DFA Applications - Automata and Complexity Theory - Lecture Slides, Slides of Theory of Automata

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Chapter Four:

DFA Applications

We have seen how DFAs can be used to define

formal languages. In addition to this formal

use, DFAs have practical applications. DFA-

based pieces of code lie at the heart of many

commonly used computer programs.

DFA Applications

• Programming language processing

• Scanning phase: dividing source file into "tokens"

(keywords, identifiers, constants, etc.), skipping

whitespace and comments

• Command language processing

• Typed command languages often require the

same kind of treatment

• Text pattern matching

• Unix tools like awk, egrep, and sed, mail systems

like ProcMail, database systems like MySQL, and

many others

More DFA Applications

• Signal processing

• Speech processing and other signal processing

systems use finite state models to transform the

incoming signal

• Controllers for finite-state systems

• Hardware and software
• A wide range of applications, from industrial

processes to video games

The Mod3 DFA, Revisited

• We saw that this DFA accepts a language of

binary strings that encode numbers divisible

by 3

• We will implement it in Java

• We will need one more state, since our

natural alphabet is Unicode, not {0,1}

1 2

1

0 0 0

0

1

1

The Mod3 DFA, Modified

• Here, Σ is the Unicode character set

• The DFA enters the non-accepting trap state

on any symbol other than 0 or 1

1 2

1

0 0 0

0

1 1

3 -{0,1} -{0,1}

-{0,1}

static private int delta(int s, char c) { switch (s) { case q0: switch (c) { case '0': return q0; case '1': return q1; default: return q3; } case q1: switch (c) { case '0': return q2; case '1': return q0; default: return q3; } case q2: switch (c) { case '0': return q1; case '1': return q2; default: return q3; } default: return q3; } }

• Reset the current state to the start state. */ public void reset() { state = q0; }

/**

• Make one transition on each char in the given
• string.
• @param in the String to use */ public void process(String in) { for (int i = 0; i < in.length(); i++) { char c = in.charAt(i); state = delta(state, c); } }

import java.io.*;

/**

• A Java application to demonstrate the Mod3 class by
• using it to filter the standard input stream. Those
• lines that are accepted by Mod3 are echoed to the
• standard output. */ public class Mod3Filter { public static void main(String[] args) throws IOException {

Mod3 m = new Mod3(); // the DFA BufferedReader in = // standard input new BufferedReader(new InputStreamReader(System.in));

// Read and echo lines until EOF.

String s = in.readLine(); while (s!=null) { m.reset(); m.process(s); if (m.accepted()) System.out.println(s); s = in.readLine(); } } }

Outline

• 4.1 DFA Applications

• 4.2 A DFA-Based Text Filter in Java

• 4.3 Table-Driven Alternatives

Making Delta A Table

• We might want to encode delta as a two-

dimensional array

• Avoids method invocation overhead

• Then process could look like this:

static void process(String in) { for (int i = 0; i < in.length(); i++) { char c = in.charAt(i); state = delta[state, c]; } }

/*

• The transition function represented as an array.
• The next state from current state s and character c
• is at delta[s,c-'0']. / static private int[][] delta = {{q0,q1},{q2,q0},{q1,q2},{q3,q3}}; /*
• Make one transition on each char in the given
• string.
• @param in the String to use */ public void process(String in) { for (int i = 0; i < in.length(); i++) { char c = in.charAt(i); try { state = delta[state][c-'0']; } catch (ArrayIndexOutOfBoundsException ex) { state = q3; } } }

Tradeoffs

• Function or table?
• Truncated table or full table?
  • By hand, a truncated table is easier
  • Automatically generated systems generally produce the full table, so the same process can be used for different DFAs
• Table representation
  • We used an int for every entry: wasteful!
  • Could have used a byte, or even just two bits
  • Time/space tradeoff: table compression saves space but slows down access