# Question about logic gate optimization and CMOS implementations

I am struggling with a conceptual question surrounding logic gates and how they're efficiently implemented in CMOS circuits.

For example, let's say I want to implement a NAND gate. If I didn't know anything about CMOS, I could build it on paper at the level of logic gates with an AND gate followed by an inverter. However, it turns out that NAND gates are efficiently "directly" implemented in CMOS; there is no need for an individual AND gate or inverter.

Similarly, some compound gates (like a 2-1 AOI) are actually simpler and more efficient to implement "directly" in CMOS than with the individual gates (an AND and a NOR gate). If I understand correctly, using a "direct" AOI gate instead of the individual gates saves me an entire level of logic!

What other complex gates are more efficiently implemented "directly" in CMOS than with the individual gates? Suppose I build a rather complex yet optimized gate/chip using only basic gates as the building blocks. How do I know there isn't a more efficient way to implement that chip "directly" in CMOS in a single level of logic?

If it turns out that my big complex chip has a simpler "direct" implementation in CMOS than with the individual gates (maybe my chip is just a 2-2 AOI!), why design at the level of individual logic gates? How are large complex chip designs (designed at the logic gate level) implemented efficiently in CMOS?

Thanks!

Boolean operators AND, OR, and NOT are the basic building blocks: any Boolean expression can be expressed with these operations. But that's not the only possible orthogonal set of Boolean operators.

NAND by itself is sufficient to express all Boolean logic expressions, because it's possible to build each of the basic AND, OR, and NOT equivalents out of nothing but NAND gates. Or to put it another way, suppose you had a technology that could efficiently realize the NAND gate: then because it's possible to make AND, OR, and NOT, that NAND technology is flexible enough to implement everything. (The same is true of NOR gates -- the Apollo_Guidance_Computer was built entirely out of 3-input NOR gates).

On the other hand, if you had a new and different technology that could only realize XOR gates, would that be enough to build arbitrary logic equations? You can get NOT by XOR with 1, but how do you build the AND and OR operators? Unless I missed something, an XOR-only technology would not be sufficient to realize arbitrary logic equations.

What other complex gates are more efficiently implemented "directly" in CMOS than with the individual gates?

CMOS technology commonly used in Integrated Circuits can efficiently implement another kind of gate called a transmission gate -- this is kind of like a switch that passes signals when "closed", or isolates signals when "open". Together with a pull-up or pull-down resistor termination, transmission gates in parallel act as a wired-AND or wired-OR logic gate. And (assuming logic high control signal makes the transmission gate "closed") a single transmission gate can be used with a pull-up resistor as a NOT gate. So a technology that has transmission gates and large-value resistors, can realize any Boolean logic equation.

(OK technically, there's always limits. A 10,000,000 input AND gate would require 10,000,000 inputs, which probably isn't practical... but that argument is like writing a C program that declares an array 100 times bigger than available memory. So there can exist C programs that can't be compiled, and there can exist Boolean expressions that can't be realized. But engineering is more concerned with realizing things that people actually want to have.)

...why design at the level of individual logic gates?

AND, OR, and NOT are fairly intuitive to understand -- these operators are named after common English words, that have compatible meaning. So it's usually straightforward to directly translate a given requirement from plain English into a Boolean expression or a circuit diagram with AND / OR / NOT. It's also not too difficult to translate a logic circuit diagram back into some meaningful and correct words. This helps verify that the circuit that was designed is designed correctly, and also verify that it solves the intended problem.

But just because a circuit is designed with AND / OR / NOT operators doesn't mean it has to be implemented that way. Commercial FPGA / ASIC design software using Verilog or VHDL, can take a design description stated in AND / OR / NOT, and translate that into whatever form the real implementation technology needs. For example, everyplace there is an AND, could be replaced with an equivalent 3-input NOR circuit, or a lookup table (LUT), or transmission gates, depending on what technology the target device supports. From that point, technology-specific optimizations can be automatically applied. (Like for example replacing two 3-input LUT and a 2:1 mux, with a single 4-input LUT). So at some point, the basic AND / OR / NOT operators just become a useful abstraction, that hides details about the underlying technology.

For that matter, Verilog and VHDL support higher-level circuit descriptions that only implicitly define the underlying Boolean expressions. So it's possible to define a state machine at a high level, thinking only about its state table and output expressions, without worrying much about how many flip-flops you will need, or how many lookup tables, or how many of the FPGA's routes will be needed. Computer-aided design is great for hiding low-level details through all these layers of abstraction. A successful designer needs to develop a good intuitive understanding of everything that's usually hidden by these layers of abstraction. (By contrast, just search for "how does electricity work" questions about the "water analogy" for resistors and capacitors, and you'll find plenty of folks getting themselves even more confused when they try to understand transistors... or wondering what happens if electrons leak out of the wires somehow, or thinking electrons get used up inside a light bulb. Sorry, but electricity just doesn't behave like anything else in nature. And it's pretty much invisible. Which makes it very hard to develop a good intuition for how it works.)

For a large FPGA holding a relatively small logic design, automatic routing tools can be pretty effective. Larger, faster, or otherwise cutting-edge designs do still require a designer to understand what's happening at the lower levels, because automatic optimizations can get stuck, and it's up to the designer to determine if there's a way to guide the tools to do the right thing, or if a different approach is needed.

If it turns out that my big complex chip has a simpler "direct" implementation in CMOS than with the individual gates (maybe my chip is just a 2-2 AOI!), why design at the level of individual logic gates?

A big complex chip is indeed complex and hard to understand, and it's incomprehensible if you try to look at the massive "sea of transistors" without having a higher-level abstraction to help make sense of it all. A functional unit made of 10 transistors may be difficult to work out from first principles, though eventually it can be understood. But 1000 transistors is incomprehensible, it's just too much to understand as a single unit. There are physical limits to how many things the human brain can pay attention to at a given time, and that's important to help prevent design errors. There's a rule of thumb in software, that a function should be no more than 1-2 screenfuls of text, otherwise it becomes incomprehensible and too costly to debug. Similarly schematics are organized so that related connections are all on the same page so it's comprehensible, otherwise if you have to look at dozens of pages just to verify or debug one functional unit, that's going to be a likely point of design failure. So a big and complex chip design has to be divided up hierarchically into smaller functional units, that can be checked and verified, that can be individually tested and understood. Then those functional units can be further subdivided until you get down to the actual technology. And you're right, there may be optimizations at the lowest level that wouldn't be obvious from the high level.

The biggest cost of developing an integrated circuit is not the cost of the silicon or the other chemicals or even the per-unit manufacturing cost, it's actually the cost of testing and verifying the design. Getting it right the first time may cost ~\$100k and 6 months (depending on process technology), not getting it right the first time costs a lot more. And although it's not impossible to probe and even patch a prototype IC to some extent (e.g. using Focused Ion Beam), it's a very expensive debugging step and essentially limited to the top layer of the chip. So high-level design verification is essential, and hierarchical abstraction is a big part of that. And ultimately that's all the AND / OR / NOT operators are, just an abstraction.

• Actually, the water analogy still holds for transistors. Consider a water hose, where turning off the flow involves pinching off the stream by applying pressure with your fingers. This is most directly analogous to a P-channel JFET; applying increasing voltage (pressure) w.r.t. the source squeezes the channel (hose) and causes reduced current (flow). There are other variations...
– Zulu
Apr 17, 2015 at 21:28

Logic synthesis can take an logic function and implement it in the logic gates provided in a standard cell library. These typically include complex gates.

The algorithmic process by which the gate-level implementation is done doesn't necessarily yield the best solution, only a solution. To quote an abstract:

obtaining an optimal (shortest length) factorization for an arbitrary Boolean function is an NP-hard problem, all practical algorithms for factoring are heuristic and provide a correct, logically equivalent formula, but not necessarily a minimal length solution.

"Not necessarily minimal length" means the the factorization, even if it uses available complex gates, might not have the fewest possible gates.