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BU CS 332 Theory of Computation Lecture 6: Reading: NFAs > - - PowerPoint PPT Presentation

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BU CS 332 Theory of Computation Lecture 6: Reading: NFAs > Regular expressions Sipser Ch 1.3, Context free grammars 2.1, 2.3 Pumping lemma for CFLs Mark Bun February 10, 2020 Regular Expressions Syntax A regular

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BU CS 332 – Theory of Computation

Lecture 6:

  • NFAs ‐> Regular expressions
  • Context‐free grammars
  • Pumping lemma for CFLs

Reading: Sipser Ch 1.3, 2.1, 2.3

Mark Bun February 10, 2020

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Regular Expressions – Syntax

A regular expression is defined recursively using the following rules: 1. , , and are regular expressions for every

  • 2. If

and are regular expressions, then so are

  • ,
  • , and
  • ∗

Examples: (over )

∗ ∗ ∗ ∗

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Regular Expressions – Semantics

the language a regular expression describes 1. 2. 3. for every

  • 6.
  • ∗
  • ∗

Example:

∗ ∗

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Regular Expressions Describe Regular Languages

Theorem: A language is regular if and only if it is described by a regular expression Theorem 1: Every regular expression has an equivalent NFA Theorem 2: Every NFA has an equivalent regular expression

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NFA ‐> Regular expression

Theorem 2: Every NFA has an equivalent regex Proof idea: Simplify NFA by “ripping out” states one at a time and replacing with regexes

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1 01*0

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Generalized NFAs

  • Every transition is labeled by a regex
  • One start state with only outgoing transitions
  • Only one accept state with only incoming transitions
  • Start state and accept state are distinct

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∗

𝑡 𝑏

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Generalized NFA Example

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𝑡 𝑏

∗

𝑡 𝑏

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NFA ‐> Regular expression

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NFA GNFA GNFA GNFA Regex 𝑙 states 𝑙 2 states 𝑙 1 states 2 states …

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NFA ‐> GNFA

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NFA

ε ε ε ε

  • Add a new start state with no incoming arrows.
  • Make a unique accept state with no outgoing arrows.
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GNFA ‐> Regular expression

Idea: While the machine has more than 2 states, rip one

  • ut and relabel the arrows with regexes to account for the

missing state

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∗

1 3 2 1 3

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GNFA ‐> Regular expression

Idea: While the machine has more than 2 states, rip one

  • ut and relabel the arrows with regexes to account for the

missing state

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∗

1 3 2 1 3

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GNFA ‐> Regular expression

Idea: While the machine has more than 2 states, rip one

  • ut and relabel the arrows with regexes to account for the

missing state

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∗

1 3 2 1 3

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GNFA ‐> Regular expression

Idea: While the machine has more than 2 states, rip one

  • ut and relabel the arrows with regexes to account for the

missing state

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1 3 2 1 2 3 4 1 3

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Example

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Context‐Free Grammars

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Some History

An abstract model for two distinct problems Rules for parsing natural languages

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Some History

An abstract model for two distinct problems Specification of syntax and compilation for programming languages

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1977 ACM Turing Award citation (John Backus) For profound, influential, and lasting contributions to the design of practical high‐ level programming systems, notably through his work on FORTRAN, and for seminal publication of formal procedures for the specification of programming languages.

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Context‐Free Grammar (Informal)

Example Grammar Derivation

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Context‐Free Grammar (Informal)

Example Grammar 𝐻 𝐹 → 𝐹 𝑈 𝐹 → 𝑈 𝑈 → 𝑈 𝐺 𝑈 → 𝐺 𝐺 → 𝐹 𝐺 → 𝑏 𝐺 → 𝑐 Derivation 𝑀𝐻

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Socially Awkward Professor Grammar

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<PHRASE> → <START><END> <PHRASE> → <FILLER><PHRASE> <FILLER> → LIKE <FILLER> → UMM <START> → YOU KNOW <END> → WHOOPS <START> → ε <END> → SORRY <END> → $#@!

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Socially Awkward Professor Grammar

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<PHRASE> → <FILLER><PHRASE> | <START><END> <FILLER> → LIKE | UMM <START> → YOU KNOW | ε <END> → WHOOPS | SORRY | $#@!

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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

Example: where

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Context‐Free Grammar (Formal)

A CFG is a 4‐tuple = variables = terminals = rules = start

  • We say

(“ yields ”) if is a rule of the grammar

  • We say

⇒ ∗

(“ derives ”) if

  • r there exists a

sequence such that

  • Language of the grammar:

∗ ⇒ ∗

Example: where

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CFG Examples

Give context‐free grammars for the following languages

  • 1. The empty language
  • 2. Strings of properly nested parentheses
  • 3. Strings with equal # of ’s and ’s

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Pumping Lemma II: Pump Harder

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Non context‐free languages?

  • Could it be the case that every language is context‐free?

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Pumping Lemma for regular languages

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

𝑗 

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

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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: Example:

∗

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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: Example:

∗

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Pumping Lemma as a game

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  • 1. YOU pick the language 𝑀 to be proved non context‐free.
  • 2. ADVERSARY picks a possible pumping length 𝑞.
  • 3. YOU pick 𝑥 of length at least 𝑞.
  • 4. ADVERSARY divides 𝑥 into 𝑣, 𝑤, 𝑦, 𝑧, 𝑨, obeying rules of the

Pumping Lemma: |𝑤𝑧| 0 and |𝑤𝑦𝑧| 𝑞.

  • 5. YOU win by finding 𝑗 0, for which 𝑣𝑤𝑗𝑦𝑧𝑗𝑨 is not in 𝑀.

If regardless of how the ADVERSARY plays this game, you can always win, then is non context‐free

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Pumping Lemma example

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Claim:

  • is not regular

Proof: Assume is regular with pumping length

  • 1. Find

with

  • 2. Show that

cannot be pumped If = with | | , then…