Lecture 8 | Sequential Logic

Sequential Logic

How can a circuit remember?

Sequential Logic

What Is Missing from Our Computer?

So far

Every chip we built computes from its current inputs: gates, adders, and the ALU.

But computers must remember

A running program needs stored values, memory locations, and the address of the next instruction.

Definition

State is information retained from an earlier time.

Clock and Time

Representing Time with a Clock

Clock signal and cycles

Clock vocabulary

Clock: A signal broadcast to all sequential chips so they agree on time.
Tick and tock: The two phases of the clock signal.
Cycle: One complete tick-tock interval; we treat it as one discrete time unit.
Commit: Stored outputs update only at cycle boundaries.

Clock and Time

What Happens During a Cycle?

Clock cycle example showing a NOT gate and how input and output values change across cycles — Clock cycles make us say exactly when an input is sampled and when an output is observed.

Reading the figure

During a cycle: Combinational logic reacts to the input values it currently receives.
At a boundary: Sequential chips commit their stored outputs.
Why it matters: Without clock cycles, phrases like "previous input" and "next output" are ambiguous.

Data Flip-Flop

Data Flip-Flop (DFF)

A Data Flip-Flop (DFF) is the primitive one-bit time delay used by the simulator.

Specification

Chip name:: DFF
Inputs:: in
Outputs:: out
Function:: out(t)=in(t-1)
Implementation:: Primitive, supplied by the simulator.

DFF timing diagram showing output as the previous cycle input — The output follows the previous cycle's input.

Sequential Logic

Combinational Logic and Sequential Logic

Book figure comparing combinational and sequential chip structure — A sequential chip contains DFF gates, possibly surrounded by combinational logic.

Combinational chipout = f(in)

Output depends only on current input values.

Sequential chipout(t) = f(in(t-1), out(t-1))

Output can depend on earlier input values and earlier stored state.

Next: understand why feedback needs a time delay.

Data Flip-Flop

Why Feedback Needs Time

Without time

If a circuit output immediately feeds back into its own input, the output depends on itself at the same instant.

With a DFF

The feedback refers to the previous cycle, so the circuit can remember a value without creating an immediate circular dependency.

Next: define the one-bit storage cell we want to build.

Registers

What Is a One-Bit Register?

A one-bit register is a storage cell that remembers one bit until we explicitly load a new bit.

Required behavior

Read: The output continuously shows the currently stored bit.
Hold: If load=0, keep the stored bit even if in changes.
Load: If load=1, store the input bit at the next clock boundary.

Terminology

The Hack platform calls this one-bit register chip Bit.

Register

Building a One-Bit Register: First Try

Invalid design with DFF output connected directly back to its input — Invalid design: direct DFF feedback

It is not clear how we will ever load this device with a new data value.
There is no means to tell the DFF when to draw its input from in and when from out.
Internal pins must have fan-in 1: each internal pin must be fed from a single source only.

Register

Correct Implementation of a One-Bit Register

Introduce a multiplexor to resolve the input ambiguity.
The Mux select bit becomes the register's load bit.
If load=1, the register starts storing the new value on in.
If load=0, the register keeps storing its internal value.

Registers

The Bit Contract

Specification

Chip name:: Bit (1-bit register)
Inputs:: in, load
Outputs:: out
Function:: If load(t-1) then out(t)=in(t-1); else out(t)=out(t-1).

Read

Probe out. This does not require load=1.

Write

Put the new bit on in and set load=1. The value appears next cycle.

Exercise 1

Trace a Bit

Use the Bit behavior: load=1 loads at the next boundary; load=0 holds the old value. Assume the stored bit starts as 0.

Cycle	in	load	out after cycle
1	1	0	0
2	1	1	1
3	0	0	1
4	0	1	0
5	1	0	0

Exercise 2

Construct the Bit Chip

1

Draw the Mux-DFF feedback circuit.

2

Label in, load, out, and the feedback wire.

3

What goes wrong if the two Mux data inputs are exchanged?

PARTS:
  Mux(a=old, b=in, sel=load, out=next);
  DFF(in=next, out=old, out=out);

The DFF output is used both as the feedback wire and as the chip output.

Registers

From One Bit to a Word

A register stores a multi-bit value. A stored multi-bit value is often called a word.

Bit0

Bit1

Bit2

...

Bit15

All sixteen Bit chips share the same load signal, so all sixteen bits load together.

Registers

The Register Contract

Specification

Chip name:: Register
Inputs:: in[16], load
Outputs:: out[16]
Function:: If load(t-1) then out(t)=in(t-1); else out(t)=out(t-1).
Comment:: The assignment is a 16-bit operation.

Example

cycle	in[16]	load	out after cycle
1	0000...0011	1	0000...0011
2	1111...1111	0	0000...0011
3	1111...1111	1	1111...1111

Exercise 3

Exit Questions

1

A Register stores 0000...0111. During a cycle, in=1111...1111 and load=0. What is out after the cycle?

2

The same Register now has load=1. When does the new input become visible at out?

3

If a Register is built from Bit chips, what signal must be shared by all sixteen Bits?

1: 0000...0111; it holds the old word.

2: At the next clock boundary, not immediately.

3: The common load signal.

Next lecture: arrange many Registers as random-access memory.

Next Lecture | Memory

Lecture 9: Random Access Memory

Selecting one word among many using addresses and direct-access logic.

Go to Lecture 9