FSM Word Problems
Notes: Review for Test #2 – Monday
Studio #8: Reading assignment is required & due next week
Today:
• First Hour: Finite string recognizer, Complex counter
– Section 8.5 of Katz’s Textbook
– In-class Activity #1
• Second Hour: Traffic signal controller, Digital
combination lock
• Section 8.5 of Katz’s Textbook
– In-class Activity #2
1
Katz Material not Covered
CoCO doesn't cover everything in Katz.
Omitted material includes:
ASM charts
The ABEL language
all of Chapter 9 is skipped
2
Word Problems
• One of the most difficult problems is making an
imprecise description of a finite state machine
into a precise one.
• Have you covered all the states?
• Omissions can cause failures, crashes, death and
destruction, etc.
• This is the Hardware equivalent of a Software
programming error.
3
Finite String Recognizer
Serial Finite State Machine
• One input: X
• One output: Z
• Description:
– Z is 1 if the 3 previous input bits are 010, and
100 has never been seen.
• Unstated assumptions:
– RESET starts the FSM at the "reset" state
– Z is asserted when the following bit is seen.
A Moore Machine implementation.
4
Example
Serial Behavior
• X: 0 0 1 0 1 0 1 0 0 1 0
• Z: - 0 0 0 1 0 1 0 1 0 0 0
• Z is 0 even though the three previous inputs are
010, because 100 was seen earlier.
5
Formal Design
State Transition Diagram
• Create sequences of states
for the strings that the
machine recognizes:
010 and 100.
• Note we reset to S0.
0
S0
[0]
Reset
1
S4
[0]
S1
[0]
1
0
S2
[0]
• Consider the unlabelled
transitions.
0
S5
[0]
0
S3
[1]
S6
[0]
0,1
6
State S3
Diagram Development
0
• Where do we go from
S3?
Reset
1
S4
[0]
S1
[0]
• A 1 means the last 3
bits are 101, so go to
S2.
• A 0 means we’ve
seen 100, so go to
S6.
S0
[0]
1
0
S2
[0]
01?
1
0
010
S5
[0]
S3
[1]
0
0,1
S6
[0]
0
100
7
States S1 and S4
Diagram Development
• Loop in S1 until we
see our first 1.
• Loop in S4 until we
see our first 0.
0
0
1
1? S4
S1
[0]
0?
1
[0]
1
0
S2
[0]
01?
S5
[0]
0
0
010
Reset
S0
[0]
S3
[1]
1
S6
[0]
0
0,1
100
8
States S2 and S5
Diagram Development
• S2 means the last 2
bits are 01, which
may be a prefix of
010.
• If the next bit is 1,
the last 2 bits are
now 11, maybe a
prefix of 100. That’s
S4.
• S5: Last 2 bits are
10. If next bit is 1,
maybe that’s a prefix
for 010. Go to S2.
0
0
S1
[0]
0?
1
1? S4
S2
[0]
0
1
S3
[1]
1
[0]
1
1
01?
010
Reset
S0
[0]
0
1
10?
S5
[0]
0
S6
[0]
0
0,1
100
9
Review of Design Steps
Katz's Method
• Write sample inputs and outputs to understand it.
• Write sequences of states and transitions for the
strings that the FSM is to recognize.
• Add missing transitions, using existing states
when possible.
• Verify that the state diagram matches the FSM.
10
Complex Counter
• Design a 3-bit counter, with one input bit, a mode, M.
• If M = 0, step to the next binary number in the
sequence 000, 001, 010, 011, 100, 101, 110, 111, …
• If M = 1, step to the next Gray code number in the
sequence 000, 001, 011, 010, 110, 111, 101, 100, ...
11
Try Some Sample Inputs
• Note that we can switch modes at any time.
Mode Input
M
0
0
1
1
1
0
0
Current
State
0 0 0
0 0 1
0 1 0
1 1 0
1 1 1
1 0 1
1 1 0
Next State
(Z2 Z1 Z0)
0 0 1
0 1 0
1 1 0
1 1 1
1 0 1
1 1 0
1 1 1
12
Formal
Representation
11
S1
[001]
1
• One state for each
output combination
Reset
S0
[000]
1
1
0
0
S2
[010]
0
S3
[011]
1
• Add appropriate arcs
for the mode control
0
S4
[100]
1
0
S5
[101]
0
S6
[110]
1
1
0
S7
[111]
0
13
Do Activity #1 Now
11
0
0
010
0
1
S2
[0]
1
0
S3
[1]
1
1
10?
0
1
S5
[0]
0
S2
[010]
0
0
S4
[100]
1
0
S5
[101]
0
S6
[0]
0
S3
[011]
1
1
01?
1
[0]
S1
[001]
1
1
1? S4
S1
[0]
0?
Reset
S0
[0]
Reset
S0
[000]
0
0,1
100
FSM String Recognizer, Z=1 if
010 is seen, but 100 not seen before
S6
[110]
1
0
S7
1
0
[111]
Complex Counter
14
Diagram of Intersection
Farmroad
C
HL
FL
Highway
Highway
HL
FL
C
Farmroad
15
Traffic Light Controller
• A busy highway is intersected by a little-used farm road.
• Detectors C sense the presence of cars waiting on the
farm road.
• With no car is on farm road, the lights remain Green in
the highway direction.
• If vehicle is on the farm road, highway lights go from
Green to Yellow to Red, allowing the farm road lights to
become Green.
• These stay Green only as long as a farm road car is
detected but never longer than a set time interval.
• When these are met, farm lights transition from Green to
Yellow to Red, allowing highway to return to Green.
• Even if farm road vehicles are waiting, the highway gets
16
at least a set interval as Green.
Available Timers
• Assume you have an interval timer that generates
a short time pulse (TS) and a long time pulse (TL)
in response to a start timer (ST) signal.
• TS is to be used for timing Yellow lights and TL
for Green lights
17
Tabulate Inputs & Outputs
Input Signal
reset
C
TS
TL
Description
place FSM in initial state
detect vehicle on farm road
short time interval expired
long time interval expired
Output Signal
HG, HY, HR
FG, FY, FR
ST
Description
assert green/yellow/red highway lights
assert green/yellow/red farm road lights
start timing a short or long interval
18
Tabulate Unique States
• Some light configurations imply others.
State
S0
S1
S2
S3
Description
Highway green (farmroad red)
Highway yellow (farmroad red)
Farmroad green (highway red)
Farmroad yellow (highway red)
19
List Assumptions
• Reset places timer in S0, highway green and farm
road red.
• Reset also starts the timer.
• Stay in S0 as long as no one is on the farm road.
• Even if there is a farm road vehicle, the highway
stays green at least long as the long time interval.
• (Unstated in Katz) There will never be a bicycle or
pedestrian on the farm road.
20
Traffic Signal State Diagram
TL + C
Reset
S0: HG
S0
TL•C/ST
S1: HY
S1
S3
S2: FG
S3: FY
S2
TL: long time interval expired
C: detect vehicle on farmroad
21
Traffic Signal State Diagram
TL + C
Reset
S0: HG
S0
TL•C/ST
S1: HY
TS
S1
S3
S2: FG
S3: FY
TS/ST
TS: short time interval expired
S2
ST: start timing a short or long
interval
22
Traffic Signal State Diagram
TL + C
Reset
S0: HG
S0
TL•C/ST
S1: HY
TS
S1
S2: FG
S3
S3: FY
TS/ST
TL + C/ST
S2
TL: long time interval expired
C: detect vehicle on farm road
TL • C ST: start timing a short or long
interval
23
Traffic Signal State Diagram
TL + C
Reset
S0: HG
S0
TL•C/ST
TS/ST
S1: HY
TS
S1
S2: FG
S3
TS
TS/ST
S3: FY
TL + C/ST
S2
TS: short time interval expired
ST: start timing a short or long
TL • C interval
24
Combination Lock
• 3 bit serial lock controls entry to locked room.
• Inputs are RESET, ENTER, 2 position switch for
bit of KEY data.
• Locks generates an UNLOCK signal when KEY
matches internal combination.
• ERROR light illuminated if KEY does not match
combination.
• Sequence is:
– (1) Press ENTER,
– (2) enter KEY bit,
– (3) Press ENTER,
– (4) repeat (2) & (3) two more times. In the last
round, it is not necessary to press ENTER.
25
Incomplete Specification
• Problem specification is incomplete:
– how do you set the internal combination?
– exactly when is the ERROR light asserted?
26
Make Assumptions
• Make reasonable assumptions, decide whether
– combination is hardwired into logic or stored
in a register?
– error is asserted as soon as an error is
detected or waits until the full combination
has been entered?
Our design: combination is stored in a register and
error is asserted after the full combination has
been entered
Why is it just possibly a bad idea to indicate an error
immediately on seeing the first bad bit ?
27
Block Diagram of Lock
Operator Data
RESET
ENT ER
Internal
Combination
UNLOCK
KEY-IN
Combi nation
Loc k FSM ERROR
L0
Inputs:
Reset,
Enter,
Key-In,
L0, L1, L2
L1
L2
Outputs:
Unlock,
Error
28
Enumerate the States
• What sequences lead to opening the door?
• Do error conditions on a second pass …
29
State
Diagram of
Lock
Enter
Enter
Comp1
Error1
KI  L1
KI = L1
Enter
Enter
Idle1
Reset + Enter
Reset
Idle1'
Enter
Enter
Start
Comp2
Reset • Enter
Error2
KI  L2
KI = L2
Comp0
KI = L0
Reset
Reset
KI  L0
Done
[Unlock]
Enter
Error3
[Error]
Enter
Idle0
Idle0'
Reset
Reset
Start
Start
30
Do Activity #2 Now
Due: End of Class Today.
RETAIN THE LAST PAGE(S) (#3 onwards)!!
For Next Class:
• Bring Randy Katz Textbook, & TTL Data Book
• Required Reading:
– Sec 11.1-11.3, skim 11.2 of Katz, omit the ABEL and
ASM descriptions
• This reading is necessary for getting points in
the Studio Activity!
31