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 1/26/2020 CS332 - Theory of Computation 2
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? 1/26/2020 CS332 - Theory of Computation 3
Machine Models • Finite Automata (FAs): Machine with a finite amount of unstructured memory Input 𝑄 𝑇 𝑄 𝑇 … Control scans left-to-right Finite control 1/26/2020 CS332 - Theory of Computation 4
A DFA for the car stereo problem 1/26/2020 CS332 - Theory of Computation 5
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 } 1/26/2020 CS332 - Theory of Computation 6
Anatomy of a DFA 1 𝒓 𝟐 0 0,1 1 𝒓 𝟏 𝒓 𝟑 0 0 1 𝒓 𝟒 1/26/2020 CS332 - Theory of Computation 7
Formal Definition of a DFA A finite automaton is a 5-tuple 𝑁 = (𝑅, Σ, , 𝑟 0 , 𝐺) 𝑅 is the set of states Σ is the alphabet ∶ 𝑅 × Σ → 𝑅 is the transition function 𝑟 0 𝑅 is the start state 𝐺 ⊆ 𝑅 is the set of accept states 1/26/2020 CS332 - Theory of Computation 8
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 } State set 𝑅 = Alphabet Ʃ = 𝑐 𝑐 Transition function 𝜀 𝜀 𝑏 𝑐 𝑏 𝑟 0 𝑟 0 𝑟 1 𝑟 1 𝑏 Start state 𝑟 0 Set of accept states 𝐺 = 1/26/2020 CS332 - Theory of Computation 9
Formal Definition of DFA Computation A DFA 𝑁 = (𝑅, Σ, , 𝑟 0 , 𝐺) accepts a string 𝑥 = 𝑥 1 𝑥 2 · · · 𝑥 𝑜 ∈ Σ ∗ (where each 𝑥 𝑗 ∈ Σ ) if there exist 𝑠 0 , . . . , 𝑠 𝑜 ∈ 𝑅 such that 𝑠 0 = 𝑟 0 1. 2. 𝜀(𝑠 𝑗 , 𝑥 𝑗+1 ) = 𝑠 𝑗+1 for each 𝑗 = 0, . . . , 𝑜 − 1, and 3. 𝑠 𝑜 ∈ 𝐺 𝑀(𝑁) = the language of machine 𝑁 = set of all (finite) strings machine 𝑁 accepts 𝑁 recognizes the language 𝑀(𝑁) 1/26/2020 CS332 - Theory of Computation 10
Example: Computing with the Parity DFA 𝑐 𝑐 Let 𝑥 = 𝑏𝑐𝑐𝑏 Does 𝑁 accept 𝑥 ? 𝑏 𝑟 0 𝑟 1 𝑏 A DFA 𝑁 = (𝑅, Σ, , 𝑟 0 , 𝐺) accepts a string 𝑥 = 𝑥 1 𝑥 2 · · · 𝑥 𝑜 ∈ Σ ∗ (where each 𝑥 𝑗 ∈ Σ ) if there exist 𝑠 0 , . . . , 𝑠 𝑜 ∈ 𝑅 such that 𝑠 0 = 𝑟 0 1. 𝜀(𝑠 𝑗 , 𝑥 𝑗+1 ) = 𝑠 𝑗+1 for each 𝑗 = 0, . . . , 𝑜 − 1, and 2. 3. 𝑠 𝑜 ∈ 𝐺 1/26/2020 CS332 - Theory of Computation 11
Automata Tutor http://automatatutor.com/ 1/26/2020 CS332 - Theory of Computation 12
Regular Languages Definition: A language is regular if it is recognized by a DFA 𝑴 = { 𝒙 ∈ 𝒃, 𝒄 ∗ | 𝒙 has an even number of 𝒃 ’s } is regular 𝑴 = { 𝒙 ∈ 𝟏, 𝟐 ∗ | 𝒙 contains 𝟏𝟏𝟐 } is regular Many interesting programs recognize regular languages NETWORK PROTOCOLS COMPILERS GENETIC TESTING ARITHMETIC 1/26/2020 CS332 - Theory of Computation 13
Internet Transmission Control Protocol Let TCPS = { 𝑥 | 𝑥 is a complete TCP Session} Theorem. TCPS is regular 1/26/2020 CS332 - Theory of Computation 14
Compilers 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. 1/26/2020 CS332 - Theory of Computation 15
Genetic Testing 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. GENETICTEST 𝒉 = {strings over {𝑩, 𝑫, 𝑯, 𝑼} containing 𝒉 as a substring} Theorem. GENETICTEST 𝒉 is regular for every gene 𝒉 . 1/26/2020 CS332 - Theory of Computation 16
Arithmetic { [ ],[ ],[ ],[ ], 0 0 0 0 LET 3 = 0 0 1 1 0 1 0 1 [ ],[ ],[ ],[ ]} 1 1 1 1 0 0 1 1 0 1 0 1 • A string over 3 has three ROWS (ROW 1 , ROW 2 , ROW 3 ) • Each ROW 𝒄 𝟏 𝒄 𝟐 𝒄 𝟑 … 𝒄 𝑶 represents the integer 𝒄 𝟏 + 𝟑𝒄 𝟐 + … + 𝟑 𝑶 𝒄 𝑶 . ∗ | ROW 1 + ROW 2 = ROW 3 } • Let ADD = { 𝑻 ∈ 𝟒 Theorem. ADD is regular. 1/26/2020 CS332 - Theory of Computation 17
Regular Operations 1/26/2020 CS332 - Theory of Computation 18
An Analogy In algebra, we try to identify operations which are common to many different mathematical structures 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 1/26/2020 CS332 - Theory of Computation 19
Regular operations on languages Let 𝐵, 𝐶 ⊆ Σ ∗ be languages. Define Union: 𝐵 ∪ 𝐶 = Concatenation: 𝐵 ∘ 𝐶 = Star: 𝐵 ∗ = 1/26/2020 CS332 - Theory of Computation 20
ҧ Other operations Let 𝐵, 𝐶 ⊆ Σ ∗ be languages. Define 𝐵 = Complement: Intersection: 𝐵 ∩ 𝐶 = Reverse: 𝐵 𝑆 = 1/26/2020 CS332 - Theory of Computation 21
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 operations yields a regular language 1/26/2020 CS332 - Theory of Computation 22
Proving Closure Properties 1/26/2020 CS332 - Theory of Computation 23
Complement 𝐵 = 𝑥 𝑥 ∉ 𝐵} Complement: ҧ Theorem: If 𝐵 is regular, then ҧ 𝐵 is also regular Proof idea: 1/26/2020 CS332 - Theory of Computation 24
Union Union: 𝐵 ∪ 𝐶 = 𝑥 𝑥 ∈ 𝐵 or 𝑥 ∈ 𝐶} Theorem: If 𝐵 and 𝐶 are regular, then so is 𝐵 ∪ 𝐶 Proof: Let 𝑁 𝐵 = (𝑅 𝐵 , Σ, 𝐵 , 𝑟 0 𝐵 , 𝐺 𝐵 ) be a DFA recognizing 𝐵 and 𝑁 𝐶 = (𝑅 𝐶 , Σ, 𝐶 , 𝑟 0 𝐶 , 𝐺 𝐶 ) be a DFA recognizing 𝐶 Goal: Construct a DFA 𝑁 = 𝑅, Σ, , 𝑟 0 , 𝐺 that recognizes 𝐵 ∪ 𝐶 1/26/2020 CS332 - Theory of Computation 25
Example 0 0 𝑵 𝑩 1 𝐵 𝐵 𝑟 1 𝑟 0 1 1 1 𝑵 𝑪 0 𝑵 = ? 𝐶 𝐶 𝑟 0 𝑟 1 0 1/26/2020 CS332 - Theory of Computation 26
Closure under union proof (cont’d) Idea: Run both 𝑁 𝐵 and 𝑁 𝐶 at the same time “Cross - product construction” 𝑅 = 𝑅 𝐵 × 𝑅 𝐶 = {(𝑟 𝐵 , 𝑟 𝐶 ) |𝑟 𝐵 ∈ 𝐵 and 𝑟 𝐶 ∈ 𝐶} ( (𝑟 𝐵 , 𝑟 𝐶 ), ) = ( 𝐵 (𝑟 𝐵 , ), 𝐶 (𝑟 𝐶 , )) 𝐵 , 𝑟 0 𝐶 ) 𝑟 0 = (𝑟 0 𝐺 = {(𝑟 𝐵 , 𝑟 𝐶 ) |𝑟 𝐵 ∈ 𝐺 𝐵 or 𝑟 𝐶 ∈ 𝐺 𝐶 } = 𝐺 𝐵 × 𝑅 𝐶 ∪ 𝑅 𝐵 × 𝐺 𝐶 1/26/2020 CS332 - Theory of Computation 27
Example (cont’d) 0 𝑵 0 𝑵 𝑩 1 𝐵 𝐵 𝑟 1 𝑟 0 1 1 1 𝑵 𝑪 0 𝐶 𝐶 𝑟 0 𝑟 1 0 1/26/2020 CS332 - Theory of Computation 28
Intersection Intersection: 𝐵 ∩ 𝐶 = 𝑥 𝑥 ∈ 𝐵 and 𝑥 ∈ 𝐶} Theorem: If 𝐵 and 𝐶 are regular, then so is 𝐵 ∩ 𝐶 Proof: Let 𝑁 𝐵 = (𝑅 𝐵 , Σ, 𝐵 , 𝑟 0 𝐵 , 𝐺 𝐵 ) be a DFA recognizing 𝐵 and 𝑁 𝐶 = (𝑅 𝐶 , Σ, 𝐶 , 𝑟 0 𝐶 , 𝐺 𝐶 ) be a DFA recognizing 𝐶 Goal: Construct a DFA 𝑁 = 𝑅, Σ, , 𝑟 0 , 𝐺 that recognizes 𝐵 ∩ 𝐶 1/26/2020 CS332 - Theory of Computation 29
Intersection Intersection: 𝐵 ∩ 𝐶 = 𝑥 𝑥 ∈ 𝐵 and 𝑥 ∈ 𝐶} Theorem: If 𝐵 and 𝐶 are regular, then so is 𝐵 ∩ 𝐶 Another Proof: ത 𝐵 ∩ 𝐶 = ҧ 𝐵 ∪ 𝐶 1/26/2020 CS332 - Theory of Computation 30
Reverse Reverse: 𝐵 𝑆 = 𝑥 1 𝑥 2 · · · 𝑥 𝑜 𝑥 𝑜 · · · 𝑥 1 ∈ 𝐵} Theorem: If 𝐵 is regular, then 𝐵 𝑆 is also regular Proof idea: Let 𝑁 = (𝑅, Σ, , 𝑟 0 , 𝐺) be a DFA recognizing 𝐵 ′ , 𝐺′ Goal: Construct a DFA 𝑁′ = 𝑅′, Σ, ′, 𝑟 0 that recognizes 𝐵 𝑆 Define 𝑁′ as 𝑁 but • With the arrows reversed • With start and accept states swapped 1/26/2020 CS332 - Theory of Computation 31
Example (Reverse) 0,1 1 0 𝑵 0 0 1 1 𝑵′ 1/26/2020 CS332 - Theory of Computation 32
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 1/26/2020 CS332 - Theory of Computation 33
Nondeterminism 1 0,1 0 0 0 1 1 A Nondeterministic Finite Automaton (NFA) accepts if there is a way to make it reach an accept state. 1/26/2020 CS332 - Theory of Computation 34
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