Truth Tables

A truth table lists every way a circuit’s inputs can be set, and what its output is for each one. It is the plan for a logic circuit: fill in the table first, then write the expression that matches it.

Contents

• Code
• Run
• Under the Hood
• Modify
• Make
• Good to Know

switch1 Enabled -> chip a
switch2 Enabled -> chip b
switch3 Enabled -> chip c
chip lit -> light Enabled

Code

Here are the four logic operators from Gates and Wires as one table. 1 means on (true) and 0 means off (false).

• && (AND) is on only in the last row, where both inputs are on.
• || (OR) is off only in the first row, where both are off.
• ^^ (XOR, “exclusive or”) is on when the inputs differ.
• ! (NOT) has one input and gives the opposite.

Two inputs make four rows. Each new input doubles the rows, so three inputs make eight. Here is a vote with three switches: the light turns on when at least two of them are on.

To turn the table into code, read it another way: the light is on when a and b are both on, or a and c are, or b and c are. Each pair is an &&, and the pairs join with ||:

@left in a: bool
@left in b: bool
@left in c: bool
@right out lit = (a && b) || (a && c) || (b && c)

The parentheses make the grouping easy to read. && binds tighter than || already, so the circuit is the same without them.

Check the code against the table: in the row 1 1 1, every pair is on. Is the light on? In how many of the eight rows is it on?

Run

1. Compile the chip and paste it in-game. Place three Switches and a Point Light.
2. Wire each Switch’s Enabled to a, b and c.
3. Wire lit to the Point Light’s Enabled.
4. Walk through the table: try all eight positions. The light is on in four of them, the four rows with two or three switches on.

Under the Hood

Open the chip. There are five gates: three ANDs, one for each pair, and two ORs that join them. Each && and || in the source is one gate.

Two expressions that give the same truth table are the same circuit, as far as anyone flipping the switches can tell. De Morgan’s laws are the two most useful pairs:

• !(a && b) is the same as !a || !b. “Not both” means “at least one is off”.
• !(a || b) is the same as !a && !b. “Neither” means “both are off”.

This chip computes both sides of the first law. Fill in the truth table for x and y, then flip the switches to check that two lights wired to them always agree:

@left in a: bool
@left in b: bool
@right out x = !(a && b)
@right out y = !a || !b

Open it. x is a single NAND gate (“not and”): the compiler joins a ! written around an && into one gate. y is three gates, two NOTs and an OR. The same happens for !(a || b), which becomes one NOR gate. When you have a choice, write the form with the ! outside.

Modify

1. Count the votes instead. A bool counts as 1 when on and 0 when off wherever a number is expected, so you can add switches:

   @left in a: bool
   @left in b: bool
   @left in c: bool
   @right out lit = a + b + c >= 2
   

   Its truth table is the same as the first chip’s. Open it: two Add gates and a Greater/Equal gate.

2. Change the rule to “every switch is on”, then to “exactly one switch is on”. Write the truth table first, then the expression.

3. Chain XOR: @right out lit = a ^^ b ^^ c. Fill in the eight rows. The light is on when an odd number of switches are on: one or three. This is a parity check, and it gives you a light that any of three switches can change, like the hall light with a third switch.

Make

Make a vote for four switches where the light turns on when at least three are on. Write out the sixteen-row table, then write it two ways: with && and || only, and by counting. Check both against the table in-game.

switch1 Enabled -> chip a
switch2 Enabled -> chip b
switch3 Enabled -> chip c
switch4 Enabled -> chip d
chip logic -> light1 Enabled
chip counted -> light2 Enabled

Good to Know

• a && b always computes both sides, even when a alone already decides: wires do not skip anything. In a handler, an action called on either side still runs, so to do it only when a is on, put an if inside an if.
• An input pin with no wire reads as off. A three-switch chip with only two switches wired treats the third as 0.
• A Switch’s Enabled stays on until it is flipped back, so a truth table is easy to walk through with switches. Button Held is on only while it is held.

Next: Choosing a Value.