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Finite State Machines 1. Finite state machines with output 2. Finite state machines with no output 3. DFA 4. NDFA. Introduction # Many kinds if machines, including components in computers, can be modeled using a structure called a finitestate machine. # Finite-state machines are used extensively in applications in computer science and data networking. For example, FSM are the basis for programs for spell checking, grammar checking, indexing or searching large bodies of text, recognizing speech, …… Definition: A finite-state machine M = {S, I, O, f, g, s0} consists of: •finite set S of states, •finite input alphabet I, •finite output alphabet O, •transition function f that assigns to each state and input pair a new state, •output function g that assigns to each state and input pair an output, •initial state s0. Remark: We can use a state table to represent the values of the transition function f and the output function g for all pairs of states and input. Example: Given a finite state machine with S ={s0 , s1,s2 , s3 } , I = {0,1}, O={0,1}. The values of the transition function f and the values of the output function g are shown in table Representation of FSM We use state diagram to represent FSM. State diagram is a directed graph with labeled edges. In this diagram, each sate is represented by a circle. Arrows labeled with the input and output pair are shown for each transition. Example: Draw the state diagram for the finite-state machine S ={s0 , s1,s2 , s3 } , I = {0,1}, O={0,1} with the following state table. S ={s0 , s1,s2 , s3 } s0 s1 s2 f(S0,0)=S1 s3 0 s0 f(S0,1)=S0 s1 1 0 s0 g(S0,0)=1 1 s1 0 ,1 s0 g(S0,1)=0 1,0 s0 s1 0 ,1 s1 1,0 0 ,1 s0 0,1 s1 s3 1,1 1,0 0 ,1 s0 s1 1,1 1,0 0,1 s3 0,0 0 ,1 s0 s1 1,1 s2 1,1 0,1 1,0 0,0 s2 1,1 s3 Example: Construct the state table for the finite-state machine with the state diagram shown in Figure Find the output string generated by the finite-state machine in Figure if the input string is 101011. Home Work Home Work Finite-State Machines with No Output One of the most important application of finite-state machines is in language recognition. This application plays a fundamental role in the design and construction of compilers for programming languages. It does not have any output. Definition: Suppose that A and B are subsets of V*, where V is a vocabulary. The concatenation of A and B, denoted by AB, is the set of all strings of the form xy, where x is a string in A and y is a string in B. Example: Let A= {0, 11} and B= {1, 10, 110}, Find AB and BA. Solution: AB is the set of all strings of the form xy, where x is a string in A and y is a string in B. x=0 and y=1, 10, or 110 then xy= 01, 010, or 0110 x=11 and y=1, 10, or 110 then xy= 111, 1110, or 11110 So, AB= {01, 010, 0110, 111, 1110, 11110} and BA={10, 111, 100, 1011, 1100, 11011}(by similar way). 1 10 110 0 01 0 10 0 110 11 11 1 11 10 11 110 0 11 1 10 111 10 100 1011 110 1100 11011 Example: Given A0= { } and An+1= An A , for n = 0, 1, 2…… Then let A={1, 00}. Find An for only n= 0, 1, 2, and 3 Solution: A0= { }, we know that An+1= An A n = 0 A0+1= A0 A= {} {1, 00} = {1, 00} n = 1 A1+1= A1 A A2= A1 A = {1, 00} {1,00} = {11, 100, 001, 0000} A3= A2 A = {11, 100, 001, 0000} {1, 00} = {111, 1100, 1001, 10000, 0011, 00100, 00001, 000000 } Home Work Let A={0,11}, and B={00, 01}. Find each of these sets. (a) AB, (b) BA, (c) A2 (d) B3 Deterministic Finite-State Automata A finite-state machine with no output is called finite-state automata. It also called deterministic finite-state automata (DFA). Definition: A DFA M=(S, I, f, s0, F ) consists of: •finite set S of states, •finite input alphabet I, •transition function f that assigns a next state, •initial or start state s0, •subset F final state. # We can represent finite-state automata using either state tables or state diagram. # Final states are indicated in state diagrams by using double circles. Example : Construct the state diagram for the finite-state automata M= (S, I, f, s0, F) , where S = {s0, s1, s2, s3}, I= {0,1}, F= {s0, s3} and the transition function f is given in the following table. f Input state 0 1 s0 s1 s2 s3 s0 s0 s0 s2 s1 s2 s0 s1 0 start s0 s1 1 s2 s3 Language recognition by DFA Definition : # A string x is said to be recognized or accepted by the machine M = (S, I, f, s0, F) if it takes the initial state s0 to the final state, that is, f(s0, x) is a state in F. # The language recognized or accepted by the machine M, denoted by L(M), is the set of all strings that are recognized by M. # Two finite-state automata are called equivalent if they recognize the same language. Example: Determine the languages recognized by the finite-state automata M1, M2, M3. 1 M1 start s0 0,1 0 s1 The only final state of M1 is s0. The strings that takes s0 to itself are those consisting of zero or more consecutive 1s. i.e {λ ,1, 11, 111, …...} So, L(M1)={1n: n=0,1 ,2, 3,…} The only final state of M2 is s2. The strings that takes s0 are 1and 01. So, L(M2)={1,01} 0 M2 start s0 0 s1 1 1 s2 0,1 0,1 s3 0 M3 start s0 1 s1 1 0 0,1 s2 s3 0,1 The final states of M3 is s0 and s3. The only strings that takes s0 to itself are ⋋, 0, 00, 000,….. i.e any string of zero or more consecutive 0s. I.e. {0n, n=0, 1, 2…} The only strings that takes s0 to s3 are a string of zero or more consecutive 0s, followed by 10, followed by any string of combination of 0 and 1. i.e. {0n10x: n=0,1, 2,… and x any string of 0 and 1} So, L(M3)={0n,0n10x: n=0,1, 2,… and x any string} Example: Construct a deterministic finite-state automaton that recognizes the set of all bit strings such that the first bit is 0 and all remaining bits are 1’s. Home Work Q1) Construct a DFA that recognized each of these languages. a) The set of bit strings that begin with two 0s. b) The set of bit strings that contain two consecutive 0s. c) The set of bit strings that do not contain two consecutive 0s. d) The set of bit strings that end with two 0s. e) The set of bit strings that contain at least two 0s. Q2) Determine the set of bit strings recognized by the following deterministic finite-state automaton. Non Deterministic Finite-State Automata In DFA for each pair of state and input value there is a unique next state given by the transition function. But in NDFA there may be several possible next states for each pair of input value and state. Definition : A NDFA, M= (S, I, f, s0, F) consists of a set S of states, an input alphabet I, a transition function f that assigns a set of states to each pair of state and input (so that f: S x I P(S)), a starting state s0,and a subset F of S consisting of the final states. Example : Find the state diagram for the NDFA with the state table shown in table. The final states are s2and s3 f Input state 0 1 s0 s1 s2 s3 s0 , s1 s0 s3 s1 , s3 s0 , s2 s1 s0 , s1 , s2 1 0 start s0 s1 1 0 s2 1 s3 Example : Find the state table for the NDFA with the state diagram 0 shown in figure. 0 0 s1 1 start s3 1 s0 0,1 0 s2 f Input state 0 1 s0 s1 s2 s3 s0 , s2 s3 s1 s4 s4 s2 s4 s3 s3 s3 1 s4 Home Work In Exercises 16–22 find the language recognized by the given deterministic finite-state automaton In Exercises 43–49 find the language recognized by the given nondeterministic finite-state automaton.