Comment by adrian_b

With NAND gates you can make any discrete system, but you can only approximate a continuous system.

This work is about continuous systems, even if the reduction of many kinds of functions to compositions of a single kind of function is analogous to the reductions of logic functions to composing a few kinds or a single kind of logic functions.

I actually do not value the fact that the logic functions can be expressed using only NAND. For understanding logic functions it is much more important to understand that they can be expressed using either only AND and NOT, or only OR and NOT, or only XOR and AND (i.e. addition and multiplication modulo 2).

Using just NAND or just NOR is a trick that does not provide any useful extra information. There are many things, including classes of mathematical functions or instruction sets for computers, which can be implemented using a small number of really independent primitives.

In most or all such cases, after you arrive to the small set of maximally simple and independent primitives, you can reduce them to only one primitive.

However that one primitive is not a simpler primitive, but it is a more complex one, which can do everything that the maximally simple primitives can do, and it can recreate those primitives by composition with itself.

Because of its higher complexity, it does not actually simplify anything. Moreover, generating the simpler primitives by composing the more complex primitive with itself leads to redundancies, if implemented thus in hardware.

This is a nice trick, but like I have said, it does not improve in any way the understanding of that domain or its practical implementations in comparison with thinking in terms of the multiple simpler primitives.

For instance, one could take a CMOS NAND gate as the basis for implementing a digital circuit, but you understand better how the CMOS logic actually works when you understand that actually the AND function and the NOT function are localized in distinct parts of that CMOS NAND gate. This understanding is necessary when you have to design other gates for a gate library, because even if using a single kind of gate is possible, the performance of this approach is quite sub-optimal, so you almost always you have to design separately, e.g. a XOR gate, instead of making it from NAND gates or NOR gates.

In CMOS logic, NAND gates and NOR gates happen to be the simplest gates that can restore at output the same logic levels that are used at input. This confuses some people to think that they are the simplest CMOS gates, but they are not the simplest gates when you remove the constraint of restoring the logic levels. This is why you can make more complex logic gates, e.g. XOR gates or AND-OR-INVERT gates, which are simpler than they would be if you made them from distinct NAND gates or NOR gates.