Introduction

Wen-Guey Tzeng

Computer Science Department

National Chiao Tung University

Formal languages and finite automata

• We have been building computing machines for solving computing problems.22222

• Computers are everywhere.• Questions

– What is a computing machine?– What problems can be solved by a computing machine?– What problems cannot be solved?

• Finite automata: abstract models of computing machines

• Formal languages: formal representation of computing problems

22017 Spring

Classification

• Finite automata: by types of memories– Finite automata

– Pushdown automata

– Turing machines

• Formal languages: by types of grammars– Regular languages

– Context-free languages

– Recursive languages (solvable)

– Recursively enumerable languages (semi-solvable)

– …

32017 Spring

Relations

• Finite automata and regular languages

• Pushdown down automata and context-free

languages

• Turing machines and recursive languages

• Complexity classes: by runtime or space used by

Turing machines– P: polynomial time computable

– NP: non-deterministic polynomial time computable

– EXP: solvable in exponential time

– …

42017 Spring

5

Computing problems

Computing machines

Finite automata

Form

aliz

atio

n

Formal languages

Ab

stra

ctio

n

Relations

Classification by: memory,

runtime, space, etc.

Classification by: grammars, machines, etc. • Regular vs. finite automata

• Context-free vs. pushdown• Recursive vs. Turing machines• Complexity classes• …

Yes/No problems

2017 Spring

Applications

• Programming languages• Compilers• Natural language processing• Algorithm design• Software engineering• Pattern recognitions• Machine learning• …

62017 Spring

72017 Spring

8

How to compile a Java program, like the above?• We need a grammar for the Java programming

language• Context-free grammar

• We need a compiler to compile it• Pushdown automata

2017 Spring

92017 Spring

Mathematical notation

• Set: a collection of elements

– |S|: the number of elements in S

– SxT={(x, y) | xS, yT}

• Function

– f: S1S2

• Binary relation

– R S1xS2

– xS1 and yS2 has relation R if (x, y)R

102017 Spring

Three Basic Concepts

1. Languages– Symbol set (alphabet) ={a, b, c, …}

– String: w=a1a2…an, n0, ai

• |w|: length of string

• String concatenation: w1w2

• String reversing: wR=anan-1…a1

• an = aa… a (repeated n times)

• Prefix(w)={, a1, a1a2, …, a1a2…an}

• Suffix(w)={, an, an-1an, …, a1a2…an}

• Empty string , – ||=0

– w= w=w112017 Spring

Alphabet={a,b}

Strings:aba, baba, …

Languages:L={aba, abbaa,

aaba, aa, …}

– A language L over is a subset of strings over

– The language consists of all strings: *

• * over {a,b} = {, a, b, aa, ab, ba, bb, …}

– The language consists of all non-empty strings +

• + = * - {}

– Operations over languages• Reverse L: LR = {wR: wL }

• Complementation of L: Lc = * - L

• Concatenation: L1L2 = {xy: xL1, yL2}

• Ln = {x1x2…xn: xiL}– L0 = {}

– L* = L0 L1 …

– L+ = L1 L2 …

122017 Spring

2. Grammar: a set of rules for generating strings

– G=(V, T, P, S)

– V: set of variables

– T: set of terminals (symbols)

– P: set of production rules

– SV: the start variable

132017 Spring

Example

G=({S,A,B}, {a,b}, {SaA, AbS, S}, S)

• String generation/derivation

• The language generated by G

– L(G) = {w | S * w}

142017 Spring

• Problem: find grammars for languages over ={a, b}

– L1 = {anbn | n0}

– L2 = {w | na(w)=nb(w)}

– L3 = {anbm | nm0}

152017 Spring

3. Automaton: an abstract model of a digital computer

– Input

– Control unit: represented by “states”

– Storage

– Output

– Operating in discrete time frame

162017 Spring

172017 Spring

Language representation

• Informal description

– L = {aa, aaa, aab, aaaa, …}

– L = {anbn | n0 }

• Formal description

– Grammar: language generator

• It tells us how a string in a language is generated

– Automata: language acceptor

• It tells us whether a given string belongs to a language

2017 Spring 18