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# Finite-State Machines CS619_683_2

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This 3 page Class Notes was uploaded by Robert Notetaker on Friday October 14, 2016. The Class Notes belongs to CS619_683_2 at Armstrong Atlantic State University taught by Rizwan in Fall 2014. Since its upload, it has received 7 views. For similar materials see Computation, Programs, and Programming in Computer Science and Engineering at Armstrong Atlantic State University.

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Date Created: 10/14/16

ICS 241: Discrete Mathematics II (Spring 2015) 13.2 Finite-State Machines with Output A ﬁnite-state machine M = (S;I;O;f;g;s ) con0ists of ▯ a ﬁnite set S of states ▯ a ﬁnite input alphabet I ▯ a ﬁnite output alphabet O ▯ a transition function f (f : S ▯ I ! S) ▯ an output function g (g : S ▯ I ! O) ▯ an initial state 0 0,1 0,1 State Input 0 1 1,0 1,1 s0 s0, 1 s1, 0 start s0 s1 s2 s1 s0, 1 s2, 1 0,1 1,0 s2 s2, 1 s1, 0 Types of Finite-State Machines ▯ Mealy machines: outputs correspond to transitions between states ▯ Moore machine: output is determined only by the state Example of a Moore Machine 0 0 start s ;1 1 s ;0 1 s ;1 0 1 2 0 1 Language Recognizer Let M = (S;I;O;f;g;s ) be a ﬁnite-state machine and L ▯ I . We say that M recognizes (or 0 accepts) L if an input string x belongs to L if and only if the last output bit produced by M when given x as input is a 1. 1 ICS 241: Discrete Mathematics II (Spring 2015) 13.2 pg. 863 # 1 Draw the state diagrams for the ﬁnite-state machines with these state tables. a ) State Input 1,1 0 1 1,0 s0 s1, 0 s0, 1 0,0 1,1 s s , 0 s , 1 start s0 s1 s 2 1 0 2 s2 s1, 0 s1, 0 0,0 0,0 b ) State Input 1,0 0 1 1,1 1,0 s0 s1, 0 s0, 0 0,0 0,1 1,1 s s , 1 s , 1 start s0 s 1 s2 s3 1 2 0 s2 s0, 0 s3, 1 s3 s1, 1 s2, 0 0,0 0,1 13.2 pg. 863 # 3 Find the output generated from the input string 01110 for the ﬁnite-state machine with the state table in a) Exercise 1(a). The state transition sequence is: s 0 s ! 1 ! s 2 s ! s1 2 1 Our output is: 01010 b) Exercise 1(b). The state transition sequence is: s 0 s ! 1 ! s 0 s ! s0 0 1 Our output is: 01000 Lecture Notes 25 Exercise Construct a ﬁnite-state machine with output that produces a 1 if and only if the last 3 input bits read are 0s. 1,0 0,1 State Input 0 1 1,0 s 0 s1, 0 s 0 0 start s0 0,0 s1 0,0 s2 s 1 s2, 0 s 0 0 s 2 s2, 1 s 0 0 1,0 2 ICS 241: Discrete Mathematics II (Spring 2015) 13.2 pg. 864 # 9 Construct a ﬁnite-state machine that delays an input string two bits, giving 00 as the ﬁrst two bits of output. 0,0 0,1 start s0 s 2 1,1 1,0 0,1 0,0 s1 s 3 1,0 1,1 s0corresponds to the last two bits having been 00, s c1rresponds to the last two bits having been 01, s2corresponds to the last two bits having been 10, s co3responds to the last two bits having been 11. 3

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