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Formal Language, chapter 4, slide 1Copyright © 2007 by Adam Webber Chapter Four: DFA Applications

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Formal Language, chapter 4, slide 2Copyright © 2007 by Adam Webber 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.

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Formal Language, chapter 4, slide 3Copyright © 2007 by Adam Webber Outline 4.1 DFA Applications 4.2 A DFA-Based Text Filter in Java 4.3 Table-Driven Alternatives

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Formal Language, chapter 4, slide 4Copyright © 2007 by Adam Webber 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

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Formal Language, chapter 4, slide 5Copyright © 2007 by Adam Webber 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

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Formal Language, chapter 4, slide 6Copyright © 2007 by Adam Webber Outline 4.1 DFA Applications 4.2 A DFA-Based Text Filter in Java 4.3 Table-Driven Alternatives

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Formal Language, chapter 4, slide 7Copyright © 2007 by Adam Webber 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}

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Formal Language, chapter 4, slide 8Copyright © 2007 by Adam Webber The Mod3 DFA, Modified Here, is the Unicode character set The DFA enters the non-accepting trap state on any symbol other than 0 or 1

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Formal Language, chapter 4, slide 9Copyright © 2007 by Adam Webber /** * A deterministic finite-state automaton that * recognizes strings that are binary * representations of integers that are divisible * by 3. Leading zeros are permitted, and the * empty string is taken as a representation for 0 * (along with "0", "00", and so on). */ public class Mod3 { /* * Constants q0 through q3 represent states, and * a private int holds the current state code. */ private static final int q0 = 0; private static final int q1 = 1; private static final int q2 = 2; private static final int q3 = 3; private int state;

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Formal Language, chapter 4, slide 10Copyright © 2007 by Adam Webber 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; } }

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Formal Language, chapter 4, slide 11Copyright © 2007 by Adam Webber /** * 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); } }

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Formal Language, chapter 4, slide 12Copyright © 2007 by Adam Webber /** * Test whether the DFA accepted the string. * @return true iff the final state was accepting */ public boolean accepted() { return state==q0; } } Usage example: Mod3 m = new Mod3(); m.reset(); m.process(s); if (m.accepted())...

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Formal Language, chapter 4, slide 13Copyright © 2007 by Adam Webber 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));

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Formal Language, chapter 4, slide 14Copyright © 2007 by Adam Webber // 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(); } } }

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Formal Language, chapter 4, slide 15Copyright © 2007 by Adam Webber C:\>type numbers 000 001 010 011 100 101 110 111 1000 1001 1010 C:\>java Mod3Filter < numbers 000 011 110 1001 C:\>

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Formal Language, chapter 4, slide 16Copyright © 2007 by Adam Webber Outline 4.1 DFA Applications 4.2 A DFA-Based Text Filter in Java 4.3 Table-Driven Alternatives

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Formal Language, chapter 4, slide 17Copyright © 2007 by Adam Webber 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]; } }

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Formal Language, chapter 4, slide 18Copyright © 2007 by Adam Webber Keeping The Array Small If delta[state,c] is indexed by state and symbol, it will be big: 4 by 65536! And almost all entries will be 3 Instead, we could index it by state and integer, 0 or 1 Then we could use exception handling when the array index is out of bounds

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Formal Language, chapter 4, slide 19Copyright © 2007 by Adam Webber /* * 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; } } }

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Formal Language, chapter 4, slide 20Copyright © 2007 by Adam Webber 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

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