Example of independent stages: Finite State Machines (FSM)

A finite-state machine (FSM) or finite-state automaton is a mathematical model of computation.
A FSM has a finite number of states
At at moment, the FSM is in one of these (finite) state
The FSM can transit (= move) from one state to another state in response to some inputs

The implementation of a FSM in digital circuit will require the use of D-flipflops (instead of D-latches)

Example of a FSM

The following FSM represents the computational model to recognize all strings of the form: a(b*)c

Strings such as ac, abc, abbc, etc will cause the FSM to reach the terminal (= accept) state

(I think FSM was covered in CS244...)

Time-controlled transitions

Many machines has a time controlled state transition
Examples:
Yes, a CPU is a Finite State Machine whose state transitions are paced (= controlled) by a clock (= time)

In the next few slides, we will build a simple time-controlled FSM as an introduction on how a CPU operates...

FSM of a traffic (stop) light

I found this FSM diagram for a traffic light on the Web:

This FSM models the normal (operational) behavior of a stop light...

FSM of a traffic (stop) light - RE-DRAWN

I have re-drawn the FSM state transition diagram:

We will show the initial state (= stop light not ready) as blinking (flashing) stop lights...

FSM of a more complex traffic (stop) light

To make the FSM more like a computing machine, we will model a more complex traffic (stop) light that can detect errors:

The signal "E" is a self-diagnosis condition
E=0 means: no error detected

When no error (i.e.: E=0) the state transitions are as before.

FSM of a more complex traffic (stop) light

To make the FSM more like a computing machine, we will model a more complex traffic (stop) light that can detect errors:

The signal "E" is a self-diagnosis condition
E=1 means: an error has been detected

When in error (i.e.: E=1) the FSM will always go to the initial state.

Constructing a FSM with digital circuits

Unlike CS244 (which only teaches FSM theory), we will build a working FSM in CS355 😃 !!
The first step in building the FSM with digital circuits is to represent (= encode) the state in (binary) numbers:
The choice of the numbering is arbitrary...

The FSM state transition diagram using the chosen representation code

The FSM's state transition diagram was:

The FSM state transition diagram using the chosen representation code

The FSM's state transition diagram using the assigned numbering is:

How to design a FSM using digital circuits

Facts to help you understand the design process:

The FSM is always in one of its (possible) states
Therefore: a FSM need to remember its current state
We must use D-flip flops to remember the current state of the FSM
Question:

How to design a FSM using digital circuits

Facts to help you understand the design process:

The FSM is always in one of its (possible) states
Therefore: a FSM need to remember its current state
We must use D-flip flops to remember the current state of the FSM

Question:

How many D-flip fops do we need to use to remember a state that is one of the following values: Answer: 2 D-ffs (2 bits)

How to design a FSM using digital circuits

We start with the 2 D-flip flops used to remember the current state of the FSM:

The output of the flip flops are called Q₁ and Q₀ to tell them apart.

How to design a FSM using digital circuits

The D-flip flops are controlled (= written) by the same clock signal:

As you have learned: the input D_i will be written to output Q_i by the clock signal

How to design a FSM using digital circuits

The FSM will make a state transition in every clock period. Therefore:

The input D₁D₁ will be written to output Q₁Q₁ and therefore, the next state = D₁D₁

How to design a FSM using digital circuits

We must compute the correct values for D₁ and D₀:

The next state D₁D₀ depends on: (1) the current state and the (2) error condition !

How to design a FSM using digital circuits

Schematically:

How to design a FSM using digital circuits

Drawn out with connecting lines:

Remaining task: design (and construct) the (computation) circuits C₁ and C₀

How to design a FSM using digital circuits

The FSM state transition diagram of the stop light:

Error Current state Next state E Q1 Q0 | D1 D0 ------------------------+--------------- 0 0 0 | 0 0 1 | 0 1 0 | 0 1 1 | 1 0 0 | 1 0 1 | 1 1 0 | 1 1 1 |

How to design a FSM using digital circuits

The FSM state transition diagram of the stop light:

Error Current state Next state E Q1 Q0 | D1 D0 ------------------------+--------------- 0 0 0 | 0 1 (red → green) 0 0 1 | 0 1 0 | 0 1 1 | 1 0 0 | 1 0 1 | 1 1 0 | 1 1 1 |

How to design a FSM using digital circuits

The FSM state transition diagram of the stop light:

Error Current state Next state E Q1 Q0 | D1 D0 ------------------------+--------------- 0 0 0 | 0 1 0 0 1 | 1 0 (green → yellow) 0 1 0 | 0 1 1 | 1 0 0 | 1 0 1 | 1 1 0 | 1 1 1 |

How to design a FSM using digital circuits

The FSM state transition diagram of the stop light:

Error Current state Next state E Q1 Q0 | D1 D0 ------------------------+--------------- 0 0 0 | 0 1 0 0 1 | 1 0 0 1 0 | 0 0 (yellow → red) 0 1 1 | 1 0 0 | 1 0 1 | 1 1 0 | 1 1 1 |

How to design a FSM using digital circuits

The FSM state transition diagram of the stop light:

Error Current state Next state E Q1 Q0 | D1 D0 ------------------------+--------------- 0 0 0 | 0 1 0 0 1 | 1 0 0 1 0 | 0 0 0 1 1 | 0 0 (flashing (fixed) → red) 1 0 0 | 1 0 1 | 1 1 0 | 1 1 1 |

How to design a FSM using digital circuits

The FSM state transition diagram of the stop light:

Error Current state Next state E Q1 Q0 | D1 D0 ------------------------+--------------- 0 0 0 | 0 1 0 0 1 | 1 0 0 1 0 | 0 0 0 1 1 | 0 0 1 0 0 | 1 1 (When error, goto state 11) 1 0 1 | 1 1 1 1 0 | 1 1 1 1 1 | 1 1

How to design a FSM using digital circuits

Designing the compute circuits C₁ and C₀:

Error Current state Next state E Q1 Q0 | D1 D0 ------------------------+--------------- 0 0 0 | 0 1 0 0 1 | 1 0 0 1 0 | 0 0 0 1 1 | 0 0 1 0 0 | 1 1 (When error, goto state 11) 1 0 1 | 1 1 1 1 0 | 1 1 1 1 1 | 1 1

We can apply the combinatorial circuit design technique learned previously...

It easy to see that:
D1 = E OR ((not Q1) AND Q0)
D0 = E OR ((not Q1) AND (not Q0))

How to design a FSM using digital circuits

The FSM is therefore (incomplete, I left out the D0 circuit):

DEMO: /home/cs355001/demo/circuits/seq-circuit1

How to design a FSM using digital circuits

This version also probe the inputs of the D-fliplops:

DEMO "how the FSM works": /home/cs355001/demo/circuits/seq-circuit1-demo

Making a FSM control a stop light

We can make the FSM control a stop light with the following combinatorial circuit:

Logic function computed by this circuit: Q1 Q0 | R Y G ---------+---------- 0 0 | 1 0 0 (Red) 0 1 | 0 1 0 (Yellow) 1 0 | 0 0 1 (Green) 1 1 | 1 1 1 (Flashing)

DEMO: /home/cs355001/demo/circuits/seq-circuit1-demo-plus-light

Viewing the stop-light FSM as staged processing

The following diagram shows an another way to draw the FSM circuit of stop-light

Viewing the stop-light FSM as staged processing

In the diagram, you can clearly see a computation stage followed by a memory stage:

A CPU is also a FSM, but will have more stages and more complex compute/forward circuits