CSC4510 AUTOMATA 2.1 Finite Automata: Examples and D efinitions Definitions

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1 CSC45 AUTOMATA 2. Finite Automata: Examples and Definitions Finite Automata: Examples and Definitions A finite automaton is a simple type of computer. Itsoutputislimitedto yes to or no. It has very primitive memory capabilities. For this chapter, consider that: The input comes in the form of a string of individual input symbols. The computer gives an answer for the current string. 2

2 A Finite Automaton (FA) or finite state machine is always in one of a finite number of states. At each step it makes a move that depends only on the state it s currently in and the input symbol it gets The move is to enter a particular state. States are either e accepting or nonaccepting Entering an accepting state means answering yes. Entering a nonaccepting state means no. An FA has an initial state, which is an accepting state if and only if the language the FA accepts includes. An FA can be described by the set of states, the input alphabet, the initial state, the set of accepting states, and a transition function. This can be described by a transition diagram or a transition table. q q q 2 q 3 q q 2 q q q 3 q q 2 q q 3 * q 3 q q

3 , states q, accepting states (F) q q 2 The program accepts a string if the process ends in a double circle. start state (q ) q 3 states 5 6 3

4 This FA accepts the language of strings that end in aa. This FA accepts strings ending with b and not containing aa

5 This FA accepts strings that contain abbaab. An FA that accepts binary representation of integers divisible by 3 States,, and 2 represent the current remainder The initial state is non accepting: at least one bit is required Leading zeros are prohibited What do we do when a prefix of abbaab has been read dbut tthe next symbol ld doesn t match? Go back to the state representing the longest prefix of abbaab at the end of the new current string. 9 5

6 FAs are ideally suited for lexical analysis, the first stage in compiling a computer program. A lexical analyzer takes a string of characters and provides a string of tokens. Tokens: reserved words, punctuation ti symbols, identifiers, operators, numeric literals, separated by spaces. Accepting states represent scanned tokens; each accepting state represents a category of token. Tokens: identifiers, semicolons, =, aa, numerical literals (digits + decimal point) D: any digit L: a lowercase letter other than a M: D or L N: D or L or a :a space All transitions not shown explicitly go to an error state and stay there 2 6

7 Definition: A finite automaton is a 5 tuple M = (Q,,, q, A, ), where: Q is a finite set of states is a finite input alphabet q Q is the initial state A Q is the set of accepting states : Q Q is the transition function From state q the machine will move to state (q, ) if it receives input symbol. Build an automaton that accepts all and only those strings that contain., q q q q 3 4 7

8 The notation *(q, x) describes the state the FA is in after starting in state q and receiving input string x. The extended transition function *: Q * Q is defined recursively as follows:. For every q Q, *(q, ) = q 2. For every q Q, every y *, and every, *(q, y ) = ( *(q, y), ) Evaluate *(q, baa) in the following FA *(q, baa) = ( *(q (q, ba), a) = ( ( *(q (q, b), a), a) = ( ( *(q, b), a), a) = ( ( ( *(q, ), b), a), a) = ( ( (q, b), a), a) = ( (q, a), a) = (q, a) ) = q 5 6 8

9 Definition: Let M=(Q,, q, A, ) be an FA, and let x *. Thenx x is accepted by M if *(q, x) A and rejected otherwise. q, The language accepted by M is L(M) = {x * x is accepted by M} L(M) = {w w is a string of s and s} q, q, L(M) = { } 7 8 9

10 q q L(M) = { w w has an even number of s} Design an FA to accept the language L={w { w has both an even number of s and an even number of s} M=(Q,, q, A, ) Q={q, q, q 2, q 3 } ={,} A={q } * q q 2 q q q 3 q q q q 2 q 3 q 2 q q 3 q 3 q q 2 9 2

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