All Models Are Wrong, Some Are Useful

“All models are wrong, some are useful.” This remark is attributed to the statistician George Box who used it as the section heading in a paper published in 1976.

Just for fun I looked up a few semiconductor statistics from 1976. Total capital spending was $238M in Japan and $306M in US and…that’s it, there was nobody else back then (at least according to SIA, I’m pretty sure something was going on in what became ST in Europe for example). It looks like Intel’s process technology had been 10um in 1974 for the 4040 (3000 transistors!) and was 3um in 1978 when the 8086 was released. So probably in 1976 the state-of-the-art was 5um, or 5000nm in today’s terminology.

Anyway, George Box went on to clarify what he meant:Now it would be very remarkable if any system existing in the real world could beexactly represented by any simple model. However, cunningly chosen parsimonious models often do provide remarkably useful approximations. For example, the law PV = RT relating pressure P, volume V and temperature T of an “ideal” gas via a constant R is not exactly true for any real gas, but it frequently provides a useful approximation and furthermore its structure is informative since it springs from a physical view of the behavior of gas molecules.

My favorite example of different kinds of models being useful for different things are these two:

The model on the left is the sort of plastic kit that I used to put together as a kid (you can still buy them apparently). If you wanted to, for example, measure the wingspan then this would be a good model. Trying to learn anything about aerodynamics, not so much. The model on the right is useless for pretty much anything other than learning about aerodynamics. It doesn’t even look like a real plane. But it flies.

I think the biggest model that we have in semiconductors is what I like to call the digital illusion. We have analog transistors and voltage levels, but we pretend they are digital gates which are 0 or 1 and have a delay that can be captured in a few parameters. When I started in EDA, we didn’t even model input and output slopes. We would characterize a gate by using SPICE and putting a step function on the input and seeing when the output reached some threshold, that was the delay. Then we went to adding slopes, so that we would measure the delay from when the input rose to 50% to when the output fell to 40% (or 60% if it was rising). It turned out some gates reached 40% on the output even before the input reached 50% so they had negative delay. So we lifted up the corner of the rug, swept the dust underneath, and set that value to zero. Then we had to start to model interconnect resistance…

But even today we still model gates as being digital with a fairly simple delay function. When we do signoff we get a bit more complex and start to model more of the analog effects to make sure our digital illusion hasn’t broken down. If we had to model a billion transistors as genuine analog devices using circuit simulation then we would never be able to design a microcontroller, never mind a smartphone application processor or a server microprocessor.

Of course we also set bounds on the environment where we expect the model to work. If you set the power supply voltage to 0.1V we don’t expect the digital illusion to tell us how the silicon would actually behave. But in the normal operating range, luckily for us, the digital illusion seems to hold and people who don’t even know how to run SPICE can confidently write thousands of lines of SystemVerilog knowing that these will accurately be transformed into real analog transistors about which they know next to nothing.

Now that’s a model that is useful.