switches
I need to plan a circuit that will convert $$-\{Y_3,Y_2,Y_1,Y_0\}$$
To it's two's complement show
$$\{S_3,S_2,S_1,S_0\}$$
I think the question is undefined well since 0 is not unique and since converting a 4 bits negative number to two's complement requires 5 bits and I only have room for 4 bits.
Is there a way to solve this?
You basically listed them all.
A Matrix of x rows and y columns. You need x+y pins.
Digital GPIO Expander IC, preferably with interrupt. I2C, SPI, even Serial are available. Interrupt pins allow you to read on interrupt instead of polling. You need to have hardware I2C/SPI/UART, or add software code. This approach is mainly used if you need a lot more GPIO than you have available on the main microcontroller. At that point, you are basically still using options 1, 3, and 4, or the direct one button per pin.
Resistor Ladder. You need an ADC, and constant polling. Better to break up into a few similar groups on multiple ADC channels, but you can make a large 20 button one if you really need to.
Charlieplexing. Like a multiplexed matrix (#1), but with \$N \times (N – 1)\$ where \$N\$ is the number of pins used. Requires as many diodes as buttons, so you are changing pin count for diodes. You could use LEDs though.
For the most part, #1 is the most common method. Every keyboard or touch tone phone you have ever used, 1000 to 1, would have used it. Hell, even cell phones use it (specifically, the Nokia 5110 I know uses it.) For 20 buttons, a 4x5 matrix will only take 9 pins, more than enough.
Of course there is.
First we write out the truth table:
Y S
000 000000
001 000001
010 000100
011 001001
100 010000
101 011001
110 100100
111 110001
We notice right away that S0 = Y0, S1 = 0, and S5 = Y1 * Y2. Let's remove those.
Y S
210 432
000 000
001 000
010 001
011 010
100 100
101 110
110 001
111 100
And now we see that S2 = Y1 * !Y0, S3 = Y0 * (Y1 ^ Y2), and S4 = Y2 * (!Y1 + Y0) = Y2 * !S2. Since you now have all 6 equations for the bits in S, building a logic circuit for it is easy, and left as an exercise for the reader.
Best Answer
A "two's complement" is just a one's complement followed by a increment. For example:
0101 original
1010 one's complement
1011 plus 1
The orginal was 0101, which is 5, and the result is 1011, which is -5 as expected. Put another way, it takes 5 increments to make the result 0.