Download Finite State Machines Lab Notes for 6.004 Spring 2009 and more Slides Computer Fundamentals in PDF only on Docsity!
L06 ā FSMs 1
(Synchronous) 2/24/
Finite State Machines Lab 2 is due Thursday
Great - Theory! Finally!Some ENGINEERING!
modified 2/23/09 09:^
6.004 ā Spring 2009^
2/24/ Our New Machine
CurrentCombinationalStateLogic
NewState
Input^
Output
StateRegistersClock
k^
k
m^
n
- ^ Acyclic graph ā¢ ^ Obeys static discipline ā¢ ^ Can be exhaustively enumerated by atruth table of 2
k+m^ rows and k+n output
columns
- Engineered cycles ā¢ Works only if dynamicdiscipline obeyed ā¢ Remembers k bits for a totalk^ of 2unique combinations L06 ā FSMs 3
2/24/ Must Respect Timing Assumptions!Questions:^ ā¢ ^ Constraints on T
for the logic?CD
- ^ Minimum clock period? ā¢ ^ Setup, Hold times for Inputs?
CurrentCombinationalStateLogic
NewState
Input^
Output
Clock^
t= ?CD,L^ t = 5nsPD,L^ t= 1nsCD,R^ t= 3nsPD,R^ t^ = 2nsS,R^ t^ = 2nsH,R^
t> 1 nsCD,L^ t= t^ + t^ S^ PD,L^ S,R
= 7 nS (^) t= t - t = 1 nSH H,R CD,L
We know how fast it goesā¦ But what can it do?
t(1 ns) + t^ CD,R^ CD,L
(?) > t(2 ns)H,R^ t^ > t^ +t^ CLK^ PD,RPD,L
6.004 ā Spring 2009^
2/24/ A simple sequential circuitā¦Lets make a digital binary
Combination Lock: Specification: ā¢ One input ( ā0ā or ā1ā) ā¢ One output (āUnlockā signal) ā¢ UNLOCK is 1 if and only if:Last 4 inputs were theācombinationā: 0110
IN^ Lock How manyregisters doI need?
U CLK
L06 ā FSMs 5
2/24/ Abstraction
du jour: Finite State Machines ā¢ A FINITE STATE MACHINE has m^ ClockedFSM
n
ā¦ S(one is āinitialā state) 1 k^
ā¦ I 1 m
ā¦ O 1 n
- ^ Transition Rules sā(s, I) for each state s and input I ā¢ ^ Output Rules Out(s) for each state s
6.004 ā Spring 2009^
2/24/ State Transition DiagramSXS0 (^0) U=0U=
S01 1 U=
S011 1 U=
S0110 0 U=
1
0 0
0
(^11)
XXXU=
NAME of state OUTPUT when in thisstate
Heavy circleMeans INITIAL^ state^0 INPUTcausingtransition
Designing our lock ā¦^ ā¢ ^ Need an initial state; call it SX.^ ā¢ ^ Must have a separate state for each stepof the proper entry sequence^ ā¢ ^ Must handle other (erroneous) entries
Why do thesego to S0 and S01?
L06 ā FSMs 7
2/24/ Yet Another SpecificationSXS0S01U=0U=0 0 U=0^1
S011S0110U=0 1 U=1^0 1
0 1 001
IN^ Current State
Next State Unlock
0 SX^
S0^0
1 SX^
SX^0
0 S
S0^0
1 S^
S01^0
0 S
S0^0
1 S
S011^0
0 S
S
1 S
SX
0 S
S
1 S
S
All state transitiondiagrams can bedescribed by truthtablesā¦Binary encodings areassigned to each state(a bit of an art)The truth table can thenbe simplified using thereduction techniques welearned for combinationallogic
000000 00 100 10 1 10 1 10 100 10^100100
00 1 000 00 10 1 100 10 101 00 000 00 10 1 1 The assignment of codes tostates can be arbitrary, however,if you choose them carefully youcan greatly reduce your logicrequirements.
6.004 ā Spring 2009^
2/24/ Valid State Diagrams
S1 ā¢ Arcs leaving a state must be: ā¢ (1) mutually exclusive ā canāt have two choices for a given input value ā¢ (2) collectively exhaustive ā every state must specify what happens for each possible inputcombination. āNothing happensā means arc back to itself.
1 S3 0 11 00 S2^01 MOORE Machine:Outputs on States
1/1 S1 S31/00/0 0/1S21/0 MEALY^ Machine:Outputs on Transitions
0
0/
L06 ā FSMs 13
6.004 ā Spring 2009^
2/24/
Whatās My Transition Diagram?
vs.
0= "
o^1 You Win!
OFF,1=ON?111" =Surprise!
- ^ If you know NOTHING about the FSM, youāre never sure! ā¢ ^ If you have a BOUND on the number of states, you can discover itsbehavior:K-state FSM: Every (reachable) state can bereached in < k steps.BUT ... states may be equivalent!
L06 ā FSMs 14
6.004 ā Spring 2009^
2/24/
FSM Equivalence
vs.
ARE THEY DIFFERENT?NOT in any practical sense! They are EXTERNALLYINDISTINGUISHABLE, hence interchangeable.FSMs
EQUIVALENT
iff every input sequence
yields identical output sequences.ENGINEERING GOAL:ā¢ HAVE an FSM which
works...
ā¢ WANT simplest (ergo cheapest) equivalent FSM.
L06 ā FSMs 15
6.004 ā Spring 2009^
2/24/
Lets build an
Ant
ā¢ ^ SENSORS: antennae L and R, each 1 if incontact with something. ā¢ ^ ACTUATORS: Forward Step F, ten-degreeturns TL and TR (left, right).
8 legs? GOAL: Make our ant smart enough to get out of a maze like:STRATEGY: "Right antenna to the wall"
L06 ā FSMs 16
6.004 ā Spring 2009^
Lost in space^? 2/24/
Action: Go forward until we hit something.^ LOSTL+RF _ _L R ālostā is theinitial state
Figure by MIT OpenCourseWare.
L06 ā FSMs 17
Bonk! 2/24/ LOSTL+RF _ _L R
L+R RCCWTL _ _L R
Action: Turn left (CCW) until we donāt touch anymore
6.004 ā Spring 2009^
2/24/ A little to the rightā¦LOSTF
L+R RCCWL+RTL R Wall1_TR,FR _ _L R
_ _L R Action: Step and turn right a little, look for wall
L06 ā FSMs 19
2/24/ Then a little to the leftLOSTF
L+R RCCWL+RTL R Wall1_TR,FR _ _L R
_ _L R^
Wall2TL,FL _ _L R _L R
Action: Step and turn left a little, till not touching (again)
6.004 ā Spring 2009^
2/24/ Dealing with cornersLOSTF
Wall2TL,F L+R RCCWL+RLTL R Wall1_TR,FR _ _L R
_ _L R^
_ _L R
_L R R _RCornerTR,F
Action: Step and turn right until we hit perpendicular wall
L06 ā FSMs 25
Ant Schematic 2/24/
6.004 ā Spring 2009^
RoboantĀ® 2/24/
Mazeselection
FSM statetable Statusdisplay
Plan viewof mazeSimulationcontrols
Featuring the new Mark-II ant: can add (M),erase (E), and sense (S) marks along its path.
L06 ā FSMs 27
2/24/ Housekeeping issuesā¦ROMorgatesNEXT inputs^ STATE
outputs s s
1. Initialization? Clear the memory?2. Unused state encodings?- waste ROM (use PLA or gates)- what does it mean?- can the FSM recover?3. Choosing encoding for state?4. Synchronizing input changes withstate update?
IN CLK
U Now, thatās a funnylooking state machine
6.004 ā Spring 2009^
2/24/ Twisting you Furtherā¦
-^ MORE THAN ANTS:Swarming, flocking, and schooling can resultfrom collections of very simple FSMsā¢ PERHAPS MOST PHYSICS:Cellular automata, arrays of simple FSMs,can more accurately model fluilds thannumerical solutions to PDEsā¢ WHAT IF:We replaced the ROM with a RAM and haveoutputs that modify the RAM?... You'll see FSMs for the rest of your life!
Did we all descend from FSMs??? I prefer to think we^ ascended ā¦
