FINITE STATE MACHINES

Where can you find applications of FSMs Computer Design Communications Linguistics Theory of Computations

Basic Concepts Most complex devices are realized by

interconnecting many simple components Only a finite number of physical symbols can

be enclosed a specified volume The symbols used to signal events between

the components are often represented by the values of physical quantities

A given signal or state requires a small but nonzero interval to measure

Basic Concepts

We convert problem solutions into sequence of steps

Designing machines one step at a time is the simplest way to make its behavior deterministic

Finite State Machines are

Finite Discrete Sequential Deterministic

Applied FSMs

Mechanical Systems Digital Electronic Systems Pneumatic and Hydraulic Systems Chemical Systems Informstion Processing Systems

Properties of Finite State Machines

INPUT CHANNEL OUTPUT CHANNEL

INITIALIZE

s(t) s(t-1)…s(1) r(t) r(t-1)…r(1)

Properties of FSM The behavior of M is defined only at the

moments t = 0, 1, 2, …

The input symbols s(t) are chosen from a finite input alphabet S

The output symbols r(t) are chosen from a finite output alphabet R

Properties of FSM The behavior of M is uniquely determined by

the sequence of input symbols presented

The behavior of M carries it through a sequence of states, each of which is a member of the state set Q

There is an initial state qi of M that describes the condition of the parts of M just before any stimulus is presented

Mathematical Description of FSMFSM consist of the following:1. The finite sets S,R,Q2. A state transition function f that gives the next

state of M in terms of the current state and the next input symbol

3. An output function g that gives the next output symbol of M in terms of the current state and next input symbol

4. A predetermined initial state q(0) = qi in which M is placed prior to instant t = 0

Machine with Transition-Assigned OutputA transition-assigned finite state machine is a six-tuple M = (Q,S,R,f,g,qi)

WhereQ is a finite set of internal statesS is a finite input alphabetR is a finite output alphabet f is the state transition function, f: Q x S Q g is the output function, g: Q x S R qi is the initial state

Representations of MachinesState Table

: .

… q’,r …

: .

q

s

s/rQ q’

Representations of Machines

State Diagram

s/rq q’

q’ = f(q,s) r = g(q,s)

Example 1

The Modulo-3 Counter

Design a FSM whose output tells the number of inout symbols modulo 3. Let a be the input symbol

Example 2

The Parity Checker (Even Parity)

Design a FSM that will accept a series of 1’s and 0’s; it should output 0 if there are even number of 1’s in the input stream otherwise it should output 1

Example 3

Language Recognizer

Design a FSM to identify if an input w is an element of the language

L ={ 11*01*}

LUNCH BREAK!

CS Entry 4 (25 points each)1. The 2-Unit DelayThe input and output alphabets of machine M are

{0,1}. The output sequence is to be a replica of the input sequence delayed by two time units

Q = {A,B,C,D}

2. Language RecognizerDesign a FSM to identify the languageL= {abk|k>=0}Q = {A,B,C}

Class Presentation

BINDING

CONTROL STRUCTURES