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

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DFA Examples Accept strings of 'a's and 'b's that begin and end with same symbol a a a a a b b b b b

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DFA Examples 2 Keep running count of total of symbols read in mod 3. Accept on Start

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

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

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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 q 0 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 c 1 c 2 …c n with c i States are q i = δ(q i-1,c i ) for i=1…n q n 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. Impossible 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! DFA to recognize strings with an equal number of strings "01" and "10"

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DFA More examples

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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 q1q1 q2q2 empty string transition q1q1 q2q2 1 q3q3 1 nondeterministic transition

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