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Languages

Languages. Given an alphabet , we can make a word or string by concatenating the letters of . Concatenation of “x” and “y” is “xy” Typical example: ={0,1}, the possible words over  are the finite bit strings. A language is a set of words. More about Languages.

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Languages

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  1. Languages • Given an alphabet , we can make a word or string by concatenating the letters of . • Concatenation of “x” and “y” is “xy” • Typical example: ={0,1}, the possible words over  are the finite bit strings. • A language is a set of words.

  2. More about Languages • The empty string  is the unique string with zero length. • Concatenation of two langauges: A • B = { xy | xA and yB } • Typical examples: L = { x | x is a bit string with two zeros } L = { anbn | n  N} L = {1n | n is prime}

  3. A Word of Warning Do not confuse the concatenation of languages with the Cartesian product of sets. For example, let A = {0,00} then A•A = { 00, 000, 0000 } with |A•A|=3, AA = { (0,0), (0,00), (00,0), (00,00) } with |AA|=4

  4. Recognizing Languages • Let L be a language  S • a machine M recognizes L if “accept” if and only if xL xS M “reject” if and only if xL

  5. Finite Automaton The most simple machine that is not just a finite list of words. “Read once”, “no write” procedure. It has limited memory to hold the “state”. Examples: vending machine, cell-phone, elevator, etc.

  6. transition rules states 0 1 1 0 q1 q2 q3 0,1 starting state accepting state A Simple Automaton (0)

  7. 0 1 1 0 q1 q2 q3 start 0,1 accept A Simple Automaton (1) on input “0110”, the machine goes: q1  q1  q2  q2  q3 = “reject”

  8. 0 1 1 0 q1 q2 q3 0,1 on input “101”, the machine goes: A Simple Automaton (2) q1  q2  q3  q2 = “accept”

  9. 0 1 1 0 q1 q2 q3 0,1 A Simple Automaton (3) 010: reject 11: accept 010100100100100: accept 010000010010: reject : reject The set of strings accepted by a DFA M is denoted by L(M), the language of the machine M. We want to build DFA for various languages and also want to understand the ones for which we can’t build a DFA.

  10. Finite Automaton (definition) • A deterministic finite automaton (DFA)M is defined by a 5-tuple M=(Q,,,q0,F) • Q: finite set of states • : finite alphabet • : transition function :QQ • q0Q: start state • FQ: set of accepting states

  11. 1 0 1 0 q1 q2 q3 0,1 M = <Q,,,q,F> • states Q = {q1,q2,q3} • alphabet  = {0,1} • start state q1 • accept states F={q2} • transition function:

  12. Recognizing Languages (definition) A finite automaton M = (Q,,,q,F) accepts a string/word w = w1…wn if and only if there is a sequence r0…rn of states in Q such that: 1) r0 = q0 2) (ri,wi+1) = ri+1 for all i = 0,…,n–1 3) rn  F

  13. Regular Languages The language recognized by a finite automaton M is denoted by L(M). A regular language is a language for which there exists a recognizing finite automaton.

  14. Examples of regular languages L1 = { x | x has an odd number of 1’s } over alphabet {0, 1} L2 = { x | x has at least one 0 and at least one 1} over alphabet {0, 1} L3 = { x | x represents a positive integer that is divisible by 3} We will show that each of these languages is regular by building a DFA.

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