Hierarchy

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Blocks inside blocks

Three small 2-to-1 MUX blocks on the left, an arrow

A modern chip has billions of gates. Nobody draws them one at a time. Instead they build a small block, verify it once, then treat it as a single part and wire copies of it into bigger blocks. That nesting is hierarchy.

You already have a block worth reusing: the 2-to-1 MUX.

Blocks inside blocks

Figure: Three small 2-to-1 MUX blocks on the left, an arrow

You already have a block worth reusing: the 2-to-1 MUX.

The 4-to-1 from 2-to-1s

Figure: A tree of three 2-to-1 multiplexers

Wire three 2-to-1 MUXes into two stages:

Stage 1, two MUXes controlled by $s_0$ :

MUX A picks between $i_0$ and $i_1$ .
MUX B picks between $i_2$ and $i_3$ .

Stage 2, one MUX controlled by $s_1$ :

MUX C picks between the outputs of A and B.

The same four-input selection as the last lesson, built entirely from a block you already trust.

Splitting the address

The two stages split the 2-bit address between them.

$s_0$ , the low bit, tells each stage-1 MUX whether to take the lower or upper of its pair.
, the high bit, tells stage 2 which pair to keep.

s_1

So address $10$ means: $s_0 = 0$ makes A output $i_0$ and B output $i_2$ ; then $s_1 = 1$ makes C keep B's value, $i_2$ . That matches the routing table exactly.

Recall the 2-to-1

This is the block being reused, the 2-to-1 MUX from Module 1. Three copies of exactly this, wired in the tree, make the 4-to-1.

Toggle s and confirm it still does one job: pick a or b. Hierarchy never changes what a block does. It just lets you stop thinking about the inside and treat it as one part with a known behaviour.

One part again

Here is the assembled 4-to-1 MUX. From the outside it is a single part with six inputs and one output. Whether it was built flat from gates or as a tree of three smaller MUXes makes no difference to anyone using it.

That is the payoff of hierarchy: each level hides the level below. A CPU is MUXes and adders; an adder is gates; a gate is transistors. You work at one level at a time.