Finite State Machines Murray Cole Finite State Machines 1 - - PowerPoint PPT Presentation

T H E U N I V E R S I T Y O F E D I N B U R G H

Finite State Machines

Murray Cole

Finite State Machines

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Embedded Systems

• Already 10’s of processor chips per person in the world
• There will “soon” be 100’s or 1000’s
• Most of these are not in “computers” but are embedded in

everyday real world systems (cars, TVs, vending machines, watches, (people?!))

• These often implement quite simple reactive systems

Finite State Machines

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Reactive Systems

• Wait to receive an input
• Respond with an output
• Response may depend upon finite history (i.e. not just the most

recent input)

Finite State Machines

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Finite State Machines (Automata)

• A conceptual tool for modelling reactive systems
• Pre-date “Computer Science”
• Force us to make precise definitions (unlike natural language)
• Can then implement in hardware or software or both

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FSM - Formal Definition

• An input alphabet,
• (“sigma”), which is a set of symbols, one for

each distinct possible input (e.g. buttons, coin slots, ...)

• An output alphabet,

(“lambda”), which is another set of symbols,

• ne for each distinct possible output (e.g. lights, can dispensed, ...)
• Some named states, each capturing some state of the system (e.g.

“coin inserted but no can dispensed”), drawn as circles, one indicated as the start state with an arrow

• A collection of transitions, each an arrow between two states,

labelled with the input which activates it and the output which results (or

(“epsilon”) indicating no input or output respectively)

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An FSM for a Simple Ticket Machine

m/ t/p r/d m/ t/ r/ 1 2

• ✁✄✂

for insert money, request ticket, request refund

• ✁
• ✁

for print ticket, deliver refund

• for convenience, label the same arrow with several i/o pairs
• for convenience, just write

as a blank

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FSM Traces

For some FSM called

• , a trace describes one possible sequence of

activities and is

• 1. A finite sequence of alternating states and transition labels

starting and ending with a state:

.

• 2. The first state,

, is the start state of

• .
• 3. For each state, label, state sequence

appearing in the sequence, there must be a transition in

• from state

to

labelled

.

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m/ t/p r/d m/ t/ r/ 1 2

For the ticket machine

• ☎

m/

m/

m/

r/d

• is a trace, but
• ☎

m/

• ☎

m/

m/

r/d

• isn’t.

The overall behaviour of a FSM is the set of all its possible traces.

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Acceptors

Acceptors are FSMs with

• empty output alphabet (all outputs are

)

• some states marked as “accepting” (double circle)

Input sequence is accepted if it can generate a trace from the start state to an accepting state. The language accepted by an (acceptor) FSM is the set of all input sequences which it accepts.

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Acceptors

1 1 1 1,0 B A C D

Accepts strings of 0s and 1s in which the number of 0s so far never exceeds the number of 1s so far by more than one and vice versa

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Determinism

• FSMs in which for every state, there is at most one transition

leaving the state for each input symbol are called determinstic.

• Other FSMs are called non-deterministic
• Sometimes easier to define non-deterministic FSMs
• Can always convert a non-deterministic FSM into an equivalent

deterministic FSM

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What FSMs can’t do

• FSM states do not have any memory inside
• The FSM “remembers” simply being in a given state, so that “Have

seen three 0’s” and “Have seen four 0’s” must be two different states, not one state with a counter variable inside

• Consequently, FSMs can’t “count” arbitrarily large histories

– accept all strings including 287 0’s - possible – accept all strings with equal numbers of 0’s and 1’s - not possible

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Summary

• Embedded and reactive systems
• Finite State Machines to model these
• FSM traces
• Acceptors and Transducers
• Determinism
• Limits to FSM power

Finite State Machines