Part-1 - Motivation Automata = abstract computing devices...

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Motivation Automata = abstract computing devices Turing studied Turing Machines (= comput- ers) before there were any real computers We will also look at simpler devices than Turing machines (Finite State Automata, Push- down Automata, . . . ), and specification means, such as grammars and regular expressions. NP-hardness = what cannot be e±ciently computed 6
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Finite Automata Finite Automata are used as a model for Software for designing digital cicuits Lexical analyzer of a compiler Searching for keywords in a file or on the web. Software for verifying finite state systems, such as communication protocols. 7
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Example: Finite Automaton modelling an on/of switch Push Push Start on off Example: Finite Automaton recognizing the string then t th the Start t n h e then 8
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Structural Representations These are alternative ways of specifying a ma- chine Grammars: A rule like E E + E specifies an arithmetic expression Lineup Person.Lineup says that a lineup is a person in front of a lineup. Regular Expressions: Denote structure of data, e.g. ’[A-Z][a-z]*[][A-Z][A-Z]’ matches Ithaca NY does not match Palo Alto CA Question: What expression would match Palo Alto CA 9
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Central Concepts Alphabet: Finite, nonempty set of symbols Example: Σ = { 0 , 1 } binary alphabet Example: Σ = { a,b,c,. ..,z } the set of all lower case letters Example: The set of all ASCII characters Strings: Finite sequence of symbols from an alphabet Σ, e.g. 0011001 Empty String: The string with zero occur- rences of symbols from Σ The empty string is denoted ± 10
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Length of String: Number of positions for symbols in the string. | w | denotes the length of string w | 0110 | = 4 , | ± | = 0 Powers of an Alphabet: Σ k = the set of strings of length k with symbols from Σ Example: Σ = { 0 , 1 } Σ 1 = { 0 , 1 } Σ 2 = { 00 , 01 , 10 , 11 } Σ 0 = { ± } Question: How many strings are there in Σ 3 11
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The set of all strings over Σ is denoted Σ * Σ * = Σ 0 Σ 1 Σ 2 ∪ ··· Also: Σ + = Σ 1 Σ 2 Σ 3 ∪ ··· Σ * = Σ + ∪ { ± } Concatenation: If x and y are strings, then xy is the string obtained by placing a copy of y immediately after a copy of x x = a 1 a 2 ...a i ,y = b 1 b 2 ...b j xy = a 1 a 2 ...a i b 1 b 2 ...b j Example: x = 01101 ,y = 110 ,xy = 01101110 Note: For any string x = ±x = x 12
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Languages: If Σ is an alphabet, and L Σ * then L is a language Examples of languages: The set of legal English words The set of legal C programs The set of strings consisting of n 0’s followed by n 1’s { ±, 01 , 0011 , 000111 ,... } 13
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The set of strings with equal number of 0’s and 1’s { ±, 01 , 10 , 0011 , 0101 , 1001 ,... } L P = the set of binary numbers whose value is prime { 10 , 11 , 101 , 111 , 1011 ,... } The empty language The language { ± } consisting of the empty string Note: ∅ 6 = { ± } Note2: The underlying alphabet Σ is always finite 14
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Problem: Is a given string w a member of a
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This note was uploaded on 11/02/2009 for the course EECS PR921 taught by Professor 顏嗣鈞 during the Fall '09 term at Norwegian University of Science and Technology.

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Part-1 - Motivation Automata = abstract computing devices...

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