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**CS 3240: Languages and Computation**

Introducing Regular Languages: Deterministic Finite Automata and Regular Expressions

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**Automata Finite Automata model many design and analysis tasks, e.g.**

Lexical analyzer in a compiler Digital cicuit design Keywork searching in texts or on the web. Software for verifying finite state systems, such as communication protocols. Your ATM, vendig machine, Etc. McCulloch&Pitts, Rabin&Scott, Moore, Huffman

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**Finite State Machine and Finite Automata**

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**Deterministic Finite Automata**

A simplest model for computing Deterministic: Machine is in a state. Upon receipt of a symbol will go to a unique state. Finite: Have a finite number of states Automata: (pl. of automaton) Self-operating machine DFA: finite-state machine without ambiguity

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**DFA and Strings DFA can recognize strings**

String is input If DFA ends at accept state, string is recognized A language is called a regular language if some finite automaton recognizes it Let us look at a few example before giving formal mathematical definition

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**DFA Examples 1 1 start state 1 2 accept state transition**

1 1 start state 1 2 accept state transition Note: The alphabet for this example is {0, 1}. Each state has a transition for every symbol in the alphabet Accept all strings that end in 1

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DFA Examples 5 3 4 2 1 a b Accept strings of 'a's and 'b's that begin and end with same symbol

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**DFA Examples 1 Start 2 1 2 1 2 2 1 Keep running count of**

1 Start 2 1 2 1 2 2 Keep running count of total of symbols read in mod 3. Accept on 0. 1

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DFA Examples Even Odd 1 Strings with an odd number of ones.

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DFA Examples '001' 1 '0' '00' 0,1 Strings containing the substring 001

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**Formal Definition of DFA**

A DFA consists of: Alphabet A set of states Q A transition function δ : Q Q One start state q0 One or more accepting states F Q Language accepted by a DFA is the set of strings such that DFA ends at an accepting state Each string is c1c2…cn with ci States are qi = δ(qi-1,ci ) for i=1…n qn is an accepting state

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**Can DFA's be designed to accept any string?**

No!

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Examples Design a DFA to recognize strings that start out with k zeros followed by k ones. Impossible Design a DFA to recognize strings with an equal number of ones and zeros. Design a DFA to recognize strings with an equal number of strings "01" and "10". Possible!

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**Actually the third one is regular!**

1 1 1 1 1 1 DFA to recognize strings with an equal number of strings "01" and "10" 1

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DFA More examples

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**DFA More examples A,B are the input into which the**

marble is dropped. The x-levers cause fall either to left or right, but lever reverses upon a marble passing Accept if marble exits through D

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**Non-Deterministic Finite Automata**

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**NFA and -NFA Nondeterministic Finite Automata **

Same input may produce multiple paths Allows transition with an empty string or transition from one state to different states given a character q1 q2 1 q3 nondeterministic transition q1 q2 empty string transition

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