Theory of Computation

Vladimir Kulyukin

www.vkedco.blogspot.com

Deterministic Finite Automata

Outline

● Blocks World Problem: A DFA for a Robotic Camera & Arm

● How DFAs Work● DFA Transition Function● Formal Languages Accepted by DFAs● Sample DFA Problems● Sample DFA Implementation

Blocks World Problem

A B

Suppose we want to design a control system for a robotic camera & arm unit that should build two-block towers: a on b or b on a.

Blocks World Problem

A

B

Goal State 01: A on B

B

A

Goal State 02: B on A

Robotic Arm & Camera Operations

• puton(X, Y) – put A on top of Y• clear(X) – verify that X's top is clear• on(X, Y) – verify that X is on top of Y

Finite Automaton for Blocks World Problem

on(A, T)on(B, T)clear(A)clear(B)

on(A, B)on(B, T)clear(A)

on(B, A)on(A, T)clear(B)

puton(A, B)

puton(A, T)

puton(B, A)

puton(B, T)

Back to Automata & Languages

The robot control for our problem can be solved with a finite state machine, i.e., it is a regular language.

Deterministic Finite Automata

• A DFA can be informally defined as a directed graph whose nodes are states and whose edges are transitions on specific symbols

• A DFA has a unique start state and a set (possibly empty) of final or accepting states

• A DFA processes the input string one symbol at a time. When the last symbol is read, the DFA reaches a state which is either final or not. If the state is final, the DFA accepts (recognizes) the string. If the state is not final, the DFA rejects the string

DFA: Formal Definition

states. (final) accepting ofset theis

state;start theis

;: function;n transitioa is

alphabet;an is

states; ofset finite a is

:where

,,,,, i.e. tuple,-5 a is DFA A

0

0

F

Qq

QQ

Q

FqQMM

Example 1: DFA M

q1q0

a

b

b

a

q0 is the start state.q1 is the final state.

Example 1: DFA M

01

100010

1

10

0

,

;,;,;, .4

; .3

;, .2

;, 1.

where,,,,,

qbq

qaqqbqqaq

qF

ba

qqQ

FqQM

How DFAs Work

Example: DFA M

a b

q0 q1 q0

q1 q1 q0

The transition function, δ, can be represented as a table:

How DFA M Works

q0

b a a b a

tape.on the

symbolleftmost thebefore placed is head The

How DFA M Works

q0

b a a b a

right. cell one move and in remain , read ,In 00 qbq

How DFA M Works

q0

b a a b a

right. cell one move and into go , read ,In 10 qaq

How DFA M Works

q1

b a a b a

right. cell one move and in remain , read ,In 11 qaq

How DFA M Works

q1

b a a b a

right. cell one move and into go , read ,In 01 qbq

How DFA M Works

q0

b a a b a

right. cell one move and into go , read ,In 10 qaq

How DFA M Works

q1

b a a b a

final. is because

ecognized,accepted/r is Thus,

1q

baaba

Observations on DFA Definition

● Q may not be empty: a DFA must have at least one state, i.e., the start state

● δ is a function, i.e., it must be for every state in Q and every symbol in Σ

● F may be empty, in which case the DFA accepts the empty language

● F can be equal to Q, in which case the DFA accepts Σ*

δ* :DFA Transition Function

The δ* Function

axqxaq

qq

,,, .2

;, .1

:statesFA tostrings maps

**

*

*

Back to Example DFA M

Fqaqabq

abbqabbq

abbqabbqbbaq

bba

100

00*

0*

0*

0*

,,,

,,,,,,,

,,,,,,

accepted?it Is . isinput theSuppose

Languages Accepted by DFAs

DFA String Acceptance

FxqFqQM

x

, ifonly and if ,,,,

DFA aby ecognizedaccepted/r is stringA

0*

0

*

DFA’s Language: L(M)

.,|Then

.,,,,such that DFA a be Let

0**

0

FxqxML

FqQMM

Regular Languages

A language L is regular if and only if there exits a DFA M such that L(M) = L.

Sample DFA Problems

Typical DFA Problems

● Given a DFA M and a set of strings, determine which strings are in the language of M, i.e. L(M)?

● Given a DFA M, state in the set former notation, state L(M)?

● Draw a DFA diagram given the formal definition of the DFA

Problem 01

q0 q1

a

a

b

Which of the following strings are in L(M)?1. b2. ε3. ab4. abba5. ababaaaaba

b

Solution 01

Fqababaaaabaq

Fqaqabqabbq

abbaq

ababq

aabbqabbaq

MLab

Fqbqbaqabq

ML

MLb

10*

0111

0*

0*

0*

0*

110*

0*

, .5

,,,,,,

,,,,

,,,

,,, .4

So, .,,,, .3

.2

.1

Problem 02

q0 q1

a,b

a,b

What is the language accepted by this DFA M?

Solution 02

.,12 and ,|

:answer Formal

odd. islength whose,over strings All

:answer Informal

* NnnxbaxxML

ba

Problem 03

q0 q1

ba

a

b

What is the language accepted by this DFA M?

Solution 03

*, and |

:answer Formal

in end that ,over strings All

:answer Informal

bayyaxxML

aba

Problem 04

odd is in s' ofnumber |, .2

,| 1.

:it acceptsthat

DFA a draw languages, following theofeach For

*

*

xabax

baxx

Solution 04.1

q0

a,b

Solution 04.2

q0 q1

b b

a

a

Problem 05

01

01

10

10

1010

,

;,

;,

;,

,,,,,,,

:DFA thisDraw

qbq

qaq

qbq

qaq

qqbaqq

Solution 05

q0 q1

a,b

a,b

Sample DFA Implementation

A DFA: Odd Numbers of Zeros

q0 q1

0

0

q2

any symbol other than 0any symbol other than 0

Java Implementation: Part 1

import java.io.*;

public class OddNumZeros {

   private static final int q0 = 0;   private static final int q1 = 1;   private static final int q2 = 2; // this is the sink

   private int state; // current state …

}

Java Implementation: Part 2

static private int delta(int s, char c) {       switch (s) {           case q0: switch (c) {               case '0': return q1;               default: return q2; // go to the sink           }           case q1: switch (c) {               case '0': return q0;               default: return q2; // go to the sink           }           default: return q2; // go to the sink       }}

Java Implementation: Part 3

public void deltaStar(String instr) {       char c;       for(int i = 0; i < instr.length(); i++) {           c = instr.charAt(i);           state = delta(state, c);       }   }

   public boolean isInputAccepted() {       return state == q1;   }

Java Implementation: Part 4public void reset() { state = q0; }

public static void main(String[] args) throws IOException {

       OddNumZeros dfa = new OddNumZeros();       BufferedReader in =               new BufferedReader(new InputStreamReader(System.in));

       String instr = in.readLine();       while ( instr != null )       {           dfa.reset();           dfa.deltaStar(instr);           if ( dfa.isInputAccepted() ) {               System.out.println(instr + " accepted");           }           else {               System.out.println(instr + " rejected");           }           instr = in.readLine();       }

}

References & Reading Suggestions

● A. Brooks Weber. Ch. 2 Formal Language: A Practical Introduction, Franklin, Beedle & Associates, Inc

● Davis, Weyuker, Sigal. Ch. 9. Computability, Complexity, and Languages, 2nd Edition, Academic Press