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Module #16 – Finite State Machines

Modeling Computation

Rosen 5th ed., ch. 11Ref: Wikipedia

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Module #16 – Finite State MachinesFinite State Machines (FSM)

Model machines (or components in a computer) using a particular structureClassification FSM with no output: determine whether the

input is accepted (recognized) or not FSM with output: generate output from the

given input

Representations (p.752) State diagram State transition table

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Module #16 – Finite State Machines

FSM with OutputDefinition: M = (S, I, O, f, g, s0) S: finite set of states I, O: input/output alphabets f: transition function, f: S I S g: output function s0: initial state

Mealy machine: g: SI OMoore machine:g: S OMealy machine: g: SI OMoore machine:g: S O

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Module #16 – Finite State Machines

Simple Example (p.753)

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Module #16 – Finite State Machines

Generating Output

Input string: x = x1x2…xk

s1 = f(s0, x1), s2 = f(s1, x2), …, sk = f(sk-1, xk) These state transition produces output y = y1y2…yk

y1 = g(s0, x1), y2 = g(s1, x2), …, yk = g(sk-1, xk)

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Module #16 – Finite State Machines

Another Example

start

Input: 101011Output: 001000

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Module #16 – Finite State Machines

Ex: vending machines

[description]State table

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Module #16 – Finite State MachinesVending Machine: state diagram

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Module #16 – Finite State Machines

Ex: Unit Delay Machine

Input: x1x2…xk; Output: 0x1x2…xk-1

state

f g

0 1 0 1

s0 s2 s1 0 0s1 s2 s1 1 1s2 s2 s1 0 0

Example & how the machine is designed (p.755)

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Module #16 – Finite State Machines

Ex: FSM for Addition

Design: s0: to remember

that the previous carry is 0

s1: to remember that the previous carry is 1

[example]

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Module #16 – Finite State Machines

FSM with No Output

Language recognition; design and implementation of compilers A string is recognized IFF it takes the

starting state to one of the final states

Def: Finite-state Automata: do not produce output, but have a set of final states

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Module #16 – Finite State MachinesEx: automaton to recognize “nice”

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Module #16 – Finite State Machines

Finite-State Automata

M = (S, I, f, s0, F) S: finite set of states I: input alphabets f: transition function, f: S I S s0: initial state F: final states, subset of S (indicated

by double circlesdouble circles)

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Module #16 – Finite State MachinesExample: construct the state diagram

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Module #16 – Finite State Machines

Recognizing Language

A string x is said to be recognized by the machine M = (S, I, f, s0, F) if f(s0, x) is a state in F

The language recognized (accepted) by the machine M, denoted by L(M), is the set of all strings accepted by M.

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Module #16 – Finite State Machines

Example

The following example is of a DFA M, with a binary alphabet, which determines if the input contains an even number of 0s

Deterministic finite

automaton

Start

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Module #16 – Finite State MachinesTransformations from/to State Diagram

It is possible to draw a state diagram from the table. Draw the circles to represent the states given. For each of the states, scan across the corresponding row and draw an arrow to the destination state(s). There can be multiple arrows for an input character if the automaton is an NFA. Designate a state as the start state. The start state is given in the formal definition of the automaton. Designate one or more states as accept state. This is also given in the formal definition.

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Module #16 – Finite State Machines

Example

L(M1) = {1n | n = 0,1,2,…}

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Module #16 – Finite State Machines

Example (cont)

L(M2) = {1, 01}

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Module #16 – Finite State Machines

Example (cont)

L(M3) = {0n,0n10x | n = 0,1,2,…, x is any string}