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BU CS 332 Theory of Computation Lecture 2: Reading: Deterministic - - PowerPoint PPT Presentation

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BU CS 332 Theory of Computation Lecture 2: Reading: Deterministic Finite Automata Sipser Ch 1.1 1.2 Regular Operations Non deterministic FAs Mark Bun January 27, 2020 Deterministic Finite Automata 1/29/2020 CS332

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BU CS 332 – Theory of Computation

Lecture 2:

  • Deterministic Finite

Automata

  • Regular Operations
  • Non‐deterministic FAs

Reading: Sipser Ch 1.1‐1.2

Mark Bun January 27, 2020

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Deterministic Finite Automata

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A (Real‐Life?) Example

  • Example: Car stereo
  • = Power button (ON/OFF)
  • = Source button (cycles through Radio/Bluetooth/USB)

Only works when stereo is ON, but source remembered when stereo is OFF

  • Starts OFF in Radio mode
  • A computational problem: Does a sequence of button

presses in

∗ leave the stereo ON in USB mode?

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Machine Models

  • Finite Automata (FAs): Machine with a finite amount of

unstructured memory

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Input 𝑄 𝑇 𝑄 𝑇 Finite control …

Control scans left‐to‐right

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A DFA for the car stereo problem

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A DFA for Parity

Parity: Given a string consisting of ’s and ’s, does it contain an even number of ’s? = = contains an even number of ’s

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Anatomy of a DFA

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𝒓𝟑 0,1 1 1 1 𝒓𝟏 𝒓𝟐 𝒓𝟒

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Formal Definition of a DFA

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is the set of states is the alphabet  is the transition function

0 

is the start state is the set of accept states

A finite automaton is a 5‐tuple 

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A DFA for Parity

Parity: Given a string consisting of ’s and ’s, does it contain an even number of ’s? = = contains an even number of ’s

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1

State set = Alphabet = Transition function Start state Set of accept states =

𝜀 𝑏 𝑐 𝑟0 𝑟1

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Formal Definition of DFA Computation

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= the language of machine = set of all (finite) strings machine accepts recognizes the language A DFA  accepts a string

  • ∗ (where each
  • ) if there exist

, . . . ,

such that 1.

  • 2.
  • for each

and 3.

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Ex Exam ample: ple: Computing with the Parity DFA

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1

A DFA  accepts a string

  • ∗ (where each
  • ) if there exist

, . . . ,

such that 1.

  • 2.
  • for each

and 3.

Let 𝑥 𝑏𝑐𝑐𝑏 Does 𝑁 accept 𝑥?

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Automata Tutor

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http://automatatutor.com/

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Regular Languages

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Definition: A language is regular if it is recognized by a DFA

𝑴 = { 𝒙 ∈ 𝟏, 𝟐 ∗| 𝒙 contains 𝟏𝟏𝟐 } is regular 𝑴 = { 𝒙 ∈ 𝒃, 𝒄 ∗ | 𝒙 has an even number of 𝒃’s } is regular

Many interesting programs recognize regular languages

NETWORK PROTOCOLS COMPILERS GENETIC TESTING ARITHMETIC

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Internet Transmission Control Protocol

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Let TCPS = { | is a complete TCP Session}

  • Theorem. TCPS is regular
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Compilers

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Comments : Are delimited by /* */ Cannot have nested /* */ Must be closed by */ */ is illegal outside a comment

COMMENTS = {strings over {0,1, /, *} with legal comments}

  • Theorem. COMMENTS is regular.
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Genetic Testing

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DNA sequences are strings over the alphabet 𝑩, 𝑫, 𝑯, 𝑼. A gene 𝒉 is a special substring over this alphabet. A genetic test searches a DNA sequence for a gene. GENETICTEST𝒉 = {strings over 𝑩, 𝑫, 𝑯, 𝑼 containing 𝒉 as a substring}

  • Theorem. GENETICTEST𝒉 is regular for every gene 𝒉.
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Arithmetic

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LET 3 =

  • A string over 3 has three ROWS (ROW1, ROW2, ROW3)
  • Each ROW

𝟏 𝟐 𝟑 𝑶 represents the integer 𝟏 𝟐 𝑶 𝑶.

  • Let ADD = {

𝟒

∗

| ROW1 + ROW2 = ROW3 }

  • Theorem. ADD is regular.

{ [ ],[ ],[ ],[ ], [ ],[ ],[ ],[ ]}

1 1 1 1 1 1 1 1 1 1 1 1

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Regular Operations

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An Analogy

In algebra, we try to identify operations which are common to many different mathematical structures

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Example: The integers are closed under

  • Addition:
  • Multiplication: ×
  • Negation:
  • …but NOT Division:

We’d like to investigate similar closure properties of the class of regular languages

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Regular operations on languages

Let

∗ be languages. Define

Union: Concatenation: Star:

∗

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Other operations

Let

∗ be languages. Define

Complement: Intersection: Reverse:

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Closure properties of the regular languages

Theorem: The class of regular languages is closed under all three regular operations (union, concatenation, star), as well as under complement, intersection, and reverse. i.e., if and are regular, applying any of these

  • perations yields a regular language

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Proving Closure Properties

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Complement

Complement: Theorem: If is regular, then is also regular Proof idea:

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Union

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Union:

  • r

Theorem: If and are regular, then so is Proof: Let

𝐵 𝐵

𝐵

  • 𝐵

be a DFA recognizing and

𝐶 𝐶

𝐶

  • 𝐶

be a DFA recognizing Goal: Construct a DFA 

  • that recognizes
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Example

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  • 1

1

  • 1

1

𝑩 𝑪

?

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Closure under union proof (cont’d)

Idea: Run both

𝐵 and 𝐶 at the same time

“Cross‐product construction”

𝐵 × 𝐶

,

𝐵 𝐶

 𝐵

𝐵  𝐶 𝐶 

  • ,
  • 𝐵
  • 𝐶

𝐵 × 𝐶 𝐵 × 𝐶

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Example (cont’d)

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  • 1

1

  • 1

1

𝑩 𝑪

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Intersection

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Intersection: and Theorem: If and are regular, then so is Proof: Let

𝐵 𝐵

𝐵

  • 𝐵

be a DFA recognizing and

𝐶 𝐶

𝐶

  • 𝐶

be a DFA recognizing Goal: Construct a DFA 

  • that recognizes
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Intersection

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Intersection: and Theorem: If and are regular, then so is Another Proof: =

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Reverse

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Reverse:

  • Theorem: If

is regular, then

is also regular

Proof idea: Let 

  • be a DFA recognizing

Goal: Construct a DFA 

  • that recognizes
  • Define

as but

  • With the arrows reversed
  • With start and accept states swapped
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Example (Reverse)

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1 1 1 0,1

𝑵 𝑵′

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Closure under reverse

is not always a DFA!

  • It might have many start states
  • Some states may have too many outgoing edges, or

none at all

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Non Nondeterminism

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1 1 1 0,1 A Nondeterministic Finite Automaton (NFA) accepts if there is a way to make it reach an accept state.