§12.2 – Finite State Machines with Output

Giving credit where credit is due:

• Most slides based on lecture notes created Most slides based on lecture notes created by by – Michael P. Frank, University of Florida Michael P. Frank, University of Florida – Longin Jan Latecki, Temple UniversityLongin Jan Latecki, Temple University– Carla Gomes, Cornell UniversityCarla Gomes, Cornell University

• Remember the general picture of a computer as Remember the general picture of a computer as being a transition function being a transition function TT::SS××II→S→S××OO??– If the state set If the state set SS is finite (not infinite), we call this is finite (not infinite), we call this

system a system a finite state machinefinite state machine..

• If the domain If the domain SS××II is reasonably small, then we can is reasonably small, then we can specify specify TT explicitly by writing out its complete explicitly by writing out its complete graph.graph.– However, this is practical only for machines that have a However, this is practical only for machines that have a

very small information capacity.very small information capacity.

Modeling Computation

Applications

• Finite-State Machines are used in a variety Finite-State Machines are used in a variety of applications.of applications.– Spell checking programsSpell checking programs– Grammar checkingGrammar checking– Indexing and searching large text filesIndexing and searching large text files– Speech/Language recognitionSpeech/Language recognition– Network ProtocolsNetwork Protocols

Types of Finite-State Machines

• Finite-State Machines with OutputFinite-State Machines with Output– MealyMealy: : Output determined by state and inputOutput determined by state and input– MooreMoore: : Output determined by state aloneOutput determined by state alone

• Finite-State Machines with No OutputFinite-State Machines with No Output– Also known Also known as finite-state automataas finite-state automata– There are two types of finite-state automataThere are two types of finite-state automata

• DeterministicDeterministic: Each state-input pair dictates a unique transition : Each state-input pair dictates a unique transition into another stateinto another state

• Non-deterministicNon-deterministic: Each state-input pair can lead to several : Each state-input pair can lead to several possible statespossible states

Vending Machine Example

• Suppose a certain vending machine Suppose a certain vending machine accepts nickels, dimes, and quarters.accepts nickels, dimes, and quarters.– If >30If >30¢ is deposited, change is¢ is deposited, change is

immediately returned.immediately returned.

• If the If the ““cokecoke”” button is pressed, button is pressed,the machine drops a coke.the machine drops a coke.– It can then accept a new payment.It can then accept a new payment.

Ignore any otherbuttons, bills,out of change,etc.

Modeling the Machine

• Input symbol set: Input symbol set: II = { = {nnickel, ickel, ddime, ime, qquarter, uarter, bbutton}utton}– We could add We could add ““nothingnothing”” or or as an additional input as an additional input

symbol if we want.symbol if we want.• Representing Representing ““no input at a given time.no input at a given time.””

• Output symbol set:Output symbol set: OO = { = {, 5¢, 10¢, 15¢, 20¢, 25¢, coke}, 5¢, 10¢, 15¢, 20¢, 25¢, coke}..

• State set:State set: SS = {0, 5, 10, 15, 20, 25, 30} = {0, 5, 10, 15, 20, 25, 30}..– Representing how much money has been taken. Representing how much money has been taken.

Transition Function Table

Old Old statestate InputInput

New New statestate OutputOutput

00 nn 55

00 dd 1010

00 qq 2525

00 bb 00

55 nn 1010

55 dd 1515

55 qq 3030

55 bb 55

Old Old statestate InputInput

New New statestate OutputOutput

1010 nn 1515

1010 dd 2020

1010 qq 3030 55¢ ¢

1010 bb 1010

1515 nn 2020

1515 dd 2525

1515 qq 3030 1010¢ ¢

1515 bb 1515

Transition Function Table cont.

Old Old statestate InputInput

New New statestate OutputOutput

2020 nn 2525

2020 dd 3030

2020 qq 3030 1515¢ ¢

2020 bb 2020

2525 nn 3030

2525 dd 3030 5¢ 5¢

2525 qq 3030 20¢ 20¢

2525 bb 2525

Old Old statestate InputInput

New New statestate OutputOutput

3030 nn 3030 55¢¢

3030 dd 3030 1010¢ ¢

3030 qq 3030 2525¢ ¢

3030 bb 00 cokecoke

Another Format: State Table

Old Old statestate

Input SymbolInput Symbol

nn dd qq bb

00 5,5, 10,10, 25,25, 0,0,

55 10,10, 15,15, 30,30, 5,5,

1010 15,15, 20,20, 30,530,5¢¢ 10,10,

1515 20,20, 25,25, 30,1030,10¢¢ 15,15,

2020 25,25, 30,30, 30,1530,15¢¢ 20,20,

2525 30,30, 30,530,5¢¢ 30,2030,20¢¢ 25,25,

3030 30,530,5¢¢ 30,1030,10¢¢ 30,2530,25¢¢ 0,coke0,coke

Each pair showsnew state,output symbol

Directed-Graph State Diagram

• As you can see, these can get kind of busy.As you can see, these can get kind of busy.

0 5 10 15 20 25 30

q,5¢

n n ndq d

q

b b b b b b

n n n

q,25¢d,10¢n,5¢

d

d,5¢

q,20¢

q,15¢

q,10¢b,coke

Formalizing FSMs

• Just like the general transition-function definition from Just like the general transition-function definition from earlier, but with the output function separated from the earlier, but with the output function separated from the transition function, and with the various sets added in, transition function, and with the various sets added in, along with an initial state.along with an initial state.

• A A finite-state machinefinite-state machine MM=(=(SS, , II, , OO, , ff, , gg, , ss00))– SS is the state set. is the state set.– II is the alphabet (vocabulary) of input symbols is the alphabet (vocabulary) of input symbols– OO is the alphabet (vocabulary) of output symbolsis the alphabet (vocabulary) of output symbols– ff is the state transition function is the state transition function– gg is the output function is the output function– ss00 is the initial state. is the initial state.

• Our transition function from before is Our transition function from before is TT = ( = (ff,,gg))..

Examples