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

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

Deterministic Finite Automata

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A (Real-Life?) Example

• Example: Car stereo
• 𝑄 = Power button (ON/OFF)
• 𝑇 = 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 {𝑄, 𝑇}∗ 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

A DFA for the car stereo problem

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A DFA for Parity

Parity: Given a string consisting of 𝑏’s and 𝑐’s, does it contain an even number of 𝑏’s? Ʃ = {𝑏, 𝑐} 𝑀 = {𝑥 | 𝑥 contains an even number of 𝑏’s}

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Anatomy of a DFA

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𝒓𝟑 0,1 1 1 1 𝒓𝟏 𝒓𝟐 𝒓𝟒

Formal Definition of a DFA

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𝑅 is the set of states Σ is the alphabet  ∶ 𝑅 × Σ → 𝑅 is the transition function 𝑟0  𝑅 is the start state 𝐺 ⊆ 𝑅 is the set of accept states

A finite automaton is a 5-tuple 𝑁 = (𝑅, Σ, , 𝑟0, 𝐺)

A DFA for Parity

Parity: Given a string consisting of 𝑏’s and 𝑐’s, does it contain an even number of 𝑏’s? Ʃ = {𝑏, 𝑐} 𝑀 = {𝑥 | 𝑥 contains an even number of 𝑏’s}

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𝑟0 𝑟1 𝑐 𝑐 𝑏 𝑏 State set 𝑅 = Alphabet Ʃ = Transition function 𝜀 Start state 𝑟0 Set of accept states 𝐺 =

𝜀 𝑏 𝑐 𝑟0 𝑟1

Formal Definition of DFA Computation

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𝑀(𝑁) = the language of machine 𝑁 = set of all (finite) strings machine 𝑁 accepts 𝑁 recognizes the language 𝑀(𝑁) A DFA 𝑁 = (𝑅, Σ, , 𝑟0, 𝐺) accepts a string 𝑥 = 𝑥1𝑥2 · · · 𝑥𝑜 ∈ Σ∗ (where each 𝑥𝑗 ∈ Σ) if there exist 𝑠

0, . . . , 𝑠 𝑜 ∈ 𝑅 such that

1. 𝑠

0 = 𝑟0

2. 𝜀(𝑠

𝑗, 𝑥𝑗+1) = 𝑠 𝑗+1 for each 𝑗 = 0, . . . , 𝑜 − 1, and

• 3. 𝑠

𝑜 ∈ 𝐺

Example: Computing with the Parity DFA

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𝑟0 𝑟1 𝑐 𝑐 𝑏 𝑏 A DFA 𝑁 = (𝑅, Σ, , 𝑟0, 𝐺) accepts a string 𝑥 = 𝑥1𝑥2 · · · 𝑥𝑜 ∈ Σ∗ (where each 𝑥𝑗 ∈ Σ) if there exist 𝑠

0, . . . , 𝑠 𝑜 ∈ 𝑅 such that

1. 𝑠

0 = 𝑟0

2. 𝜀(𝑠

𝑗, 𝑥𝑗+1) = 𝑠 𝑗+1 for each 𝑗 = 0, . . . , 𝑜 − 1, and

• 3. 𝑠

𝑜 ∈ 𝐺

Let 𝑥 = 𝑏𝑐𝑐𝑏 Does 𝑁 accept 𝑥?

Automata Tutor

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http://automatatutor.com/

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

Internet Transmission Control Protocol

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Let TCPS = { 𝑥 | 𝑥 is a complete TCP Session}

• Theorem. TCPS is regular

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.

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. GENETICTEST𝒉is regular for every gene 𝒉.

Arithmetic

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LET 3 =

• A string over 3 has three ROWS (ROW1, ROW2, ROW3)
• Each ROW 𝒄𝟏𝒄𝟐𝒄𝟑 … 𝒄𝑶 represents the integer

𝒄𝟏 + 𝟑𝒄𝟐 + … + 𝟑𝑶𝒄𝑶.

• Let ADD = {𝑻 ∈ 𝟒

∗ | ROW1 + ROW2 = ROW3 }

• Theorem. ADD is regular.

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

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

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 ℤ = {… − 2, −1, 0, 1, 2, … } are closed under

• Addition: 𝑦 + 𝑧
• Multiplication: 𝑦 × 𝑧
• Negation: −𝑦
• …but NOT Division: 𝑦 / 𝑧

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

Regular operations on languages

Let 𝐵, 𝐶 ⊆ Σ∗ be languages. Define Union: 𝐵 ∪ 𝐶 = Concatenation: 𝐵 ∘ 𝐶 = Star: 𝐵∗ =

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Other operations

Let 𝐵, 𝐶 ⊆ Σ∗ be languages. Define Complement: ҧ 𝐵 = Intersection: 𝐵 ∩ 𝐶 = Reverse: 𝐵𝑆 =

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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 𝐵 and 𝐶 are regular, applying any of these

• perations yields a regular language

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Proving Closure Properties

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Complement

Complement: ҧ 𝐵 = 𝑥 𝑥 ∉ 𝐵} Theorem: If 𝐵 is regular, then ҧ 𝐵 is also regular Proof idea:

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Union

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Union: 𝐵 ∪ 𝐶 = 𝑥 𝑥 ∈ 𝐵 or 𝑥 ∈ 𝐶} Theorem: If 𝐵 and 𝐶 are regular, then so is 𝐵 ∪ 𝐶 Proof: Let 𝑁𝐵 = (𝑅𝐵, Σ, 𝐵, 𝑟0

𝐵, 𝐺𝐵) be a DFA recognizing 𝐵 and

𝑁𝐶 = (𝑅𝐶, Σ, 𝐶, 𝑟0

𝐶, 𝐺𝐶) be a DFA recognizing 𝐶

Goal: Construct a DFA 𝑁 = 𝑅, Σ, , 𝑟0, 𝐺 that recognizes 𝐵 ∪ 𝐶

Example

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𝑟0

𝐵

𝑟1

𝐵

1 1

𝑟0

𝐶

𝑟1

𝐶

1 1

𝑵𝑩 𝑵𝑪

𝑵 = ?

Closure under union proof (cont’d)

Idea: Run both 𝑁𝐵 and 𝑁𝐶 at the same time “Cross-product construction” 𝑅 = 𝑅𝐵 × 𝑅𝐶 = {(𝑟𝐵, 𝑟𝐶) |𝑟𝐵 ∈ 𝐵 and 𝑟𝐶 ∈ 𝐶} ( (𝑟𝐵, 𝑟𝐶), ) = (𝐵(𝑟𝐵, ), 𝐶(𝑟𝐶, )) 𝑟0 = (𝑟0

𝐵, 𝑟0 𝐶)

𝐺 = {(𝑟𝐵, 𝑟𝐶) |𝑟𝐵 ∈ 𝐺𝐵 or 𝑟𝐶 ∈ 𝐺𝐶} = 𝐺𝐵 × 𝑅𝐶 ∪ 𝑅𝐵 × 𝐺𝐶

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Example (cont’d)

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𝑟0

𝐵

𝑟1

𝐵

1 1

𝑟0

𝐶

𝑟1

𝐶

1 1

𝑵𝑩

𝑵

𝑵𝑪

Intersection

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Intersection: 𝐵 ∩ 𝐶 = 𝑥 𝑥 ∈ 𝐵 and 𝑥 ∈ 𝐶} Theorem: If 𝐵 and 𝐶 are regular, then so is 𝐵 ∩ 𝐶 Proof: Let 𝑁𝐵 = (𝑅𝐵, Σ, 𝐵, 𝑟0

𝐵, 𝐺𝐵) be a DFA recognizing 𝐵 and

𝑁𝐶 = (𝑅𝐶, Σ, 𝐶, 𝑟0

𝐶, 𝐺𝐶) be a DFA recognizing 𝐶

Goal: Construct a DFA 𝑁 = 𝑅, Σ, , 𝑟0, 𝐺 that recognizes 𝐵 ∩ 𝐶

Intersection

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Intersection: 𝐵 ∩ 𝐶 = 𝑥 𝑥 ∈ 𝐵 and 𝑥 ∈ 𝐶} Theorem: If 𝐵 and 𝐶 are regular, then so is 𝐵 ∩ 𝐶 Another Proof: 𝐵 ∩ 𝐶 = ҧ 𝐵 ∪ ത 𝐶

Reverse

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Reverse: 𝐵𝑆 = 𝑥1𝑥2 · · · 𝑥𝑜 𝑥𝑜 · · · 𝑥1 ∈ 𝐵} Theorem: If 𝐵 is regular, then 𝐵𝑆 is also regular Proof idea: Let 𝑁 = (𝑅, Σ, , 𝑟0, 𝐺) be a DFA recognizing 𝐵 Goal: Construct a DFA 𝑁′ = 𝑅′, Σ, ′, 𝑟0

′ , 𝐺′

that recognizes 𝐵𝑆 Define 𝑁′ as 𝑁 but

• With the arrows reversed
• With start and accept states swapped

Example (Reverse)

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1 1 1 0,1

𝑵 𝑵′

Closure under reverse

𝑁’ 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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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.

Example

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1

𝑴(𝑵) =

𝜻 𝜻

Example

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0,1 0,𝜻 1 0,1 1

𝑴(𝑵) =

Formal Definition of a NFA

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𝑅 is the set of states Σ is the alphabet  ∶ 𝑅 × Σ𝜁 → 𝑄(𝑅) is the transition function 𝑟0  𝑅 is the start state 𝐺 ⊆ 𝑅 is the set of accept states

An NFA is a 5-tuple 𝑁 = (𝑅, Σ, , 𝑟0, 𝐺)

𝑁 accepts a string 𝑥 if there exists a path from 𝑟0 to an accept state that can be followed by reading 𝑥.

Example

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1 𝜻 𝜻 (𝒓𝟒, 𝟐) = 𝑵 = (𝑹, 𝚻, , 𝑹𝟏, 𝑮) 𝑹 = {𝒓𝟏, 𝒓𝟐, 𝒓𝟑, 𝒓𝟒, 𝒓𝟓} 𝚻 = {𝟏, 𝟐} 𝑮 = {𝒓𝟓} (𝒓𝟑, 𝟐) = 𝒓𝟐 𝒓𝟑 𝒓𝟒 𝒓𝟓 𝒓𝟏

Example

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0,1 0,𝜻 1 0,1 1 𝒓𝟐 𝒓𝟑 𝒓𝟒 𝒓𝟏 𝑶 = (𝑹, 𝚻, , 𝒓𝟏, 𝑮) 𝑹 = {𝒓𝟏, 𝒓𝟐, 𝒓𝟑, 𝒓𝟒} 𝚻 = {𝟏, 𝟐} 𝑮 = {𝒓𝟒} (𝒓𝟏, 𝟐) = (𝒓𝟑, 𝟏) = (𝒓𝟏, 𝟏) = (𝒓𝟐, 𝜻) =

Nondeterminism

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Ways to think about nondeterminism

• (restricted)

parallel computation

• tree of possible

computations

• guessing and

verifying the “right” choice Deterministic Computation Nondeterministic Computation accept or reject accept reject