about CFLs

• 1. Which languages are not

context‐free?

• 2. How do we recognize

whether 𝑥 ∈ 𝑀?

• 3. What are the closure

properties of CFLs?

Pushdown Automata

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Regular Expressions : Finite Automata :: Context‐Free Languages : ???

Pushdown Automata

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Input 𝑏 𝑐 𝑏 𝑏 Finite control … Input 𝑏 𝑐 𝑏 𝑏 Finite control … 𝑦 𝑦 𝑧 Memory: Infinite Stack

Pushdown Automata (PDAs): Machine with unbounded structured memory in the form of a stack Finite Automata (FAs): Machine with a finite amount of unstructured memory

Pushdown Automaton (the idea)

• Nondeterministic finite automaton + stack
• Stack has unlimited size, but machine can only manipulate (push,

pop, read) symbol at the top

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Input 𝑏 𝑐 𝑏 𝑏 Finite control … 𝑦 𝑦 𝑧 Memory: Infinite Stack

Transitions of the form: 𝑏, 𝑦 → 𝑦′

Example: Even Palindromes

• ∗

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Input 𝑏 𝑐 𝑐 𝑐 𝑐 𝑏 Finite control … Memory: Infinite Stack

Example: Even Palindromes

• ∗

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Input 𝑏 𝑐 𝑐 𝑐 𝑐 𝑏 Finite control … Memory: Infinite Stack

Algorithmic Description

1. Place the marker $ on the stack 2. Nondeterministically, either a) Read a character and push it to the stack, or b) Go to the next step 3. Nondeterministically, either a) Pop the stack if it matches the next character or b) Go to the next step 4. Accept if the top of the stack is $

Example: Even Palindromes

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𝜁, 𝜁 → $ 𝑏, 𝜁 → 𝑏 𝜁, 𝜁 → 𝜁 𝑏, 𝑏 → 𝜁 𝜁, $ → 𝜁

𝑔

𝑐, 𝜁 → 𝑐 𝑐, 𝑐 → 𝜁

Pushdown Automaton (formal)

A PDA is a 6‐tuple (sorry)

• 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

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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 .

Example: Even Palindromes

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𝜁, 𝜁 → $ 𝑏, 𝜁 → 𝑏 𝜁, 𝜁 → 𝜁 𝑏, 𝑏 → 𝜁 𝜁, $ → 𝜁

𝑔

𝑐, 𝜁 → 𝑐 𝑐, 𝑐 → 𝜁 𝜀 : 𝑅 Σ Γ

→ 𝑄 𝑅 Γ

• 𝜀 𝑞, 𝑐, 𝜁

𝜀 𝑟, 𝑏, 𝑏 𝜀 𝑟, 𝑏, 𝑐