Over lunch John clarified a few things for me about the nomenclature used for transfer functions.
A "transfer function" is the model of how a gate/amplifier behaves. Given an input level (voltage for an electrical device or molarity for a chemical one) the model describes the equilibrium (or steady-state) output level. The above graph illustrates a hypothetical transfer function.
The main point of confusion for me was "What exactly is the definition of 'gain'?" and "By what convention are logic levels defined?"
John pointed out that the word "gain" is an over-used / abused word. Many people over-simplify the transfer function graph above and use 'gain' to mean different things. The gain is the slope -- but as you can see the slope of the function is different at different input levels so these is no such thing as "the" gain for a gate.
In the middle, linear range, the slope is roughly constant over an input domain. When building analog devices it is this roughly-linear region that is of interest and so an analog engineer would probably refer to the approximately-constant slope in this linear region as "the gain".
However a digital engineer uses the wider non-linear range to encode a binary variable. In this case, we must now have a convention that defines the logic levels. The electrical convention for this is that the two places where the slope, aka the "incremental gain", are equal to 1 are the places that define the inside bounds of the logic levels. Anything outside of these bounds are considered valid logic levels. Anything inside of them are considered "undetermined". The nominal values (the desired levels to be obtained by any gates) are defined by a "noise margin" outside of these inc. gain=1 points.
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Digital engineers love circuits with "infinite gain". (Though you're right, the word gain can mean transistor beta in casual discussion.) Infinite gain means the slope in the transfer function is vertical, thus the switching time from 0->1 or 1->0 is zero. There are a variety of ways to hook up transistors so that they amplify like crazy (without latch-up). The audio engineers, of course, design their circuits with the complete opposite criteria - completely linear response with no sharp edges. (It's also worth looking at vacuum tube response, which behaves differently than transistors at the sharp edges, since tubes saturate rather than clip, both giving a good Jimi hendrix sound for audio engineers and frustrating digital vacuum tube engineers.)
What genetic part has infinitely fast runaway reaction, acting as an infinite gain component? Once engaged, the part is "on" in a big way. Silicon-controlled rectifiers (SCRs) behave this way. To turn off an SCR, power has to be removed. I'd guess biology doesn't usually behave this way because status quo with biofeedback is better darwinism.. unless there's a tumor gene involved.
Logic levels have come a long way and it was decades of incompatible transistor circuits before settling on CMOS. Two old transistor circuits connected together would not be able to switch each other on or off or maybe only one way, because the output levels of circuit #1 didn't match the input levels of circuit #2. Leading, of course, to a great business in fixing this problem with level-shifting circuits placed in the middle.
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