Automata and Formal Language Quan Thanh Tho qttho@dit.hcmut.edu.vn Course Overview  An introduction to the fundamental theories and algorithms for computing on digital computer – – Automation: A model producing input from acceptable output based on self-made decision Formal Language: An abstraction of programming language syntax Course Outline      Chapter 1: Introduction   Chapter 6: Simplification of Context-Free Grammar Chapter 2: Finite Automata Chapter 3: Regular Language and Regular Grammar Chapter 4: Properties of Regular Language Chapter 5: Context-Free Grammar Chapter 7: Pushdown Automata Reading Materials • Giáo trình lý thuyết automat ngơn ngữ hình thức • • Hồ Văn Qn An introduction to formal languages and automata Peter Linz Introduction to automata theory, languages, and computation John Hopcroft & Jeffrey Ullman Assessment  Assignment: 30%  Final Exam: 70% Related Issues  Digital Circuit Design  Compiler  Programming Languages Required Background  Set and Graph Theory  Induction and Contradiction-based Methods Three Basic Concepts  Languages  Grammars  Automata Languages • Alphabet: a finite and nonempty set of symbols Σ = {a, b} • Example 1.1 • • Roman Alphabet: A,B,C, ,Z Greek Alphabet: α,β,… Language (cont’d) • String: finite sequence of symbols from Σ λ: empty string Σ*: the set of all strings on Σ (Σ+ = Σ* − {λ})  Example 1.2: Given Σ = {a,b} => Σ* = {λ, a, b, aa, ab, ba, aaa, } w = abaaa is a string of Σ L = {anbn | n ≥ 0} = {λ, ab, aabb, } is a set of strings of Σ Formal Grammar • Formal grammar: G = (V, T, S, P) V: finite set of variables T: finite set of terminal symbols S∈V: start variable P: finite set of productions Formal Grammar (cont’d) • Productions: x → y, x∈(V ∪T )+, y∈(V ∪T )* • Example 1.6: G = ({S}, {a, b}, S, P) P: S → aSb S→λ Directly Derive • w = uxv derives z = uyv if xy is a production w⇒z w ⇒* w (w ⇒ w ⇒ ⇒ w | w = w ) n n n w ⇒+ w n Directly Derive (cont’d) • Example 1.7: Take into consideration the grammar given in Example 1.6: S ⇒ aSb ⇒ aaSbb ⇒ aabb S ⇒* aabb Generated Language G = (V, T, S, P) L(G) = {w∈T* | S ⇒* w} • Example 1.8: Take into consideration the grammar given in Example 1.6: L(G) = {anbn | n ≥ 0} Derivation  S ⇒ w1 ⇒ w2 ⇒ ⇒ wn ⇒ w∈L(G)  Example 1.9: The sequence of directly deriving given in Example 1.6 is a derivation S ⇒ aSb ⇒ aaSbb ⇒ aabb Sequential Forms • S, w1, w2, , wn (containing variables) with S ⇒ w1 ⇒ w2 ⇒ ⇒ wn ⇒ w is a derivation • Example 1.10: S ⇒ aSb ⇒ aaSbb ⇒ aabb aabb: sentence aaSbb: sentential form Automata  “An automaton (plural: automata) is a self-operating machine The word is sometimes used to describe a robot, more specifically an autonomous robot Used colloquially, it refers to a mindless follower.” (Wiki)  An abstract model of digital computer: Automaton Structure Input file Control unit Output Storage Input File  Input file: is  Divided into squares or cells, each of which holds a symbol of the alphabet  The symbols are to be read from left to right  The end of input file is detectable  The automaton cannot change the contents of the input file Storage  A device consisting of unlimited cells  Each cell hold a symbol from an alphabet (it is not necessary to be the same alphabet as the input one)  The automaton can read and change the storage cells Control Unit  Having a finite set of infernal states  Can be any one of its infernal states  Can change from one infernal state to other Automation Operations ã Transition function: current state ì input symbol × storage info → next state Output may be produced Info in the storage may be changed • Configuration: current state ì input symbol ì storage info ã Move: current configuration → next configuration Automaton Types • Accepter: yes/no output Transducer: string of symbols as output • Deterministic: single possible move at one point Non-deterministic: multiple possible moves Homework • Exercise 4, 5, 6, 8, 9, 12, 15, 17 of Section 1.2 - Linz’s book ... automat ngơn ngữ hình thức • • Hồ Văn Qn An introduction to formal languages and automata Peter Linz Introduction to automata theory, languages, and computation John Hopcroft & Jeffrey Ullman Assessment... Programming Languages Required Background  Set and Graph Theory  Induction and Contradiction-based Methods Three Basic Concepts  Languages  Grammars  Automata Languages • Alphabet: a finite and. .. Grammar Chapter 2: Finite Automata Chapter 3: Regular Language and Regular Grammar Chapter 4: Properties of Regular Language Chapter 5: Context-Free Grammar Chapter 7: Pushdown Automata Reading Materials