This is the kind of weird (ab)use of floating point instructions that I can imagine some DRM solution using as a means to obfuscate a VM of some kind.
The next step would be to use these properties to write a compiler to run normal source code as floating point integers, maybe with some kind of FFI to call regular OS APIs.
One day as we were getting coffee a coworker just casually drops that his buddy from school made some programs called learnfun and playfun for a YouTube video (which is my favorite Tom7 video). Tech is a surprisingly small world.
This is a proof by construction that the Intel MMU's fault handling mechanism is Turing complete. We have constructed an assembler that translates 'Move, Branch if Zero, Decrement' instructions to C source that sets up various processor control tables. After this code has executed, the CPU computes by attempting to fault without ever executing a single instruction. Optionally, the assembler can also generate X86 instructions that will display variables in the VGA frame buffer and will cause control to be transferred between the native (display) instructions and 'weird machine' trap instructions.
That whole channel (suckerpinch / Tom 7) is incredible. Extremely nerdy, thoughtful and funny content, and I just love his delivery. I cannot recommend it enough, especially to the HN crowd.
That implements conversion from an IEEE-754 double to a pair of two such doubles whose values are integers of the low and high 32 bits of the bitwise representation of the argument double, implemented only with double add/sub/mul.
To be entirely precise, it is functionally complete in combination with access to the constant false (-0.0). If not given access to this constant it is not functionally complete, unlike e.g. NAND which can produce false from any value. I shall clarify that in the article.
The point of the article was more to illustrate that using nothing but signed zeros and floating point subtraction you can simulate arbitrary circuits, and 'functionally complete' was the most concise term for that I could think of, even if it is bending the rules a little bit when strictly looking at the truth table.
It's a matter of definitions, but it always bothered me that functional completeness is defined so that it only requires the ability to produce any non-nullary function, not any function including nullary ones. That is, the set { NAND } is considered functionally complete, even though it can only produce a function that maps any input x to TRUE, and can't produce the value TRUE itself. As soon as you care about constants, which I think you should, { NAND, FALSE } or whatever isn't any more magical than { AND, NOT } or { XOR, TRUE }.
> { NAND } is considered functionally complete, even though it can only produce a function that maps any input x to TRUE, and can't produce the value TRUE itself.
When you're reducing formulas, those are the same thing.
The expression "(p 🡑 (p 🡑 p)) 🡑 (p 🡑 (p 🡑 p))" is just a particularly longwinded name for the constant "0".
I don't see why you're comparing {NAND, FALSE} to {AND, NOT} - how do you produce TRUE from {AND, NOT} by a standard that {NAND} by itself doesn't also meet? The normal way to produce TRUE from {AND, NOT} is
Yeah, listing {AND, NOT} was a mistake — you’re right, you do need a constant.
My problem with p 🡑 ¬p ≡ 1 is simply that you need some (arbitrary) value p from somewhere. It’s not 1, it’s a unary function that returns 1. That just bothers me.
> It’s not 1, it’s a unary function that returns 1.
How do you know it's a unary function? For all you know, the function in question is f(p,q,r,s,t) = p 🡑 ¬p.
In fact it's a nullary function, because you don't need any input. There is no difference between the functions f(p,q,r,s,t) = p 🡑 ¬p and f(p,q,r,s,t) = r 🡑 ¬r. They have exactly the same behavior in every respect. And for the same reason, there is also no difference between those functions and the functions f(p) = p 🡑 ¬p, f(q) = p 🡑 ¬p [sic], and f() = p 🡑 ¬p.
If you want to be able to implement nullary functions, then you need a nullary function to begin with. You are not really implementing anything besides maybe negating the constant you got. You would also have to extend { AND, NOT } with a constant. The best you could do would change from one binary function to one binary function and a constant.
Hey, not related to the post but since you're here: your domain name has an AAAA IPv6 record but the server doesn't respond to IPv6 requests. The problem most probably is that the web server is not binded to the IPv6 address of the system.
Thanks for letting me know. I just double-checked and the AAAA IPv6 record does have the right IP, port 80 is open in the VPS firewall for both IPv4 and v6 and my nginx config does listen on both as well:
listen 80;
listen [::]:80;
I'm by no means a networking expert, so I'm a bit puzzled. I'll investigate more in a couple of days, not particularly excited to mess with the system while serving a post on the front page.
That's the HTTP config, but the website is served over HTTPS and the HTTP version redirects to it. My bet would be that the HTTPS settings does not bind to IPv6.
Do you have:
listen [::]:443 ssl;
somewhere in the server {} block where the certificate is declared?
My mobile phone carrier uses IPv6 so I cannot access your website from my phone (except if I connect to a wifi network that uses IPv4).
> Maybe the second line should be "listen 443 ssl;" (without the colon, like in the non-HTTPS version)? That's how it is in my config.
That's a clerical error while copying to Hacker News, it is without the colon in my config as well. I've edited the post.
I think I figured it out, Hetzner lists 2a01:4f8:c012:175e::/64 as the IPv6 for my VPS, so I put 2a01:4f8:c012:175e:: in the DNS record. However it seems it only actually listens on 2a01:4f8:c012:175e::1. Probably just me being an idiot and fundamentally misunderstanding how IPv6 addresses work. I've updated it, although it will probably take some time before the DNS cache refreshes.
> Hetzner lists 2a01:4f8:c012:175e::/64 as the IPv6 for my VPS
Yup, that's the address prefix, 64 bit long as indicated by the /64. Your VPS can therefore be configured with 2^(128-64)=2^64 IP addresses, as long as they start with that prefix.
The actual IP is chosen by your VPS itself, so I guess it has assigned itself prefix::1. You can see that address with `ip -6 a`. And add new ones if you want: `ip -6 address add 2a01:4f8:c012:175e::2 dev yournetworkcard0`. You can technically add one IP address per service and bypass the reverse proxy by having the services listen on their dedicated IPs. That makes it easy to migrate services to another host (change the AAAA record).
I dont really know what you mean by truth-preserving here, but maybe a hint is thats its not ONLY subtraction which is functionally complete, it's subtraction and the constant symbol 0. From subtraction and 0 he makes false (as -0.0), and then he has the functionally complete set found in wikipedia [1] as {->, _|_ } (my attempt at rendering rightarrow and bottom).
Truth-preserving means that it maps T to T. In fact, the Wikipedia's article you link to has Post's theorem about five Post's classes of Boolean functions with all of their definitions: monotonic, affine (which has a funny definition in this article: I was taught the definiton via ANF, "is just a XOR of (some) of variables"), self-dual, truth- and false-preserving. They're all closed but functionally incomplete (in fact, they're functionally pre-complete: adding any single function outside of a class to that class produces a functionally complete set, — and there are no other pre-complete classes).
Subtraction is truth preserving on the sign bit. It's not truth-preserving in the actual subtractive bits.
(I disagree with their claim that the subtractive bit is functionally complete on its own - you're right, since it's truth-preserving, it clearly is not functionally complete)
Below the truth table for implication (with arguments reversed) they claim
> It turns out this truth table is functionally complete [1]
yet the linked Wikipedia article clearly states that
> every two-element set of connectives containing NOT and one of { AND, OR, IMPLY } is a minimal functionally complete subset of { NOT, AND, OR, IMPLY, IFF }. I.e. IMPLY by itself is not functionally complete.
Why would truth preservation prevent it from being functionally complete? How can you even tell a truth table is truth preserving? A truth table is not a logical argument.
Truth-preserving in this context means that the function value is true if all function arguments are true. If you only have truth-preserving functions available, then you can not output false if all inputs are true, hence you can not build all possible functions. An analog argument applies to falsehood-preserving functions.
If functionally complete means that any logic circuit can be constructed, doesn't this imply that IEEE-754 floating point subtraction is effectively Turing complete? (Or not?)
No, functionally complete is mussing the looping functionality to be Turing complete.
Turing complete is often misused to say functionally complete, either because people mistake the two or because it makes for a more appealing blog post / article:
- homomorphic encryption systems are functionally complete but not Turing complete (since looping leaks the number of operations done, break the encryption)
However, the movfuscator as implemented does still require a sigaction(2) syscall (via an INT 0x80 or similar) to set up a signal handler, under the justification that "it is not actually part of the program" and that "if we were in ring 0, we wouldn't need help from the kernel" [0]. The latter part seems a little dubious to me: without the help of the bootloader running non-MOV instructions, you'd never be able to escape from 16-bit real mode into 32-bit protected mode, since you wouldn't be able to load a valid GDT with the LGDT instruction.
You also need an infinite memory to be called "Turing complete". It doesn't make sense to say that a bunch of operations are or are not Turing complete. It's a property of a whole machine, not just a set of operations. And no real machine can possibly be Turing complete, because they don't have infinite memory or time.
> In computability theory, a system of data-manipulation rules (such as a model of computation, a computer's instruction set, a programming language, or a cellular automaton) is said to be Turing-complete or computationally universal if it can be used to simulate any Turing machine (devised by English mathematician and computer scientist Alan Turing).
And Turing machines have unbounded memory. That's usually ignored when talking about Turing completeness, but it's nevertheless true that physical computers cannot simulate all Turing machines.
> it's nevertheless true that physical computers cannot simulate all Turing machines
Right, I wasn’t arguing that physical computers can be Turing machines, but instead that sets of operations can be Turing complete. There are sets of operations with which one can compose a program that perfectly simulates a Turing machine.
The problem is that physical computers cannot always run these programs accurately, due to memory constraints. But the set of operations is itself Turing complete.
> It doesn't make sense to say that a bunch of operations are or are not Turing complete.
and the article’s first sentence says that a “system of rules” such as a computer’s instruction set can be Turing complete.
The article matches my understanding, which is that Turing completeness is a property describing the expressive power of a bunch of operations. You don’t need a computer with infinite memory, or even any physical computer at all, for a bunch of operations to be Turing complete.
To quote someone from reddit (substitute NAND gates with subtraction)
> You can bulid a turing-complete machine out of NAND-gates, but to say that a NAND-game is turing-complete is like saying that you can live in a brick. You can't, but you can bulid a house out of bricks and live in that.
I posted this on the thread in /r/programming a while ago, but I might as well post this here too. It's possible to implement the adder in "only" 11 subtractions:
fn adder(a: Bit, b: Bit, c: Bit) -> (Bit, Bit) {
let r0 = c - b;
let r1 = c - r0;
let r2 = ZERO - r0;
let r3 = b - r1;
let r4 = r2 - r3;
let r5 = a - r4;
let r6 = r4 - a;
let r7 = ZERO - r5;
let r8 = r7 - r1;
let r9 = r7 - r6;
let r10 = ZERO - r8;
(r9, r10)
}
"If both of the addends have the same sign, the output must have that sign. However, for x−y that means if x and y have different signs the output must have the sign of x."
This is trivially wrong, or mixing two different meanings of "signs".
Given variables x and y with values 5 and 10, ie both having the same positive sign, x-y will produce a result -5, that has negative sign.
Even if we assume that the sign of the y variable is actually inverted, it's still trivial by choosing say -3 and -6, the latter which has now inverted to 6, and the result is +3, which has different sign than x.
Direct your attention to the first line "If both of the addends have the same sign, the output must have that sign"
This is absolutely not true, as already shown. x=5 y=10 z=x-y=-5, which has different sign from x.
If we assume sign of y inverts because of the operation, then direct your attention to the second line "However, for x−y that means if x and y have different signs the output must have the sign of x" x=-3 y=-6=>6 these now have different sign, so result should have sign of x, but z=x+y=3, which again has different sign from x.
> Direct your attention to the first line "If both of the addends have the same sign, the output must have that sign" This is absolutely not true, as already shown. x=5 y=10 z=x-y=-5, which has different sign from x.
The first sentence is referring to addition, with addends, not subtraction. x - y is not an addition, it is a subtraction, so the first sentence does not directly apply. It does apply however if you treat x - y as the sum x + (-y), which the second sentence clarifies.
In other words, the first sentence applies directly to additions, and applies to subtractions if you flip the sign of the second argument. The second sentence applies to subtractions directly without any sign flips, but obviously does not apply to additions.
> If we assume sign of y inverts because of the operation, then direct your attention to the second line "However, for x−y that means if x and y have different signs the output must have the sign of x"
> x=-3 y=-6=>6 these now have different sign, so result should have sign of x, but z=x+y=3, which again has different sign from x.
No, x=-3 and y=-6 both have the same sign, they're both negative.
Direct your attention to the first line "If both of the addends have the same sign, the output must have that sign"
This is absolutely not true, as already shown.
x=5
y=10
z=x-y=-5, which has different sign from x.
If we assume sign of y inverts because of the operation, then direct your attention to the second line "However, for x−y that means if x and y have different signs the output must have the sign of x"
x=-3
y=-6=>6
these now have different sign, so result should have sign of x, but
z=x+y=3, which again has different sign from x.
Cute. Next steps? If you fill out your soft-hardware instruction set in the straightforward way, you could rig up a compiler to target it. (There’s probably no practical use beyond obfuscation like hiding malware from decompilers.)
The next step would be to use these properties to write a compiler to run normal source code as floating point integers, maybe with some kind of FFI to call regular OS APIs.