Topics Automata Theory Grammars and Languages Complexities Formal Languages and Automata Theory1.

TopicsAutomata TheoryGrammars and LanguagesComplexities

Formal Languages and Automata Theory 1

Why Automata Theory?To study abstract computing devices which are closely related to today’s computers. A simple example of finite state machine:


off onstart

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1

There are many different kinds of machines.

Another Example


start

0

1on

0

0

1

offoff

1

When will this be on?Try 100, 1001, 1000, 111, 00, …

Grammar and LanguagesGrammars and languages are closely related to automata theory and are the basis of many important software components like:Compilers and interpretersText editors and processorsSearch enginesSystem verification components


ComplexitiesStudy the limits of computations. What kinds of problems can be solved with a computer? What kinds of problems can be solved efficiently?


PreliminariesAlphabetsStringsLanguagesProblems


AlphabetsAn alphabet is a finite set of symbols.

Usually, use to represent an alphabet.

Examples:

= {0,1}, the set of binary digits.

= {a, b, … , z}, the set of all lower-case letters.

= {(, )}, the set of open and close parentheses.


Strings A string is a finite sequence of symbols from

an alphabet.

Examples:

0011 and 11 are strings from = {0,1}

abc and bbb are strings from = {a, b, … , z}

(()(())) and )(() are strings from = {(, )}


StringsEmpty string:

Length of string: |0010| = 4, |aa| = 2, ||=0

Prefix of string: aaabc, aaabc, aaabc

Proper prefix of string: aaabc, aaabc

Suffix of string: aaabc, aaabc, aaabc

Proper suffix of string: aaabc, aaabc

Substring of string: aaabc, aaabc, aaabc


StringsConcatenation: =abd, =ce, =abdce

Exponentiation: =abd, 3=abdabdabd, 0=

Reversal: =abd, R = dba

k = set of all k-length strings formed by symbols in

e.g., ={a,b}, 2={ab, ba, aa, bb}, 0={}

What is 1? Is 1 different from ? How?


StringsKleene Closure * = 012… = k0 k

e.g., ={a, b}, * = {, a, b, ab, aa, ba, bb,

aaa, aab, abb, … } is the set of all strings

formed by a’s and b’s.

+ = 123… = k>0 k

i.e., * without the empty string.


LanguagesA language is a set of strings over an alphabet.

Examples:

={(, )}, L1={(), )(, (())} is a language over .

={a, b, c, … , z}, the set L of all legal English

words is a language over .

The set {} is a language over any alphabet.

What is the difference between and {}?


LanguagesOther Examples:

={0, 1}, L={0n1n | n1} is a language over

consisting of the strings {01, 0011, 000111, … }

={0, 1}, L = {0i1j | ji0} is a language over

consisting of the strings with some 0’s (possibly

none) followed by at least as many 1’s.


ProblemsIn automata theory, a problem is to decide

whether a given string is a member of some particular language.

This formulation is general enough to capture the difficulty levels of all problems.


Finite Automata ( or Finite State Machines)

This is the simplest kind of machine.We will study 3 types of Finite Automata:

Deterministic Finite Automata (DFA)Non-deterministic Finite Automata (NFA)Finite Automata with -transitions (-NFA)


Deterministic Finite Automata (DFA) We have seen a simple example before:


off onstart

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1

There are some states and transitions (edges)between the states. The edge labels tell whenwe can move from one state to another.

Definition of DFA A DFA is a 5-tuple (Q, , , q0, F) where

Q is a finite set of states is a finite input alphabet is the transition function mapping Q to

Q q0 in Q is the initial state (only one)

F Q is a set of final states (zero or more)


Definition of DFA For example:


off onstart

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1

Q is the set of states: {on, off} is the set of input symbols: {1} is the transitions: off 1 on; on 1 offq0 is the initial state: offF is the set of final states (double circle): {on}

Definition of DFA Another Example:


start

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1q2

0

0

1

q1q0

1

What are Q, , , q0 and F in this DFA?

Transition Table For the previous example, the DFA is

(Q,,,q0,F) where Q = {q0,q1,q2}, = {0,1}, F = {q2} and is such that


StatesInputs

0 1q0

q1

*q2

q1 q0

q2

q0q1

q0

Note that there is one transition only for each inputsymbol from each state.

Language of a DFA Given a DFA M, the language accepted (or

recognized) by M is the set of all strings that, starting from the initial state, will reach one of the final states after the whole string is read.

For example, the language accepted by the previous example is the string that ends with 00


DFA Example Consider the DFA M=(Q,,,q0,F)

where Q = {q0,q1,q2,q3}, = {0,1}, F = {q0} and is:


StatesInputs

0 1q0

q1

q2

q3

q2

q2q1

q1

q3

q3q0

q0

q0

q2 q3

q1

1

1

1

1

0 000

Start

We can use a transition table or a transition diagram to specifythe transitions. What input can take you to the final state in M?

OR

Topics Automata Theory Grammars and Languages Complexities Formal Languages and Automata Theory1.

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