Analog computers don't get the respect they deserve. There's one more computer, the FCC. The Flight Control Computer is an analog computer in the Saturn V that controlled the rocket gimbals. It's a two-foot cylinder weighing almost 100 pounds.
the only reason they have the same name is that they were both originally built to replace people cranking out calculations on mechanical desk calculators, who were also called 'computers'
the flight control 'computer' has more in common with an analog synthesizer module than it does with a cray-1, the agc, an arduino, this laptop, or these chargers, which are by comparison almost indistinguishable
ENIAC, for example, was not a stored-program computer. Reprogramming required rewiring the machine.
On the other hand, by clever use of arithmetic calculations, https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.37.... says the Z3 could perform as a Universal Computer, even though, quoting its Wikipedia page, "because it lacked conditional branching, the Z3 only meets this definition by speculatively computing all possible outcomes of a calculation."
Which makes me think the old punched card mechanical tabulators could also be rigged up as a universal machine, were someone clever enough.
"Surprisingly Turing-Complete" or "Accidentally Turing Complete" is a thing, after all, and https://gwern.net/turing-complete includes a bunch of them.
probably numerous eddies in natural turbulent fluid flows have been digital turing-complete computers, given what we know now about the complexity of turbulence and the potential simplicity of turing-complete behavior. but is there an objective, rather than subjective, way to define this? how complicated are our input-preparation and output-interpretation procedures allowed to be? if there is no limit, then any stone or grain of sand will appear to be turing-complete
a quibble: the eniac was eventually augmented to support stored-program operation but not, as i understand it, until after the ias machine (the johnniac) was already operational
another interesting question there is how much human intervention we permit; the ias machine and the eniac were constantly breaking down and requiring repairs, after all, and wouldn't have been capable of much computation without constant human attention. suppose we find that there is a particular traditional card game in which players can use arbitrarily large numbers. if the players decide to simulate minsky's two-counter machine, surely the players are turing-complete; is the game? are the previous games also turing-complete, the ones where they did not make that decision? does it matter if there happens to be a particular state of the cards which obligates them to simulate a two-counter machine?
if instead of attempting to measure the historical internal computational capability of systems that the humans could not perceive at the time, such as thunderstorms and the z3, we use the subjective standard of what people actually programmed to perform universal computation, then the ias machine or one of its contemporaries was the first turing-complete computer (if given enough memory); that's when universal computation first made its effects on human society felt
Sure. One of the "Surprisingly Turing-Complete" examples is that "Magic: the Gathering: not just TC, but above arithmetic in the hierarchy ".
See https://arxiv.org/abs/1904.09828 for the preprint "Magic: The Gathering is Turing Complete", https://arstechnica.com/science/2019/06/its-possible-to-buil... for an Ars Technica article, and https://hn.algolia.com/?q=magic+turing for the many HN submissions on that result.
An HN comment search, https://hn.algolia.com/?dateRange=all&page=0&prefix=false&qu... , finds a few more lay examples, with https://news.ycombinator.com/item?id=21210043 by dwohnitmok being the easiest for me to somewhat make sense of.
I think the idea is, suppose you have an oracle which tells you if a Turning machine will halt, in finite time. There will still halting problems for that oracle system, which requires an oracle from a higher level system. (That is how I interpret "an oracle that magically gives you the answer to the halting problem for a lower number of interleavings will have its own halting problem it cannot decide in higher numbers of interleavings").
https://risingentropy.com/the-arithmetic-hierarchy-and-compu...
but then we have to ask thorny ontological questions: does a card game count if it requires a particular configuration to be turing-complete, but nobody ever played it in that configuration? what if nobody ever played the game at all? what if nobody even knew the rules?
> Prior to this work, no undecidable real games were known to exist. Demaine and Hearn (2009) [10] note that almost every real-world game is trivially decidable, as they produce game trees with only computable paths. They further note that Rengo Kriegspiel {Rengo Kriegspiel is a combination of two variations on Go: Rengo, in which two players play on a team alternating turns, and Shadow Go, in which players are only able to see their own moves.} is “a game humans play that is not obviously decidable; we are not aware of any other such game.” It is conjectured by Auger and Teytaud (2012) [1] that Rengo Kriegspiel is in fact undecidable, and it is posed as an open problem to demonstrate any real game that is undecidable.
> The approach of embedding a Turing machine inside a game directly is generally not considered to be feasible for real-world games [10].
Regarding your ontological question, that we don't know if something is Turning complete doesn't mean it isn't.
People explored the Game of Life before it was proven to be Turing complete. The 1970 SciAm article says "Conway conjectures that no pattern can grow without limit." so Martin and Gardner didn't even know about gliders then. People don't say GoL wasn't Turning complete in 1970.
I pointed to a 1998 paper claiming the Z3 machine from the 1940s was Turing complete, that author clearly believes that a particular physical configuration is not required.
Nor did Turing construct a physical representation for his paper.
FWIW, the MtG preprint gives a concrete example of a 60-card initial deck. I would be surprised if neither the authors nor anyone else has ever tried it.
The ontological question is even more fully resolved because no one has ever created a real Turing machine. We always have the proviso "if given enough memory".
Similarly, the video game "Minesweeper" is NP-complete as the size increases, and with an infinite board - clearly not physically realizable - is Turing complete. https://en.wikipedia.org/wiki/Minesweeper_(video_game)#Compu...
well, maybe one thing: if it doesn't matter whether anyone played the game or knew the rules, then magic: the gathering was turing-complete before the first magic deck was printed, before the first human was born, before the first star was formed, perhaps before the big bang
If you go down that route you'll realize there are an uncountable number of games that have never been created, and will never be created, which are Turing-complete.
And start wondering if mathematics is created or discovered.
If we could optimize a set of programs down to the FPGA bitstream or even Verilog level, that would approach the kind of programs analog computers run.
I can't tell anything about Turing completeness though. It's a fully discrete concept, and analog computers operate in the continuous signal domain.
Turing completeness is a tar pit that makes your code hard to analyse and optimise. It's an interesting challenge to find languages that allow meaningful and useful computation that are not Turing complete. Regular expressions and SQL-style relational algebra (but not Perl-style regular expressions nor most real-world SQL dialects) are examples familiar to many programmers.
Programming languages like Agda and Idris that require that you prove that your programs terminate [0] are another interesting example, less familiar to people.
[0] It's slightly more sophisticated than this: you can also write event-loops that go on forever, but you have to prove that your program does some new IO after a finite amount of time. (Everything oversimplified here.)
There is still active research in the area, eg. https://www.lix.polytechnique.fr/~bournez/i.php?n=Main.Publi...
you can't simulate an 11-integrator general-purpose analog computer or other differential analyzer with a 10-integrator differential analyzer, and you can't simulate a differential analyzer with 0.1% error on a (more typical) differential analyzer with 1% error, unless it's 100× as large (assuming the error is gaussian)
the ongoing research in the area is of course very interesting but a lot of it relies on an abstraction of the actual differential-analyzer problem in which precision is infinite and error is zero
given these hypothetical abilities, you can of course simulate a two-counter machine, but a bigger question is whether you can compute anything a turing machine cannot; after all, in a sense you are doing an infinite amount of computation in every finite interval of time, so maybe you could do things like compute whether a turing machine will halt in finite time. so far the results seem to support the contrary hypothesis, that extending computation into continuous time and continuously variable quantities in this way does not actually grant you any additional computational power!
this is all very interesting but obviously not a useful description of analog computation devices that are actually physically realizable by any technology we can now imagine
do you want to call them all 'computers' now?
When the arithmetic circuits, i.e. the "central arithmetical part", as called by von Neumann, are coupled with a "central control part", as called by von Neumann, i.e. with a sequencer that is connected in a feedback loop with the arithmetic part, so that the computation results can modify the sequence of computations, then this device must be named as a "computer", regardless whether the computations are done with analog circuits or with digital circuits.
What defines a computer (according to the definition already given by von Neumann, which is the right definition in my opinion) is closing the feedback loop between the arithmetic part and the control part, which raises the order of the system in comparison with a simple finite state automaton, not how those parts are implemented.
The control part must be discrete, i.e. digital, but the arithmetic part can be completely analog. Closing the feedback loop, i.e. the conditional jumps executed by the control part, can be done with analog comparators that provide the predicates tested by the conditional jumps. The state of an analog arithmetic part uses capacitors, inductors or analog integrators, instead of digital registers.
Several decades ago, I had to debug an analog computer during its installation process, before functioning for the first time. That was in a metallurgic plant, and the analog computer provided outputs that controlled the torques of a group of multi-megawatt DC electric motors. The formulae used in the analog computations were very complex, with a large number of adders, multipliers, integrators, square root circuits and so on, which combined inputs from many sensors.
That analog computer (made with op amps) performed a sequence of computations much more complex than the algorithms that were executed on an Intel 8080, which controlled various on-off execution elements of the system, like relays and hydraulic valves and the induction motors that powered some pumps.
The main reason why such analog computers have become obsolete is the difficulty of ensuring that the accuracy of their computations will not change due to aging and due to temperature variations. Making analog computers that are insensitive to aging and temperature raises their cost much above modern digital microcontrollers.
you can even include multiplexors in your analog 'computer', even with only adders and multipliers and constants; x · (1 + -1 · y) + z · y interpolates between x and z under the control of y, so that its output is conditionally either x or z (or some intermediate state). but once you start including feedback to push y out of that intermediate zone, you've built a flip-flop, and you're well on your way to building a digital control unit (one you could probably build more easily out of transistors rather than op-amps). and surely before long you can call it a digital computer, though one that is controlling precision linear analog circuitry
it is very commonly the case that analog computation is much, much faster than digital computation; even today, with microprocessors a hundred thousand times faster than an 8080 and fpgas that are faster still, if you're doing submillimeter computation you're going to have to do your front-end filtering, upconversion or downconversion, and probably even detection in the analog domain
I agree that this kind of "analog computers" does not deserve the name of "computer", because they are equivalent only with the "registers + ALU" (RALU) simple automaton that is a component of a CPU.
Nevertheless, there is no reason why a digital control part cannot be coupled with an analog arithmetic part and there have existed such "analog computers", even if they have been rarely used, due to high cost and complexity.
It is not completely unlikely that such "analog computers", consisting of a digital control part and an analog arithmetic part, could be revived with the purpose of implementing low-resolution high-speed machine learning inference.
Even now, in circuits like analog-digital converters, there may be analog computing circuits, like switched-capacitor filters, which are reconfigurable by the digital controller of the ADC, based on various criteria, which may depend on the digital output of the converter or on the outputs of some analog comparators (which may detect e.g. the range of the input).
this is sort of like how biologists try to convince people to stop calling jellyfish 'jellyfish' and starfish 'starfish' because they aren't fish. the difference is that it's unlikely that someone will get confused about what a jellyfish is because they have so much information about jellyfish already
my quest to get people to call cellphones 'hand computers' is motivated by the same values but is probably much more doomed
Sometimes it has been applied to the kind of computers mentioned by me, with a digital control part and a completely analog arithmetic part.
However it has also been frequently used to describe what were hybrid arithmetic parts, e.g. which included both digital registers and digital adders and an analog section, for instance with analog integrators, which was used to implement signal processing filters or solving differential equations.
IMO, "hybrid computer" is appropriate only in the second sense, for hybrid arithmetic parts.
The control part of a CPU can be based only on a finite state automaton, so there is no need for any term to communicate this.
On the other hand, the arithmetic part can be digital, analog or hybrid, so it is useful to speak about digital computers, analog computers and hybrid computers, based on that.
in some sense almost any circuit in which a digital computer controls an analog multiplexer chip or a so-called digital potentiometer could qualify. and cypress's psoc line has a bit of analog circuitry that can be thus digitally reconfigured
By doing so, you get to make a point—perhaps via analogy, perhaps via precision, perhaps via pedantry—which is illuminating for you but now confusing for your reader. And to explain yourself, you must swim upstream and redefine a term while simultaneously making a different point altogether.
Much has been written about jargon, but a primary benefit of jargon is the chance to create a domain-specific meaning without the baggage of dictionary-correct associations. It’s also why geeks can be bores at dinner parties.
By analogy to HCI: words are affordances. Affordances exist because of familiarity. Don’t make a doorknob that you push on, and expect people not to write in telling you to use a door-bar on that door instead.
Unilaterally changing language is not forbidden, but if The Culture Wars™ has thought us anything, it is that people are allergic to talking about what they see as mandated changes to their language, even if it is reasonable and you can explain it.
Colour me stoked, but you could still just do it unilaterally and wait till somebody notices.
However my caveat with viewing everything as computation is that you fall into the same trap as people in the ~1850s did when they wanted to describe everything in the world using complex mechanical devices, because that was the bleeding edge back then. Not everything is an intricate system of pulleys and levers it turned out, even if theoretically you could mimic everything if that system was just complex enough.
> The Flight Control Computer (FCC) was an entirely analog signal processing device, using relays controlled by the Saturn V Switch Selector Unit to manage internal redundancy and filter bank selection. The FCC contained multiple redundant signal processing paths in a triplex configuration that could switch to a standby channel in the event of a primary channel comparison failure. The flight control computer implemented basic proportional-derivative feedback for thrust vector control during powered flight, and also contained phase plane logic for control of the S-IVB auxiliary propulsion system (APS).
> For powered flight, the FCC implemented the control law $ \beta_c = a_0 H_0(s) \theta_e + a_1 H_1(s) \dot{\theta} $ where $ a_0 $ and $ a_1 $ are the proportional and derivative gains, and $ H_0(s) $ are the continuous-time transfer functions of the attitude and attitude rate channel structural bending filters, respectively. In the Saturn V configuration, the gains $ a_0 $ and $ a_1 $ were not scheduled; a discrete gain switch occurred. The Saturn V FCC also implemented an electronic thrust vector cant functionality using a ramp generator that vectored the S-IC engines outboard approximately 2 degrees beginning at 20 seconds following liftoff, in order to mitigate thrust vector misalignment sensitivity.
https://ntrs.nasa.gov/api/citations/20200002830/downloads/20...
Unless you typically salvage digital computers from the wreckage of a failed rocket test and stick it in the next prototype. If the FCC is wrong, kaboom.
https://en.wikipedia.org/wiki/Core_rope_memory input wires were energized, and they were coupled (or not) to the output wires depending on if they shared a magnetic ring (or not).
Much less than “rebuilding”.
There have been some hybrids too.
Who said women can't do math?
https://www.smithsonianmag.com/science-nature/history-human-...
Asimov.
https://literature.stackexchange.com/questions/25852/where-d...
The straw man?
Nobody
https://www.dailymail.co.uk/news/article-524390/The-women-ad...
Granted, that same search will show you many examples of content accusing unnamed other people of having this attitude.
https://www.science.org/content/article/both-genders-think-w...
It’s an antiquated notion in my mind, but I don’t think it is a thing of the past.
Women being bad at advanced math would make sense as an antiquated notion, but those in charge of hiring decisions until about 50 years ago evidently thought women were great at basic math.
The study you linked showing that women lag behind men in math to a degree proportional to some gender disparity metric is also interesting, but doesn't really tell us how we got here.