Quine, Hofstadter, Gödel again

Filed under: math, philosophy — jlm @ 20:25

During some recent traveling, I re-read some sections of Douglas Hofstadter’s Gödel, Escher, Bach. Of particular interest was a step in the proof of Kurt Gödel’s first incompleteness theorem which involved substituting a variable of a formula with the formula itself. Hofstadter calls this peculiar operation “quining” after the logician W. V. Quine, who wrote extensively about self-reference. As with the previous times I read through that part, I noticed that the operation didn’t specify a variable for substitution like substitutions generally do, but instead performed substitution on all free variables, which is something I haven’t encountered anywhere else. This oddity wasn’t even mentioned, much less justified. Unlike after my previous reads, this time I decided to figure out why it was performed that way.

Now, the more interesting parts of Gödel’s Incompleteness Theorem involve what we now call Gödel coding to demonstrate that classes of logical systems can represent quining their own formulas and verifying their own proofs, the latter of which was very surprising at the time. But those parts turn out to be unnecessary for dealing with this small facet of Gödel’s proof, so let’s just deal with the proof’s final proposition, which is comparatively simple and straightforward: given that a logical system supports the operations of quining and proof verification (and logical conjunction and negation, and an existential quantifier), that logical system is incomplete (or inconsistent).



Sleet again

Filed under: sfba — jlm @ 13:00

Having learned from my experience on Monday, I didn’t take my bike out shopping as usual, despite it being 48°F / 9°C (so probably warmer) and midday instead of evening. Then, in due course, ting-ting-ting-ting on my car’s body. I guess I need to save my biking in the rain for warmer parts of the year, even though it’s not at present cold per se.



Filed under: biking, sfba — jlm @ 19:09

I biked home from work this day, as usual.
The bicycle has been my preferred mode of
personal transportation almost all my life.

It was raining.
I learned to bike in Portland.
I’m used to biking in the rain.

It was raining hard.
I was drenched. But one quickly learns that
once soaked, you can’t get any more wet.

The air was cool. High 40’s, I think.
That’s a good temperature to bike in. Not
uncomfortably cold, but cold enough to
pull your body heat away without sweating.

The rain was cold.
I’m from Portland. I can handle cold rain.
Suddenly, the rain stang!
What is this sorcery?
The rain beat a tattoo on my helmet.
Augh! Never mind the wet, this is sleet!
The weather forecast said nothing about sleet!
I have yet to enjoy being outdoors with sleet or hail.


Less is more, CPU edition

Filed under: general, programming — jlm @ 17:11

I fixed an interesting bug recently. By way of background, I work on industrial robotics control software, which is a mix of time-critical software which has to do its job by a tight deadline (variously between 1 and 10 ms) or a safety watchdog will trigger, and other software that does tasks that aren’t time sensitive. We keep the timing-insensitive software from interfering with the real-time software’s performance by having the scheduler strictly prioritize the deadline-critical tasks over the “best effort” tasks, and further by massively overprovisioning the system with far more CPU and memory than the tasks could ever require.

The bug (as you may have guessed) is that the deadline was sometimes getting missed, triggering the watchdog. What makes the bug interesting is that it was caused by us giving the system too much excess CPU and memory!

The deadline overruns happen when the computer runs best-effort tasks. It only runs those tasks a small fraction of the time, but no matter, they shouldn’t be interfering with the deadline completion at all (and having our software fail its deadlines is unacceptable, of course). The real-time tasks’ peak usage is only about a quarter of the computer’s CPU capacity, and the system gives them that always: 100% of the time they want CPU, they get CPU. They are never delayed, and never have to wait on a resource held by a best-effort task. When they miss the deadlines, it’s always because they’ve gotten jobs that take the usual amount of work to complete, and had their usual amount of time to do it, yet the jobs somehow just take more time to finish. There’s no resource contention, nor are the caches an issue.

Yet when the best-effort tasks execute, the calculations done by the real-time tasks run slower on the same CPU for the same job with the same cache hit rate and with no memory or I/O contention. What’s going on? After hitting some false leads, I discovered that they’re going slower because the CPUs are running at a lower frequency. It turns out the CPUs’ clocks have been stepped down because they’re getting too hot. (Yes, the surrounding air is much warmer than it “should” be, but we don’t have a choice about that.)

The computer is hot because all the CPUs are going at full blast, because all of the best-effort tasks are executing because the computer can fit them all in memory. Half a minute later, the tasks are done, the computer cools off, the CPU frequency gets stepped back up, and the deadline overruns cease. An hour later, the hourly background tasks all go again, the fans spin up to full, but they’re not enough and the CPU frequency steps down, hence the watchdog alerts about missed deadlines.

Okay, maybe we should change the background tasks so they execute in a staggered fashion. But before thinking about how I might do that, I tried disabling ⅔ of the computer’s CPUs instead. The hourly processing now takes 200s instead of 30, but it’s still far below 3600, and that’s what matters. Now the computer stays nice and cool, so the CPUs stay nice and quick, and the deadlines all get met. We were using too many CPUs to get the calculations we needed to get done fast, done fast. Who’d’a thunk?

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