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# Finite Automata

Finite Automata

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### Finite Automata

1. 1. Finite Automata Mukesh N. Tekwani Elphinstone College Mumbai, India 2006 Finite Automata Finite Automata
2. 2. Nondeterministic Finite Automata A nondeterministic finite automata (NFA) is collection of 5 things or 5 tuple: A set of states S. A set of input symbols ∑ (alphabet) A transition function δ that maps state-symbol pairs to sets of states. A state S 0 (sometimes denoted by Q 0 ) called as the start state or initial state . A set of states F called as the accepting states or final states . An accepting state is denoted by a double circle.
3. 3. <ul><li>Test if a string matches some pattern. </li></ul><ul><li>Scan for virus signatures. </li></ul><ul><li>Process natural language. </li></ul><ul><li>Search for information using Google. </li></ul><ul><li>Search for markers in human genome. </li></ul><ul><li>Access information in digital libraries. </li></ul><ul><li>Search-and-replace in a word processors. </li></ul><ul><li>Filter text (spam, NetNanny). </li></ul><ul><li>Validate data-entry fields (dates, email, URL, credit card). </li></ul>Why study regular expression and DFA?
4. 4. Deterministic Finite Automata <ul><li>Theoreticians have developed a number of theoretical models to describe &quot;computing&quot; </li></ul><ul><li>Simplest model is known as a DFA </li></ul><ul><li>Deterministic : Machine will be in a state. Upon receipt of a certain symbol, it will go to a known state </li></ul><ul><li>Finite : The machines only have a certain number of states </li></ul><ul><li>Automata : Machine, robot </li></ul>
5. 5. DFA's <ul><li>DFA's recognize strings. </li></ul><ul><li>If the input ends and the DFA is in an accept state then the string is &quot;recognized&quot; </li></ul><ul><li>A &quot;language&quot; can be described as a set of strings </li></ul><ul><li>A language is called a regular language if some finite automaton recognizes it. </li></ul><ul><li>There is a precise mathematical definition of exactly what is meant by a finite automaton </li></ul>
6. 6. Parts of a DFA 1 0 1 1 0 accept state transition start state The alphabet for this example is {0, 1}. Each state has a transition for every symbol in the alphabet 2
7. 7. DFA Examples Example. 1 Accept all strings that end in a 1 q 0 q 1 0 1 1 0 Start
8. 8. DFA Examples Strings with an odd number of ones. Even Odd 0 0 1 1 Start
9. 9. DFA Examples Strings containing the substring 001 '001' 0 0 1 1 '0' '00' 0 1 0,1
10. 10. Finite State Machine (DFA) 0 1 2 3 4 start h e 5 6 8 9 7 ACCEPTED State Machine that recognizes the strings “ he”, “hers”, “his”, and “she”
11. 11. Finite State Machine (DFA) State Machine that recognizes the strings “ he”, “hers”, “his”, and “she” 0 1 2 3 4 start h e 5 6 8 9 7 ACCEPTED r s
12. 12. Finite State Machine (DFA) 0 1 2 3 4 start h 5 6 8 9 7 ACCEPTED i s State Machine that recognizes the strings “ he”, “hers”, “his”, and “she”
13. 13. Finite State Machine (DFA) 0 1 2 3 4 start 5 6 8 9 7 ACCEPTED s h e State Machine that recognizes the strings “ he”, “hers”, “his”, and “she”
14. 14. Finite State Machine (DFA) 0 1 3 2 start 4 A DFA that recognizes the strings “ and”, & “any” a n d ACCEPTED y
15. 15. Finite State Machine (DFA) 0 1 3 2 start 4 A DFA that recognizes the strings “ and”, & “any” a n ACCEPTED
16. 16. Nondeterministic Finite Automata 0 start A NFA that recognizes the strings “and”, & “any” n 1 2 3 4 5 6 d n y a a
17. 17. Examples <ul><li>Design a DFA to recognize strings that start out with k zeros followed by k ones. </li></ul><ul><li>Design a DFA to recognize strings with an equal number of ones and zeros. </li></ul><ul><li>Design a DFA to recognize strings with an equal number of strings &quot;01&quot; and &quot;10&quot;. Impossible? </li></ul><ul><ul><li>1 yes </li></ul></ul><ul><ul><li>2 No </li></ul></ul>
18. 18. Actually the third one is regular! DFA to recognize strings with an equal number of strings &quot;01&quot; and &quot;10&quot; 0 0 0 0 0 1 1 1 1 1 1 0 1 0